presented by Diplom-Physiker Andreas Hartl born in Menden

Transcription

1 Dissertation submitted to the Combined Facuties for the Natura Sciences and for Mathematics of the Ruperto-Caroa University of Heideberg, Germany for the degree of Doctor of Natura Sciences presented by Dipom-Physiker Andreas Hart born in Menden Ora examination: 5 February 27

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3 Tomographic Reconstruction of 2-D Atmospheric Trace Gas Distributions from Active DOAS Measurements Referees: Prof. Dr. Urich Patt Prof. Dr. Bernd Jähne

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5 Zusammenfassung Die Anwendung des tomographischen Prinzips auf die aktive DOAS Fernerkundung stet eine neue Meßtechnik von atmosphärischen Spurengasverteiungen dar, für die anaytische Standardmethoden zur Rekonstruktion von skaaren Federn aus ihren Wegintegraen wegen der wenigen (1-5), unregemäßig angeordneten Strahen nicht anwendbar sind. Stattdessen wird die Verteiung durch eine begrenzte Zah von okaen, hier stückweise konstanten oder inearen, Basisfunktionen parametrisiert und das diskrete ineare inverse Probem durch einen Least Squares - Minimum Norm Ansatz geöst. Für räumich stark begrenzte 2-D Konzentrationsspitzen wird in Abhängigkeit ihrer Ausdehnung gezeigt, wie die Rekonstruktion durch optimierte Wah der Parametrisierung, d.h. Zah und Art der Basisfunktionen, sowie des A Prioris erhebich verbessert werden kann. Die Reguarisierung der Lösung spiet eine untergeordnete Roe. Vorschäge zur Rekonstruktion von Konzentrationsspitzen durch Kombination verschiedener Parametrisierungen werden systematisch untersucht, wobei deren Erfog stark von der am meisten interessierenden Eigenschaft der Verteiung abhängt. Vergeich verschiedener 2-D Strahgeometrien ergibt, daß ineare Unabhängigkeit des resutierenden Systems entscheidend ist. Eine detaiierte Anayse des Rekonstruktionsfehers stet Besonderheiten der Tomographie mit wenigen Strahen heraus und argumentiert, daß dessen Abschätzung ohne A Priori nicht mögich ist. Unter dieser Voraussetzung wird ein numerisches Schema zur Berechnung des Rekonstruktionsfehers vorgestet. Die Methoden werden auf ein Innenraumexperiment zur Simuation von Emissionsfahnen, sowie auf 2-D Modeverteiungen über einer Straßenschucht angewendet, wobei für etztere gezeigt wird, wie Modeevauation trotz weniger Lichtstrahen mögich sein kann. Anders as für räumich begrenzte Maxima besteht hier starke Abhängigkeit vom Grad der Reguarisierung. Abstract Appying the tomographic principe to active DOAS remote sensing eads to a nove technique for the measurement of atmospheric trace gas distributions. Standard anaytica methods for the reconstruction of a scaar fied from its ine integras cannot be used due to ow numbers of ight paths (1-5) and their irreguar arrangement, so that the concentration fied is expanded into a imited number of oca (piecewise constant or inear) basis functions instead. The resuting discrete inear inverse probem is soved by a east squares-minimum norm principe. For sharp 2-D concentration peaks it is shown systematicay with respect to their extension how the optima choice of parametrisation (in terms of number and kind of basis functions) and a priori can tremendousy improve the reconstruction. Reguarisation pays a minor roe. Proposas for retrieving peak distributions by combining different parametrisations are again examined systematicay showing that their usefuness heaviy depends on the features one is most interested in. Comparison of different 2-D ight path geometries reveas that inear independency within the associated systems is pivota. A detaied anaysis of the reconstruction error points out specia issues of tomography with ony few integration paths and argues that a compete error estimation is not possibe without a priori assumptions. Based on this discussion a numerica scheme for cacuating the reconstruction error is suggested. The findings are appied to an indoor experiment simuating narrow emission puffs and 2-D mode distributions above a street canyon, respectivey. For the atter case it is demonstrated how mode evauation can be possibe even with a reativey sma number of ight paths. Contrary to the reconstruction of peak distributions reguarisation becomes crucia.

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7 Contents Symbos, Acronyms and Notation Introduction 14 I. Basics Trace Gas Distributions in the Atmosphere The need for measured trace gas distributions Seected trace gases Dispersion by turbuent diffusion The Gaussian pume mode Urban trace gas distributions Tomography and Remote Sensing of Atmospheric Trace Gases The principe of tomographic measurements Atmospheric remote sensing IR spectroscopy LIDAR DOAS Tomographic appications in the atmosphere First tomographic DOAS measurements Tomography and Discrete, Linear Inverse Probems Forward mode and inverse probem Continuous inversion methods Transform methods Remark on existence and uniqueness The discrete, inear inverse probem Discretisation by quadrature Finite eement discretisation - oca basis functions The question of i-posedness The singuar vaue decomposition The east squares probem Least squares and east norm soutions Weighted east squares and weighted east norm with a priori Reguarisation and constrained east squares probems

8 8 Contents Resoution matrix and averaging kernes Remark on the norm Statistica approach to the east squares probem The optima estimate Degrees of freedom and information content Iterative soution of the east squares probem Iterative versus direct methods Typica convergence behaviour Brief comparison of some iteration methods ART, SART and SIRT Other reconstruction methods Maximum entropy and maximum ikeihood Backus-Gibert method Exampe of goba optimisation fitting Gaussian exponentias II. Methodoogy The Error of the Reconstructed Distribution The probem in defining the reconstruction error Composition of the tota error The idea discretisation and the discretisation error The inversion error The measurement error Numerica estimation of the reconstruction error Overa reconstruction errors and quaity criteria Reconstruction Procedure Reconstruction principe and inversion agorithm Reconstruction grid The idea reconstruction grid Combining grids for the reconstruction A priori and constraints Choice of the a priori Background concentrations Additiona constraints Fitting peak maxima Finding optima settings from simuations Designing a Tomographic Experiment Requirements for the setup number of ight paths Light path geometry degrees of freedom and infuence of the a priori Incuding point measurements and profie information Aspects of mode evauation

9 Contents 9 III.Numerica Resuts D Simuation Resuts for Gaussian Peaks Test distributions and ight path geometry Parametrisation Grid dimension Basis functions Grid transation Agorithms: ART versus SIRT Optima iteration number and reconstruction quaity Sensitivity to noise Light path geometry Singuar vaue decomposition of the geometry Comparison of different geometries Point measurements Scaing with ight path number Background concentrations Discussion D Reconstruction of NO 2 Peaks from an Indoor Test Experiment The experiment Sampe reconstruction Discussion of the reconstruction error Some aspects of atmospheric measurements of emission pumes D Reconstruction of Mode Trace Gas Distributions above a Street Canyon Mode system, set-up and resuts Sampe reconstruction for NO Discussion of mode evauation Estimation of the reconstruction error using the optima estimate Estimation of the reconstruction error using a random ensembe Case studies Concusion and Outook Concusions Outook IV.Appendices 184 A. Atmospheric Stabiity Casses and Dispersion Coefficients 185 B. Generation of Random Test Distributions 187 C. Auxiiary Cacuations 189

10 1 Contents D. Software & Numerics 194 References 195

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12 Symbos, Notation and Acronyms p. 54 A = (a 1,...,a m ) T system matrix of the discretised tomographic probem with coumn vectors a i c (i) (r,t) atmospheric concentration fied (of chemica p. 25 species i) ĉ(r,t) reconstruction error fied p. 86 ĉ,u (r,t) ower/upper bounds of the reconstruction error p. 85 fied ĉ ± (r,t) positive/negative reconstruction error (under- p. 94 /overestimation by the reconstruction) d = (d 1,...,d m ) T vector of coumn densities ( data vector ) pp. 49, 57 ǫ = (ǫ 1,...,ǫ m ) T measurement error pp. 5, 9 E set (ensembe) of concentration fieds pp. 93, 17 i usuay index in discrete data space (ight paths) I index for concentration fieds of a set E or index of subgrids for grid combination schemes p. 13 j usuay index in discrete state space (parameters of the discretisation) J index of grid knots for grid combination schemes p. 13 k iteration number usuay ength of reconstruction area L usuay ight path ength m dimension of data space (number of ight paths) M number of subgrids for grid combination schemes p. 13 n dimension of state space (number of parameters of the discretisation) N number of grid knots for grid combination schemes p. 13 R resoution matrix p. 69 S x,ǫ,a covariance matrix of the a posteriori, measurement pp. 65,72 error and apriori S j coumn density from DOAS fit for chemica species p. 42 j σ x,y,z pume dispersion coefficients p. 28 Σ = diag(σ 1,...,σ rank[a],,...) matrix of singuar vaues σ i p. 6 Σ r = diag(σ 1,...,σ rank[a] ) U = (u 1,...,u m ) matrix of singuar vectors u i associated with the p. 6 data space V = (v 1,...,v n ) matrix of singuar vectors v j associated with the p. 6 state space φ = (φ 1,...,φ n ) T vector of basis functions φ j pp. 54,57 x = (x 1,...,x n ) T vector of discretisation parameters x j ( state vector ) pp. 54,57 x a a priori of the state vector pp. 66,72 X = (X 1,...,X J ) T vector of concentration vaues on knots for grid combination schemes p

13 13 average (of concentration fieds) ( ) tempora or spatia mean or impicit transformation by (S)ART, SIRT p. 78 ( ) reconstructed quantities ( ) turbuent fuctuations or components in the system of the singuar vectors p. 25 p. 79 ( ) transformation to diagona covariance matrices p. 73 BG background LP ight path N M B normaised mean bias (accuracy of reconstructed/modeed spatia mean concentrations) NEARN nearness (speciay normaised root mean square error of the reconstructed concentration fied) P P A peak prediction accuracy (accuracy of reconstructed/modeed peak maxima) PIPA peak integra prediction accuracy (accuracy of reconstructed tota emissions) p. 97 p. 96 p. 97 p. 97

14 1. Introduction Trace gases form ony a tiny part of ambient air their voume fractions range from ess than 1 9 to some 1 6 but their actua amounts can be both vita and threatening to the ife-forms of our ecosystem. On the one hand, it is their direct impact on the radiation budget of the earth which makes them so important, the so-caed greenhouse effect, e.g. of water vapour, CO 2, O 3 or CH 4. On the other hand, it is their chemica reactivity which, besides making themseves or their reaction products a risk to creatures and pants, can change the composition of the atmosphere and thus indirecty affects its natura radiation baance. A these effects coud or sti can be observed since anthropogenic activities have started to increase trace gas eves ocay or gobay beyond their natura vaues: smog caused by harmfu amounts of industriay emitted SO 2, photochemica smog as a noxious mixture of trace gases such as NO x, O 3 and VOCs (voatie organic compounds), acid rain formed from emissions of SO 2 and NO x, destruction of the ozone ayer, e.g. by CFCs (chorofuorocarbons), and finay with deepest impact and on a goba scae the anthropogenic greenhouse effect. It has taken decades to understand some of these phenomena because of their compex chemistry. In view of the rapid industriaisation taking pace in some densey popuated parts of the word and with the most serious consequences of the anthropogenic greenhouse effect yet to emerge, it is beyond doubt that measurement and understanding of atmospheric trace gas concentrations is of paramount interest. In contrast to the principa constituents of the air, these concentrations depend on emission sources and the chemica environment, so that, in fact, we are speaking of trace gas concentration distributions, varying spatiay and temporay. Depending on the trace gas species and the part of the atmosphere one is concerned with, the concentration distributions are more or ess we understood or reevant, so that motives to dea with them may be, e.g., to understand their chemistry, to aocate emissions, for pure monitoring purposes or to evauate modes and so forth. These reasons have gained significance on a wide range of spatia scaes. For exampe, goba CO 2 distributions in the context of wordwide emission trading have to be monitored just as air poution at different sites on a communa eve. Goba chemica transport modes designed to utimatey predict future trends have to be vaidated by measurements just as disperion modes aowed by EU reguations for monitoring air poution on a oca scae down to traffic hot spots [e.g. Schatzmann et a., 26; Trukenmüer et a., 24]. Measurements of trace gas concentrations are either in situ, i.e. oca, or remote sensing, which here means using the interaction of moecues with eectromagnetic waves to infer atmospheric parameters aong the propagated path. A singe measurement of the former kind provides the vaue of the concentration fied at the measurement ocation. 1 A singe measurement of the atter kind gives a ine integra aong the propagated path over a functiona of the concentration fied ideay over the fied itsef. Either method bears more or ess apparent drawbacks. Interpreting in situ measurements 1 Measurements that restrict the wave propagation to a sma voume by mutipe refections are regarded as point measurements in this context. 14

15 15 aways raises the question of their representativeness, especiay if the variabiity of the concentration fied is unknown. Remote sensing methods give information ony on ine integras invoving the desired concentration distribution. Whie getting spatia information from point measurements simpy needs the experimenta act of taking enough sampes, in order to obtain the same from any number of remote sensing measurements additionay requires mathematica techniques to retrieve the concentration fied from the integras. The combination of taking the right remotey sensed sampes and subsequent inversion techniques to unrave the integras is commony referred to as tomography. Naturay, it has become an extremey eaborate technique in discipines where taking sampes of the object of investigation is impossibe or prohibitivey expensive. Medica tomography benefits from sma regions of interest and it is reativey easy to hande source-detector configurations, so that arge numbers (of the order 1 5 ) of integras can be measured. Huge numbers of data ( ) for acoustic trave time tomography of the earth s interior are convenienty provided by earthquakes or exposions. There is a variety of spectroscopic techniques appied to the remote sensing of trace gases in the atmosphere, differing either in the waveength range or the physica principe they expoit for detection, with natura or artificia radiation sources (passive or active remote sensing). The most important ones are infrared spectroscopy (IR), the ight detection and ranging (LIDAR) technique and the differentia optica absorption spectroscopy (DOAS). Athough these methods are abe to detect various trace gases with high sensitivity for certain spatia ranges, in genera, they have not been used for tomographic measurements of atmospheric trace gas distributions. Byer and Shepp [1979] put forward the idea to reconstruct 2-D atmospheric trace gas distributions near ground from tomographic absorption LIDAR measurements with a rather eaborate setup. About the same time Feming [1982] proposed tomographic sateite measurements of IR and microwave emissions for 2-D reconstruction of temperature and trace gas distributions by increasing the number of viewing directions from one to five. Whie the atter proposa for passive measurement has been reaised in a smiiar manner in aircraft DOAS measurements using sunight [Heue, 25], so far no active experiment has been performed in the atmosphere that comes near the number of integration paths considered by Byer and Shepp [1979]. Comparabe optica setups for 2-D tomographic reconstruction of indoor air contaminants have been studied mosty in simuations ight path numbers in actua gas chamber experiments using the Fourier transform IR (FTIR) spectroscopy were between around 3 and 2. 2 Active DOAS tomographic measurements aong 16 ight paths were used by Laeppe et a. [24] to reconstruct the NO 2 distribution perpendicuar to a motorway. Beotti et a. [23] reconstruct vocanic CO 2 from horizonta open-path IR measurements aong 15 paths. On the one hand, active DOAS measurements aow precise detection of a variety of important trace gas species for ight paths between a few and up to 1 2km at the same time. On the other hand, amost 3 years after first proposas of tomographic remote sensing, this technique sti has not been used for genuine tomographic DOAS measurements motivation enough to advance both experimenta techniques and theoretica insight. As a theoretic contribution to these efforts, this thesis investigates the possibiity to reconstruct 2-D distributions of tropospheric trace gas concentrations from active DOAS measurements aong a moderate number of about 1 to 4 ight paths. The mathematics of computerised tomography (CT) with reguar geometries and arge numbers of integration paths, as they appear, for exampe, in medica image processing, is we understood. The appication at hand 2 See sec. 3.3 for references.

16 16 1. Introduction with ony few ight paths which are unikey to be arranged in a reguar fashion requires consideration beyond the existing iterature because anaytic inversion methods that are standard in CT cannot be appied (due to the irreguar geometry), estimation of the error of the reconstruction resut is not standard and contributions that are negigibe esewhere become important, smooth atmospheric distributions may differ from absorption patterns in the human body or ayer structures in the earth, discipines ike computerised tomography, geophysics and atmospheric profiing have deveoped powerfu concepts for their inverse probems and sometimes quite different retrieva techniques. It is not aways obvious why this is the case and which methods can be adopted. Existing studies with sma numbers of integration paths are scarcey systematic. The ack of a standard iterature that our methods coud refer to is refected by the structure of this work, being spit into three parts. The first part reviews basic facts on trace gases, their remote sensing and the mathematics of tomography, whie the second part contains the methodoogy party deveoped in this thesis for the cacuation of the reconstruction error, for the reconstruction itsef and the panning of a tomographic experiment. It aso contains a anaytica resuts. The third part consists in numerica appications. Not ony is there no standard approach to atmospheric tomography, emissions, transport and chemica transformation make the situation aso far too compex to admit a standard picture of atmospheric trace gas distributions. Therefore, after presenting some of the most important trace gases reevant for DOAS, chapter 2 considers two scenarios of trace gas distributions that wi be reevant for the third part: dispersion from a point source and the compexity of urban trace gas distributions. Chapter 3 gives an introduction to the tomographic principe and the most important remote sensing methods that potentiay coud make use of it. The DOAS technique is discussed in more detai. Finay, previous tomographic measurements of atmospheric parameters are presented and put into the context of standard tomographic appications. Chapter 4 justifies the discretisation and reformuation of the tomographic reconstruction probem as a east-squares east-norm probem. Apart from the very common box discretisation, inear parametrisation is considered for the 2-D case. Specia attention is paid to the question how known instabiities of this probem (so-caed i-posedness) affect different kinds of soutions. The optima estimate that is commony used to sove inverse probems in atmospheric science is presented both as a reguarisation method (to stabiise the east-squares soution) and as a probabiistic approach. The iterative soution of the east-squares east-norm probem that has been adopted from computerised tomography is discussed in comparison with other iterative agorithms before the chapter coses with brief sketches of aternative reconstruction principes. The detaied discussion of the reconstruction error in chapter 5 incudes the error reated to the finite representation of the soution that is usuay not covered in the tomographic iterature. We present a method for numerica estimation of the reconstruction error from test distributions that differs in severa points from the one empoyed by Laeppe et a. [24]. The fina section reviews and compares quaity criteria used in image processing and atmospheric modeing. The discussion up to then wi be irrespective of the concentration distributions or the dimensions of the reconstruction. The schemes

17 17 proposed in chapter 6 use severa grids for parametrisation and reconstruction and are especiay intended for 2-D reconstruction of peak distributions as they arise from point sources. The discussion of a priori information and background concentrations is aso specific for atmospheric trace gas distributions. Chapter 7 deas with experimenta design issues ike the signa to noise ratio, incuding additiona point measurements and how tomographic measurements can be used to evauate mode resuts. Concepts and methods of the first two parts are appied to the scenarios introduced in chapter 2: to 2-D Gaussians peaks representing sections through atmospheric emission puffs (chapters 8 & 9) and to mode distributions of a highy pouted city street canyon (chapter 1). The simuations of chapter 8 investigate the 2-D reconstruction of concentration peaks systematicay with respect to the parametrisation of the probem and the reconstruction scheme. Two iterative agorithms are compared to study the effect of reguarisation. Finay, different kinds of path geometries are investigated in detai and the roe of background concentrations is discussed. Chapter 9 appies the simuation resuts to data from a tomographic indoor experiment [Mettendorf et a., 26] and contains a detaied estimation of the compete reconstruction error. The fina chapter 1 highights the chaenge of reconstructing trace gas concentration fieds in a compex urban environment by means of high resoution mode distributions and shows how mode evauation coud be carried in a rea experiment even with ow spatia resoution of the reconstructed distributions.

18 Part I. Basics 18

19 2. Trace Gas Distributions in the Atmosphere This chapter starts with a brief motivation of the experimenta efforts to measure the spatia distribution of trace gases. The discussion is not as genera as the chapter tite may suggest but, ike the compete work, it is restricted to near ground trace gases detectabe by DOAS and to spatia (and tempora) scaes reevant to a specia kind of this technique, the ong-path DOAS, i.e. scaes from a few 1m up to around 1km. The remaining sections try to give an idea of how distributions actuay arise on these scaes from chemica and maybe more important transport processes. We concentrate on two specia scenarios. Point emission (sec. 2.4) is important for practica appications ike stack emission from industria compexes or power pants or, in fact, for any distinct source. But because it is such a simpe scenario, it is aso idea to study the nature of turbuent concentration fuctuations. Urban trace gas distributions, on the other hand, are of great pubic interest but far too compex to aow a universa picture. Therefore sec. 2.5 highights some aspects of urban mode deveopment. The reconstruction of point source emissions wi be the subject of simuations in chap. 8 and of a tomographic experiment in chap. 9, whie resuts from the mode system M-SYS designed for urban micro environments wi be used in chap The need for measured trace gas distributions Apart from pure scientific curiosity directy measured trace gas concentration distributions are desirabe in the foowing situations: Emission sources are uncear. Chemica transformation processes are not we known. For mode evauation and input. For air poution monitoring. The first point can mean that a source is suspected but not identified or characterised for a atmospheric conditions. Exampes might be the yet not competey understood chemistry of haogen oxides in maritime areas [Von Gasow et a., 24; Peters et a., 25], the aocation of biogenic emissions [Kessemeier and Staudt, 1999] or exporation of geoogica sources such as vocanoes or ava fows [Bobrowski et a., 23]. As emission into the atmosphere aways impies transport, the easiest way to get a comprehensive picture of emission sources is to produce a concentration map for the region of interest. A second reason for uncear emissions might be that, athough the source is known, it 19

20 2 2. Trace Gas Distributions in the Atmosphere may be hard to measure directy under rea word conditions. This is the case for traffic emissions on a motorway [e.g., Corsmeier et a., 25b], take-off and reguar emissions of air panes are another exampe. Certainy, the chemica transformation of reactive species is extremey compex and can be studied in the aboratory ony to a imited extent, especiay if precipitation or surfaces pay a roe. An important area of ongoing research is deaing with the detais of the chemistry of photo smog as it occurs in heaviy pouted urban areas and cities during summer. The situation concerning both the emissions and the effects of transport and turbuent mixing are too compicated to be simuated in a smog chamber. Compicated fow patterns in urban areas due to buidings make aocation of point-ike samped concentration vaues extremey difficut and time series of concentration fieds combined with wind data woud increase the information content tremendousy. The verification of modes that in their most compex form are designed to simuate emission, transport and chemica evoution of trace gases and partices on different atmospheric scaes is important. Even if emissions and reaction paths are modeed with sufficient precision there sti remains the transport cacuation. Whie describing dispersion of poutants over even terrain and on mesoscaes ( O(1 2 )km) might be a reativey easy task if one does not ook into spatia detais, compex terrain and fow patterns on microscaes ( O(1 2 )m) are more of a probem both conceptuay and computationay. Boundary conditions for poution transport modes are pre-processed by meteoroogica modes. But whie nobody wi ask what the weather is ike next street, concentration variations on a street scae are important, for exampe when it comes to accumuation of traffic emissions in a street canyon or whether the NO x pume of a power pant meets a near surface VOC pume and ocay forms ozone or not. In view of the fact that buiding effects can change concentration vaues by an order of magnitude [Schatzmann and Leit, 22], both meteoroogica and chemica transport modes have to meet rather chaenging requirements on urban scaes (see sec. 2.5). The experimenta proof that a mode predicts concentrations with enough accuracy is thus indispensabe and itsef an intricate matter as discrepancies have to be traced back to either wrong emission data, incompete chemica mechanisms, fauty transport cacuations or insufficient parametrisation of atmospheric turbuence. Point measurements are in genera not suited due to high concentration fuctuations both with time and ocation whereas concentration maps with sufficient tempora and spatia resoution give information on the ocation of emissions, their transport and concentration variabiity. The discussion of mode evauation wi be resumed in sec Addressing the ast point, air poution monitoring, it can be remarked that despite technica progress in reducing emissions factors that is the average amount of a poutant per amount of fue or for a specific process for exampe, by improved fitering of industria discharge or by cataytic converters for cars, absoute amounts of emission can sti be high in regions with extensive industry. They can be even growing if the absoute numbers of cars increase to a higher degree or if countries radicay intensify their industria production. Within the EU, air poution and its monitoring on a nationa eve is reguated by EU aws. As a matter of fact, the European Air Quaity Guideine 96/62/EU and its daughter directives reach down to the municipa eve by requiring concentration maps from members not ony for the state in tota and arger conurbations (more than 25 habitants) but aso for heaviy pouted micro environments [Schatzmann et a., 26]. If provided by point measurements, the EU directives reguate that measurements at urban background sites shoud be representative for an area of severa km 2 and at traffic-orientated sites for at east 2m 2 [Trukenmüer et a., 24].

21 2.2. Seected trace gases 21 poutant quaity indicator mode quaity objective SO 2, NO 2, NO x houry mean 5-6% daiy mean 5% annua mean 3% Ozone 8 h daiy maximum 5% 1 h mean 5% Benzene annua mean 5% CO 8 h mean 5% Tabe 2.1: Modeing quaity objectives estabished by the EU directives 1999/3/EC, 2/69/EC and 22/3/EC. Quaity objective is defined as the maximum deviation of measured and cacuated concentration vaues within the period given by the quaity indicator (to which the imit vaues refer, too). Appying an obstace resoving chemica transport mode with high spatia resoution (up to 1 m) to a heaviy pouted street canyon, Schünzen et a. [23] show that point measurements at ground eve barey fufi these requirements even under stationary meteoroogica conditions. Effective experimenta techniques to obtain concentration maps rather than point sampes woud be invauabe for air poution monitoring. Legisation acknowedges the current state of measurement technique and the costs of monitoring stations by permitting modeed concentration maps for monitoring purposes (see aso tabe 2.1) which brings us back to the evauation of these modes by measurements. Even if chemica dispersion modes eventuay reach a eve aowing them to be used for operationa air quaity predictions, experimenta concentration distributions become by no means obsoete. One reason is that modes (at east in the short run) depend on the quaity of the input, i.e. initia vaues and boundary conditions. Another reason of universa significance ies in the fact that modes usuay aow a arge choice of parameters and different settings. For transport modes this coud be the grid size, grid spacing, turbuence cosure scheme etc. and this gives the user additiona degrees of freedom and makes it unikey that two users appying the same compex numerica mode to the same probem wi produce the same resuts. This was demonstrated in the European project, Evauation of Modeing Uncertainty (EMU) (Ha, 1997) in which four experienced user groups predicted the dispersion of dense gas reeases around simpy shaped buidings by using the same commerciay avaiabe CFD [computationa fuid dynamics] code. The variabiity between different modeer s resuts was shown to be substantia. Depending on the quantity under consideration, differences up to an order of magnitude were reported. [Schatzmann and Leit, 22] 2.2. Seected trace gases Trace gases, occurring with number densities way beow the main constituents of unpouted air nevertheess pay an important roe because of their chemica reactivity, for the radiation budget of the earth (greenhouse effect) and for the human heath. Foowing up the preceding discussion, here we are interested ony in species that emerge in near ground air poution and which can be detected with DOAS measurements reevant for this work. Some of these trace gases are isted in tabe 2.2

22 22 2. Trace Gas Distributions in the Atmosphere trace gas major sources major sinks mixing ratios [ppb] remote rura-urban pouted-urban O 3 photochem. production, deposition, hydroysis, up to 3 free radicas photoysis, NO 2 fossi fue, soi emission, oxidation by OH or O biomass burning, ightning to HNO 3 SO 2 fossi fue, vocanoes, dry deposition, up to a few 1 suphide oxidation oxidation to H 2 SO 4, wet deposition HONO het. formation on photoysis surfaces CH 2 O, traffic, incompete photoysis (CH 2 O), Benzene, combustion, photochem. reaction with OH, & Touene production (CH 2 O) dry/wet depos Tabe 2.2: Sources, sinks and approximate mixing ratios for seected trace gases occurring in near ground air poution and that are detectabe with ong-path DOAS. Mixing ratios are taken from [Finayson- Pitts and Pitts, 2, sec. 11.A.4]. with typica concentrations 1 in pouted and unpouted air. 2 Starting to investigate the tomographic reconstruction of trace gas distributions from DOAS measurements, we concentrate ony on the most important atmospheric processes responsibe for the spatia patterns of the concentrations. Transport processes are addressed in the next section, very basic textbook facts on the chemica behaviour of important species are given now. Further detais can be found, for exampe, in [Seinfed and Pandis, 1998; Finayson-Pitts and Pitts, 2]. Nitrogen oxides and ozone The argest source of NO and NO 2 is fossi fue combustion in industria processes, power generation and traffic. Reative amounts of NO and NO 2 in the exhaust depend on the detais of the combustion process. For power pants the ratio of NO 2 to NO varies between ess than 1% and 4%, with typica vaues between 5 1% [Band et a., 2]. For traffic emissions, the ratio takes vaues around 5 1% [e.g. Koher et a., 25; Bäumer et a., 25; Stern and Yamartino, 21; Berkowicz et a., 1997]. NO quicky reacts with ozone to NO 2, so that they are usuay combined to NO x by [NO x ] = [NO] + [NO 2 ]. NO 2 photodissociates for waveengths 42nm, eading to the formation of ozone. In tota NO + O 3 NO 2 + O 2 (R 2.1) NO 2 hν λ 42 nm NO + O(3 P) (R 2.2) O( 3 P) + O 2 M O3, (R 2.3) where the coision partner M enabes the recombination. In the absence of any other chemica transformations, reactions (R 2.1) (R 2.3) ead to an approximate steady state described by the so- 1 To be correct, concentrations (amounts of trace gas per voume) shoud be distinguished from so-caed mixing ratios which are amounts of trace gas per amount of air, both referring to voume, moe or mass and the corresponding units being parts per miion (ppm) or biion (ppb) by voume, mass or moe. But we wi make a distinction ony if crucia and use the same symbo c for both quantities. 2 Foowing the definition in [Seinfed and Pandis, 1998, p. 49], a substance benign or harmfu is regarded as poutant, if norma ambient concentrations are exceeded due to anthropogenic activities with measurabe effect on the environment or humans.

23 2.2. Seected trace gases 23 caed photostationary state reation [NO] [NO 2 ] = j (R2.2) k (R2.3) [O 3 ]. Here k (R2.3) is the rate constant of reaction (R 2.3) and j (R2.2) the photoysis frequency of NO 2. For typica O 3 mixing ratios of a few 1ppb, at daytimes the ratio [NO]/[NO 2 ] takes vaues of the order one. The above system of reactions offers ony an insufficient description if stronger oxidants pay a roe. In fact, these are usuay present in the form of radicas, most importanty the hydroxy (OH) and the hydroperoxy (HO 2 ) radicas. They are photochemicay produced, predominanty from ozone, HONO and formadehyde (CH 2 O). On the one hand, the oxidation of NO by HO 2 NO + HO 2 NO 2 + OH instead of reaction (R 2.1) eads together with reactions (R 2.2) and (R 2.3) to a net production of O 3. On the other hand, there is a cyce between OH and HO 2 in which the radica HO 2 is formed by the reaction of OH with oxidisabe gases ike CO or CH 2 O. The reaction of HO 2 back to OH via O 3 + HO 2 OH + 2 O 2 pays a roe for ow NO x concentrations and resuts in a net destruction of ozone. Whether the couped NO x and HO x (= OH + HO 2 ) cyces eventuay produce or destroy O 3 depends on the NO x concentrations. For higher NO x, eves O 3 production exceeds the destruction by radicas. Lower eves resut in net destruction of O 3. Apart from its roe in the HO x cyce, the highy reactive OH radica constitutes aso a day time sink of NO 2 by the reaction NO 2 + OH M HNO 3, which is important for high NO x eves, whie at night the oxidation by O 3 becomes a NO 2 sink. In both cases the product is HNO 3, which is finay removed by precipitation and dry deposition. Voatie organic compounds The arge cass of voatie organic compounds (VOCs) comprises hydrocarbons such as akanes, akenes and akines, aromatic hydrocarbons and oxidised compounds (adehydes, ketones, acohos, organic acids). Some occur naturay, ike methane (CH 4 ) or terpenes which are produced by certain trees. The dominant anthropogenic source for VOCs ike benzene (C 6 H 6 ), touene (C 7 H 8 ) or formadehyde (CH 2 O) is (incompete) combustion in traffic, domestic fue or industry. Some VOCs, ike benzene, pay prominant roes in air poution because of their severe toxic character. But in any case, VOCs are important for the production of HO 2 and thus for the ozone production,

24 24 2. Trace Gas Distributions in the Atmosphere as iustrated here for the group of akanes (written as RH) OH + RH R + H 2 O R + O 2 M RO2. In pouted areas, the reaction of the peroxy radica RO 2 with NO is most ikey so that RO 2 + NO RO + NO 2 RO + O 2 R OHO + HO 2. Simiary, other VOCs drive the HO x NO x system to produce ozone, whie the VOCs themseves are reacted to adehydes and finay to CO 2. The interpay between the HO x and NO x cyces resuts in an intricate dependency of the O 3 production on inita VOCs and NO x concentrations (for detais and the so-caed VOC- and NO x -imited ozone production see the references mentioned in the introduction). Suphur dioxide Suphur compounds enter the atmosphere both naturay (vocanoes, bioogica sources) and anthropogenicay. In the atter case they are emitted predominanty as SO 2 when burning suphur containing coa or fue. In the context of air poution, the reaction of SO 2 to suphuric acid (H 2 SO 4 ) and subsequent emergence of acid rain has been a major issue. Concentrating on emissions in the form of SO 2, the most important gas phase reaction is oxidation by OH SO 2 + OH M HOSO 2 HOSO 2 + O 2 SO 3 + HO 2 SO 3 + H 2 O M H 2 SO 4, but SO 2 is aso removed from the atmosphere by iquid phase oxidation and dry deposition. Depetion by the above oxidation process is quite moderate. For a reativey cean urban environment and a coudess summer day, the 24h averaged rate of SO 2 oxidation amounts to.7%/h assuming [OH] moecues cm 3 [Seinfed, 1986, p.167]. In concusion, it has to be emphasised that the picture of the reactions of trace gases just given does not even contain a basic mechanisms for exampe, the NO 3 radica and the nitrous acid HONO were omitted, the suphur compounds were reduced to SO 2 and so forth. But nevertheess, besides conveying insight into the compexity of air poution, it expains important characteristic features of trace gases ike diurna cyces for some of them and gives an idea about when certain simpifying assumptions may be vaid and when not. For exampe, on time scaes of a few hours the evoution of a SO 2 pume may be approximated by wind transport and dispersion, negecting its chemica transformation. In constrast, freshy emitted NO NO 2 in the emissions, for instance, of a power pant undergo rapid transformation during their dispersion unti they eve off at some ratio more or ess given by the photostationary state reationship. As there are no point sources for O 3 as such, spatia variations of the ozone concentration can ony resut from high oca production rates, as in the so-

25 2.3. Dispersion by turbuent diffusion MACRO LOCAL MESO MICRO 1 mm 1 m 1 km 1 km CFCs N 2 O CH yr ong-ived characteristic time scae [s] sma cumuus turbuence induced by a forest sma-scae turbuence cycone hurricane oca winds thunderstorm arge cumuus tornado jet stream anticycone CO Aerosos trop. O 3 SO 2 H 2 O 2 NO x C 3 H 6 C 5 H 8 CH 3 O 2 HO 2 NO 3 OH 1 yr 1 d 1 h residence time 1 min 1 s moderatey ong-ived short-ived boundary ayer mixing time characteristic horizonta distance scae [m] Figure 2.1: Tempora and spatia scaes of turbuent atmospheric processes and ifetimes of some species in the atmosphere. Transport of the atter is characterised by spatia scaes according to the turbuent scaes. The precise definition of the terms micro-, meso- and macroscae is not aways the same in the iterature but this of no concern here [combined from Stu, 1988; Wayne, 2]. caed photo-smog, or depetion, for exampe by reaction with freshy emitted NO. Ozone distributions show itte spatia variation at ow concentration eves in unpouted background air, but enhanced vaues can occur downwind of urban centres where the O 3, graduay formed in the NO x and VOC enriched city pume, reaches its maximum. The ife time of a species in the atmosphere is of great interest for transport and dispersion processes. Figure 2.1 gives an idea about the time scaes for some trace gases. But whie these average vaues might be usefu for species with ony few reaction paths (as seen for SO 2 ), they become ess meaningfu for compex systems where not ony emissions vary, but aso factors ike sunight or coud cover affect chemica transformations. For NO 2 Neophytou et a. [25] specify diurna variations of the transformation time scae between a few minutes and up to 1 4 minutes Dispersion by turbuent diffusion The tempora evoution and spatia distribution of a scaar quantity ike the concentration c i (r,t) of a trace gas species i is governed by a continuity equation of the form c i t + (u c i) = D i c i + R i (c 1,...,T) + S i (r,t), (2.1)

26 26 2. Trace Gas Distributions in the Atmosphere where u denotes the veocity fied, D i the moecuar diffusion constant for species i, R i its rate of generation or destruction by chemica reaction and S i is a source term. Here and in the foowing it is assumed that this equation for c i is independent of the equations for u, the temperature T and a other meteoroogica quantities. But even in the most simpe case of a chemicay inert gas without source and sink (the moecuar diffusion term is of the order smaer than the advection term ( )), eq. (2.1) cannot be soved because u contains an unknown, stochastic component reated to turbuences in the atmosphere. Turbuent eddies can be thermay generated by the heating of the atmosphere or mechanicay caused when air moves past an obstruction. Turbuence occurs under a circumstances and on the whoe time and ength scae of atmospheric phenomena (fig. 2.1). The veocity fied u is of course not totay erratic so that it makes sense to write it as a sum of the deterministic mean ū and a random part u u = ū + u. As a consequence the concentration fied, too, consists of a deterministic and a random contribution in a sense iustrated in fig. 2.2a. The (hypothetica) average c i (r,t) for species i can be regarded as the ensembe mean of a measurabe, rea distributions c i (r,t). It can be obtained by time averaging over measurements if the meteoroogica conditions for ū are stabe and the chemica generation and destruction processes are more or ess stationary compared to the turbuent time scaes for the veocity fied. Put in other words, these time scaes give a cue to what degree the distribution c i (r,t) is representative for the mean meteoroogica conditions. For this reason and without going into the detais of the statistica representation of turbuence, I woud ike to point out the foowing with regards to the turbuent variations of the veocity and concentration fieds. The time scae for the veocity turbuence can be defined by the time integra of the Lagrangian autocorreation function of the veocity fied, that is the autocorreation in a system moving aong with the veocity fied. This function is hard to measure and an exact reation with the corresponding Euerian function in usua fixed-point coordinates is not known. Often they are assumed to be of the form e t/t with different times T L, T E for the Lagrangian and Euerian function. Based on the Tayor (frozen eddy) assumption one finds the estimates T L (repetition rate around dominant eddies of size ) 1 T E (repetition rate of dominant eddies) 1 ū T L = C ū T E σ v σ v (2.2a) (2.2b) (2.2c) with mean veocity ū, standard deviation σ v of the vertica veocity component and C.7 inferred from observations [e.g., Backadar, 1997]. Empirica vaues for T L and σ α = σ v /ū can be found in appendix A. Time scaes for the concentration turbuence can be defined in a simiar manner. But there is no reason why they shoud match the veocity time scaes. On the contrary, for the concentration fuctuations in time series obtained from a number of smoke pume experiments, Hanna and Insey [1989] find that the time scae of the concentration spectra is 2 5 times ess than for the wind speed and 1 2 times more than that of the vertica veocity. Turbuent patterns

27 2.4. The Gaussian pume mode 27 z c(r, t) c(r, t) q y h ū x (a) (b) (c) Figure 2.2: (a) A concentration fied c(r, t) under turbuent conditions as a member of a stochastic ensembe with mean c(r, t) which is assumed to be a Gaussian distribution here. (b) Emission from a point source at an effective height h with source strength q [kg s 1 ]. (c) Evoution of a pume depending on the eddy size [adapted from Seinfed and Pandis, 1998]. in the time series transate into variances σ c of the concentration around the ensembe mean (see next section). As a deterministic approach to turbuence is not possibe, the aim must be to describe the mean fieds for the concentration c i (r,t) as accuratey as possibe. Inserting the decompositions for u and the corresponding one into eq. (2.1) and averaging over the turbuent reaisations eads to c i t c i (r,t) = c i (r,t) + c i(r,t) (2.3) + (ū c i ) + u c i = D i c i + R i ( c 1 + c 1,...,T) + S i (r,t) (2.4) with two unknown quantities. The fact that the unwanted fux u c i cannot be substituted without introducing combinations of higher order in u j and c i is known as the cosure probem of turbuence.3 There are ways of varying compexity to sove the cosure probem eading to more or ess invoved numerica modes. The most simpe of them is to assume a diffusion ike reationship for the turbuent fux and the gradient of the mean fied, i.e. u c i = K c i with eddy diffusity tensor K, here K jk = K jj δ jk. This approach, known as K-theory, mixing-ength theory or gradient transport theory, resuts in equations that describe the turbuent dispersion of a poutant as a diffusive process parametrised by the eddy diffusities K jj (if moecuar diffusion is again negected and the mean veocity fied is taken to be incompressibe). They are referred to as atmospheric diffusion equations The Gaussian pume mode Assuming a constant mean veocity fied ū and negigibe chemica reaction, it is straightforward to show that the atmospheric diffusion equation has soutions in the form of Gaussian distributions. To 3 More precisey the cosure probem of the Euerian approach as in eq. (2.1).

28 28 2. Trace Gas Distributions in the Atmosphere 1 4 horizonta dispersion coefficient 1 5 vertica dispersion coefficient σy [m] P.-G. A 1 2 D CB 1 1 Briggs Urb. A=B FE E=F DC distance x downwind from source [m] (a) σz [m] distance x downwind from source [m] (b) Figure 2.3: (a) Horizonta dispersion coefficients (σ x = σ y) for the Pasqui-Gifford (thin ines) and the urban Briggs (thick ines) parametrisation. For the atter the dispersion coefficients for stabiity casses A and B and the ones for E and F are identica. (b) The same as (a) for the vertica dispersion. The functiona forms and further detais can be found in appendix A. get more specific, consider the situation depicted in fig. 2.2b, i.e. a point source at the origin, emitting at a rate q (in units kg s 1 ) at an effective height h and a wind fied ū = (ū,,). For instantaneous reease of the tota amount Q = q dt one gets the Gaussian puff formua c(x,y,z,t) = Q ( 8(π t) 3/2 (K 11 K 22 K 33 ) exp 1/2 [ exp ( (x ūt)2 4K 11 t (z h)2 4K 33 t ) y2 4K 22 t ) + exp ( (z + ) ] h)2, z, 4K 33 t whie the stationary concentration fied resuting from continuous reease is described by the Gaussian pume equation c(x,y,z) = q ( 8 π (K 22 K 33 ) 1/2 x exp ū 4x [ ( exp ū 4x y 2 ) K 22 (z h) 2 K 33 ) + exp ( ū 4x ] (z + h) 2 ), z, (2.5a) K 33 provided that the dispersion in the direction of the fow can be negected, i.e. that the distance x a partice has traveed since its reease is arge compared to the spread at that time K 11 t or K 11 ū x (so caed sender pume approximation). Both formuas are for boundary conditions corresponding to tota refection of the pume at ground (z = ). Simiar expressions can be derived for tota or partia absorption at ground and additiona refection at an inversion ayer above the pume [e.g., Seinfed and Pandis, 1998]. Based on arge numbers of observations (ike the egendary prairie grass experiment [Barad, 1958] and the urban dispersion study in St. Louis [McEroy and Pooer, 1968]) severa parametrisations of the pume dispersion coefficients σ x = σ y = 2K 22 x/ū and σ z = 2K 33 x/ū have been proposed. They

29 2.4. The Gaussian pume mode 29 usuay refer to a cassification of atmospheric stabiity proposed by Pasqui [1961] and distinguish six (sometimes seven) casses ranging from extremey unstabe (A) to moderatey stabe (F) (or extremey stabe (G)). Appendix A contains detais about this cassification in terms of meteoroogica observabes. For the widey used Pasqui-Gifford scheme [Gifford, 1961] fig. 2.3 shows σ y, σ z for the casses A F as a function of the downwind distance x. Briggs [1973] gave different parametrisations for open country and urban conditions (fig. 2.3). Expicit formuas can be found in appendix A. Athough these schemes have been introduced more than 4 years ago and new ones are being proposed [e.g., Hanna et a., 23] the origina curves are sti in use [Barratt, 22]. When appying the Gaussian pume or puff mode, apart from the steady state conditions assumed above, one shoud notice: The modes are not fit for compex terrain. It is the effective emission height h that enters the formua (important if the gas is emitted with initia momentum, see [Barratt, 22; Seinfed and Pandis, 1998] for correction terms). The parametrisations are not reiabe near the source [see aso Venkatram et a., 24], too far away from the source ( 1km), i.e. the uncertainty of σ y, σ z increases with x, for eevated sources higher 1m [Barratt, 22]. Stricty speaking, the σ y, σ z are vaid ony for the averaging times for which they have been derived (3 1min and 1h for the parametrisations mentioned above, see appendix A for detais). A singe measurement over a imited period wi not see a Gaussian concentration distribution, c.f. fig. 2.2a. A few remarks on the ast point which refers again to the context of fig. 2.2a and is especiay critica for the meandering pume in fig. 2.2c. Loosey speaking this is the case if the effective eddy size is much arger than the size of the pume. Turbuent eddies of a size comparabe to the pume change its size and disperse it diffusion-ike whie very sma eddies mix the pume within. The size of the dominant eddies can be estimated with eqs. (2.2) by consuting empirica vaues for T L and σ α. For exampe, using Draxer s specification for T L in appendix A one gets: For a surface source and stabiity cass C (sighty unstabe) 8 < 6 s atera T L = : 1 s vertica, σ α = 15 eq. (2.2c) T E = 8 < 22 s atera : 7.5 s vertica and for a wind speed of ū = 5 m s thus roughy 1 m. If the measurement time T is shorter than necessary to reproduce the ensembe mean, the expected deviation of the mean concentration c T during T at a specific ocation from the ensembe mean c at the same pace is according to Venkatram [1979] given by E [ ( c T c ) 2] var[ c 2 ] c c 2 2 2(Γ 1) TE T T T E ese

30 3 2. Trace Gas Distributions in the Atmosphere where Γ = cmax c with c max being the maximum concentration detected at the ocation and T E the Euerian time scae for the concentration turbuence. For the moderate exampe Γ = 1, T/T E = 1 one gets E [ ( c T c ) 2] 1.4 c 2, i.e. the deviation is of the same size as the ensembe mean vaue. For the specia case of a Gaussian pume the author gives a further theoretica expression for the expected deviation at the centreine of the pume in [Venkatram, 1984]. Ma et a. [25] give an estimate for the same quantity by anaysing experimentay obtained crosswind sections through pumes. They find that the crosswind fuctuation σ c (y)/c(y) aong y increases at the edges of the pume and gets smaer towards the centreine. For distances further away from the source and averaging times around 1 h, vaues at the edges of the pume amount to σ c (y)/c(y) 1 and at the centreine to σ c ()/c().1. Near the source the fuctuations increase dramaticay. In fact, the authors propose a fit of the form σ c ()/c() a + b x for the fuctuation at the centreine depending on the downwind distance to the source x. In principe, the Gaussian pume mode can be appied to ine and area sources, too [e.g., Barratt, 22; Seinfed and Pandis, 1998]. Simpe anaytic corrections of the Gaussian soution concern the initia pume size, pume rise and exponentia chemica decay of the emitted substance. More invoved chemica transformations have to be impemented numericay [e.g. Ocese and Tosei, 25; Song et a., 23; Von Gasow et a., 23]. The imitation to stabe atmospheric conditions can be ifted by using a series of puffs instead of a continuous pume. The frequenty empoyed numerica mode SCIPUFF [Sykes and Gabruk, 1997] is based on this approach. Other numerica muti-pume dispersion modes incorporating more or ess directy the Gaussian pume mode are mainy appied to impact assessment of reguar or accidenta emissions from industria compexes, power pants etc. There is a arge variety of these dispersion modes empoyed by environmenta institutions, nationa agencies and companies. The cassic industria source compex (ISC) mode 4 was deveoped by the US Environmenta Protection Agency (EPA) for source types from point to voume sources. It describes the dispersion by the Pasqui-Gifford parametrisation and accounts for deposition when concentration hits the ground. The further deveoped mode AERMOD incudes cacuation of vertica profies for meteoroogica parameters and treats dispersion under convective conditions more accuratey. Apart from the input of emission and meteoroogica data it requires specification of the topography to correct fow patterns. Likewise other state of the art modes ike the often mentioned Atmospheric Dispersion Mode System (ADMS) have modues for processing terrain data, meteoroogica parameters (e.g. the mesoscae meteoroogica mode MM5), pume rise (e.g. PRIME) and so forth. The Lagrangian Atmospheric Dispersion Mode (LADM) from the CSIRO Division of Atmospheric Research (Austraia) is capabe of modeing photochemica formation and dispersion of gases such as O 3 and NO 2, but in genera mode systems designed to predict atmospheric dispersion do not incude reaction mechanisms. Short descriptions for some of the atest dispersion mode systems can be found in [Barratt, 22] which aso contains a ist of reated internet sites Urban trace gas distributions Here, we are interested in concentration distributions from a tomographic point of view, i.e. mainy in their spatia patterns, not so much in their chemistry. The atter aspect has been very briefy 4 The mode is avaiabe in the ong term version ISCLT and the sort term version ISCST, the current version being ISCST3.

31 2.5. Urban trace gas distributions 31 addressed above in sec. 2.2 and we restrict ourseves here to the foowing observations concerning emission source types. Traffic is an important primary source for NO x, CO and VOCs (possiby SO 2 ). It can show strong tempora variabiity in terms of daytime or rush hours. The simpest way is to treat it as a ine source, a more invoved approach incudes traffic induced turbuences (which according to Tsai and Chen [24] can in street canyons be as important as wind induced turbuences for the mixing of the poutants). The source strength does not ony depend on the traffic frequency but aso on the emission factors. These are far from we-known as pointed out by Corsmeier et a. [25b] who report underestimations for the CO and NO x emission factors used by modes of around 2%. Domestic heating constitutes a seasona source of SO 2, CO and CO 2 (with diurna variations, see [Rippe, 25] for measurements in Heideberg) and can be regarded as an area source, at east in densey popuated downtown and residentia areas. Factories, industria compexes as we as power pants or bock power stations are sources for a variety of species ike NO x, SO 2, CO, VOCs and CH 2 O. As gases are discharged via more or ess high chimneys they represent the most typica case for the Gaussian pume mode. Impact of the emission can be estimated once the effective stack height and the stabiity cass have been specified. According to Barratt [22] the stack height is usuay chosen at east 2.5 times the height of any buiding within a radius of twice the stack height to make sure that downdraught of the emissions does not occur. The rue of thumb quoted in this context that the airfow is disturbed up to a height of 2.5 times the buiding height agrees with the observation that mechanicay induced turbuences usuay cause neary neutra stabiities over urban areas up to heights around two times the average buiding height [Hanna et a., 23]. Not ony emissions from within the town and its suburbs make up the urban trace gas distributions. Pouted air or air with a specia composition ike ozone rich air imported from outside add to the concentrations and react in a way they might not do in unpouted air. Turning from the emissions to the immissions: What kind of measurements are there to give insight into the spatia distribution of urban concentration distributions? We woud ike to characterise these efforts in the foowing manner: Intensive measurements aiming at the concentration distributions on a mesoscae, e.g. the chemica composition for particuar meteoroogica or topographica conditions or the chemica deveopment of a city pume etc. This kind of campaigns typicay comprises a variety of measurement techniques ike point or mobie in situ samping, remote sensing measurements and possiby airborne measurements and they usuay address very specific questions. Exampes are the BERLIOZ study that investigated the evoution of the Berin pume [e.g., Voz-Thomas et a., 23], the FORMAT project focussing on formadehyde in and around Miano in the heaviy pouted Po vaey [Hak et a., 25] or a series of measurements in the extremey pouted megacity of Mexico City to earn more about oxidation processes in the atmosphere [Shirey et a., 26].

32 32 2. Trace Gas Distributions in the Atmosphere Figure 2.4: Different scaes invoved in urban modeing: from mesoscae modes for background concentrations and meteoroogica boundary conditions to modes on the scae of a street canyon (shown here with a typica rotor ike fow pattern) [simpified from Fisher et a., 26]. Long-term in-situ measurements, especiay operationa air quaity monitoring. They provide important statistica data for individua sites but detectors are usuay not instaed in a way that admits concusions on the spatia distributions. The representativeness of these measurements, even for a oca neighbourhood, are questioned by [Schünzen et a., 23] on the grounds of mode cacuations with very high spatia resoution. Hot spot measurements on a microscae ike the measurement of concentrations and airfow within a street canyon [e.g. Schatzmann et a., 26; Vardouakis et a., 25] or around an industria compex. On its proper scae, each of these setups might give a comprehensive picture of the concentration fieds, but very often they do not: Too few point measurements, too short time series of data, not a reevant species measured and so on. Not east because data interpretation might be impossibe without mode assumptions, some aspects of modeing trace gas distributions in an urban environment are now introduced. The probem with urban distributions is that any mode describing them satisfactoriy has to be mutiscae, see fig On the spatia microscae buiding effects become important (and, as aready said, according to Schatzmann and Leit [22] can easiy change concentrations by more than an order of magnitude). Mesoscae conditions drive transport of airmasses to the town and away from it. A resut from ong-term measurements at a heaviy pouted street canyon in Hannover iustrates this: For a period of two years it was found that NO x concentrations at roof eve in the street canyon are to 3% due to regiona transport [Schäfer et a., 25]. But mesoscae conditions can aso be responsibe for more compex patterns as shown in fig According to Fisher et a. [26] the compexity of urban meteoroogy is not fuy acknowedged by modeers and [...] there seems to be a reuctance from mode deveopers to move away from famiiar concepts of the boundary ayer even if they are not appropriate to urban areas. Time scaes reevant for the mode are not just given by transport times but rather by chemica transformation times. These may range from 1 4 s for the OH radica to 1 8 s for CH 4 (see fig. 2.1) and can vary for one and the same species with mixing ratios of other trace gases or physica parameters ike radiation and temperature. Among the different concepts for describing urban trace gas concentrations we are here interested ony in methods that can at east in principe reproduce spatia (and tempora) variabiity on scaes

33 2.5. Urban trace gas distributions 33 Figure 2.5: Exampe of how mesoscae meteoroogica conditions can show up in the urban airfow patterns. This iustrates both the compexity of urban meteoroogy and the necessity of appying modes on different scaes [adopted from Fisher et a., 26]. reevant for ong-path DOAS measurements. This excudes a kinds of statistica modes, for exampe receptor modes [e.g., Seinfed and Pandis, 1998], or diagnostic modes which use empirica functions to simuate fow around buidings and eaves us mainy with chemica transport modes (CTM) that have a high enough resoution within the urban area. The probem of mutipe scaes can be tacked by either of the foowing two ways: A singe (Euerian) mode nested at different scaes [e.g., Pieke et a., 1992]. A system of modes, each for a different scae [e.g., Trukenmüer et a., 24; Souhac et a., 23; Kesser et a., 21]. Both approaches rey on further components that provide emissions, meteoroogica data and the topography within each mode/nesting domain as input, as sketched in the cartoon fig Here the meteoroogica modes provide fow fieds (where the mesoscae mode processes meteoroogica observations) that enter the (on- or offine) cacuation of concentration fieds by the chemica transport modes. Any detais depend on the specific mode system. For exampe, the emission component can depend on the meteoroogica conditions or contain a more or ess compex mode to simuate traffic emissions. Measured background concentrations or vaues interpoated from monitoring stations coud be used as immission constraints for the chemica modes and so on. The scaes regiona, urban and street in fig. 2.6 are not universay defined. 5 Domain size and grid resoution at each scae (nesting) depend on the mode and, of course, on the probem. A typica horizonta resoution for an area 2 2km 2 (regiona scae) is around 2 5km. To be more specific, mode domains and and grid resoutions for the first two mode systems mentioned above are as foows (the system in [Kesser et a., 21] ies somewhere between the former two). Souhac et a. [23] use a square area of 28 28km 2 with a grid spacing of 4km for the mesoscae mode around Lyon and a resoution of 5m within an 5 Britter and Hanna [23] suggest the definition of street scae for ess than 1 2 m, neighbourhood for scaes up to 1 2 km, city up to 1 2 km and regiona up to 1 2 km.

34 34 2. Trace Gas Distributions in the Atmosphere immission data meteoroogica data CTM regiona meteoro. mode emission data CTM urban meteoro. mode topographic data CTM street industria meteoro. mode Figure 2.6: Simpified sketch of an urban mode system. The scaes street, urban and regiona do not have a fixed meaning. The industria scae is given by the range of typica industria emissions, e.g. up to 1 km. The corresponding transport mode coud be a Gaussian dispersion mode. Mode systems can have more or ess scaes than shown. Meteoroogica and chemica cacuations can be done by the same modes on different scaes or by different modes for different scaes. Immission data coud enter at any eve of the system to nudge the CTM components towards observations. Emission data are suppied by inventories, modes for point and ine sources and possiby emission modes for vegetation, traffic etc. Topographic data incude surface eevation, buidings and any other obstaces. area of 32 32km 2 centred on the city. The vertica extension of the domains is 1km and 2.5km, respectivey, with 22 grid points in the first case and spacing between 2m (bottom) and 4m (top) in the second. The mode system M-SYS presented in [Trukenmüer et a., 24] has been deveoped to provide concentration distributions on a scaes reevant for EC air poution reguation. For the mode setup described in this pubication a mode domains are centred on a street canyon in the city of Hannover with the argest domain being 2 2km 2 with grid spacing 16km and a domain size of the innermost mode of 1 1km 2. In this sma area the grid is chosen non-uniformy such that the highest resoution (1.5m) is within the street canyon and the grid spacing at the boundary is 15m. Input of meteoroogica and emission data as we as forcing of the ower scae modes happens on an houry basis in both mode systems. The street canyon cacuations in [Trukenmüer et a., 24] use a stationary fow fied. There are, of course, many other mesoscae transport modes and appications on the regiona scae and quite a few on the street scae, bur far ess simuations on an urban scae as comprehensive and highy resoved as M-SYS. In particuar, the Euerian CTM CALGRID [Yamartino et a., 1992] shoud be mentioned at this point for its popuarity. Originay deveoped for regiona and urban scaes, it has been modified for use within the EU framework directives for appications on street scae, micro- CALGRID or MCG. For a first evauation point measurements of O 3, CO, NO and NO 2 in a street canyon (Schidhornstrasse, Berin) were compared with highy resoved cacuations (horizonta domain of 1 3m 2 with grid ces of 12 3m 2 and vertica spacing from 3m to 24m up to an height of 1m) and showed good agreement [Stern and Yamartino, 21]. It shoud be noted though that the mode was adjusted to measured background concentration during the 5 day mode run. Finay, for the sake of competeness, I woud ike to mention that concepts from computationa fuid dynamics (CFD) have proven increasingy usefu to describe fow around buidings and in street canyons [e.g., Puen et a., 25; Ridde et a., 24; Tsai and Chen, 24], in particuar arge eddy simuation modes that are capabe of describing at east arger turbuent structures. However, these modes are computationay too expensive to be appied on urban scaes [Schatzmann and Leit, 22].

35 3. Tomography and Remote Sensing of Atmospheric Trace Gases This chapter describes the experimenta foundations of this thesis. After a very basic introduction of the tomographic principe, common remote sensing methods are briefy reviewed with respect to their potentia for tomographic measurement of atmospheric trace gases. Here it has to be emphasised that instrumenta deveopment is ongoing in a areas and we do not attempt an outook. DOAS is covered in more detai not east to make the origin of the detection imits and measurement errors cearer athough the theoretica and simuation resuts in the foowing basicay appy to any remote sensing technique with we defined optica paths. A straightforward appication of the tomographic principe to measurements with scattered sun ight is not possibe and a proper discussion unfortunatey beyond the scope of this work. Whie the advantages of tomographic remote sensing are evident and have ed to detaied proposas for atmospheric measurements (sec. 3.3), none of the existing appications can be caed a we estabished technique. Finay, sec. 3.4 describes first tomographic DOAS experiments and sums up a comparison with other tomographic appications in tabe The principe of tomographic measurements To start with an iustration consider the foowing Exampe An observer sends out two ight beams A and B that trave on straight ines to two mirrors at different height above ground. From the spectra of the refected ight beams he can by some cever method deduce average concentrations of a trace gas aong the ight paths. What are the average concentrations in the two ayers 1, 2 in fig. 3.1a? mirror ayer 2, c 2 L B ayer1, c 1 amp+detector L A (a) (b) Figure 3.1: The principe of tomography. Loca properties of a fied are retrieved from path integrated quantities. (a) From average concentrations aong the beams, the mean concentrations in two ayers can be inferred. (b) In the same way, two dimensiona properties can be reconstructed. 35

36 36 3. Tomography and Remote Sensing of Atmospheric Trace Gases source detector source detectors (a) Parae scanning (b) Fan-beam scanning (c) eft: image pixes, right: Figure 3.2: X-ray tomography. (a) and (b) show two kinds of scanning in transmission tomography [Adopted from Natterer, 21]. (c) Axia CT scan of the brain from around 1975 and a state of the art scan using a scanning method as sketched in (b) [Siemens company]. Typica sizes of image pixes are around.1 1 mm. Detector arrays consist of about 5 1 detectors of size 1 mm. A fan is made of about 5 rays and, depending on the appication, the source-detector system is rotated for O(1) O(1) projection anges. The average concentrations aong the ight paths are c I = 1 I ZL I ds c(s) I = A, B, and if Ai and Bi are the engths of the ight paths in ayer i = 1, 2, respectivey, then one gets a simpe system of equations for the average concentrations c i in box i c A = A1 A c 1 c B = B1 B c 1 + B2 B c 2. Despite its simpicity this exampe reveas the basic principe of a tomographic reconstruction. Extending it to two dimensions works obviousy by adding integration or ight paths in further directions and asking for the two dimensiona dependency of the desired quantity as in fig. 3.1b. Three dimensiona reconstructions can be obtained correspondingy or by adding sices of two dimensiona reconstructions. 1 The principe of tomography can be characterised by either of the two ways as: A non-invasive measurement technique that from the propagation of waves aong a sufficient number of paths through a medium reconstructs oca properties of this medium. The reconstruction of a function from a set of integra equations given by the ine integras of the function aong a sufficient number of integration paths. In practice, tomographic measurements are made using either acoustic waves or eectromagnetic fieds (mainy X-ray, microwaves, radiowaves). Depending on the phenomenon under investigation and the mechanica/eectrica properties of the medium, the acoustic/eectromagnetic waves can be emitted, absorbed or transmitted and possiby be refected and diffracted. The associated tomographic techniques are caed emission tomography, transmission tomography and so forth. The cassic exampe of 1 This is where tomography got its name from: tomos meaning sice and graphia describing.

37 3.2. Atmospheric remote sensing 37 a tomographic measurement is the X-ray absorption tomography appied to the human body, fig.3.1. Passage of X-rays through the body can be described by straight ines and an intensity attenuation aong the way x obeying Lambert-Beer s aw I/I = f(x) x, where the attenuation coefficient f(x) depends on the tissue, bone structure etc. In its integrated form n(i /I ) = dx f(x) this can L readiy be seen to be a ine integra aong a straight ine L. Parae scanning as shown in fig. 3.2a was introduced by A. Cormack in 1963 and aso appied in the first commercia scanner. Later fan-beam scanners were used (fig. 3.2b). The fied of X-ray tomography has coined the expression computerized tomography (aso computed tomography) (CT) for computer assisted tomographic reconstruction in the widest sense and (together with ater deveopments ike the singe partice emission CT (SPECT) or the positron emission tomography (PET)) it has produced numerous techniques and agorithms for tomographic reconstruction (see chapter 4, especiay sec. 4.9). Another highy deveoped tomographic discipine is seismoogy that tries to infer the earth s interior from trave time differences of earth quake signas due to different materia aong their way (trave time tomography)[e.g., Noet, 1987]. This technique is aso pursued activey by acoustic or eectromagnetic emitters in borehoes or on the surface as we as in oceans [e.g., Munk et a., 1995] using either varying transmittance or refectivity to earn something about the structures within the medium. There are numerous other experimenta methods ike the nucear magnetic resonance imaging (NMRI) or the eectrica impedance tomography (EIT) deveoped for medica use and tomographic appications in areas different from medicine and earth science, e.g. bioogy, materia testing or pasma physics. Even a short survey of experimenta techniques is beyond scope, therefore we wi concentrate on atmospheric appications in sec Remark on the effect of measurement errors in the exampe on page 35 Trivia as they may be, the exampe reveas another characteristic feature of tomographic measurements. Suppose that the observer intends to make a very precise measurement for a very thin ayer 2, i.e. B2 / B1 1. If the reative measurement errors for the average concentrations c A and c B are ǫ c A / c A c B / c B then straightforward error propagation yieds the reative errors for the reconstructed concentrations c 1 = ǫ c 1 c 2 2 ǫ with = A B, c 2 i.e. the reative error of the reconstructed concentration for the thin ayer gets magnified by / 1. Anticipating the discussion in sec. 4.4, we remark here that the instabe behaviour occurs aso in two and three dimensiona probems and can in genera not be avoided by specia choices of the integration paths Atmospheric remote sensing Among the many experimenta techniques to anayse the chemica composition of the atmosphere some methods ike spectroscopic methods (gas spectroscopy, mass spectroscopy, optica spectroscopy) are fairy universa whie others can be ony appied to a group of species or just one species. Foowing up the preceding section, the most important remote sensing methods are now reviewed, having in mind

38 38 3. Tomography and Remote Sensing of Atmospheric Trace Gases (a) Vertica scanning of a NO 2 pume. (b) Horizonta scan of O 3 over Berin at 1 m height. Figure 3.3: Exampes for the range resoving capacity of LIDAR measurements (Courtesy of K. Stemaszczyk, FU Berin). their potentia to provide maps of concentration distributions either directy as a range resoving technique or by tomographic measurements. The roe of in-situ, i.e. point ike measurements for examining concentration distributions wi be briefy addressed in sec A remote sensing methods to detect atmospheric trace gases use optica spectroscopy differing either in the waveength range or the physica phenomenon they expoit. The method is denoted as active if it provides an artificia radiation source, passive if it uses a natura one. Current experiments use waveengths from the infrared (IR) to the utravioet (UV). Active microwave sensing does not pay a roe for the detection of trace gases (at east up to now) IR spectroscopy Infrared spectroscopy was originay used to detect CO 2. The deveopment of the Fourier transform IR spectroscopy (FTIR) aows the measurement of path averaged concentrations for trace gases ike NO, CH 4, HNO 3, CH 2 O, HCOOH and H 2 O 2. This technique was used recenty by Beotti et a. [23] with spatiay separated source and detectors to make a tomographic measurement of vocanic CO 2 emissions. The maximum ength of the optica paths for this technique is roughy 1km. Another deveopment is the use of tunabe diode aser spectrometers (TDLS) for measurements of NO, NO 2, HNO 3, CH 2 O and H 2 O 2. Both FTIR and TDLS have been used in a variety of tomographic chamber experiments to study indoor gas dispersion by means of tracer gases such as SF 6 and CH 4 [Fischer et a., 21; Samanta and Todd, 1999; Drescher et a., 1997] LIDAR The ight detection and ranging (LIDAR) technique works simiar to the probaby better known RADAR (radiowave detection and ranging). By sending out short puses of (aser) ight into the atmosphere and comparing them to the back-scattered signas, in principa, the spatia distribution of aerosos and trace gases aong the ight path can be retrieved. Because the back-scattered signa is too weak for confident detection of trace gases, measurement at two waveengths is necessary: at

39 3.2. Atmospheric remote sensing 39 SO 2 O 3 NO NO 2 λ on [nm] λ off [nm] ower (upper) DL [ppm] ([ppb]) 4 (27) 4 (25) 2 (65) 1 (8) accuracy [ppm] ([ppb]) ±1 (±1) ±1 (±1) ±5 (±5) ±15 (±15) max. spatia resoution [m] 3 (5) 3 (5) 3 (25) 3 (1) best range [km] 1 (1) 1 (1).5 (1.2) 3 (19) mean range [km] 3 (-) 3 (-).5 (-) 5 (-) Tabe 3.1: Specifications for the mobie LIDAR given in [Kösch, 199]. Vaues in brackets ( ) refer to measurements with topographica targets. DL stands for detection imit, see sec Waveengths are isted for competeness ony. λ on for which the species absorbs and at λ off for which it does not (so-caed differentia absorption idar (DIAL)). If mirrors or topographic targets are used to refect the aser signa, the sensitivity of the method is greaty enhanced but the range resoving power gets ost. The pure back-scattering DIAL can be used for measurement of strong emissions (with a imited range) or vertica immissions. Horizonta and vertica sections of SO 2 pumes from power pants with high spatia resoution have been obtained with a mobie instrument in [Beniston et a., 199; Kösch, 199]. Simutaneous measurement of NO and NO 2 from an artificia source (maximum vaues in the pume around 1ppm for both species) was reported in [Kösch et a., 1989], see fig. 3.3a. The same system was used to measure horizonta path averaged concentrations of SO 2, NO and NO 2 in various European cities and horizonta distributions of O 3 at 1m above ground over the city centre of Berin, see fig. 3.3b. Tabe 3.1 gives an idea about ranges and resoutions for the instrument used. Vertica profies of O 3 have been obtained in simiar experiments [e.g., Ducaux et a., 22]. In concusion it can be said that the current state of LIDAR technoogy aows measurements with high spatia resoution in the metre range for distances up to 1km ony for high trace gas concentrations of the order ppm, ong range (up to 1km) and poor spatia resoution for concentrations on the ppb eve DOAS The differentia optica absorption spectroscopy (DOAS) [e.g., Patt, 1994] is based on Lambert- Beer s aw retrieving integrated concentrations of trace gases aong a ight path from their narrow band absorption structures. In principe, it can be appied if these have a smaer width than 1nm which is the case for many trace gases ike O 3, SO 2, NO 2, HONO, CH 2 O and BrO. The technique is used both with direct or scattered sunight and with artificia ight sources in the UV, visibe and near IR waveength range. The focus wi be on the active DOAS method as the experimenta resuts for this thesis were obtained by this technique. Method The Lambert-Beer s aw appied to the case of a singe gaseous absorber with homogeneous concentration c describes the attenuation of radiation at waveength λ after passing the distance L by

40 4 3. Tomography and Remote Sensing of Atmospheric Trace Gases L σ(λ) c I(λ,L) = I (λ) e (3.1a) where I (λ) is the initia intensity and σ(λ) the waveength-dependent absorption cross section of the trace gas. Knowedge of the distance, the cross section and measurement of I and I aows to determine the concentration as c = τ σ(λ)l, where τ = n I (λ) I(λ) (3.1b) is the so-caed optica density of the ayer of thickness L. For the waveengths of interest here the UV and visibe trace gases in the atmosphere exhihibit absorption bands due to rotationa, vibrationa and eectronic excitations, so that a variety of species contribute to absorption at a given λ. Furthermore the ight is attenuated aso by scattering at the air moecues (essentiay through Rayeigh scattering with waveength dependency λ 4 ) and at aeroso partices (usuay described by Mie scattering λ n with n often taken around 1.3). Finay atmospheric turbuences ead to an intensity oss by widening the beam. Taking into account spatia dependencies, Lambert Beer s aw in the atmosphere thus takes the form I(λ,L) = I (λ)a(λ) exp [ L ds j σ j (λ,p,t)c j (r(s)) + ǫ R (λ,r(s)) + ǫ M (λ,r(s)) ] (3.2) where I(λ,L) is the intensity at the detector, c j (r(s)) the number density of species j at position r(s) aong the ight path and σ j (λ,p,t) its absorption cross section depending aso on air pressure p and temperature T. ǫ R (λ,r) and ǫ M (λ,r) are the extinction coefficients for Rayeigh and Mie scattering, respectivey and the factor A(λ) stands for intensity oss due to turbuences. The Lambert-Beer s aw describes the statistica behaviour of photons aong a straight ine from the source to the detector and does not appy straightforwardy to photons that reach the detector via scattering processes, as is the case in scattered sun ight experiments. In any case, to infer the c j from eq. (3.2) in addition to the absorption cross sections at east I and the extinction coefficients have to be known, which is extremey difficut to achieve. Aeroso concentrations are usuay not known. I for passive measurements is not known exacty, for artificia ight sources it might fuctuate in time, and so forth. The basic principe common to a appications of DOAS reies on the observation that the unknown or unwanted quantities vary ony sowy with λ whie the absorption cross sections have characteristic narrow band structures unique for each species, see fig Dividing the absorption cross sections into broad band and narrow band contributions σ j, σ j such that for the exponentias in eq. (3.2) exp[...σ j ] = exp[...σ j ] exp[...σ j ], one gets L I(λ,L) = I (λ)a(λ)exp [ ds σ j (λ,p,t)c j (r(s)) + ǫ R (λ,r(s)) + ǫ(λ,r(s)) ] j } {{ } I (λ,l,p,t) exp [ L ds j σ j(λ,p,t)c j (r(s)) ].

41 3.2. Atmospheric remote sensing 41 Figure 3.4: Iustration of the characteristic structures of the differentia absorption cross sections σ for seected trace gases that can be measured by DOAS. Detection imits of the mixing ratios for the given ight paths ength L are ony exampes (as the detection imits, eq. (3.6), depend amongst others on the instrumenta detais). [Adopted from Vokamer, 21]. The ast exponentia in this equation represents intensity oss by narrow band absorption ony, socaed differentia absorption, whie I (λ,l,p,t) is the intensity as it woud be without differentia absorption. Assuming that the absorption cross sections in the ower troposphere, where measurements take pace, do not vary with pressure and temperature (which is not vita but simpifies things) eads thus to I(λ,L) = I (λ,l) exp [ j σ j(λ) c j L ] where c j = 1 L L ds c j (r(s)). (3.3) Inference of the path-integrated concentrations c j as in eqs. (3.1) is sti not possibe because I (λ,l) is not avaiabe and the species are not separated. But if the waveength dependency of I(λ, L) is measured within the range where the trace gases of interest show absorption structures, the distinct functiona form of the differentia cross sections σ j (λ), see fig. 3.4, aows to find the concentrations c j by fitting the σ j (λ) for the reevant trace gases simutaneousy to I(λ,L). However, to carry out this fit the impact of the measurement system on I(λ,L) that up to now refers to the intensity as it woud be before entering the detector has to be taken into account. First, the waveength anaysis, typicay by a grating spectrograph, has ony imited spectra resoution. This can be expressed mathematicay by convoution with a so-caed instrument function H(λ), i.e. I(λ,L) H(λ) = dλ I(λ λ,l)h(λ ). Second, the continuous waveengths get mapped to pixes or channes i of the recording system. The

42 42 3. Tomography and Remote Sensing of Atmospheric Trace Gases further procedure for finding c j from the recorded spectrum I (i,l) depends argey on the foowing factors: The size of the so-caed differentia optica density τ = L j σ j (λ) c j deciding whether the exponentia in eq. (3.3) can be Tayor expanded, the spectra resoution of the detector compared to the width of the trace gas absorption bands and finay, structures in the spectrum of the ight source I (λ,l) compared to the resoution of the instrument. In the case of a weak absorber, i.e. sma optica densities, the exponentia can be expanded and, assuming a smooth spectrum of the ight source, one gets (after a number of steps omitted here) J(i) = ni (i) = S j σ Hj(i) + broad band variations + noise + pixe sensitivity (3.4) with S j = L c j (3.5) and σ H j (i) being the differentia cross section convouted with the instrument function and mapped to the channes. Broad band variations mean contributions from the amp, from Rayeigh and Mie scattering etc. Pixe sensitivity refers to variations of the spectra sensitivity of the detector within the pixes or channes. The S j are caed coumn densities (more precisey coumn number densities or coumn mass densities). The fit probem in the specia case of weak absorbers is now inear in the S j and coud be soved, for exampe, by a east squares principe. In practice, the fit is compicated by a number of uncertainties. We mention just two of them. The broad band variations in eq. (3.4) can be taken care of by an additiona poynomia in a way that preserves the inearity of the probem. Misaignment of the waveengths on both sides of eq. (3.4) is a more awkward matter as even a mere shift between the zero positions of the waveengths enters nonineary. It can be caused by inaccurate positioning of the grating, unwanted dispersion or wrong caibration of the absorption cross sections and has to be corrected. In the end, the fit probem is inear with respect to the S j and the parameters of the broad band correction poynomia, but noninear in the parameters for the waveength aignment. Correspondingy, the fit agorithm consists of a east squares fit for the inear parameters and a noninear fit for the waveength aignment, where for the atter the Levenberg-Marquardt iteration has proven usefu. The case of weak absorption and smooth spectrum of the ight source described above is the one most reevant within this work. A genera discussion can be found in [Patt, 1994], but as the tomographic reconstruction ony depends on the coumn densities as end products, it is of minor importance here. We woud just ike to point out the rigorous formuation of DOAS in terms of operators (for the convoution with the instrument function, the discretisation of the spectrum etc) in [Wenig et a., 25] which becomes usefu to keep track of various approximations, especiay for passive measurements. Appications DOAS is appied on various patforms ranging from ground based experiments over airpane and baoon measurements to sateite observations. Measurements on ground empoy both artificia ight sources and the sun, whie air and space borne appications use either direct or scattered sun ight (we negect sources as the moon or stars). Among the passive techniques notaby the so-caed muti-axis (MAX) measurements have found ap-

43 3.2. Atmospheric remote sensing 43 pication on ground [e.g., Hecke et a., 25; Hönninger et a., 24a,c,b; Wagner et a., 24] and on airpanes [e.g., Heue et a., 25; Petritoi et a., 22]. Here, scattered sun ight is detected in severa ines of sight and aows thus concusions on the vertica distribution of trace gas, in particuar the distinction of tropospheric and stratospheric contributions, or on concentration and extension of emission pumes [e.g., Bobrowski et a., 23]. Active DOAS measurements reevant in our context typicay consist of a teescope emitting ight from a broad band ight source (e.g., a Xenon arc amp) and retro-refectors that send the ight beam back to the instrument where the receiving part of the optics transmits the ight to the detector. As the detection of trace gases ike O 3, NO 2 or SO 2 typicay requires ight paths of the order of a few hundred m up to 1 2km this kind of technique is referred to as ong-path (LP) DOAS [e.g., Stutz and Patt, 1997]. Among other things, it has been very important for the detection of radicas ike OH and NO 3 [Perner et a., 1976; Patt et a., 198; Geyer et a., 21], the important point being that it is a non-contact measurement. Active measurements have the advantage of being conceptionay simpe as the ight traves aong a defined path but they have the disadvantage of of being ogisticay expensive and being restricted by the setting up of the very sensitive teescope and the arrangement of the retro-refectors. Passive measurements ike MAX-DOAS, on the other hand, are easiy set up with the instruments getting smaer and cheaper. But the technique itsef is ess sensitive than the active method and data interpretation is in genera not possibe without empoying numerica modes that simuate the transport of soar radiation through the atmosphere, so-caed radiative transport modes (RTMs). Measurement errors and detection imits The smaest amount of trace gas detectabe with a given method, the detection imit, is a key vaue for the sensitivity of this method. As ony the combination σ j c j L, i.e. the optica density τ, enters the argument of the exponentia in eq. (3.4) there is no universa concentration imit, but ony detection imits for specific species and engths of the ight path. A proper estimate of the detection imit has to be based on actua measurement errors, which are either of statistic or systematic nature. Random noise can be caused by the photon statistics or the eectronic components of the instrument etc. Whie it is trivia to cacuate its propagation from the recorded spectrum to the coumn densities in the case of the inear absorption east squares fit for the inear case eq. (3.4) (namey as in eq. (4.4)), the effect of the noninear waveength aignment has to be found from simuations. The foowing estimate for the detection imit of the retrieved S j = c j L takes into account the inear fit ony and defines that a species is detectabe if for the reative error S j /S j = c j / c j >.5 [Stutz and Patt, 1996]. The detection imit δs j for the S j is then δs j 6 m 1 σ j (3.6) where σ 2 j var[s j] and m is the number of degrees of freedom for the measurement, i.e. the number of channes or pixes of the detector. The order of the detection imit for some of the trace gases important for DOAS are contained in fig The impact of systematic errors is more difficut to estimate in a genera fashion. Systematic errors can be introduced by fauty reference cross sections, but often their origin remains uncear. Common

44 44 3. Tomography and Remote Sensing of Atmospheric Trace Gases sources are stray ight in the spectrograph, an offset of the measured intensities etc. Systematic errors wi not be considered anywhere in the foowing Tomographic appications in the atmosphere 1 3km sateite km (a) Ionospheric tomography 1 sateite 1 (b) Proposa for sateite emission tomography s (c) Tomographic AMAX-- DOAS h LP (d) Proposa for tomographic MAX-DOAS 16 m 4 m motorway (e) Vertica LP-DOAS tomography of motorway emissions (f) Horizonta IR tomography of vocanic CO 2 (g) Indoor gas tomography (h) Indoor gas tomographic study Figure 3.5: Geometry of optica paths for some tomographic appications in the atmosphere. (a) Ionospheric tomography, three ground receivers are shown. (b) Proposa for IR or microwave emission tomography from space by [Feming, 1982], here with three ines of sight. (c) Muti-axis airpane measurements [Heue, 25], shown three downwards and two upwards ooking teescopes with arbitrary ines of sight. For s, h see text. (d) Principe of tomographic measurements by (passive) muti-axis DOAS with defined ight paths LP as suggested in [Frins et a., 26]. (e) Vertica 2-D projection of the geometry with two teescopes and eight refectors used for ong-path DOAS measurements at a motorway [Laeppe et a., 24]. (f) Tomographic FTIR measurements of vocanic CO 2 [Beotti et a., 23]. (g) Measurement geometry for a gas chamber FTIR experiment [Fischer et a., 21] and (h) a theoretica study [Verkruysse and Todd, 24]. Having briefy sketched the principe of tomography and presented the most important atmospheric remote sensing methods, we turn now to the combination of both techniques. The reconstruction of atmospheric parameters ike temperature, pressure or concentrations from tomographic remote sensing instead of conducting, for exampe, a corresponding number of point measurements can be motivated by various reasons. The advantages of the remote sensing method can be used. This coud be contactess measurement of reactive gases or the simutaneous measurement of species for the specia case of DOAS etc. Point measurements providing the same spatia resoution and/or quaity are too expensive. The best exampe here are probaby goba maps of trace gas coumns from sateite observations. Air poution monitoring on a microscae has been addressed in sec In-situ samping is difficut or impossibe. Vocanic emissions have been mentioned, wood fire or noxious gas are further exampes.

45 3.3. Tomographic appications in the atmosphere 45 Athough not primariy intended, tomography in the atmosphere profits from the fact that stochastic variations of the concentration fied due to turbuent fuctuations are reduced by integrating, i.e. averaging, over the optica path (see aso sec. 7.4). Some measurements and proposas using tomography with different remote sensing techniques are now being presented in the order of the spatia scae invoved. DOAS tomographic appications wi be introduced in the next section. An appication on the argest scae is the ionospheric tomography that tries to reconstruct the spatia distribution of the eectron number density in the ionosphere, i.e. the atmospheric ayer roughy between 8 1 km, by its infuence on the propagation of radio waves through changes in the diffraction index [e.g., Kunitsyn et a., 1995]. Frequency dispersion, Dopper shifting, ange of arriva or signa attenuation can be used to measure the tota eectron content, that is the integra of the eectron density aong the ine between emitter and receiver as in the exampe in fig. 3.5a [Fehmers, 1996]. Here, an orbiting sateite sends out radio signas to ground receivers and the tota eectron content aong a ines of sight derived by the differentia Dopper technique aows reconstruction of two dimensiona vertica distributions. Whie the first proposa for ionospheric sateite measurements was made in 1986 [Austen et a., 1986], Feming [1982] simuates (in a to my opinion extraodinariy inteigibe paper) the two dimensiona reconstruction of temperature distributions from IR (or microwave) radiation measured by sateite and suggests simiar measurements to retrieve concentration fieds, see fig. 3.5b. He compares temperature profies reconstructed from hypothetica measurements aong three and five viewing anges (zero degree zenith, ±45 and additionay ±6 ) to conventionay retrieved profies and finds improvements of the accuracy up to 34%. It shoud be noted that he assumes measurements at five different frequencies (channes) for each ine of sight. To my knowedge, sateite observations of this kind have not been reaised so far. Somewhat earier than these trave time and emission tomographic sateite measurements Byer and Shepp [1979] proposed absorption LIDAR for (horizonta) two dimensiona tomographic measurement of air poution. The refector-lidar system assumed for the simuations in [Wofe and Byer, 1982] consists of a aser source and mirrors that refect fan beams 2 through the area of interest ( 3km) to a detector array. With around 1 virtua sources and 7 detectors, the authors find that a spatia resoution of ess than 2m becomes reaistic (an adjective that might not appy to the setup as far as I know, this setup has not been reaised with anything near the number of optica sources and detectors assumed here). Horizonta temperature distributions on an area of about 2 2m 2 obtained by acoustic trave time measurements are compared in [Weinbrecht et a., 24] with highy resoved resuts from arge eddy simuations. The measurements used a moderate number of six acoustic sources and five receivers at a height of 2m above ground. The tomographic IR measurements in an active vocanic region have aready been mentioned [Beotti et a., 23]. They were performed on an area of 1 1m 2 measuring CO 2 absorptions successivey aong 15 optica paths given by the aser transmitter/receiver system and a retro-refector (fig. 3.5f). The detection imit for the coumn density is stated as 1ppm m and the samping time per path was chosen between 5 1min. Reconstructed CO 2 varies between 35ppm (background) 2 Buidings, trees or topographic objects coud aso be used at the expense of intensity.

46 46 3. Tomography and Remote Sensing of Atmospheric Trace Gases and 3 7 ppm. Finay, the indoor gas dispersion studies aready mentioned try to reconstruct gas distributions on a metre scae from IR measurements. Athough not reay being atmospheric appications they are among a studies isted the most reevant from a geometrica point of view as some of their ight path configurations are very simiar to what can be expected from tomographic LP-DOAS experiments (figs. 3.5g, 3.5h). For the same reason their approach to reconstruction of concentration fieds is worth ooking at. Todd and Leith [199] start ambitiousy with a theoretica study to detect contaminants ike ammonia, benzene, tri- and perchoroethyene by FTIR spectroscopy for parae projection ike beam geometries with as many as 4 paths. This bois down to 136 paths actuay reaised in an experiment [Samanta and Todd, 1999] with four spectrometers by rotating the sources successivey towards different mirrors and eventuay down to 4 rays in a series of theoretica studies on path configurations and reconstruction agorithms [Todd and Leith, 199; Todd and Ramachandran, 1994a,b; Todd and Bhattacharyya, 1997]. For simiar experiments Drescher et a. [1996] present an approach especiay suited for the reconstruction of point emissions (SBFM, see sec ) and find from a detaied time series anaysis that the evoution of an indoor gas emission shows far more variabiity than expected [Drescher et a., 1997]. The method is adopted in [Phiip, 1999; Hashmonay et a., 1998] to investigate the possibiity of reconstructing point sources by radia scanning with one rotating optica source ony. SBFM is given up in [Fischer et a., 21] for the reconstruction of CH 4 tracer distributions reeased in a chamber experiment for being too sow (see footnote 3, p. 11) First tomographic DOAS measurements The idea to infer information on the spatia distribution of trace gases from DOAS remote sensing aong more than one ight path has been reaised in profie measurements with retro-refectors on mountain sites [Patt, 1978] or mounted on towers [e.g., Stutz et a., 24], c.f. fig. 3.1a on p. 35, and aso in baoon experiments [Veite et a., 22]. A genuine tomographic experiment aimed at the 2-D vertica reconstruction of the exhaust pume perpendicuar to a motorway from LP-DOAS measurements [Pundt et a., 25]. The experiment was part of a joint campaign incuding in-situ measurements and mode studies [Corsmeier et a., 25a]. It utiised two LP teescopes each sending successivey one ight beam aong the motorway to one of eight retro-refector arrays mounted on two towers next to the carriage way (fig. 3.5e). Scanning a refectors to achieve the compete geometry of 16 ight paths took about 45 min. With optica path engths around 8m, NO 2 concentrations coud be measured most reiaby and were thus considered for tomographic reconstruction in [Laeppe et a., 24]. For a time averaging interva of 3 h, the pure DOAS error on the coumn densities was about 2 5%, whie the error due to the successive scanning was estimated from time series of the coumn densities about to be 2%. Maximum reconstructed NO 2 concentrations in the pume ie around 2 ppb and the maximum of the mean reconstruction error was estimated to be roughy 5ppb which is about as much as the maximum deviation from mode predictions. The motorway experiment pointed out a major probem of active DOAS tomography: To generate as many ight paths as possibe within an acceptabe time one has to either empoy as many ight

47 3.4. First tomographic DOAS measurements 47 sources (teescopes) as possibe or direct the teescopes successivey at the refectors ( scanning ) or a combination of both. The first is (at east at the moment) prohibitivey expensive, the second not easy to accompish as conventiona LP instruments are rather heavy and have to be instaed stabe. This ed to the construction of a teescope that emits up to four ight beams at once [Mettendorf, 25; Pundt and Mettendorf, 25]. Three instruments have been tested in a tomographic indoor experiment [Mettendorf et a., 26] that used one or two cyindric containers fied with NO 2 to mimic puff emissions. A geometry corresponding to 36 integration paths was reaised in three steps à 4 beams per teescope. Whie spatia dimensions can be scaed up to atmospheric experiments this is not true for the time scaes of the measurement. As the reconstruction from the coumn densities of this experiment was part of this thesis, further detais wi be presented in chap. 9. The instruments are currenty used for 2-D horizonta measurements over the centre of Heideberg. The tomographic principe can be appied straightforwardy to DOAS measurements of sun ight if scattering can be negected, i.e. for observations of direct or directy refected sun ight, which restricts possibiities somehow, uness some kind of mirrors are used. For tomographic measurements of scattered sun ight depending on the degree of homogeneity assumed 1-D, 2-D or 3-D simuations have to mode the ight paths. These radiative transport modes are predominanty 1-D as cacuations in two and three dimensions are quite invoved. The Monte-Caro agorithm TRACY-II [Wagner et a., 26] is one of the very few RTM capabe of both 2-D and 3-D simuations. First passive tomographic measurements have been attempted by instaing muti-axis DOAS instruments onto an airpane (AMAX-DOAS) with three teescopes ooking upwards and seven ooking downwards [Heue, 25]. Thus sun ight from above the airpane and refected from ground can be used for tomographic reconstruction, see fig. 3.5c. Fying at heights h between 5 2 m above ground at a speed of about 2 km h together with integration times around 3s for the visibe waveength range and 11s in the UV resuts in measurement points every s 1.7km (VIS) and 6m (UV). The method was used to examine the emission pume of a power pant in the Po vaey. Preiminary 2-D vertica recontructions of the NO 2 pume were obtained from 2-D RTM cacuations. A very interesting idea put forward in [Frins et a., 26] amounts to using passive MAX-DOAS in a way simiar to LP-DOAS, thus combining the benefits of the former (simpe setup, no refector arrays necessary) and the atter (simpe concept, especiay for tomographic measurements). As for LP-DOAS, the objective is to measure near-surface concentrations and this is achieved by pointing the teescope at a target in a way that the ight refected from it passes the region of interest. This part of the ight path is we defined and a one has to do is to subtract the remaing part from the top of the atmosphere to the target (fig. 3.5d), which can be done amost routiney. With severa (reativey cheap) mini MAX-DOAS instruments and suitabe targets tomographic experiments identica to LP-DOAS measuremets can be set up. Drawbacks coming aong with the passive technique are that measurements depend, of course, on the sun ight, they might not be as sensitive as LP- DOAS measurements and if scattering between target and instrument cannot be negected, radiative transport modeing gets invoved.

48 48 3. Tomography and Remote Sensing of Atmospheric Trace Gases integration paths mode params. area time reconstruction m n m/n [km 2 ] [min] method a Axia X-ray O(1 5 ) O(1 5 ) m > n O(1 2 ) cm 2 transf. meth. typica 5 rays FBP ( ) proj. pixes Seismoogy ( ) ( ) m n (non)in. LS (LSQR, CG) 2-D Sateite 2 1 constrained tomography /4... LS, of ionosphere b 6 2 constr. opt. AMAX-DOAS c LP-DOAS O(1) O(1) ( ) O(1) ( )... O(1) motorway d m 2 45 SIRT Indoor FTIR e m 2 7 s genera. LS f m > n 6 7 m 2 6 SBFM a See chap. 4. b [Fehmers, 1996] c For the fight around a power pant reported in [Heue, 25] and a singe scattering approximation. d [Laeppe et a., 24] e Simutaneous measurement with 28 instruments by Fischer et a. [21]. f Sequentia measurement with one source and fix mirrors [Drescher et a., 1996]. Tabe 3.2: Dimensions of some tomographic appications.

49 4. Tomography and Discrete, Linear Inverse Probems The previous chapter has ceary shown that tomographic DOAS measurements ie beyond cassica appications of tomography. Consequenty, there is no standard method to gain the concentration distribution from the measured ine integras. Indeed, it is not exaggerated to say that there are as many suggestions for reconstruction methods as there are potentia or actua appications. It wi become cear in parts II and III that the choice of the reconstruction procedure for a imited set of measurement data is a deicate matter and we prefer to discuss it on the background of conventiona methods. This impies jargon, concepts and agorithms from computerised tomography, image processing, inverse theory and atmospheric profiing and, as I am not aware of a suitabe synoptic reference, I have tried to treat the matter as concise as possibe in a common, uniform context. Sti the coverage is by no means compete and especiay the discussion of reguarisation ony scratches the surface. First we reate the tomographic probem to other inverse probems given by integra equations and, by considering continuous tomographic inversion methods, are then ed to a discretisation of the probem. The reconstruction method chosen here within the discrete approach is essentiay east squares minimisation which on the one hand has the advantage of eading to inear systems of equations. On the other hand, the east squares formaism arises naturay if the underying statistics are assumed to be normay distributed. We briefy review soution strategies for the east squares probem and eaborate on a famiy of iterative agorithms we known in computerised tomography and image reconstruction, but not common in atmospheric sciences. If not sef evident or stated expicitey in this chapter, justification of the approach wi be given in sec The chapter concudes by brief descriptions of aternative approaches for ater reference Forward mode and inverse probem The formuation and soution of the tomographic reconstruction probem is carried out for the concentration distribution of each species seperatey and we write d S j, see eq. (3.5), for the coumn densities obtained by the DOAS anaysis for the species under consideration (d standing for the input data of the inverse probem). As starting point we assume that an idea ong-path (LP) DOAS measurement aong a ight path LP with parametrisation LP : r = r(p) R 3, p [a,b], provides a coumn density d measured at time t which simpy is the ine integra of the concentration 49

50 5 4. Tomography and Discrete, Linear Inverse Probems distribution c at the same time. That is d(t) = b a = LP dp dr(p) dp c(r(p),t) dsc(r,t), (4.1a) (4.1b) where in the shorthand notation eq. (4.1b) ds denotes the ine eement aong LP and r = r(s). The mode eq. (4.1) represents a measurabe cause-effect reationship between a given concentration fied and the coumn densities that resut from it in contrast to the pure mathematica backward inference of the unknown concentration fied from the measured integras so that it is commony referred to as a forward mode. It reies on the assumptions: The ight path is known. The beam diameter is negigibe. There is no significant time deay between absorption process and the recording of the absorption spectrum. In this work we do not take into consideration any modifications that arise from reaxing the above assumptions except for errors that utimatey can be expressed as random noise on the coumn densities: d ǫ = d + ǫ. (4.2) Here d ǫ is the error afficted measured coumn density, d the idea one and ǫ an (unbiased) random quantity. Time averaging does not cause troube as ong as it acts consistenty on both sides of eq. (4.1) d(t) T = ds c(r,t) T, (4.3) LP T indicating the time averaging interva T = [t;t+ t]. This is the case, for exampe, when averaging over subsequent spectra to reduce stochastic noise. If data is taken by the same instrument (e.g. the muti-beam instrument, see sec. 3.4, that emits severa ight beams simutaneuousy) the question of correated measurement errors has to be addressed. The (ideaised) tomographic inverse probem can now be stated as foows. p-doas tomographic reconstruction probem Given the coumn densities d i of LP-DOAS measurements aong m ight paths LP i and a compact set Ω R 3 containing the LP i, find a function c(r) over Ω satisfying the forward mode d i = dsc(r), i = 1,...,m, (4.4) LP i where the coumn densities refer to a particuar time: d i = d i (t), or time averaging interva: d i = d i (t) Ti that is the same for a measurements (T i T ).

51 4.2. Continuous inversion methods 51 q p θ Figure 4.1: Parae beams in the pane for the projection ange θ k. The circe indicates the region of interest. d... i = 1 i = i = Continuous inversion methods The most eegant way to sove probem eq. (4.4) woud be anaytic inversion of the integra equations. This is indeed possibe under certain conditions. But before addressing these transform methods, it is instructive to compare the tomographic inverse probem with another arge cass of inverse probems given by the integra equation g(q) = 1 dpk(q,p)f(p), p 1. (4.5) Here the integra vaue is a function of the parameter q of the kerne function k (see [Groetsch, 1993] for an eementary introduction). These so caed Fredhom equations of the first kind appear in numerous appications. A prominent exampe in atmospheric research is the integra equation given by the soution of the atmospheric radiative transfer equation and the two textbooks [Twomey, 1997; Rodgers, 2] practicay excusivey dea with the inversion of this equation. If the discrete ight path index i in eq. (4.1) can be transformed into a continuous parameter q, the tomographic probem becomes a Fredhom equation. Consider the 2-D exampe of parae ight paths in fig Their parameter representation for the projection ange θ is ( cos θ r i (p) = i d sin θ ) ( sin θ + p cos θ ), i =, ±1, ±2,..., p R. (4.6) If the distance d of the beams is sma against a reevant scaes (i.e. the ength of the area of interest, the ength of typica structures within the area etc.), q = i d effectivey becomes a continuous parameter and the ine integras are now d θ (q) = dp c(x,y), with ( x(p,q) y(p, q) ) ( q cos θ psin θ = q sinθ + pcos θ ). (4.7) Apart from the fact that the unknown function depends on two parameters instead of one as in eq. (4.5) the kerne is trivia so that inference from the one dimensiona d θ (q) to the two dimensiona c(x,y) is impossibe. Providing further projections, it turns out that the kerne is both probem and cure as its

52 52 4. Tomography and Discrete, Linear Inverse Probems specia form aows the use of very efficient transform methods. Discrete methods treat c(x, y) as a one dimensiona object and the resuting discrete system is formay identica to the system obtained from discretising a Fredhom equation but ony formay, as wi be discussed in sec Transform methods Transform methods rey on a quasi-continuous representation of the measurement resuts whie the parametrisation itsef depends cruciay on the configuration of the integration paths, i.e. the kind of projection. We resume the exampe of the parae projection, fig. 4.1, for its simpicity, athough this kind of projection is not reaistic for a DOAS tomographic measurement. Other cases ike fan beam projections (fig. 3.2b) and incompete projections can be found in the appication-oriented introduction by Kak and Saney [21] and in the extensive, mathematica coverage [Natterer, 21]. The transform from the 2-D space for the (concentration) fied to the space of projections of c with its two dimensions θ and q in eq. (4.7) is caed Radon transform. If the projections aong the q- axis are measured for sufficienty many anges θ, inversion of the Radon transform becomes feasibe. Straightforward Fourier transformation does the job, but does not take into account the geometry of the probem to simpify the soution. There is a number of formuations of the inverse Radon transform that do expoit the nature of the projection, but they are numericay not very stabe. By far the most widespread way to invert the Radon transform in practica and commercia appications is the so caed fitered backprojection (FBP). It was aso used in the theoretica study on the reconstruction of ambient trace gas distributions from tomographic LIDAR measurements [Wofe and Byer, 1982] that was mentioned in sec The fitered backprojection can be motivated by considering the 2-D Fourier transform c(k x,k y ) = + dx dy e i2π(kxx+kyy) c(x,y) which in the case k y =, i.e. no dependence in y, takes the form c(k x,) = + dx e i2πkxx[ + ] dy c(x, y). The term in brackets is the ine integra aong the y-axis d θ= (q). Thus one gets the reation c(k x,) = d θ= (k x ) for the Fourier transforms. Obviousy this resut cannot depend on the choice of the coordinate system and by rotating the axis one gets more generay the so caed Fourier sice theorem c(k cos θ,k sin θ) = d θ (k). In other words, the 2-D transform of c aong the projection is given by the 1-D Fourier transform of the projection. This reduces the number of integrations in the inverse Fourier transform for c(x, y) to ony two: c(x,y) = π dθ dk k e i 2π q k dθ (k) = π dθ D(q). (4.8) where as above q = xcos θ + y sinθ. D is up to the so caed fiter factor, here k, the inverse Fourier

53 4.3. The discrete, inear inverse probem 53 transform of d and hence caed fitered projection. For a given x,y in the image pane there is a unique tupe θ,q, but eq. (4.8) shows that in fact a points on the ine with equa distance q to the origin contribute the same for a given θ. The reconstruction seems to project each fitered projection D(q) back into the image pane aong the ine of constant q. This is why the transform is caed fitered backprojection. In practice, the theoretica fiter k does not work and has to be modified. One reason is the inherent instabiity of the inverse probem discussed in sec Remark on existence and uniqueness The questions of existence and uniqueness of the soution for the tomographic inverse probem eq. (4.4) are mathematicay rigorousy discussed by Natterer [21, chap. 2]. He shows irrespectivey of the inversion method that for an arbitrary function c and a compact region there is aways another function c for which any finite number of ine integras have the same vaues on this arbitrary sma region. As this hods even with both functions being C ( ) this sounds rather discouraging. However, c is a highy osciating, artificia function and if its variation is restricted in terms of some norm the agreement of c and c cannot aways be achieved and the soution becomes unique. In the framework of the transform method just sketched the question of uniqueness is addressed by samping theorems simiar to the we-known Nyquist theorem. The conditions for the norm of the function of c correspond now to bounds on the frequency range of c, i.e. it has to be band-imited [Natterer, 21; Kak and Saney, 21, chap. 2]. Further discussion is beyond the scope of this work, as it wi dea ony with the discretised version of the reconstruction probem The discrete, inear inverse probem It is cear from the preceding discussion that we are not addressing the kind of discretisation that is eventuay necessary for the computationa soution by any numerica agorithm, but the reformuation of the probem in terms of a imited number of parameters. For either way of ooking at it, there are essentiay two ways of discretising integra equations [Hansen, 1998] Discretisation by quadrature Appying some quadrature rue ike the midpoint rue, Simpson s rue etc. eads to the foowing representation N i d i = dsc(r) = w ij c ij + J i (c), (4.9) LP j=1 i where the weights w ij depend on the detais of the quadrature (ine segments s in the simpest case) and c ij = c(r i (s j )) are the concentration vaues on the interpoation nodes s j aong path i. The functiona J i stands for the error due to the discretised evauation of the integra. To make use of the tomographic character of the measurement concentration, vaues for different rays have to be set to equa vaues so that the individua equations (4.9) get couped. This can be done by identifying concentrations in certain regions, e.g. cubes in three dimensions, and adjusting the interpoation nodes to them. For the most simpe case that approximates the integra as dsc(r(s)) N j=1 s j c(r(s j ))

54 54 4. Tomography and Discrete, Linear Inverse Probems and uses n cubes in which concentrations are identica one gets: n d i = A ij c j + disc J i (c). j=1 Here the sum is over the cubes and the weight A ij is the ength of ray i in cube j. The number disc J i (c) represents the error that comes with the approximations of the integra and the function c. Obviousy, for a given eve of discretisation, i.e. number of cubes, appying a very sophisticated quadrature is somewhat pointess, if concentration vaues over a arge region are forced to equa vaues. Refined schemes invoving 2-D or 3-D interpoation woud be necessary, however, we do not pursue this issue any further as there is a more eegant and consistent way of discretising the probem Finite eement discretisation - oca basis functions Instead of approximating the integra one can discretise the function c first by representing it on a finite set of so caed basis functions 1 {φ j } n j=1 : n c(r) = x j φ j (r) + disc (c(r)), (4.1) j=1 where x j R are the n parameters of the representation and disc stands for the error reated with it. Inserting this into the forward mode eq. (4.4) yieds n d i = A ij x j + I i j=1 with A ij = dsφ j (r) and I i = ds disc (c(r)). (4.11) LP i LP i Depending on the iterature consuted this approach is mosty referred to as Gaerkin s method or finite eement method (FEM). The atter term impies that the basis functions have compact support. Severa aspects are reevant for the choice of the basis functions: continuity, smoothness etc. computationa expense (m n integras A ij have to be cacuated) reguarisation behaviour (this wi be discussed ater in sections and 8.2.2) other mathematica reasons (e.g., do derivatives appear in the probem?) For arge probems computationa economy wi dominate over the aim to minimise discretisation errors. In fact, at one end of the scae of this trade-off there are appications of computerised tomography for which use of any other than piecewise constant basis functions is too expensive [Kak and Saney, 21]. For moderate numbers of ight paths, DOAS tomography finds itsef at the other end 1 This term is sighty miseading as, in genera, the φ j do not form a basis of the function space of the concentration fieds c.

55 4.3. The discrete, inear inverse probem 55 1 φ j (x) φ 1 j (r).5 x j x j φ 1 j (x) y j+1.5 x j 1 y j x j 1 x j x j+1 x j y j 1 x j+1 (a) (b) Figure 4.2: (a) The piecewise constant and inear basis function, or spines of degree and 1, respectivey. The support (finite eement) is defined by two and three nodes. (b) The (2-D) biinear basis function given by eq. (4.14) with contour ines in the bottom graph. of this scae where computation time does not pay any roe at east not for cacuating the integras eq. (4.11) and more expensive basis functions may ead to a better representation and soution of the probem. Turning to their expicit form, the basis functions are divided into two categories: A set of basis functions {φ j } is caed oca if the φ j have compact support Ω j Ω. It is caed goba if Ω Ω j. Legendre and Chebyshev functions are exampes of goba poynomia basis functions in one dimension, another exampe that reates cosey to the tomographic reconstruction probem wi be met in sec A most simpe exampe of a set of oca basis functions is the aforementioned piecewise constant basis (over the disjoint union of the Ω j ) 1 if r Ω j φ j (r) =, with Ω j Ω k = for j k and j Ω j = Ω. ese A widey used cass of oca basis functions consists in functions buit from piecewise poynomias, motivated by the observation that smooth functions can be approximated to arbitrary accuracy by poynomias of the right degree. In the widest sense, piecewise poynomia functions used for interpoation and/or smoothing are caed spines. Fig. 4.2a shows basis functions constructed from the most simpe spines of degree and 1 for the case of one spatia dimension. The functiona form φ j (x) for the spine of degree reads 2 φ 1 if x [x j,x j+1 [ j(x) = (4.12) ese. The two parameters in bx + a for the representation of φ 1 j (x) are given by demanding φ1 j (x j) = 1 and 2 The nodes x j here must not be confused with the parameters x j in eq. (4.1). There wi be no cause for confusion in the foowing.

56 56 4. Tomography and Discrete, Linear Inverse Probems φ 1 j (x j 1) = = φ 1 j (x j+1) : x x j 1 x j x if x [x j 1,x j [ j 1 φ 1 j(x) = x j+1 x x j+1 x if x [x j,x j+1 [ j ese. (4.13) Evidenty, representation by the piecewise inear basis functions (aso caed hat or tent functions) is equivaent to inear interpoation between two nodes. Higher order poynomias can be constructed by incuding nodes other than the neighbouring ones or by providing conditions on the derivatives. The next higher order basis function emerging naturay in this context is the cubic basis function, basicay because the four parameters in d x 3 +cx 2 +bx+a can be determined by the function vaues pus sopes on two neighbouring nodes. An aternative construction eading to the so caed natura cubic spine with continuous first and second derivatives can be found in [Press et a., 1992]. A simpe way to buid higher dimensiona basis functions is to form tensor products of ower dimensiona ones. In two dimensions this eads for the above cases to the piecewise constant φ j(r) = φ j(x)φ j(y) and biinear basis functions (fig.4.2b) φ 1 j(r) = φ 1 j(x)φ 1 j(y), (4.14) with support [x j,x j+1 [ [y j,y j+1 [ and [x j 1,x j+1 [ [y j 1,y j+1 [, respectivey, i.e. rectanguar areas for both functions. As in the 1-D case, the piecewise constant basis is orthogona (supports do not overap) whereas a piecewise inear basis function does overap with its nearest neighbours. Representation by this biinear basis corresponds to 2-D interpoation (on a square grid). Piecewise constant and triinear basis functions in three dimensions with their supports being cubes can be constructed simiary. Of course, there is a far arger variety of basis functions, e.g. piecewise Hermite poynomias or B- spines (a generaisation of the Bézier curve) just to mention poynomias. And discipines ike digita image processing or the finite eement method for soving differentia equations use basis functions with support chosen to optimay suit the probem at hand, e.g. in two dimensions trianguar and quadriatera eements instead of the rectanguar above. On the other hand, many of the exampes of tomographic reconstruction cited in chapter 3 that take a discrete approach do not use particuary eaborate discretisation schemes. One reason might be the computationa expense mentioned before, especiay if the spatia resoution given by the size of the meshes of the discretisation grid is high anyway. For appications with ow spatia resoution ike the LP-DOAS tomography a different argument becomes important: Contrary to a probem of best function approximation where the number of parameters is imited by computer memory or cacuation time, for a discrete inverse probem every free parameter coming aong with the discretisation has to be determined from the imited amount of data. And it is by no means cear that estabished discretisation schemes work here as we as for their origina appication. Fig. 4.3 iustrates a very simpe case. To parametrise a concentration fied on 2-D finite eements given by rectanges as shown in the figure, one parameter per eement is required for the piecewise constant basis functions whereas the four nodes of the biinear basis carry four parameters. In three dimensions, a cubic finite eement sti ony needs one parameter, the triinear basis

57 4.3. The discrete, inear inverse probem 57 Figure 4.3: Left: For the 2-D parametrisation on a rectange the biinear representation needs four concentration vaues on the nodes ( ) as parameters. Taking the same rectange as support for the piecewise constant basis, ony one paramater occurs. With four parameters, i.e. four finite eements ( ), gradients within the rectange can be described by the piecewise constant basis, too. Right: The same for three dimensions, where the representation by triinear basis functions on a cube requires eight parameters. eight. Of course, the bi-or triinear representation, in principe, aows a far better approximation of smooth fieds but ony if the parameters can be reconstructed correcty. The remainder of this thesis wi amost excusivey dea with 2-D tomographic reconstruction and we wi systematicay compare the piecewise constant and the biinear basis functions, having in mind the points just mentioned, to tacke the question whether more invoved discretisation schemes are worth considering. Using the discretisation by oca basis functions and combining into vectors ike 3 d = (d 1,...,d m ) T R m, x = (x 1,...,x n ) T R n, (4.15) φ = (φ 1,...,φ n ) T R n the discrete probem is now stated expicity in two (or three) dimensions. Discretised 2-d(3-d) p-doas tomographic reconstruction probem Given a vector of coumn densities d R m of LP-DOAS measurements aong m ight paths LP i with time dependency as in the continuous probem eq. (4.4), a compact set Ω R 2 (R 3 ) containing the LP i and a set of n basis functions {φ j (r)} with supports Ω j R 2 (R 3 ) so that j Ω j = Ω, then the reconstructed concentration fied represented by these basis functions is defined as ĉ(r) = [φ(r)] T x, (4.16a) where x R n is a soution of d = Ax (4.16b) and A R m n is given by x and φ are defined in eq.(4.15). A ij = LP i dsφ j (r). (i) Existence and uniqueness of the soution: In contrast to the conditions for the soution of the 3 The in this case discrete spaces of a vectors d, x are usuay referred to as data and mode space, respectivey. The vector x is caed state vector.

58 58 4. Tomography and Discrete, Linear Inverse Probems origina reconstruction probem eq. (4.4), existence and uniqueness for the discretised probem are now ceary defined by the inear system of equations (4.16b): soution exists rank[a] = rank[(a d)] unique soution exists n = rank[(a d)] where (A d) denotes the augmented matrix formed by adding d as a coumn to A. In practice, cassifying the system as even-determined m = n no soution or infinitey many or exacty one over-determined m > n no soution (or exacty one) under-determined m < n infinitey many soutions (or none) is more hepfu. Whie a soution of the discretised probem wi in genera not be the same as a soution of the origina reconstruction probem eq. (4.4) due to the discretisation error reated with the finite representation (c.f. eq. (4.1)), the coumn densities can be reproduced exacty, if ony eq. (4.16b) is satisfied. (ii) Interpretation of A: A is sometimes caed weighting (function) matrix or kerne (function) matrix. Its eement A ij is the contribution of basis function with index j to the ine integra aong LP i and in the case of the piecewise constant basis functions this is just the ength of LP i within Ω j. For the construction of the 2-D(3-D) basis functions used here, Ω j is a rectange (cube), in the foowing caed box in both cases. The expressions pixe (picture eement) and voxe (voume eement) adopted from image processing are aso commony used for the 2-D and 3-D ces. The piecewise constant basis functions wi aso be referred to as box basis functions The question of i-posedness In the preceding section it was not assumed that the number of free parameters x j equas the number of ight paths, i.e. m = n. But even if one cared to design a discretisation in a way that assures a square matrix A whose inverse exists, the soution A 1 d might be useess because it is numericay unstabe. This means that very sma changes in the data (measurement errors or even numerica inaccuracies) have arge effects on the reconstruction resut. Before tracing its origin this instabiity is iustrated by an Exampe from [Twomey, 1997]. Let d 1 = x 1 d 2 = A 21 x 1 + A 22 x 2 with A 21 =.99995, A 22 =.1 d 1 = 1, d 2 = 1.995, (4.17) where a units are suppressed. The exact soution is x 1 = 1, x 2 = 1. Now assume an error on the data around one percent: d 1 d 1.1 d 2 d Then the resuting change of x 2 amounts to or amost 2%. The instabiity in this exampe can be understood by the amost inear dependency of the row vectors in the above system (1, ) and (.99995,.1). Appying Cramer s rue to sove the system a determinant

59 4.4. The question of i-posedness 59 Box 2 Box 2 Box 3 Box 4 LP 2 LP 3 LP 2 LP 3 LP 2 Box 1 Box 1 Box 2 LP 1 Box 1 LP 1 LP 1 (a) (b) Figure 4.4: (a) Two ight paths and boxes ( ) as in the exampe on p. 58 (but with different proportions to make the figure cearer). Dividing into two boxes as indicated by the dashed ine (--) makes the probem perfecty we behaved. (b) Finer discretisation eads to under-determined systems of equations but with inear independent rows. appears in the denominator which gets sma for amost inear dependent vectors (In two dimensions this determinant is the (area) 2 of the paraeogram the two vectors span, in genera basicay the (voume) 2 of the corresponding paraeopiped). Sma perturbations get thus ampified by a arge factor. A nice geometrica expanation of this behaviour can be found in [Twomey, 1997, section 5.3]. An immediate question is whether this exampe with its near-singuar system matrix A is at a reevant for any inverse probem. We first show that it is and afterwards turn to the case of tomographic inverse probems. Assume for the moment that the system of equations was derived from discretising (by whatever method described above) a probem ike the Fredhom equation (4.5) with a smooth kerne function k. For exampe, processes invoving radiation attenuation in the atmosphere, i.e. processes that can be described by the radiative transfer equation, woud ead to an exponentia kerne something very smooth. Uness the discretisation is very coarse, the row vectors of the system matrix A wi show a high degree of interdependency. In fact, the paradoxica situation arises that the finer the discretisation, in other words the more accurate you try to be, the more unstabe your soution gets. Of course this is ony a very superficia, yet pausibe, argument and one coud try to work around the probem of instabiity by a more cever discretisation or choice of basis functions or by orthogonaising the kerne functions etc. (see [Twomey, 1997] for a more eaborate, but sti not mathematicay rigorous discussion). Nevertheess some inverse probems remain inherenty unstabe. These are caed i-posed, a term going back to Hadamard [192], who defined a probem as we-posed if 1.) A soution exists, 2.) The soution is unique and 3.) The soution depends continuousy on the data. There are more and ess cever ways to approach an i-posed probem but in the end usuay a reformuation aso known as reguarisation (c.f. section 4.6.3) of the probem is necessary. Here, we do not dea with genera criteria for we- or i-posedness of continuous and the reated discrete inverse probems (As one might expect they are cosey reated to the mathematica properties of the kerne: compactness, its smoothing behaviour etc. [Groetsch, 1993; Hansen, 1998]). Instead, we focus on the tomographic inverse probem. Take again the numerica exampe from above, this time understood as a tomographic measurement (fig. 4.4a). The amost inear dependent rows in the matrix A are due to a very inappropriate discretisation: A = ength LP 1 in box 1 ength LP 1 in box 2 ength LP 2 in box 1 ength LP 2 in box 2! = !

60 6 4. Tomography and Discrete, Linear Inverse Probems Dividing the area differenty into two boxes yieds a we behaved system. Choosing ony one box eads to an over-detemined system with exacty inear dependent rows. This is further iustrated in fig. 4.4b. If the square area has a ength of 2 in arbitrary units, one gets A 2 boxes = C B A, A 4 boxes C A. Coarse discretisation eads to over-determined systems with unstabe soutions as the rows of A become inear dependent. Finer discretisation resuts in under-determined systems of equations which are iposed in the sense that the soution is not unique, but uness ight paths and discretisation do not suit each other and the rows in A become inear dependent there is no reason why the resuting system has to be unstabe. What if both the area coverage by the ight paths and the discretisation become very fine? As suggested by the continuous approach in section 4.2, the kerne now turns into a smooth function and the argument of interdependency becomes effective again. Indeed, it turns out that tomographic measurements which provide such a high ray density ead to i-posed inverse probems. To quote Natterer [21, p. 85]: A probems in Computerised Tomography are i-posed, even if to a very different degree. Assuming reasonabe smoothness of the unknown function and further properties of the operator (or matrix) norm of A, he shows that compared to other i-posed inverse probems tomographic probems are ony midy to modesty i-posed (The exact definition of these categories is beyond the scope of this work). To sum up, the tomographic reconstruction probem is in principe i-posed. Parametrisations for measurements with an abundance of data that make use of high spatia resoution wi ead to unstabe soutions. For measurement with ony few data, ike the LP-DOAS tomography, i-posedness in the sense of instabiity is not given per se. Instabiity cruciay depends on the detais of the discretisation and is ess ikey to occur for under-determined systems. The next section introduces a too that aows to quantify the degree of instabiity of a discrete inverse probem The singuar vaue decomposition The Singuar Vaue Decomposition (SVD) is a generaisation of the ordinary expansion of a square symmetric matrix in terms of orthogona eigenvectors. In fact, one way to motivate it is to ook at the eigensystems of the square symmetric matrices AA T and A T A. We do not derive it here (see [Goub and van Loan, 1996] for a standard reference) and just mention its importance both theoreticay and practicay for numerous areas such as signa and image processing, time series anaysis, pattern recognition etc. The usefuness of the SVD for the discrete tomographic probem wi become evident from frequent appications in the remainder of this thesis. It states as foows. The singuar vaue decomposition Let A R m n be a matrix of rank r = rank[a]. Then there exist orthonorma matrices U,V, UU T = U T U =½m and V V T = V T V =½n, such that ( ) A = UΣV T Σ r, Σ = R m n, (4.18)

61 4.5. The singuar vaue decomposition 61 where Σ r = diag(σ 1,σ 2,...,σ r ) and σ 1 σ 2... σ r >. The σ i are caed singuar vaues of A. Without proof, we ist some properties of the SVD for ater use. (i) Singuar vectors: Writing U = (u 1,u 2,...,u m ), V = (v 1,v 2,...,v n ) the orthonorma coumn vectors u i and v j are associated with σ i, i = 1,...,r, via Av i = σ i u i. (4.19) The SVD can be written as r A = σ i u i vi T. (4.2) i=1 (ii) Reated eigenvaue decomposition: In genera, the singuar vaues for square A do not agree with the (ordinary) eigenvaues. This is the case ony for symmetric A. However, the singuar vaues are eigenvaues of the foowing eigenvaue probems AA T u i = σ 2 i u i, A T Av i = σ 2 i v i. (4.21a) (4.21b) (iii) Uniqueness: The σ i are unique. Two singuar vectors u i, v i are unique up to a common sign, except for vectors associated with mutipe singuar vaues. Here ony the spaces spanned by these vectors are unique. Thus U and V are unique up to inear combination of the corresponding coumn vectors. (iv) Fundamenta subspaces: A coumn vectors of U and V form a basis of the data and mode space, respectivey. Defining the range (or coumn space) of A as R(A) = {y = Ax x R n } (4.22) and the nuspace (or kerne) as N(A) = {x R n Ax = }, (4.23) the coumn vectors of V associated with the zero singuar vaues are a basis of N(A). More competey, for the four fundamenta subspaces of A : N(A) = span[v r+1,...,v n ], R(A) = span[u 1,...,u r ], N(A T ) = span[v 1,...,v r ], R(A T ) = span[u r+1,...,u m ]. (4.24) The importance of these seemingy abstract spaces ies in the fact that components of atmospheric states that beong to the nuspace of the matrix A describing the measurement go unnoticed by the experiment. Simiary, components of the data ying outside the range of A cannot be reproduced exacty by any retrieva x. (v) Characteristic features: The foowing two features are very common for discrete i-posed probems [Hansen, 1998] and wi aso appear ater in the tomographic appications of this work. The singuar vaues decay graduay to zero without particuar gaps in between. Increasing the dimensions of A wi increase the number of sma singuar vaues.

62 62 4. Tomography and Discrete, Linear Inverse Probems The singuar vectors have more sign changes in their components with increasing index i, i.e. decreasing σ i. If AA T and A T A are totay positive, then u i and v i have exacty i 1 sign changes. Taking eq. (4.2) and (v) together aows the interpretation that high frequency components of the state get damped by sma singuar vaues so that they add itte information to the measurement. On the other hand, the inversion wi be dominated by these osciating components (in the same way the soution of a square system is unstabe because of a sma system determinant). Sma singuar vaues ie at the heart of the i-posed probem making it effectivey rank-deficient. Obviousy the absoute size of the σ i is irreevant otherwise rescaing of A woud sove the probem of i-posedness. Instead one expects some reative expression for the instabiity. It can be derived by considering the perturbed system A(x + x) = d + ǫ (4.25) and assuming for the moment that A 1 exists. Then (choosing the induced norm of the vector norm) from x A 1 ǫ and d A x it foows for the reative error of the soution x x A 1 A ǫ d. (4.26) The factor A 1 A measures the reative sensitivity of the soution against perturbations and is caed condition number. Using the 2-norm one gets cond[a] = A 1 2 A 2 = σ 1 σ r (4.27) where the ast expression defines the condition number for any matrix A. The matrix in section 4.4 has singuar vaues σ 1 = , σ 2 =.71, thus a condition number of about 2, meaning that simpe inversion magnifies the reative error on the data at worst by a factor 2. Typica condition numbers for appications in CT are around 1 6! 4.6. The east squares probem As pointed out in the preceding section, the discrete tomographic probem is i-posed in the itera sense: The formuation in the form Ax = d causes the troube, probems with the soution are ony secondary. Reformuation depends on the origin of the i-posedness instabiity or ack of information and the physica nature of the underying probem in terms of hard, empirica or statistica constraints for admissibe soutions. The soution method, i.e. the agorithm, depends on the mathematica and numerica properties of the probem (inear/noninear, sma/arge systems of equations, sparse/dense systems etc). 4 The east squares approach is a mathematicay we covered formaism which aows simpe formuation and soution of basic i-posed probems irrespective of the cause of i-posedness. An extensive reference for a variety of east squares probems with focus on their numerica treatment is [Björck, 1996]. Least square criteria in a wider conceptiona context of inverse theory can be found 4 And the working fied. It seems that in geophysics, atmospheric profiing and image reconstruction peope traditionay empoy different agorithms for the same probems, sometimes mathematicay very simiar ones, yet with different names. The mathematica iterature offers a more comprehensive treatment.

63 4.6. The east squares probem 63 in [Tarantoa, 1987, 25]. We start our discussion with the most simpe cases Least squares and east norm soutions There are certainy many ways to motivate the use of east squares methods, ike the top-down derivation from statistica properties of the mode, here we take a rather pragmatic point of view and distinguish two cases: (i) Least squares minimisation (m > n): Having more equations than unknowns cannot ead to unique soutions if measurement errors are present. Instead, minimum discrepancy between data and forward cacuated data in terms of a vector norm is demanded. Choosing the 2-norm (see the discussion in section 4.6.5) gives the foowing reformuation of the discrete inverse probem. For given d R m and A R m n, find x R n such that Derivation eads to the norma equations min x d Ax 2 or min x (d Ax) T (d Ax). (4.28) A T Ax = A T d (4.29) and the soution x = ( A T A ) 1 A T d, (4.3) provided the inverse of the m m matrix A T A exists. This is the case if and ony if rank[a] = n which can easiy be shown using the SVD. (ii) Least norm minimisation (m < n): In the case where there are more parameters to determine than measurements avaiabe, conditions have to be added to the measurement equations in a way that gives rise to a unique soution. Picking the vector of smaest norm from a admissibe soutions makes sense from an economic point of view (uness you expect the soution to be very fuctuating). Aso this east norm soution has attractive mathematica properties (see beow). But nevertheess it remains arbitrary and if it can be repaced by physica constraints or statistica information, a the better. The mathematica impementation is such that the equations Ax = d are regarded as constraints for the norm minimisation probem, i.e. min x x 2, d = Ax. (4.31) With a Lagrange mutipier λ j for each of the n equations and writing λ = (λ 1,...,λ n ) T, one gets the norma equations of the second kind AA T λ/2 = d, x = A T λ/2 (4.32) with unique soution x = A T ( AA T) 1 d (4.33) if rank[a] = m. One might be famiiar with the characteristics of the east square method as it is the too for any kind of regression, but the behaviour of the east norm soution may not be equay cear.

64 64 4. Tomography and Discrete, Linear Inverse Probems Therefore, we woud ike to present a simpe Exampe based on fig. 4.4b, p. 59. Assume a constant concentration in a four square boxes of ength 1 (again we suppress a units for the moment): c = 1. Negecting any measurement errors yieds the coumn densities d = (2, 2 2, 2) T and together with this gives the east norm soution A = C A x LN = (1.5,.5,.5,.5) T, i.e. overestimation in the box with three ight paths, underestimation in the others. Why not a uniform soution (1, 1, 1, 1) T? Because x LN is compatibe with the measurements and has a smaer norm: x LN 2 = 3 < 2 = (1, 1, 1, 1) T 2. In the foowing case, a concentration peak is ocated in one of the boxes which are crossed by one ight beam, here the upper eft one in fig. 4.4b: c = (1, 1, 1, 1) T d = (2, 2 2, 11) T and x LN = (3.75, 1.75, 7.25, 1.75) T, i.e. underestimation of the peak. In this case negative concentrations compatibe with the data ead to smaest norm. We finay address the cases of rank deficiency, i.e. rank[a] < min(m, n). To prepare the forma soution, notice that the inverse of a square matrix A with rank[a] = m = n can easiy be expressed in terms of the SVD eq. (4.18) A 1 = V Σ 1 r U T. (4.34) A simiar expression hods for the fu-rank over-determined case eq. (4.3) ( A T A ) 1 A T = V ( Σ 1 r and the fu-rank under-determined case eq. (4.33) A T ( AA T) 1 = V ( Σ 1 r ( Σ 1 r It is tempting to consider the expression V deficient case. Indeed, the foowing theorem hods: Least squares-minimum norm soution ) U T, rank[a] = m (4.35) ) U T, rank[a] = n. (4.36) ) U T as a generaised inverse in the rank Let A R m n be of rank r min(m,n) and the SVD of A be given by eq. (4.18) then the genera

65 4.6. The east squares probem 65 east squares probem has the unique soution min x 2, S = {x R n Ax d 2 = min} (4.37a) x S x = A d, (4.37b) where the matrix A = V ( Σ 1 r ) U T (4.37c) is caed the pseudoinverse of A. See [Björck, 1996] for a proof. Depending on the dimensions m, n the pseudoinverse (aso caed generaised inverse or Moore-Penrose inverse) automaticay generates a east squares, east norm or the exact soution if A is fu rank. Otherwise the resut is a mixture referred to as east squaresminimum norm soution Weighted east squares and weighted east norm with a priori The east squares and east norm principes of the preceding section might need modification to incorporate the underying physics. To make the point cearer we assume rank[a] = min(m,n). (i) Data weighting: Imagine that the uncertainty differs substantiay within the measured data d i (something not too hard to imagine). This means, for exampe, that a arge data vaue which is arge because of a arge measurement error enters the east squares minimisation probem eq. (4.28) in the same way as a arge vaue with a sma measurement error. Therefore, it is common practice to weight the data according to their errors, usuay by the inverse of the error variances σ ǫ i or more genera, the inverse of the error covariance matrix S ǫ = E[(d ǫ d)(d ǫ d) T ]. Here E[ ] denotes the expected vaues and d = E[d ǫ ] according to eq. (4.2) with unbiased errors. The weighted east squares minimisation repaces eq. (4.28) by 5 min x (d Ax)T S 1 ǫ (d Ax). (4.38) S ǫ is positive semidefinite. If none of the σ ǫ i equas zero it is positive definite and the inverse exists. It is straightforward to obtain the soution x = ( A T Sǫ 1 A ) 1 A T Sǫ 1 d (4.39) and, empoying the reation S x = BS y B T for the covariances that hods for any inear reation x = By, the covariance of x is (ii) A priori: S x = ( A T S 1 ǫ A ) 1. (4.4) Data weighting has no effect on the east norm soution (apart from unintended numerica impications) as the equations Ax = d are fufied exacty. But the minimisation principe for x 2 might not be a good choice for cases ike the foowing. Assume that the concentration fied shows oca variabiity 5 Another way to ook at it is that the 2-norm gets repaced by a norm defined as 2 S 1 ǫ = ( )S 1 ǫ ( ) T.

66 66 4. Tomography and Discrete, Linear Inverse Probems on an otherwise constant but high background that is more or ess known. Instead of demanding that the norm is sma, it woud make far more sense in this case to demand that the deviation from this background is minima, i.e. min x x a 2, where x a stands for this known, estimated or assumed concentration. It is usuay referred to as a priori because whatever information is used for its construction, in genera, it has to be known beforehand and cannot be gained as a product or byproduct of the reconstruction process. Inserting a weighting matrix as in eq. (4.38) the minimisation principe with a priori reads now min x (x x a ) T S 1 a (x x a ), d = Ax. (4.41) There is another drawback of the east norm principe eq. (4.31) which becomes apparent from eq. (4.36). Whenever a component of the retrieva ies in the nuspace of A it is set to zero. One coud argue that naturay the principe cannot produce vaues that are not given by the data, but zero is as arbitrary as any other guess. Soving eq. (4.41) makes this cearer: x = x a + S 1 a A T ( AS 1 a A T) 1 (d Axa ). (4.42) Obviousy, eq. (4.33) is a specia case of this with x a =, i.e. it contains an a priori, too. In principe, a soution simiar to eq. (4.37b) coud be given for the modified east squares - minimum norm probem min (x x a) T Sa 1 (x x a ), S = {x R n (d Ax) T Sǫ 1 (d Ax) = min}, (4.43) x S with a priori x a R n, measurement error covariance S ǫ and an a priori weighting matrix S a, but for the remainder ony the expicit form of the soutions eqs. (4.39,4.41) for the fu-rank case is important Reguarisation and constrained east squares probems The east squares-minimum norm principe soves the probems of non-uniqueness and non-existence but does not make any difference for the instabiity of the soution because of sma singuar vaues. This is very cear for the soutions written in the form of eqs. ( ) (see aso [Twomey, 1997, chap. 6.2]). Athough i-posedness due to instabiity is not the main issue for the under-determined reconstruction probems focused on in this work, severa strategies to reguarise (as this stabiising is caed) the east squares-east norm soution are briefy sketched in the foowing for two reasons. Firsty these methods are widey used, indispensabe reference and starting points for any discussion and future deveopment. Secondy, whie some reguarisation methods are purey mathematica, others aim to invove physica constraints reated to the actua measurement to generate a stabe and unique soution. These atter approaches are attractive aternatives to methods that contain arbitrary ad hoc parameters and are especiay interesting for under-determined probems provided that there is enough information to formuate these physica constraints. (i) Tikhonov reguarisation: Notice that the instabiity of the east squares and east norm soution ies in the expressions ( A T A ) 1

67 4.6. The east squares probem 67 and ( AA T) 1, respectivey. Making the substitution ( A T A ) 1 ( A T A + λ 2½n ) 1 (4.44a) or ( AA T ) 1 ( AA T + λ 2½m ) 1, (4.44b) i.e. adding a term proportiona to the unit matrix, changes the singuar vaues in the decompositions eqs. (4.35,4.36) to σ 1 i σ i. (4.45) + λ2 For the right choice of λ sma singuar vaues are suppressed as desired. Foowing Tikhonov a choice α = α(λ) is said to resut in a reguar soution x λ of Ax = d if σ 2 i λ α(λ), x λ A d with A as in eqs. (4.37). There is a minimisation probem that eads to the same resut as the substitutions eqs. (4.44): min x Ax d λ 2 x 2 2 (4.46) (Mind that λ is not a Lagrange-mutipier here). The reasoning for this ansatz in the over-determined case is that minimisation of the discrepancy Ax d 2 aone eads to osciating soutions whie the addition of x 2 favours smooth soutions. Aowing additionay weighting and an a priori as in section one has the more genera reguarised (aso caed damped or generaised 6 ) east squares principe min x with the straightforward soution (Ax d) T S 1 ǫ (Ax d) + λ 2 (x x a ) T D T D (x x a ) (4.47) x = x a + ( A T Sǫ 1 A + λ 2 D T D ) 1 A T Sǫ 1 (d Ax a ) (4.48a) = ( A T Sǫ 1 A + λ 2 D T D ) 1 ( A T Sǫ 1 d + λ 2 D T ) D x a (4.48b) = x a + ( D T D ) 1 A T ( A(D T D) 1 A T + λ 2 S ǫ ) 1 (d Axa ). (4.48c) Here D R n n can be the identity matrix, a diagona weighting matrix or a discrete approximation of a derivative operator to infuence the smoothness of x etc. In the ast step it was assumed that D T D is reguar. Foowing from the norma equations, the soution is unique if N(S 1/2 ǫ A) N(D) = {}. Cacuation of the covariance S x can be carried out as in eq. (4.4). The generaised east squares soution formay contains the east squares and east norm soution in the imits generaised east squares λ east squares λ 2 S ǫ east norm. (4.49) This method is generay attributed to Tikhonov [1963] but sometimes aso referred to as Twomey- Tikhonov approach [Twomey, 1963]. Whie there exist estimations for the effect of the reguarisation, i.e. bounds on how much the soution changes compared to the unreguarised (see sec ), there 6 The term constrained east squares used by Twomey [1997] is not common in this context. The nomencature is far from uniform.

68 68 4. Tomography and Discrete, Linear Inverse Probems is nothing teing what exact vaue the reguarisation parameter λ ought to take. If λ is chosen too sma there is no reguarisation, if it is too arge the soution is oversmoothed. There is a variety of suggestions for the parameter choice, some of them are very intuitive ike the discrepancy principe that chooses λ such that the perturbation bounds are smaer than the residua norm Ax d. This is not pursued any further, instead we refer to [Hansen, 1998, chap. 7] and quote Linz [1984]: There are no genera criteria by which different agorithms can be compared. Consequenty, many methods are proposed which, on the evidence of a few specia cases, are caimed to be effective. (ii) Constrained east-squares: The foowing east squares probems with quadratic constraints are dua to each other in that for the right choice of parameters δ,ǫ R their soutions are identica to the Tikhonov-reguarised soution min Ax d 2, S = {x R n D(x x a ) 2 δ} (4.5a) x S min D(x x a) 2, S = {x R n Ax d 2 ǫ}. (4.5b) x S This time the constraints can be taken into account by Lagrange mutipiers as the inequaities are amost aways fufied as equaities for quadratic constraints. The advantage of this formuation ies in the fact that it aows to express the reguarisation in terms of bounds on physica quantities ike gradients in the case of eq. (4.5a) or measurement errors for eq. (4.5b). For some inverse probems, ike the tomographic probem, a further inequaity constraint is necessary to assure nonnegativity of the soution: min D(x x a) 2, S = {x R n Ax d 2 ǫ x }. (4.51) x S Whie there are standard methods to sove east squares probems with inear inequaity constraints or quadratic constraints ike eqs. (4.5a,4.5b) [e.g., Björck, 1996, chaps.5.2,3], soving the optimisation probem eq. (4.51) is rather more invoved [e.g., Fehmers, 1998]. (iii) Truncated singuar vaue decomposition and iterative reguarisation: Both methods rey on the SVD. The truncated SVD expoits the fact that the generaised inverse eq. (4.37c) has an expansion simiar to A eq. (4.2): A = r i=1 σ 1 i u i v T i (4.52) with r = rank[a] and increasing contributions as i increases. So cutting off the sum at some i c < r wi stabiise the inverse. Basicay it amounts to repacing the i-conditioned, fu-rank matrix A by a we-conditioned, rank-deficient matrix. Because u i, v j are orthonorma, the soution wi aso have a smaer norm than the unreguarised. Again there is no canonica choice for the cut-off parameter. It coud be given by an upper bound f ǫ for the reative error magnification eq. (4.27), i.e. σ 1 /σ ic < f ǫ or σ ic > σ 1 /f ǫ. Reguarisation by means of iterative agorithms that are for reasons to be discussed ater used to generate east squares soutions works very simiar to the cut-off of sma singuar vaues. If the agorithm initiay picks up components in the SVD that beong to arge singuar vaues or converges faster for these, then stopping the iteration prematurey wi suppress sma singuar vaues. The roe

69 4.6. The east squares probem 69 of the reguarisation parameter is payed by the iteration number. Reevant detais wi be discussed in chap (iv) Parametrisation of the continuous probem: There are other factors that infuence the stabiity of the probem athough it might be hard to formuate them according to the mathematica definition of reguarisation given by Tikhonov. For exampe, in section 4.4 it was argued that the number of basis functions infuences the conditioning of the system matrix. The same hods for the kind of basis functions. We woud ike to mention a thorough study by Doicu et a. [24] who theoreticay investigated the reconstruction of 1-D temperature and concentration profies from far infrared airborne observations and infrared imb emission spectra, respectivey. Apart from using a reguarised agorithm, the authors show that different basis functions (Chebyshev, Hermite poynomias and B-spines in this case) ceary have different reguarising properties. Another approach totay ignored here is to reguarise before discretising. In fact, this approach is reated to the choice of basis functions [see Groetsch, 1993, section 5.2]. And one coud sti think of other reconstruction schemes invoving a finite number of paramaters that circumvent the probem of instabiity and ead to possiby not idea, yet stabe soutions Resoution matrix and averaging kernes Assume that the retrieva is given by the inear map A inv, so that x = A inv d. If a errors in the discrete forward mode are negected, the coumn densities can be cacuated from the true state of the system x true as d = Ax true. Inserting this into the previous equation yieds x = A inv Ax true = Rx true. (4.53) R is caed resoution matrix as it quantifies how we a state is resoved by the measurement pus retrieva process. Its coumn vectors describe to what degree the components x truei are averaged over, i.e. for the tomographic reconstruction: how the origina box concentrations are spread over the retrieva grid and finay give rise to the concentration x i in box i. The rows of R are caed averaging kernes. R takes a simpe form for the unweighted east norm soution (i.e. m < n) eq. (4.33): R = A T (AA T ) 1 A = r=rank[a] j=1 v j v T j. (4.54) This is just the projection matrix onto the orthogona compement of the nuspace of A, that is the subspace where the retrieva is competey given by the measurement data. The matrix P =½n R =½n A T (AA T ) 1 A (4.55) projects onto the nuspace of A.

70 7 4. Tomography and Discrete, Linear Inverse Probems In the same way, inserting d = Ax true into the generaised east squares soution eq. (4.48a) eads to x = x a + ( A T Sǫ 1 A + λ 2 D T D ) 1 A T Sǫ 1 A(x x a ) (4.56a) } {{ } R = Rx true + (½n R)x a (4.56b) = A inv d + (½n R)x a (4.56c) with simiar expressions for eq. (4.48c). This formuation shows nicey that the information from the measurement is mapped to the retrieva x by R whie anything ese (½n R) has to come from the a priori. For the over-determined case (m > n) the projector in eq. (4.54) becomes R =½n if r = rank[a] n. In this case definition of a corresponding data resoution matrix woud be appropriate. But as we do not make use of it, we eave the matter to it Remark on the norm Finay, a justification for our choice of norm is mandatory, even if ony very briefy. In principe, any norm coud be used in the above minimisation probems, e.g. any member of the p norm famiy defined by x p = ( n x i p) 1/p, p 1. i=1 But the choice of a norm very different from the standard ones requires firm knowedge of the anaytic and agebraic impications. For exampe, on the one hand the probem min x Ax d p, rank[a] = n has a unique soution for p > 1 ony, whie on the other hand soutions with norms near p = 1 have more stabe numerica properties [Scaes and Gersztenkorn, 1988]. Furthermore, the norm is cosey reated to the statistics of the probem (see [Tarantoa, 1987] for an extensive discussion). Here, we assume a reevant quantities to be Gaussian distributed which naturay eads to the 2-norm. In other words, the choice is dictated by the statistica nature of the measurement, not by the detais of the retrieva method. There might be norms which are better suited for the inversion of certain probems or concentration fieds, even if the statistics behind is Gaussian. Then a mathematica toos that are based on the 2-norm, e.g. the SVD, woud have to be modified. The choice of the norm is a fundamenta matter under severa aspects and with no indication against the conventiona 2-norm at hand, a reconsideration is beyond the scope of this work Statistica approach to the east squares probem The statistica approach to the inverse probem describes a quantities invoved by probabiity densities, i.e. P(x) and P(d) for x and d in our notation (see [Tarantoa and Vaette, 1982] for a concise yet genera formuation of the approach). The forward mode and thus ideay the measurement is expressed as conditiona probabiity P(d x), the reconstruction by the a posteriori conditiona prob-

71 4.7. Statistica approach to the east squares probem 71 P(x) P(d x) a) b) c) d) Figure 4.5: Schematic picture of how the measurement can update the a priori information about the system. (a) The measurement P(d x) increases the knowedge about the a priori P(x) but does not confine the a posteriori estimate competey. (b) The measurement constrains the a priori competey. (c) The measurement does not provide any new information and the retrieva distribution is basicay given by the a priori estimate. (d) A priori and measurement are inconsistent in some user-defined sense. abiity P(x d). As both are reated via Bayes theorem P(x d)p(d) = P(d x)p(x), (4.57) the statistica approach is aso referred to as Bayesian approach. To infer the a posteriori conditiona probabiity P(x d) from eq. (4.57) both P(d x) and P(x) have to be known. P(d x) is given by the forward mode, the measurement data and its statistics. The a priori probabiity density is a more intricate matter as it asks for (statistica) knowedge about the system before the measurement. If the Bayesian approach is intended to increase the information on the actua state of the system after the measurement, the a priori has to be based on either theoretica insight or empirica ground that is firm enough to aow for a statistica description of it. This coud be certain knowedge ike concentrations are aways positive or a cimatoogy, for exampe confining the temperature profie of the atmosphere from past measurements etc. Whether the a priori is usefu for the reconstruction is a different matter, as sketched in the cartoon in fig Another way to use the Bayesian method is the foowing: An assumption about the state mode is formuated as a probabiistic statement which serves as a priori probabiity for the Bayesian inversion. Here the point is to constrain the a priori mode assumption more tighty by the a posteriori estimate (see aso fig. 4.5). The measurement data can aso hep to construct the a priori but then the uncertainties have to be corrected. This method is known as empirica Bayes approach [e.g., Carin and Louis, 1998]. As we wi see, yet another way is to (mis)use the statistica inversion as a reguarisation method. An attractive feature of the statistica inversion is that by its nature it provides an uncertainty for the retrieva, i.e something ike an error estimate. The Bayesian method is widey used for any of the questions just mentioned in geophysics [e.g., Scaes and Snieder, 1997; Tarantoa, 25] and atmospheric research, here mosty in the very popuar form of the optima estimate for Gaussian probabiity densities [Rodgers, 2] The optima estimate To get a specific retrieva one has to choose a state from the a posteriori distribution. The optima estimate is the soution that maximises the a posteriori probabiity. It is therefore more correcty caed maximum a posteriori (MAP) or ess correcty maximum ikeihood (ML) soution. Assuming

72 72 4. Tomography and Discrete, Linear Inverse Probems a Gaussian distribution P(d x) e (d Ax)T S 1 ǫ (d Ax) for the measurement is pausibe (here S ǫ denotes again the measurement error covariance). The statistics of the a priori is ess evident. If it is obtained from some kind of cimatoogy, then a Gaussian distribution makes sense. But if, for exampe, it consists in an educated guess based on mode resuts, choice of Gaussian statistics is more than questionabe. Nevertheess we assume a Gaussian probabiity density P(x) e (x xa)t S 1 a (x xa) where x a is again the a priori (state vector) and S a its covariance. The MAP soution coincides with the expected mean E[P(x d)] and takes the foowing forms [Rodgers, 2; Tarantoa and Vaette, 1982]: with covariance x = x a + ( A T S 1 ǫ = ( A T S 1 ǫ A + Sa 1 ) 1 A T Sǫ 1 (d Ax a ) (4.58a) A + S 1 ) 1 ( A T Sǫ 1 d + Sa 1 ) x a (4.58b) a = x a + S a A T ( AS a A T + S ǫ ) 1 (d Axa ). (4.58c) S x = ( A T S 1 ǫ A + S 1 ) 1 (4.59a) a = S a S a A T ( AS a A T + S ǫ ) 1 ASa. (4.59b) The soution is formay identica with the generaised east squares soution eq. (4.48) if one sets S 1 a = λ 2 D T D. And this is exacty how the optima estimate reguarises or can be used to reguarise the east squares soution, as becomes especiay cear for diagona S a = σa½n. 2 σa 1 pays now the roe of the reguarisation parameter λ. For competey non-committa a priori σ a (see fig. 4.5b) or infinitey precise measurement, the optima estimate turns into the unreguarised east squares soution. If the a priori state of the system is known exacty σ a (fig. 4.5c) or the data error is extremey arge, the soution is just the a priori x a (oversmoothing). For the standard case (fig. 4.5a) the optima estimate can be seen as the weighted mean of what is known from the measurement and the a priori. In fact, treating the a priori as an additiona measurement by buiding the augmented measurement vector, system matrix d = ( d x a ) R m+n, A = ( A½n ) R (m+n) (m+n) (4.6a) and covariance S 1 = ( S 1 ǫ S 1 a ) R (m+n) (m+n) eqs. (4.58) can be obtained from the weighted east squares probem (4.6b) min (d Ax) T S 1 (d Ax). x Conditions for the uniqueness of the soution are as for eqs. (4.48).

73 4.7. Statistica approach to the east squares probem Degrees of freedom and information content A further benefit of the Bayesian approach is that it aows to adopt concepts from any other context that uses a stochastic description ike statistica physics, information theory etc. Two quantities are introduced here for ater use without deriving them or putting them into proper context [c.f. Rodgers, 2, chap. 2.4]. The optima estimate minimises the expression (x x a ) T Sa 1 (x x a )+ǫ T Sǫ 1 ǫ where ǫ = d Ax. The first term provides information about x, the second contribution is just noise. Their actua average vaues are interpreted as degrees of freedom for the signa d s and noise d n, respectivey. For the optima estimate one gets d s = tr[r] = m = d s + d n, rank[ã] i=1 σ 2 i /(1 + σ 2 i ), (4.61a) (4.61b) where à = S 1/2 ǫ ASa 1/2 and σ i are the singuar vaues of Ã. The resoution matrix R for the optima estimate is given as in eq. (4.56), again with the substitution λ 2 D T D Sa 1 : R = ( A T S 1 ǫ A + S 1 ) 1 A T Sǫ 1 A. (4.62) The d s and d n add up to the degrees of freedom of the measurement, as expected. Identifying d si = σ i 2/(1 + σ2 i ) as the degree of freedom associated with the ith singuar vaue, one woud say that it increases the information about the system if d si > d ni or σ i > 1. Otherwise its contribution to the measurement cannot be distinguished from noise. Consequenty, these quantities can be used to evauate the measurement setup and reconstruction for different eves of noise. Expressing the resoution matrix with the a posteriori covariance S x (eq. 4.59) gives a or R =½n S x S 1 a n = d s + tr(s x S 1 a ), (4.63a) (4.63b) where the trace term can be interpreted as the number of parameters resoved by the a priori. Up to now, expressions ike increase of information or knowedge have been used ony in a very oose sense. They can now be given a specific meaning in terms of entropy S. In thermodynamics S is given by the ogarithm of the number of microstates compatibe with a given macrostate. 7 So the difference between the entropy before and after the measurement agrees perfecty with what intuitivey woud be described as the effect of the measurement: Excude states of the system that, in principe, are possibe. The information content of the measurement is thus defined as H = S (P(x)) S (P(x d)). The probabiity densities repace probabiities in the forma definition of the entropy and for Gaussian 7 Provided that a microstates are equay probabe.

74 74 4. Tomography and Discrete, Linear Inverse Probems density functions this yieds H = 1 2 n ½n R = 1 2 rank[ã] which agrees with a cacuation from the difference S (P(d)) S (P(d x)). i=1 n ( 1 + σ 2 i ). (4.64) d s and H represent two different ways to cacuate a singe number with concrete physica meaning from the rather unhandy resoution matrix or averaging kernes Iterative soution of the east squares probem After formuating a number of east squares probems and presenting the forma soution for the unconstrained cases, we turn now to the actua computation of these soutions Iterative versus direct methods Numerica agorithms for east squares probems are based on either type of the norma equations (4.29,4.32), not on matrix inversion as in x = (A T A) 1 A T d. (The SVD for which cacuation of the inverse is trivia, may be regarded as an exception). They are either direct methods, i.e. they generate a soution in a finite number of steps, or iterative, i.e. converging towards the soution in a sequence {x (k) }, k = 1,...,. A direct method coud be any method for soving systems of equations that eads to stabe soutions of the norma equations, ike the QR or Choesky decomposition or the Househoder transformation. For i-conditioned probems the system with a reguarised matrix has to be soved. Iterative agorithms are chosen mainy for two reasons. First, it usuay suffices to know the action of the matrix during the iteration, one does not have to store the whoe matrix itsef. This is particuary attractive if the matrix is very arge or its action easiy generated. The second appication concerns cases where the initiay favourabe form of the matrix is not utiised but instead worsened by a direct method, e.g. fi-in of a sparse matrix. This is the case especiay for reconstruction probems where A is randomy sparse as in tomographic probems [Björck, 1996, 7.1.1]. Whie for sma systems there is sti no urgent need to resort to iterative sovers for the norma equations, the situation may change if constraints have to be taken into account. The east squares probem becomes now a constrained optimisation probem min f(x), x S where S is the set of feasibe soutions (e.g. positive soutions etc.) and the so caed objective or cost function is quadratic here. As mentioned in sec , there are direct soutions for simpe inequaity constraints, but their impementation is much easier for a certain cass of iterative agorithms that has been extensivey studied in image reconstruction and signa processing, the so caed projections onto convex sets (POCS) [see Byrne, 24, and references therein]. This is now iustrated for the under-determined case, where the east squares-east norm soution satisfies Ax = d. Each of the m equations can be regarded as a set S i (hyperpane in this case) inên. If their intersection S = i S i is nonempty then the sequence P k of the product of the projection operators P i onto the sets P = i P i wi converge towards a member of the feasibe set S, fig. 4.6a. The detais of convergence have to be

75 4.8. Iterative soution of the east squares probem 75 x 2 x 2 P 2 x (1) x () P 1 x P + x x 1 x 1 (a) (b) Figure 4.6: (a) The projection onto convex sets (POCS) for two panes d i = P j Aijxj, i = 1, 2 in the case of a unique soution. The cyce of orthogona projections P 1, P 2, P 1,... shown here is equivaent to the agebraic reconstruction technique (ART). Notice that the smaer the ange between the panes (sma singuar vaues) the sower the convergence wi be. (b) The projection P + onto {x Ên x } acts by setting negative components to zero. specified, of course, aso this does not hod for arbitrary sets (it does for convex sets). Furthermore, the operator P does not have to be a projection operator [for detais Byrne, 24] but the preceding exampe is genera enough for our purpose. The important point in favour of these iterative agorithms is now that the impementation of inequaity constraints ike nonnegativity becomes trivia by just inserting the corresponding operator into the above sequence of projections (e.g., fig. 4.6b for the positivity constraint) Typica convergence behaviour Before comparing some important iterative methods used to sove east square probems, we mention some characteristics common to their convergence behaviour when appied to i-conditioned systems. (i) Data residuum: For the iterative soution x (k) the residua d (k) 2 = d Ax (k) 2 (4.65) wi typicay be monotonicay decreasing during the iteration because the iterative agorithms considered here are constructed to reduce d (k) 2, see fig. 4.7a. In the consistent under-determined case d (k) 2 can reach zero. (ii) Convergence rates: Without any errors x (k) wi converge to the exact east-squares soution (or east squares-minimum norm to be more precise) x as k, fig. (4.7b). Convergence rates are convenienty anaysed with the hep of the SVD. Simiar to the decomposition of the generaised inverse A = σ 1 i u i vi T, eq. (4.34), one writes x (k) = i f (k) (σ i,d) σ 1 i u i v T i d (4.66) where the fiter factors or response functions f (k) 1 as k. For many iteration schemes they do not depend on d. The fiter factors quantify the behaviour necessary for iterative reguarisation as suggested in 4.6.3(iii), that is, they give information on the convergence rates of the frequency modes

76 76 4. Tomography and Discrete, Linear Inverse Probems d Ax (k) 2 x x (k) 2 x x (k) ǫ 2 (a) k (b) k (c) k Figure 4.7: (a) Data residua versus iteration number k for the consistent ( ) and inconsistent (--) case. (b) Without errors the iterative agorithm converges to the exact (east squares) soution. (c) Typica behaviour of iterative agorithms if errors are present ( semiconvergence). The soid ine ( ) corresponds to the ART agorithm, (--) to the CG method and (- -) to the Landweber iteration in the exampe [Hansen, 1998, p. 165]. of the soution. (iii) Reguarisation: In the presence of errors the convergence of i-conditioned equations turns into a behaviour denoted as semi-convergence. In the beginning x (k) approaches the exact soution but deteriorates again for arger iteration numbers, see fig. 4.7c. The soution is reguarised by stopping the iteration before this happens ideay at the optima iteration number Brief comparison of some iteration methods Deaing with iterative agorithms can be sighty confusing. For exampe, the very simpe Landweber iteration discussed extensivey by Twomey [1997] is known under (at east) five different names in the iterature [Hansen, 1998]: Landweber, Richardson, Fridman, Picard and Cimino iteration. Athough imited to a minimum, a short, quaitative overview seems in order. The cassica methods for east squares probems ike the mentioned Landweber, the Jacobi and Gauß-Seide agorithms are a based on different spittings A T A = M N, M non-singuar, eading to the sequences Mx (k+1) = Nx (k) +d. Ony one component of x is updated in one step which makes it easy to introduce constraints, e.g. in the form of a projection P as discussed above. As their convergence is rather sow, the Gauß- Seide agorithm has been endowed with a so-caed reaxation parameter, yieding the successive over-reaxation (SOR) scheme. Thanks to its simpicity and fexibiity it is being empoyed in appications for which this property is essentia [e.g., Sauer and Bouman, 1993; Fesser, 1994]. Another cassica group of agorithms is the cass of projection methods (not to be confused with the projections onto convex sets from above. The projections here have different meaning and target space). Widespread projection methods are the gradient-based methods ike the steepest descent and the conjugate gradients (CG) agorithms. They can be viewed as optimisation schemes based on a oca search where the search directions are not random but depend on the oca properties of the supposed resut, ike its gradients. In the geophysica iterature they appear to be the standard too to sove inverse probems, but they are not very stabe for i-posed probems, especiay not for higher iteration numbers. This ed to the mathematicay equivaent but more stabe LSQR method [see Björck, 1996]. Furthermore, these methods do not easiy accommodate inequaity constraints [Fesser, 1994], see

77 4.8. Iterative soution of the east squares probem 77 aso [Cavetti et a., 24] for a short discussion. Finay, there are the ART ike methods, named after the first agorithm the agebraic reconstruction technique, which was visuaised in fig. 4.6a. These agorithms beong to the POCS and, depending on whether the projections are performed successivey as in P i P i or at once P i P i, they are further cassified as sequentia or simutaneous. Because their computation is very cheap in terms of computer memory space, these agorithms have been widey used in image reconstruction. In passing, we note that the origina ART is equivaent to the Gauß-Seide agorithm for the norma equations of the second kind, whie the simutaneous version SART is a specia case of the Landweber iteration. Without giving proofs, we add some reevant facts concerning the convergence and fitering to get the foowing picture SOR aows simpe impementation of constraints. Sma singuar vaues (high spatia frequencies) converge faster than arge singuar vaues [Sauer and Bouman, 1993; Fesser, 1994]. CG cannot easiy be combined with inequaity constraints. Large eigenvaues converge faster. It converges quicky with sma residuas, but it is not apt for i-posed probems. In the case of semi-convergence it converges fast to the optimum, but deteriorates quicky past this point, too [Björck, 1996]. ART ike methods can easiy be augmented with constraints. The fitering behaviour depends on the specific agorithm. ART favours smaer singuar vaues, the simutaneous methods arge singuar vaues. The convergence behaviour of SOR is suspicious, in the sense that the faster converging sma singuar vaues are more susceptibe to noise. Indeed, it wi be shown that the simiary behaving ART performs worse than simutaneous methods. Requesting simpicity and fexibiity for the impementation of constraints rues out CG. Thus the next chapter wi have a coser ook at the ART ike methods ART, SART and SIRT The origina method of these row acting methods, as they are aso caed because they act ony on one row of the matrix A at the same time, goes back to Kaczmarz [1937]. It was rediscovered and generaised by Tanabe [1971], appied to image reconstruction by Herman et a. [1973] where it has been widey used and further deveoped. A simutaneous version of ART, the simutaneous image reconstruction technique, SIRT, was proposed by Gibert [1972]. SART [Andersen and Kak, 1984], a sighty different simutaneous anaogue of ART, improved reconstructed images in medica appications significanty. Numerous modifications of these agorithms concern the acceeration of convergence speed, the appication to constrained east squares probems and their formuation as bock-iterative (i.e. ony parts of the matrix act simutaneousy, the rest sequentiay) and other aspects. We refer to [Censor and Herman, 1987] for a brief review. Athough being much sower than direct methods ike the fitered back projection (sec ), it appears that with increasing computer power these iterative agorithms regain interest thanks to their fexibiity [Leahy and Byrne, 21], especiay for cases with inconsistent or incompete data [e.g., Mueer and Yage, 2].

78 78 4. Tomography and Discrete, Linear Inverse Probems Iteration step: Introducing the row vectors of the m n matrix A = (a 1,...,a m ) T, ART in its simpest form can be written as x (k+1) = P i x (k) = x (k) + ω a i a T i a (d Ax (k) ) i, k =,1,2,... (4.67a) i i = k mod m + 1, ω >, where ω is a reaxation parameter and x () a user defined starting vector. P i corresponds to the projection onto the ith hyperpane, for ω = 1 iustrated in fig. 4.6a. More expicity, the iteration cyce ooks ike this: x (1) = P 1 x () x (m+1) = P 1 x (m) x (2) = P 2 x (1).... (4.67b) x (m) = P m x (m 1) SIRT reads as x (k+1) = Px (k) = x (k) + ω i a i a T i a (d Ax (k) ) i. (4.68) i SART was in [Andersen and Kak, 1984] experimentay motivated in order to reduce the impact of noise and derived in the foowing form x (k+1) j = Px (k) = x (k) j + ω 1 A ij (d Ax i 1 A i1j j (k) ) i. (4.69) 2 A ij2 i In both cases the correction to the update is a weighted mean of the projections onto the individua hyperpanes. The positivity constraint can be incorporated by inserting the projection P + (fig. 4.6b) for every iteration step in ART and SIRT/SART by substituting P i P + P i with P + x j = max(,x j ), (4.7a) P P + P. (4.7b) Convergence: None of the three agorithms actuay soves Ax = d or the reated norma equations but rescaed versions thereof. This can be seen [Van der Suis and van der Vorst, 1987; Trampert and Lévêque, 199] by introducing the weighting matrices m L φ = diag[ (A ij ) α ] Ên n, (4.71a) L = diag[ i=1 n (A ij ) 2 α ] Êm m j=1 (4.71b)

79 4.8. Iterative soution of the east squares probem 79 with nonnegative parameter α and performing the transformation The iteration steps now take the form: ART (α = ) SIRT (α = ) and SART (α = 1) Ā = L 1/2 AL 1/2 φ, (4.71c) x = L 1/2 φ x, (4.71d) d = L 1/2 d. (4.71e) x (k+1) = x (k) + ω ĀT ( d Ā x(k) ) i e i, with (e i ) j = δ ij (4.72a) x (k+1) = x (k) + ω ĀT ( d Ā x(k) ). (4.72b) Eqs. (4.72) show that a soution x (k) x satisfies the rescaed norma equations Ā T Ā x = ĀT d or A T L 1 Ax = AL 1 d (4.73) that correspond to a weighted east squares probem, c.f. eq. (4.39). A correct anaysis using the SVD yieds: Convergence of sirt and sart The one parameter famiy of simutaneous agorithms eqs. (4.72b) with a nonnegativity constraint as in eq. (4.7b) converges for α 2, < ω < 2 to the soution of the east square-minimum norm probem min (x x S x() ) T L φ (x x () ), S = {x R n (d Ax) T L 1 (d Ax) = min, x }. (4.74) This means The iteration start x () is the a priori of the east squares probem. Impicit weighting of the a priori by SIRT (α = ): L φ = m½n, i.e. no weighting SART (α = 1): L φ = diag( i A ij), sum of a ight path engths in box j, i.e. samping of box j. Impicit weighting of the data by SIRT (α = ): L 1 = diag( j A2 ij ) 1, inverse sum of (box LP engths) 2 for LP i SART (α = 1): L 1 = diag( j A ij) 1, inverse ength of LP i, where the simpe geometrica interpretation hods ony for the box basis. Finay, the fiter factors, see eq. (4.66), of the nonzero singuar vaues for the transformed system are f (k) ( σ i ) = ( 1 (1 ω σ i 2 ) k), i rank[ā], (4.75) where for the singuar vaues of Ā hods σ i 1. For the components of x in the system of the singuar vectors, i.e. of x = V T x, the convergence rate is given by x (k) j x ( ) j = (1 ω σ j) 2 k (x () j x ( ) j ), j rank[ā]. (4.76)

80 8 4. Tomography and Discrete, Linear Inverse Probems As the convergence is faster for arger vaues of ω σ i 2, the reaxation parameter ω can be used to speed up the iteration process. Eq. (4.76) estabishes the above statement for the frequency fitering of the simutaneous ART methods. For ART the simpe SVD approach does not work because of the series of operator products. Basicay the method converges simiary to the simutaneous versions except for the inconsistent case. Here the agorithm may not converge to a singe fix point but to a cycic subsequence. Comparing eqs. (4.67a), (4.68) it foows an impicit weighting of ART ike SIRT. For detais see [Kak and Saney, 21] where a geometrica iustration is given and [Jiang and Wang, 23] for further mathematica references Other reconstruction methods One coud think of various other approaches to the tomographic inverse probem eq. (4.4) ike genetic agorithms, neura networks or Monte Caro methods. Whie some of these approaches indeed occur in the iteratue and might be we justified, they are nevertheess back box methods that may not bear great potentia for further insight. This chapter concudes by sketching some important methods that are aso empoyed within the context of tomographic or discrete inverse probems Maximum entropy and maximum ikeihood The concept of entropy was appied in sec to cacuate the information content of a measurement under the assumption that a quantities invoved can be described by Gaussian probabiity density functions. But entropy can aso be used constructivey as a reconstruction principe based on the assumption that the most probabe reconstruction among a that reproduce the data (and possiby further constraints) is the one with maximum entropy. This principe has the important property that it does not introduce any new correation in the reconstruction resut that goes beyond the data [Gu and Danie, 1978]. Stricty speaking, the maxium entropy method MEM (or MaxEnt) requires probabiity density functions but usuay the configurationa entropy S(x) = j x j nx j is computed directy in state space. The most simpe formuation of the reconstruction probem then reads max S(x), x S S = {x R n (d Ax) T (d Ax) = min, x }. (4.77) The maximum entropy method has found numerous appications, sometimes rigorousy founded, but in many cases on a mere empirica basis [see Censor and Herman, 1987, for references]. As for the quadratic objective function severa iterative agorithms have been proposed to sove eq. (4.77) for the consistent, i.e. equaity constrained case which is especiay important in image reconstruction. The mutipicative agebraic reconstruction technique MART was introduced by Gordon et a. [197] and has found a number of modifications incuding its simutaneous version SMART (see [Byrne, 24; Reis and Roberty, 1992] and references therein). The iterative step is as simpe as for ART ike methods but the correction is appied mutipicativey, not additativey. Another maximum entropy method sometimes used is MENT [Minerbo, 1979]. The entropy objective function in image reconstruction has been especiay successfu for noisy images. The maximum ikeihood ML method appied in this context tries to estimate parameters x from an

81 4.9. Other reconstruction methods 81 incompete set of data d by maximising the ikeihood L(x) = P(d x), i.e. the probabiity that a given x eads to the observed data d. Using the og-ikeihood its most simpe form states thus max np(d x) x Ên A universa iterative agorithm converging towards the ML estimate, the expectation maximisation EM, was deveoped by Dempster et a. [1977]. It was appied to the specific case of emission tomography under the assumption that the measurement can be described as a Poisson process in [Shepp and Vardi, 1982; Vardi et a., 1985]. The resuting iterative agorithm, usuay referred to as EMML but aso as MLEM or EM, is frequenty used in image reconstruction. It is aso a very simpe agorithm but it seems to be quite sensitive to data noise [Censor and Herman, 1987; Byrne, 21]. Athough deveoped independenty, the entropy maximising SMART and the ikeihood maximising EMML (and in fact further agorithms) can be obtained by minimising a functiona of the form KL(x,y) = j x j n (x j /y j ) + y j x j [Byrne, 21]. The Kuback-Leiber distance KL is cosey reated to the cross entropy which is used in information theory to measure the overa difference between two probabiity distributions Backus-Gibert method The Backus-Gibert method (see [Backus and Gibert, 197] for the origina paper) is often quoted in geophysica and atmospheric science, as it represents a diagnostic formuation of the inverse probem that aso eads to a east squares soution. Hansen [1998] refers to in the context of moifier methods. Basicay, it attempts to construct the inverse from a imited amount of data in a way that eads to optima resoution. Usuay the presentation is for a forward mode in the form of ordinary integra equations d i = dr K i (r) c(r), here it is the discrete system Ax = d. First, it is assumed that the reconstructed or estimated concentration ĉ at some r is given by a inear combination of the observed data ĉ(r ) = i a inv i (r ) d i. (4.78) Second, assume that there is a function, the averaging or resoving kerne R, reating the estimated to the true concentration fied c ĉ(r ) = dv R(r,r ) c(r). (4.79) Ideay, R woud be the δ-functiona, in practice it is some smoothing or averaging functiona (c.f. sec ). The key idea is now to choose R as narrow as possibe whie sti reproducing the measurement data. To reate eqs. (4.78, 4.79) reca that c(r) j x j b j (r) where from now on merey for simpicity the basis functions are taken to be orthonorma. Then aso R(r,r ) = j r j (r ) b j (r) (4.8) and repacing d i in eq. (4.78) by (Ax) i it foows r j (r ) = j A T ji a inv i (r ). (4.81)

82 82 4. Tomography and Discrete, Linear Inverse Probems The third assumption concerns the functiona that measures the width of the averaging kerne R(r,r ). Its generic form is to some extent arbitrary but typicay something ike J(r ) = α dv R(r,r ) 2 r r β. Inserting eq. (4.8) eads with eq. (4.81) to a quadratic functiona of a inv = (a inv 1,...,a inv m ) T and thus to the east squares probem min J. a inv The soution of this probem is simpy a particuar soution of the under-determined inverse probem [see aso Tarantoa, 25, pp ]. It might have to be modified (reguarised) not ony if it is unstabe, but aso for the foowing reason. The variance of the estimated concentration ĉ due to measurement errors can with eq. (4.78) easiy be seen to equa a inv (r ) T S ǫ a inv (r ), i.e. it aso depends on the a inv i. In order not to risk too arge variances whie tightening up the resoution, it might be necessary to minimise a weighted mean of J and the variance. This amounts to a Tikhonov reguarisation Exampe of goba optimisation fitting Gaussian exponentias So far the fit probem min d i ds c(r;x 1,x 2,...) 2 x 1,x 2,... i with mode parameters x j has been considered ony for parametrisations that ead to discrete quadratic minimisation probems which in turn yied inear equations. In cases where the functiona form of the concentration fied is known to some extent it can make more sense to expoit this knowedge rather than to use a competey unspecific parametrisation. If the resuting equations cannot be soved by standard methods, goba optimisation agorithms have to be empoyed. The disadvantage that a back box agorithm is totay ignorant to the tomographic origin of the probem is not as critica as the convergence behaviour and speed of the genera purpose optimisation agorithm. An exampe for this approach can be found in [Drescher et a., 1996] for the 2-D reconstruction of indoor gas concentrations. Assuming that the indoor dispersion can be approximated by a Gaussian diffusion equation, the authors make the foowing ansatz for the resuting concentration fied in the case of a imited number of point sources ĉ(x,y;c 1,x 1,y 1,σ x1,σ y1,φ 1,C 2,...) = k C k e 1 2 ( ) (cos φ k (x x k )+sin φ k (x x k )) 2 /σ 2 x + ( x y ) k, (4.82) i.e. for each Gaussian six parameters for peak height, ocation, orientation and variances. The optimisation probem arising is highy noninear and soved using the Amebsa routine from [Press et a., 1992], a combination of the simpex and the simuated anneaing method. A standard search for one reconstruction takes about 2h according to the authors. As another way to ook at the method is that the discrete (oca) basis functions have been repaced by smooth (goba) basis functions, the approach was named the smooth basis function minimization (SBFM) in the origina work.

83 4.9. Other reconstruction methods 83 An approach based on the fitting of symmetric Gaussians is aso proposed by Giui et a. [1999] and appied to the reconstruction of vocanic CO 2 distributions in [Beotti et a., 23], c.f. sec. 3.3.

84 Part II. Methodoogy 84

85 5. The Error of the Reconstructed Distribution This chapter deveops a cosed methodoogy for the compete reconstruction error of the continuous inverse probem eq. 4.4, p. 5. For reasons that wi become cear from the first section, there is hardy a standard treatment of the fu error of the reconstruction resut. Whie errors reated to the discrete system Ax = d are commony covered (especiay if the optima estimate is empoyed, otherwise very often merey in form of sensitivity studies) the error associated with the discretisation is rarey addressed. I am not aware of my approach to it in sec for oca basis functions appearing in the iterature on tomography with sma numbers of ight paths. The theoretica discussion of the different error contributions in sec. 5.2 is neither restricted to tomographic inverse probems nor does it depend on any specific inversion method. Sec. 5.3 treats the purey numerica estimation of the tota reconstruction error as it was more or ess used in [Laeppe et a., 24]. Finay, the ast section on error norms (or overa errors) connects quaity evauation measures from image reconstruction and atmospheric modeing The probem in defining the reconstruction error c(r) Figure 5.1: The probem of quantifying the reconstruction error, iustrated here for the case where the 2- D section through c(r) shows structures that cannot be resoved by a measurement with the path geometry shown. If the variabiity of c(r) is not confined, the reconstruction error can become arbitrary arge. Using the definitions of sec. 5.2 it is the sum of discretisation and inversion error that is unbounded if c(r) is not constrained. x y The aim of an error cacuation is to derive bounds for the deviation ĉ(r) of the reconstructed concentration fied ĉ(r) from the true state c(r): ĉ(r) c(r) ĉ(r) } {{ } ĉ(r) u ĉ(r). (5.1) These bounds can be hard constraints or probabiistic statements. An exampe for the former is the 85

86 86 5. The Error of the Reconstructed Distribution remainder term of the Tayor expansion, an exampe for the atter the 1σ decaration of a measurement resut. Negecting a measurement errors for the moment, ideay, it shoud be possibe to cacuate these bounds soey from the measured data d and the detais of ight path geometry and inversion agorithm. It was pointed out in remark that a finite number of measurements cannot pin down the continuous function c(r). Unfortunetay, the proof in [Natterer, 21] is not constructive. That is, it does not contain a recipe for the functions that give rise to exacty the same coumn densities and I am not aware of any other work that does. Evidenty the ambiguity of the soution is aggravated for decreasing number of ight paths and coarser discretisations (see fig. 5.1). On the other hand, it was noticed in the same remark that the space of soutions can be reduced by further assumptions on the properties of c(r). Thus, we find The reconstruction error cannot be estimated (at east not within this work) from the measurement data d and reconstruction procedure ony. Further information about the true concentration c(r) is necessary or assumptions have to made. In the case of statistica inversion this information consists in the a priori probabiity density function. In other words, the decaration of a reconstruction error as in eq. (5.1) ony makes sense in combination with the assumptions on the true soution c(r) it is based on. It is important to reaise that this ambiguity is not an effect of the inverse probem s instabiity. This is taken care of by reguarisation and the resuting change of the soution can be cacuated and expressed in terms of perturbation bounds (sec ). A remedy of this dissatisifying situation, where not ony the soution of the i-posed probem invoves more or ess arbitrary assumptions but aso the estimate of its uncertainty, is beyond the scope of this thesis. But not a contributions to the tota reconstruction error ĉ(r) impy hidden assumptions as shown in the foowing Composition of the tota error Going through the ogica steps of the reconstruction again, there appears first the approximation of the continuous c(r) by a finite set of basis functions, c.f. eq. (4.1). We ca this discrepancy the discretisation error. Even if c(r) can be expressed exacty as a inear combination of the basis functions, the inversion of Ax = d is in genera not possibe in a unique and/or stabe way. This inversion error is defined in R n. Finay, the measurement error ǫ wi affect the fit resut x ǫ and propagate to the reconstructed fied. The effect of the measurement error is controed by reguarising the soution x x λ with reguarisation parameter λ, giving rise to the reguarisation error. The deviation of the perturbed from the error free reguarised soution is caed perturbation error. In formuas x inv x reg x pert { }} { { }} { { }} { ĉ(r) = c(r) [φ(r)] T x id +[φ(r)] T ( x id x) +[φ(r)] T ( x x λ ) +[φ(r)] T ( x λ x ǫ ), (5.2) } {{ } } {{ } } {{ } } {{ } c disc (r) ĉ inv (r) ĉ reg (r) ĉ pert (r)

87 5.2. Composition of the tota error 87 discretisation inversion measurement unknown continuous c(r) [φ(r)] T x id x id discreteên known from recontruction... if ǫ x x x λ x λ x ǫ [φ(r)] T [φ(r)] T tota reconstruction error c disc (r) + ĉ inv(r) + ĉ reg(r) + ĉ pert(r) Figure 5.2: The composition of the tota reconstruction error ĉ(r). see aso fig Here x id is the idea representation of the continuous c(r) for a given basis, i.e. the x that eads to the smaest discretisation error in terms that yet have to be defined. [φ(r)] T x is the reconstruction resut as it woud be without measurement errors and reguarisation (e.g. a eastsquares or east-squares east-norm soution). [φ(r)] T x λ is the reguarised soution without errors, [φ(r)] T x ǫ with. Before discussing the individua contributions in turn, we observe that: The different errors are usuay reated to each other. For exampe, choosing a coarser discretisation that makes the probem over-determined wi ead to sma or vanishing inversion errors, but to arger discretisation errors and higher sensitivity to noise. Fine discretisation that makes the probem under-determined impies arger inversion errors but the reguarisation and perturbation error may become ess important (see sec. 4.4). The significance of each contribution to the tota error argey depends on the appication. For tomographic measurements with arge numbers of integration paths ike X-ray tomography the discretisation error is usuay negected (or not discussed). Likewise the error anaysis for atmospheric inversion probems is predominanty inên for the x inv, x reg and x pert ony. Statistica inversion inên (this is how the optima estimate, sec , is usuay used) provides an estimate of x inv + x reg + x pert in terms of the a posteriori probabiity density function (or the a posteriori covariance in the case of the optima estimate where this function is a Gaussian distribution) but it has to be specified, whether the discretisation error is contained in the a priori uncertainty or negected The idea discretisation and the discretisation error The idea parameters are given as minimum of some objective function measuring the misfit between continuous and parametrised fied. Foowing remark 4.6.5, again the 2-norm is chosen so that x id is given by min x c(r) [φ(r)] T x 2, x (5.3)

88 88 5. The Error of the Reconstructed Distribution 2 15 Wind 1 NO2 [ppb] 1 c(x) [a.u.] ength x of the section to the motorway [m] (a) x [a.u.] (b) Figure 5.3: (a) Horizonta profies at 5 m above ground of the NO 2 concentration fied perpendicuar to a motorway [Bäumer et a., 25] and its biinear discretisation on grid of 4 3 boxes as used for reconstruction in [Laeppe et a., 24]. indicates the position of the motorway. The maximum discretisation error is 3 ppb. (b) Box discretisation of a Gaussian function with σ = 7 a.u.. The paraboa shows the quadratic Tayor expansion around the peak maximum used in the text to estimate the width of a genera peak. with 2 2 = dv ( ) 2 (5.4) Ω and Ω being the voume or area for two and three dimensions, respectivey. The norma equations can be written as Φx id = c (5.5a) where Φ = dv φ(r)[φ(r)] T, c = dv φ(r)c(r). (5.5b) Ω Ω For the orthonorma box basis cacuation of these quantities and inversion of the norma equation is trivia: Φ ij = V i δ ij, c i = V i c(r) i (5.6a) where V i is the area or voume of box i and ( ) i denotes the spatia mean in it. Thus x idi = c(r) i, (5.6b) i.e. the idea discretisation parameters x id for the box basis are the box averages of c(r). For the box basis it is easy to see that if Ax = d has a unique soution, it is x id. This is not obvious neither for a more genera oca basis nor for the specia case of biinear basis functions. Aso the inversion of the norma equations for the biinear basis is not so simpe because Φ is not diagona. Fig. 5.3a is the resut of a numericay cacuated idea biinear discretisation for the NO 2 distribution perpendicuar to a motorway modeed by Bäumer et a. [25]. It shows horizonta profies of the mode concentration and the ideay parametrised c id (r) = [φ(r)] T x id for a discretisation grid as it was used by Laeppe et a. [24] to reconstruct the motorway pume from measurement data. Before trying to estimate the discretisation error, we iustrate which properties of c(r) might be important by considering one dimension for simpicity. The gradient of c is a critica property for the box basis functions. Ony if the reative change c/c within a box is much smaer than 1, we woud say

89 5.2. Composition of the tota error 89 that the box basis is a good approximation of c. Thus for a box ength x: c/c c x x/c 1 or x c c x 1. Linear basis functions can easiy mode gradients but not changes of the sope. The change of the sope, essentiay the curvature, characterises peaks (or vaeys). Assuming that the peak can ocay be approximated by a quadratic Tayor expansion (c.f. fig. 5.3b), the peak width at the baseine is 2 q 2 c 2 c x 1, where the function vaues refer to the maximum. The oca basis 2 functions can qmimic the peak appropriatey ony if the box ength is sma compared to the peak width, thus x c 2 c x 1. 2 As the above definition eq. (5.3) of the idea discretisation invoves coocation not interpoation, simpe estimates for interpoation errors cannot be used. Instead, we take the definition of the discretisation error (ignoring the nonnegativity constraint for the moment) c disc (r) = c(r) [φ(r)] T Φ 1 c (5.7) and approximate c(r) in c by a Tayor expansion to second order. For the 2-D box basis functions it is shown in appendix C that where c disc (r) c(r) n i=1 ( ) ( c(r i ) x 2 2 c i x + 2 c ) y2 2 i (ri ) φ y i(r) 2 (5.8a) r i : centre of box i, r : chosen such that Tayor expansion around r i is vaid, x i : ength of box i in x direction, y i the same for y and especiay in the centre of the box c disc (r i ) 1 24 ( x 2 2 c i x + 2 c) y2 2 i (ri ). y 2 (5.8b) Fig. 5.3b shows the exampe of a Gaussian peak with σ = 7 a.u. for the piecewise constant discretisation with 5 boxes. At the maximum the second derivative of a Gaussian peak is given by σ 2. Inserting this and the box ength of 2 a.u. into eq. (5.8b) yieds for one dimension c disc.3 a.u. which roughy argrees with the actua vaues. A simiar estimate for the biinear basis is compicated by the cacuation of the inverse Φ 1 and not pursued further 1. 1 Trying to estimate discretisation errors from the textbook error bounds for interpoating quadrature in box i, again for one dimension, in the form 1 x i 1 x i Z dx c I (c) max c box i x box i Z dx c I 1 (c) 1 2 x i max 2 c box i x 2 box i (with I, I 1 the integras over the interpoation poynomias of degree and 1, respectivey) did not prove very usefu in that these bounds are too high.

90 9 5. The Error of the Reconstructed Distribution The inversion error The inversion error was defined as ĉ inv (r) = [φ(r)] T (x id x) for an error free retrieva x and can ony be estimated for appropriate x id. If the reconstruction is given by a inear map and we regard x id as the true vector state of the atmosphere for a given discretisation, then by recaing the definition of the resoution matrix eq. (4.56) x x a = R (x id x a ) one has x inv = x id x = (½n R )(x id x a ). (5.9) and x inv 2 ½n R 2 x id x a 2 = n rank[a] x id x a 2 for the generaised inverse. (5.1) The difference x inv is caed smoothing error in [Rodgers, 2, sec ] for reasons discussed in sec Mind that the inversion error is proportiona to the discrepancy between R and the idea resoution matrix½n and the difference of the guess x a and the true state. If x a happens to be the true x id, the inversion error becomes zero despite a resoution matrix deviating from the idea one. If the reconstruction method is noninear ĉ inv (r) can ony be inferred numericay The measurement error Random noise Ony unbiased random noise is considered here. It appears on the right hand side of Ax ǫ = d ǫ and propagates through the inversion to the soution x ǫ and further to ĉ(r). Reguarisation contros the effect of errors, e.g. by some parameter λ as in the Tikhonov method, λ = σa 1 or the iteration number k in the case of iterative reguarisation. We consider the case where the inversion is performed by a inear map A inv λ. Then in the optima estimate with covariance x x ǫ = A inv λ (d d ǫ ) = A inv λ ǫ S x = E[(x x ǫ )(x x ǫ ) T ] = A inv λ S ǫ A inv λ T (5.11a) (5.11b) (mind that for Bayesian east squares, e.g. the optima estimate, eq. (4.59) hods instead) and variance for the concentration fied σĉ(r) 2 = var[ĉ(r)] = [φ(r)] T S x φ(r). (5.11c) Thus in principe, the impact of random noise can be estimated without any assumptions on the unknown concentration c(r). But in practice the reguarisation depends on the concentration fied at hand. For iterative soution the optima iteration number, for exampe, takes quite different vaues depending on the dimensions of the system Ax = d and the type of distribution, e.g. peaks or smooth background etc. We ook at the individua contributions x reg and x pert in more detai. Assuming uncorreated measurement errors S ǫ = σǫ½m 2 to simpify the formuas, a straightforward cacuation

91 5.2. Composition of the tota error (1 σ 2 ) 2k σ =.5 σ.1 σ =.5 σ iteration number k ((1 (1 σ 2 ) k )/ σ) 2 σ =.5 σ.1 σ =.5 σ iteration number k 1 4 (a) (b) Figure 5.4: (a) Contribution of the singuar vaue σ to the reguarisation error x reg 2 2, eq. (5.13a), as a function of the iteration number for four seected vaues of σ. (b) The same for x pert 2 2 in eq. (5.13b). by means of the SVD yieds in genera [Van der Suis and van der Vorst, 1987] x reg 2 2 = i x i2 (f(σi,d) 1) 2 (5.12a) E [ x pert 2 ] ( 2 = σ 2 f(σi,d) ) 2 ǫ (5.12b) σ i i E [ x x ǫ 2 2] = xreg E [ x pert 2 2], (5.12c) where f(σ i,d) are the fiter factors of the reguarised soution, c.f. eq. (4.66), and x = V T x the vector x in the system of singuar vectors. Inserting the fiter factors eq. (4.75) for SIRT ike agorithms eq. (4.72b) gives for the transformed system, eq. (4.71), Ā = L 1/2 AL 1/2 φ, x = L 1/2 φ x, d = L 1/2 d the bounds x reg 2 2 = i x i2 (1 ω σ 2 i ) 2k (5.13a) E [ x pert 2 ] ( 2 = σ 2 1 (1 ω σ 2 i ) k ) 2 ǫ (5.13b) σ i i with x i now as in eq. (4.76). Their dependence on the iteration number k is sketched in fig Ceary, sma singuar vaues have more impact on both errors than arger singuar vaues (remember that σ 1). But whie the contribution to the reguarisation error decreases in the course of the iteration (it has to vanish for k, i.e. no reguarisation), the perturbation error is monotonicay increasing. The perturbation error depends ony on σ, i.e. the measuring system, whie cacuation of the reguarisation error requires the reconstructed state or the unperturbed coumn densities. Eqs. (5.12,5.13) are the quantitative expanation for the semiconvergence of iterative agorithms in the presence of noise that was mentioned in sec

92 92 5. The Error of the Reconstructed Distribution d(t) t Figure 5.5: Time series of a coumn density showing stochastic variations that cannot be expained by the measurement error. The deviation around the mean can be used as an estimate for the error d associated with the source of this error. Errors if measurements do not refer to the same point or period of time If not a ight paths finay used for reconstruction measure at the same time, this wi introduce an error if the concentration distribution changes during the measurement interva. Exampes of such non-simutaneous measurements are setups where the compete geometry is obtained by scanning the retro-refectors in severa steps (see secs. 3.4, 9.1) or aircraft and sateite measurements where the geometry arises through the motion of the detector. Tempora changes of the concentration fied can be due to 1. varying sources, sinks or chemica transformation, 2. turbuent fuctuations, 3. changing meteoroogica conditions, 4. transport. Concentrating on LP-DOAS measurements using mutibeam instruments (see sec. 3.4), the measurement cyce is of the order of minutes [Mettendorf et a., 26] detais depending, of course, on the number of steps necessary to generate the compete geometry, integration times etc. Variations in that are sow compared to this time scae can be negected. If the variations are of stochastic nature, ike turbuences or traffic emissions in an urban environment, they wi party cance when averaging over the path. Otherwise, the error on the coumn densities reated to them can be estimated from coumn density time series, if a other factors remain stabe (fig. 5.5). Errors due to changing meteoroogy are best avoided by seecting stabe periods. For the tomographic reconstruction of concentration puffs that wi be studied in part III changes of the concentration fied caused by wind transport are especiay troubesome. The spatia scaes, given by s = ū t and thus of the order of severa hundred metres for wind speeds ū of a few metres per second and measurement times t of a coupe of minutes, are comparabe to the dispersion coefficients for trave distances of a few hundred metres (c.f. fig 2.3). To quantify the error on the coumn densities,

93 5.3. Numerica estimation of the reconstruction error 93 we consider two ight paths which, if measured at the same time, woud yied the coumn densities d 1 (t) = dsc(r,t), LP 1 d 2 (t) = dsc(r,t). LP 2 If the actua measurement for LP 2 takes pace at t + t, its coumn density is two first order in t given by d 2 (t + t) LP 2 dsc(r,t) + t LP 2 ds Assuming that the dominant contribution to c t is given by advection with constant wind vector, eq. (2.4) gives c t = ū c = ū ūc, where ū is the directiona derivative aong the wind direction. Writing the integras as average quantities aong the ight path, one gets with s = ū t d 2 (t + t) L c L s ūc. c t. The second term can be treated as sma error d, if s ūc c or, in other words, if the reative average change of c aong ū over the distance s is much smaer than one. 1, 5.3. Numerica estimation of the reconstruction error Even if the discretisation error can be negected, an estimation of ĉ(r) according to eq. (5.2) invoves assumptions about the unknown true state x id = x true. Whie the statistica impact of noise can easiy be estimated either anayticay by eqs. (5.11) or by Monte-Caro simuations, the cacuation of the inversion error is often reduced to a sensitivity anaysis of the form: If my concentration profie is ike this, what does my retrieva ook ike? Here, we refer to a statistica estimate of ĉ(r) incuding the discretisation error based on a sufficient number of simuated reconstructions for concentration fieds c I (r), I = 1,2,...,N, from a suitabe set (ensembe) E = {admissibe c I (r), I = 1,2,...,N}. (5.14a) If the c I are very simiar, the sign of the individua reconstruction errors ĉ I (r) = c I (r) ĉ I (r) wi behave simiar within the reconstruction area, too, and it makes sense to take the ensembe mean ĉ I (r) E. Otherwise one coud use a positive quantity ike ( ĉ I (r)) 2 E, but then a information about systematic over- or underestimation within the area gets ost. Instead of the mean reconstruction error used by Laeppe et a. [24], we consider the error fieds + ĉ(r) and ĉ(r) as measures of

94 94 5. The Error of the Reconstructed Distribution under- and overestimation, respectivey, defined by ĉ(r) = + ĉ(r) + ĉ(r), ĉ(r) if ĉ(r), where ± ĉ(r) =. ese (5.14b) Without measurement errors one gets thus the scheme: c 1 (r) E forw. mode d R d i= ds c(r) i reconstruct 1 ĉ 1 (r) ĉ 1 (r) = c 1 (r) ĉ 1 (r), c 2 (r) E d 2 ĉ 2 (r) ĉ 2 (r) = c 2 (r) ĉ 2 (r),.... ĉ(r) ĉ I (r) E, or ĉ u/ (r) ± ĉ I (r) E, (5.14c) where ĉ,u (r) are the ower and upper bound of the reconstruction error as in eq. (5.1). In principe, an ensembe of smooth, very simiar random distributions can give rise to the same mean reconstruction errors as an ensembe of highy fuctuating distributions (a ess dramatic exampe wi occur in chap. 9, sec. 9.3). But the standard deviation wi be much higher in the atter case. Therefore, we define the ower and upper bounds of the reconstruction error as ĉ u/ (r) = ± ĉ I (r) E ± std E [ ± ĉ I (r)]. (5.14d) For simuated measurements, where the error free coumn densities d are known, the statistic contribution of an unbiased measurement error to the reconstruction error fieds ± ĉ is given by the standard deviation std ǫ [ĉ(r)], with ĉ retrieved from d. For inear reconstruction the standard deviation is given by eqs. (5.11). The error bounds eq. (5.14d) take now the form ĉ u/ = ± ĉ I (r) E ± std E [ ± ĉ I (r)] ± std ǫ [ĉ(r)]. (5.14e) For actua measurements, where ony the error afficted coumn densities d ǫ are known, one way to get an estimate for the impact on the unknown coumn densities d is to cacuate the mean impact within the ensembe E by repacing the above simuation step with c I (r) E d I +ǫ ĉ I (r) ( ) ǫ = ĉ I (r) ǫ ± std ǫ [ĉ I (r)] or ± ĉ I (r) ǫ ± std ǫ [ĉ I (r)] (5.14f) to get the average ĉ ± (r) ĉ ±I (r) ǫ E ± std ǫ [ĉ I (r)] E. (5.14g) This approach was foowed by Laeppe et a. [24]. But uness it can be shown that the synthetic coumn densities obtained from the c I are a better estimate of d, there seems itte reason to prefer them to the measured coumn densities d ǫ for the estimation, especiay for sma errors. Besides, the ensembe estimate is computationay much more expensive if the standard deviations cannot be

95 5.4. Overa reconstruction errors and quaity criteria 95 cacuated anayticay. In the foowing, the approach resuting in the bounds eq. (5.14e) wi be taken. Stricty speaking ony distributions consistent with the actuay measured coumn densities d = (d 1,...,d m ) T shoud be taken into account, i.e. E = {admissibe c I (r) d i ds c I σ ǫi, i = 1,...,m}, (5.15) LP i but this is rather invoved in practice. Instead it is assumed in the foowing that the ensembe means (±) ĉ I (r) E for suitaby chosen E are good estimates for the reconstruction error of the concentration fied ĉ retrieved from experiment, without demanding consistency of the coumn densities. The test concentrations c I can be provided by mode cacuations, be based on the assumption that the reconstruction is correct within certain bounds (see appendix B and aso sec. 7.4) etc. One shoud notice that, in genera, the ensembe wi be biased in the sense that ĉ I (r) E ĉ(r). (5.16) 5.4. Overa reconstruction errors and quaity criteria The spatia dependency of the reconstruction error is not aways of interest. A typica situation where the error fied is an inconvenient quantity is the foowing. You want to find optima parameters of the reconstruction procedure for a possiby arge number of test concentration fieds which hopefuy come cose to the unknown rea one. These parameters are then used for the actua reconstruction from the measurement data (see sec. 6.4). For the processing of arger amounts of data you therefore want a simpe criterion, best a number Q, teing that if for two sets of parameters p 1 and p 2 Q 1 > Q 2 then reconstruction with p 1 is better (or worse) than with p 2. Denoting the true (simuated) quantities as c, d, the reconstructed ones as ĉ, ˆx and defining ˆd = Aˆx, then two cases can be distinguished. (i) Quaity criteria based on the measurements, i.e. Q = Q(d, ˆd): The most simpe number constructed from d, ˆd is the data residua, c.f. eq. (4.65), i.e. the overa agreement between measured and reconstructed data d 2 = d ˆd 2 (5.17) or the reative data residuum if it is normed by the coumn densities. Another quantity that has the advantage of being normaised is the correation coefficient between the d i and ˆd i (see beow). However from the discussion so far it foows: In the under-determined case of an i-posed probem, quaity criteria based on the agreement between measured (simuated) and reconstructed path integras are not decisive as the measured data can be reproduced by (in genera) infinitey many soutions. A reguarised soution with higher residua can easiy be better than an unreguarised soution with vanishing residua. In fact, for a measurement error ǫ it does not reay make sense to strive for a residuum smaer than ǫ 2. If the soution of the over-determined probem invoves d Aˆx 2 = d ˆd 2 in the cost

96 96 5. The Error of the Reconstructed Distribution function, as it does in the east squares fit, the residua can be regarded as a measure for the quaity of the soution provided that the measurement errors are not too arge. If they are, sma residua means fitting to noise. For the reguarised soution the data residua is in genera not a usefu indicator. We refer here to the discussion of the choice of the reguarisation parameter in [Twomey, 1997] and [Hansen, 1998, chap. 7]. In any case, forcing the data residuum to be smaer than ǫ 2 is questionabe. To sum up, quaity criteria based on data agreement possess the attractive feature that they do not rey on the unknown concentration fied. But for the (inear) i-posed inverse probem they are of no use in the under-determined case. In the over-determined case their significance depends on kind and degree of reguarisation. (ii) Quaity criteria based on unknown concentrations, i.e. Q = Q(c,ĉ) or Q(x, ˆx) : We consider criteria for the continuous c(r), ĉ(r) first. The average reconstruction error V 1 Ω dv ĉ(r) is not suitabe because over- and underestimation cance each other within the area or voume Ω. Instead one coud use Ω dv ĉ(r) or Ω dv[ ĉ(r)]2 = ĉ(r) 2 2. The integrated square of the reconstruction error is mathematicay more appeaing. Especiay E [ ĉ(r) 2 2] = cdisc (r) ĉ inv (r) ĉ reg (r) E [ ĉ pert (r) 2 2]. (5.18) This hods ony if the discretisation error is defined as in eqs. (5.5). We note in passing that if S x as in eq. (5.11b) is known, it foows immediatey that E [ ĉ ǫ (r) 2 2] = tr [Sx Φ]. These equations impy that the different contributions to the overa error defined by the 2-norm can be cacuated individuay and added up ater. And it means that one does not have to vary a parameters at once, but can vary e.g. parameters for the discretisation first, then vary measurement errors and so forth. The quasi normaised key figure nearness, buit from ĉ(r) 2, was introduced by [Herman et a., 1973; Herman and Rowand, 1973] into tomographic image reconstruction: NEARN = c(r) c 1 2 ĉ(r) 2, with c = V 1 Ω dv c(r). (5.19) If the reconstruction ĉ(r) = [φ(r)] T ˆx equas the spatia mean c of the unknown concentration fied (i.e. ˆx = const) then NEARN = 1. Vice versa, if NEARN = 1 then the retrieved fied has the same overa reconstruction error in terms of nearness as the spatia mean of c(r) woud have. If the rea and reconstructed concentrations are compared not on the entire area or voume V but on a subset of points {r i } N i=1, then evauating the agreement between the rea concentrations c i = c(r i ) and the reconstructed ĉ i = ĉ(r i ) corresponds to the probem of atmospheric mode evauation from N measurements or from simuations for a fixed time. An exceeding number of measures (aso caed indices or metrics) is used for this purpose, depending on whether the spatia evauation concerns meteoroogica parameters, chemica transport or air poution modes. In the context of dispersion modeing a set of mosty statistica and reative 2 measures suggested by Hanna [1989] frequenty 2 Air quaity assessment usuay uses reative numbers to evauate mode performance whie for meteoroogica modes absoute numbers are prefered. The reason being that the significance of a reative error can be very different for

97 5.4. Overa reconstruction errors and quaity criteria 97 metric Q anaytica or statistica expression range best overa error 2 nearness NEARN = `σ c ( ĉ) 2 1/2 NEARN normaised mean square error NMSE = ( c ĉ) 1 ( ĉ) 2 NMSE index of agreement IOA = 1 ( c c + ĉ ĉ ) 2 1 ( ĉ) 2 IOA 1 normaised mean absoute error NMAE = c 1 ĉ NMAE bias normaised mean bias NMB = c 1 ĉ NMB fractiona bias FB = 1 2 ( ĉ + c) 1 ĉ 2 FB 2 correation r = (σ c σĉ) 1 (c c)(ĉ ĉ) 1 r 1 1 other fraction of factor 2 FA2 = %-age of data with 1 ĉ 2 2 c FA2 1% 1% peak prediction accuracy PPA = c 1 max ĉ max PPA 1 peak integra prediction accuracy PIPA = ( R R Peak c) 1 ĉ Peak PIPA 1 Tabe 5.1: Quaity measures considered in this thesis. For evauation of the tomographic reconstruction c is the true, i.e. simuated fied, ĉ the reconstructed. For mode evauation c refers to measured concentrations and ĉ to the modeed ones. The mean ( ) can be understood as ( ) = V R 1 dv ( )(r) Ω or N 1 P i ( )i, both being identica for N. In the same sense σ2 ( ) = `( ) ( ) 2. c max, ĉ max are the absoute maximum vaues of c(r), ĉ(r) within Ω. R is the integra within the area of a Peak peak (eq. (5.2)) so that PIPA measures the precision of reconstructed tota emissions (see text). appears in the iterature, some of which are isted in tabe 5.1 and wi be used ater. The normaised mean square and absoute error NMSE and NMAE measure the overa discrepancy between the measured or simuated concentrations c i and the vaues ĉ i predicted by the mode. Normaised mean and fractiona bias NMB and FB indicate how much the mode average concentration vaue differs from the observed mean. r is the common (Pearson) correation coefficient so that a vaue of 1 means perfect correation, 1 perfect anticorreation and a vaue of no correation. F A2 is sef-expanatory. The index of agreement IOA was defined by Wimott [1981] to incude the variances between mode and measurement and between mode, measurements and the mode mean in one quantity. In contrast to r it is sensitive to mode and observed means. Vaues of mean perfect agreement, 1 poor agreement. The non-statistic peak prediction accuracy PPA refers to the (unpaired) absoute maxima of origina and reconstructed distributions within Ω. For the 2-D simuations in part III one further index is introduced. Thinking of concentration peaks as emission pumes or puffs, a measure for the precision of reconstructed tota emissions suggests itsef. The peak integra prediction accuracy PIPA PIPA = da ĉ(r) Peak da c(r) (5.2) Peak different meteoroogica parameters. Just think of a 1% error on a wind speed of 1 m/s (1 m/s) versus the same reative error on a temperature of 3 K (namey 3 K!).

98 98 5. The Error of the Reconstructed Distribution compares the concentration integrated over the area of the reconstructed peak with the one of the origina. For the evauation of the integras in the case of Gaussian peaks, 3 max(σ x,σ y ) is taken as the radius of the origina peak distribution (at 3 σ the peak vaue has faen to 1%). For the reconstructed peak, one pixe ength of the reconstruction grid is added to this radius to take account of the spreading within the pixes for the box basis and the interpoation between neighbouring grid points for the inear basis functions, respectivey. This, especiay in the case of severa arge distributions with overap and for arger pixe sizes, wi ony be a rough estimate. If the concentration peak originates from point source emission, then by writing the emission rate q (in units kg s 1 ) ike q = da uc(r) u da c(r), it can be seen that PIPA = ˆq/ q = ˆq/q provides a measure for the precision of emission rates or tota emissions (if the wind speed has a component perpendicuar to the reconstruction area). For horizonta cuts through the atmosphere this is the case, e.g., if the area contains a source with vertica transport. For horizonta transport of a pume this number can be reated to tota emissions ony with further assumptions on the vertica dispersion since its reease. Finay, the case Q = Q(x, ˆx) can be obtained by choosing the sampe points r i such that they coincide with the boxes for the piecewise constant discretisation or the grid nodes for the inear discretisation, i.e. c i = x i, ĉ i = ˆx i. We concude this chapter by some remarks Pume Evidenty, one measure is not enough to grasp the continuous fied ĉ(r) competey. Therefore, depending on the appication a variety of more or ess specific criteria exists apart from the ones in tabe 5.1. For higher resoved fieds far more refined evauation methods can be appied, especiay for time dependent anaysis. Quaity criteria can differ argey in their degree of sensitivity to error patterns. For exampe, the correation coefficient is insensitive to an overa offset between measured and modeed/reconstructed vaues, no matter how arge this offset is, whereas it woud give a arge contribution to, e.g., NMSE. Above a, the significance of most quaity measures is a reative one, because its rough size argey depends on the concentration fieds. For exampe, the best possibe modeing or reconstruction of a sharp concentration peak wi in genera have arger r vaues than a poory modeed or reconstructed smooth background concentration.

99 6. Reconstruction Procedure The focus of this work with respect to the reconstruction method ies on improved parametrisation of the discrete inverse probem as we as on the choice and impementation of a priori assumptions as they might typicay occur in atmospheric appications and not on agorithm deveopment for the inversion of the discrete system of equations itsef. Therefore, after a brief summary of the reconstruction method as far as it was described in chap. 4 and has been appied before in the theoretic study [Feming, 1982] for sateite tomography of trace gases, by Todd and Ramachandran [1994a] for indoor gas concentrations 1 and by Laeppe et a. [24] for the reconstruction of a motorway emission pume in sec. 6.1, sec. 6.2 presents generaisations of the so-caed grid transation method suggested by Verkruysse and Todd [24] that reies on reconstructions from severa grids. The subsequent sec. 6.3 introduces a priori constructions for the (under-determined) east-squares probem based on the kind of concentration distributions discussed in chap. 2. After a discussion of additiona inear (in)equaity constraints, the grid transation scheme is revisited in sec for peak concentrations to introduce the maximum reconstructed vaue as constraint. Finay, sec. 6.4 reviews a variabes of the reconstruction method as it wi be used in part III and how they are optimised through simuations Reconstruction principe and inversion agorithm It was argued in sec. 4.2 that ong-path DOAS tomography is not fit for the appication of transform methods ike the fitered back projection (sec ). The aternative formuation in terms of a finite number of parameters can ead to a inear or noninear probem. The former aows powerfu concepts from inear inverse theory ike the resoution matrix (averaging kernes) and simper discussion of reconstruction errors (see secs , 5.2.3), the atter in genera requires goba optimisation agorithms, except for specia cases where inearisation is possibe or where particuar agorithms exist (c.f. sec ). For these reasons I have chosen the parametrisation by oca basis functions described in sec for this first systematic study of the possibiity to reconstruct various forms of atmospheric trace gas distributions from reaistic ong-path DOAS tomographic measurements. This does not mean that for certain appications specia schemes ike the fit of Gaussian distributions (SBFM, sec ) may not ead to better resuts, but we want to keep the discussion genera and among other drawbacks, the ast exampe introduces unavoidabe functiona a priori that is not aways appropriate (for exampe not in the case of the motor way emission pume in [Laeppe et a., 24] 2, see aso fig. 5.3a, p. 88). The resuting inear discrete i-posed probem is repaced by a east-squares east-norm principe as in secs , Demanding a minimum norm of the soution is mathematicay appeaing. Aternativey, often the weighted sum of the residua and the norm of the soution or discrete approximations 1 Both ony with piecewise constant basis functions. 2 Private communication with T. Laeppe. 99

100 1 6. Reconstruction Procedure of its gradients or higher derivatives serves to reguarise (Tikhonov reguarisation, sec ). It can thus be used as a seection criterion to smooth the soution for the right choice of the ad hoc weight. For the appications in part III neither of these reasons appeared to be critica so that this approach was not considered here. 3 The east-squares east-norm probem pus nonnegativity constraint is soved iterativey as discussed in secs , by row acting methods (sec ), where ony ART and SIRT wi be considered as representatives for the sequentia and simutaneous agorithms. The simutaneous SART was found to be ess appropriate for the reconstruction of Gaussian peaks in [Todd and Ramachandran, 1994a]. The impicit weighting of these agorithms according to eq. (4.74) wi not be corrected and ignored in the subsequent discussion because for ART and SIRT it ony acts on the norm of d Ax and in the under-determined case, which is the most reevant in the foowing, this kind of weighting is not or ess effective. Furthermore, the reaxation parameter ω, usuay used to speed up the convergence for arge systems of equations, wi not be varied and set to one as the systems in the foowing are rather sma. Reguarisation, if necessary, is carried out iterativey, as discussed in sec Reconstruction grid The deveopment of the reconstruction procedure in chap. 4 tacity assumed that the parameters of the discretisation are the same as the ones entering the east-squares fit, in other words, that the discretisation grid coincides with the reconstruction grid. For the over-determined case it is easy to think of schemes for which this is not the case, especiay if the discretisation is carried out by quadrature as sketched in sec (see [Doicu et a., 24] for a 1-D exampe). In this work they are identica. Piecewise constant and inear basis functions are constructed by tensor products of one dimensiona functions (sec ) so that the supports of the piecewise basis functions are rectanguar (boxes). For discretisation that is fine reative to the whoe area this is not a probem, but for very coarse discretisation this may become critica, e.g., if the region defined by the ight paths is trianguar. Reconstruction grid means in the foowing the set of points consisting in the corner points of the boxes (which are the discretisation nodes in the case of the bi(tri)inear basis functions). The number of parameters, i.e. the number of boxes for the piecewise constant or the number of grid nodes in the case of the inear basis functions, is refered to as grid dimension. The region of interest, the reconstruction area or voume, is aways such that it is defined by the outermost grid points.

101 6.2. Reconstruction grid 11 Figure 6.1: The idea reconstruction grid. How has the box ength x of the ower box to be chosen so that for the two ight paths shown the singuar vaues of A, and therefore their information content, become as equa as possibe? x 2 1 LP 2 LP The idea reconstruction grid We assume for the moment that no additiona information enters the probem, i.e. there are as many integration paths as parameters to be determined: m = n. The idea parametrisation of c(r) by the basis functions woud be such that A becomes diagona with entries of the same size. Then these coincide with the singuar vaues which are equa to the roots of the eigenvaues of A T A or AA T, see eqs. (4.21), and thus carry a the same information content according to eq. (4.61). For typica ight path geometries as in fig. 3.5, p. 44, and basis functions with rectanguar support diagona A is in genera not feasibe, but one can try to choose the grid such that singuar vaues do not become too different. To see how this can be accompished Gerschgorin s theorem 4 is appied to the eigenvaue system of A T A to give the foowing bounds on any singuar vaue σ i a 2 ij σ 2 i i a 2 ij a ij a ij σ 2 j j i i a 2 ij + j j a 2 ij a ij a ij i (6.1a) (6.1b) for some j. For the box basis i a2 ij measures the tota square ength of a ight paths added up in box j. j j i a ija ij is basicay the overap of the ight paths. As tr[aa T ] = i,j a2 ij = j σ2 j eqs. (6.1) impy that the grid has to be chosen such that the engths of the ight paths within the boxes have to be simiar and that the overap has to be sma compared to it. Conversey, a singuar vaue can become sma if a ight path s ength goes to zero (eq. (6.1a)) or, for arbitrary ength, if the grid is chosen in a way that resuts in arge overap with other ight paths (eq. (6.1b)). To study the question of the right grid quantitativey, we consider the foowing simpe Exampe: A = ` a 11 a 12 a 21 a 22 with box basis functions and ight paths of engths 1, 2 as in fig. 6.1, i.e. a 11 = x, a 12 = 1 x and a 21 = x, a 22 = 2 x if x 2, a 21 = 2, a 22 = otherwise. What vaue x shoud the box ength have to minimise the difference between the two singuar vaues σ 1, σ 2? The singuar vaues are σ 2 1/2 = 1 2 tr[aat ] ± 1 2 p tr[aat ] 2 4det[A] 2. (6.2) The genera expression for x minimising the root is quite engthy, so I give three exampes instead for 1 = 2 a.u. and the foowing vaues for 2: 3 Price et a. [21] have appied the Tikhonov approach minimising the discrete approximation of the third derivative to the reconstruction of indoor gas concentrations. The method hardy needs the nonnegativity constraint and is therefore inear. Unfortunatey, this study is not systematic with respect to the shape of the distribution and does not compare its method to other soutions ike the minimum norm soution. 4 Gerschgorin s theorem sais that for each eigenvaue λ of a matrix M there is an i such that λ m ii X i i m ii.

102 12 6. Reconstruction Procedure J j y x x/(2 + 1) origina grid subgrids (m x = m y = 2) combined grid Figure 6.2: 2-D grid transation scheme proposed by Verkruysse and Todd [24] for a rectanguar reconstruction area and grid, depicted here for a square area with identica spacing in x and y of the grid which is shifted twice in both directions. The dimension of the fina composite grid is N = [3 2+1] 2 = 49. Mind that the subgrids have different dimensions. Subgrids with sma boxes near the boundaries cause smaer singuar vaues according to sec [a.u.] x [a.u.] σ 2 [a.u.] σ 2 [a.u.] This may be not what one expects and iustrates the intricate interpay between the box-engths of the ight paths (the trace term in eq. (6.2)) and their overap (which eads to sma determinants in eq. (6.2)). For compex irreguar ight path geometries the design of an idea grid from the singuar vaues point of view wi become quite compicated and a constructive method needs further thought. But one shoud be aware of the fact that the reconstruction method finay wi not use the exact singuar vaues of A, if a priori (or constraints etc.) enters the probem Combining grids for the reconstruction Whie the type of basis functions φ j and the ocation of the grid nodes may be ony of minor importance for very fine discretisations, the shape of the reconstruction is mainy given by the φ j (r) if c(r) is represented ony by a few basis functions. In this case, especiay if the ight path geometry is irreguar, the exact position of the grid nodes is to a arge extent arbitrary, which suggests to take into account representations based on different grids (provided that these are equay sensibe). Motivated by the fact that a reconstruction underestimates the maximum vaue of a sharp peak if the position of the origina peak ies unfavourabe reative to the boxes of the reconstruction grid, Verkruysse and Todd [24] suggested to use severa grids shifted against each other in the pane as shown in fig The reconstruction resut is then defined on the grid that arises from putting these subgrids on top of each other and attributing to a box of this higher resoved grid the concentration that corresponds to the nearest of the boxes among a subgrids (nearest is defined by the grid ines and if there are severa equay near, the average concentration is taken). This grid transation scheme was defined for box basis functions and a reguar origina grid that was shifted m x times in x-direction and m y times in y-direction ike this: The first grid is generated by shifting the origina nodes by x/(m x + 1) in x-direction, where x is the x-distance between two nodes of the origina

103 6.3. A priori and constraints 13 grid. This is performed m x times, then the initia grid is shifted by y/(m y +1) in y and the procedure is repeated, and so forth. The idea was adopted in [Hart et a., 26] for biinear basis functions. The reconstruction area is assumed to be rectanguar, as the appications in part III are a on rectanguar areas. The shape of the reconstruction area is not essentia but, especiay for the biinear basis, the design of reguar shifted grids for irreguar areas becomes difficut or impossibe uness the number of basis functions is increased significanty. As before, the fina, higher resoved grid is given by the union of the nodes from a subgrids and the parameter vector X = (X 1,...,X N ) T for this grid is defined in two different ways: (i) In a composite scheme the component X J for grid node r J is taken from the Ith subgrid that has a coinciding node there: X J = ĉ I (r J ). (6.3) The boundary grid ines do not change and here the average of coinciding nodes is taken. This scheme corresponds to the suggestion by Verkruysse and Todd [24] for the box basis. (ii) In the avaraging scheme X J is taken as the average of a subgrid reconstructions evauated on grid node r J : X J = 1 M M ĉ I (r J ), (6.4) I=1 where M is the number of subgrids. This scheme is competey inear. If the subgrids are generated by the transation of an origina grid as sketched above and shown in fig. 6.2, this resuts in M reconstructions for the subgrids and a tota of N nodes for the composite fina grid given by M = m x m y + m x + m y + 1 N = [(m x + 1)(n x 1) + 1][(m y + 1)(n y 1) + 1], (6.5a) (6.5b) where n x and n y are the numbers of nodes in x- and y-direction. Both schemes wi be discussed in detai in A priori and constraints A priori knowedge is any information avaiabe prior to the reconstruction, in terms of probabiities, equaities, inequaities etc. Here as often done in the iterature we refer to a priori as an initia guess x a of the parameter vector x with or without additiona information about its uncertainty. Such an a priori is aways present for an under-determined east-squares probem and corresponds to the iteration start x () for the iterative row acting agorithms discussed in sec We are now going to specify it for the reconstruction of atmospheric concentration distributions in part III.

104 14 6. Reconstruction Procedure Choice of the a priori According to eqs. (4.56) the contribution of the a priori to the east-squares east-norm soution is given by (½n R)x a, where R is the resoution matrix of the inear inversion. 5 We assume throughout the foowing an a priori that is competey noncommitta with respect to the spatia distribution, i.e. x a = c a (1,...,1) T [(½n R)x a ] j = c a (1 j R jj ). (6.6) Foowing the discussion in chapter 2, two scenarios for the concentration c a are distinguished. (i) Locay enhanced (or reduced) concentrations (e.g. emission puffs or pumes) where the concentration peaks or sinks are on confined areas within the tota reconstructed region. A typica exampe for a negative peak woud be oca ozone depetion due to strong emission of NO. c a is set to zero if the background concentration of the trace gas of interest is negigibe. The case of substantia background concentrations is addressed in the next section. (ii) Smooth concentration distributions without significant peaks. In this case an average concentration is taken as a priori that is obtained from the measurements as where L i is the ength of LP i. c a = 1 m m c i = 1 m i=1 m i=1 d i L i, (6.7) Using the definitions of chap. 5, an incorrect a priori is reated with the inversion error according to eq. (5.1). An estimated error c a of c a eads to a contribution ( x) j = (1 j R jj ) c a (6.8) to the inversion error Background concentrations We refer to the peak scenario in sec (i). Background is understood in a graphica sense as what remains of the distribution after subtracting the peak(s) and it does not have to agree with the natura background or any other atmospheric offset concentration. Taking into account the imited resoution of the tomographic experiments here, ony constant background is considered c BG (r) c BG, i.e. x BG = c BG (1,...,1) T. (6.9) (i) Fitting the background: Inserting c(r) = c (r) + c BG (6.1a) 5 And in the unweighted case P = (½n R) is the projector onto the nuspace of A, see eq. (4.55).

105 6.3. A priori and constraints 15 into the forward mode d i = dsc(r) eads to the discrete system i and = (L 1,...,L m ) T is the vector of the ight path engths. d = B y, with B = (A ) (6.1b) y = (x, c BG ) T (6.1c) In the over-determined case a east-squares fit with the additiona parameter c BG eads to (see appendix C) x = (A T LA) 1 A T Ld = A L d (6.11) c BG = T T (½m R L )d, where L = T ½m T and R L = AA L is the resoution matrix of the weighted inverse in eq. (6.11). In the under-determined case the east-norm principe can be appied to the variation x aone, i.e. to x 2. Minimisation of both variation and background can be obtained by y 2, whie j (x j c BG) 2 minimises the deviations from the background. A three cases can be written as y T Hy with suitabe matrix H, and the soution of the corresponding east-norm probem min y y T Hy, d = B y can be expressed as weighted east-norm soution for x with a priori x a (c.f. appendix C). For the three specia choices one gets H = 1 1 H = H = : x = A T (AA T + T ) 1 d c BG = T (AA T + T ) 1 d, n : x = AT c BG = (½m (AAT ) 1 T T (AA T ) 1 T T (AA T ) 1 (AAT ) 1 d, : x = (A T AT (AA T ) 1 T 1 2 T T (AA T ) 1 c BG = 1 T 2 T (AA T ) 1 (AAT ) 1 d. ) (AA T ) 1 d ) (AA T ) 1 d (6.12a) (6.12b) (6.12c) The sum x = x + c BG (1,...,1) T is the same for the ast two cases. The background in eq. (6.12b) can be written as T c BG = T (AA T ) 1 (AAT ) 1 1 d = x j rank[a], with x = A d and for more or ess constant c(r) c this is roughy the same as c a in eq. (6.7). 6 j 6 In fact, c BG in eq. (6.12b) can aso be written as c BG = rank[a] 1 P d i i e i2 where d, and e are the vectors d, i and e = (1,..., 1) T in the system of singuar vectors. As the transformation to this system is orthonorma, the

106 16 6. Reconstruction Procedure (ii) Subtracting the background: The idea is to subtract the background as an offset on both sides of eq. (6.1b) and to sove the reduced system d = d c BG = Ax. (6.13) The rationae behind subtracting c BG rather than using it as a priori is that the reduced system eq. (6.13) with zero a priori and positivity constraint aows finer discretisation as wi be shown in sec In principe, c BG can be taken from a fit as in (i), but the choice c BG = min( d1 L 1,, dm L m ) (6.14) assures positive d i. The case of a oca sink can be treated accordingy, if after subtraction of the background the reduced system is written as d = d = Ax Additiona constraints Additiona constraints to the system Ax = d can arise from further measurements (c.f. sec. 7.3), from mode estimates or they may occur in simuations. Linear inequaity constraints can be impemented in the same way as the positivity constraint through iterative projection (secs , 4.8.4). Linear equaities of the form Fx = c, F Êf n (6.15) ead to the augmented system ( ) ( ) A d x =, (6.16) F c } {{ } } {{ } A Ê(m+f) n d Ê(m+f) where the rows shoud have simiar weights, i.e. be mutipied with suitabe factors if necessary. The associated east-squares and east-norm probems can be soved iterativey for the system Ax = d in the noninear constrained case or expicity in the inear case as shown in appendix C: x = (A T A + F T F) 1 (A T d + F T c) if m + f n, (6.17) see aso eqs. (4.6), and x = A T (AA T ) 1 d + P A x a = x a + P F A T( AP F A T) 1 (d Axa ) + P A F T( FP A F T) 1 (c Fxa ) if m + f n, (6.18) with P as in eq. (4.55).

107 6.4. Finding optima settings from simuations 17 1 Figure 6.3: Exampe showing how the reconstructed peak height gets reduced by averaging over shifted grids. The initia reguar grid carries five inear interpoation nodes and is shifted twice in x-direction. The Gaussian peak is as in fig. 5.3b. The soid ines show the idea discretisation (sec ) for the shifted grids, the dashed ine is their average. c(x) [a.u.] x [a.u.] Fitting peak maxima The grid averaging scheme (sec ) wi inevitaby smear narrow peaks and reduce the maximum vaue, c.f. fig Looking at the individua subgrid reconstructions of Gaussian peaks (chap. 8) it was observed that the grid node of the absoute maximum of a subgrids correates we with the grid node next to the actua peak maximum. Furthermore, the grid nodes and the vaues of the reconstructed absoute maximum among the subgrids agrees we with what can be expected from an idea parametrisation of the Gaussian peak (sec ). It thus makes sense to regard this absoute maximum c max = ĉ I (r Jmax ) beonging to subgrid I and the node J max of the composite grid as a better representation of the true maximum than the averaged vaue X Jmax according to eq. (6.3). Therefore, the grid averaging scheme is run a second time with the additiona constraint x I Jmax = c max (6.19) for a subgrids I = 1,...,M as described in the preceding section. Resuts for this procedure wi be presented in sec In principe, severa oca maxima coud be treated in the same manner Finding optima settings from simuations The variabes for the reconstruction procedure sketched so far are summarised in fig A specific choice of these parameters eads to a reconstruction resut and an error estimate, e.g. by the numerica scheme eqs. (5.14), reated to this particuar setting. The aim is, of course, to find the optima setting in the sense that it eads to smaest possibe reconstruction errors. Simiar to the numerica error estimation this can ony be done by simuated reconstruction for suitabe test distributions. As now the reconstruction error averaged over the ensembe has to be compared for possiby many reaisations of the parameter set P, scaar quaity measures Q as discussed in sec. 5.4 are evauated rather than the error fied ĉ(r). For a choice of parameters p P a simuation step for the test distribution c I from the ensembe E takes the form c I (r) E forward mode d R d i= ds c(r) i reconstruct I ĉ(r) Q(c I,ĉ I,p). (6.2) p atter expression shoud be comparabe to c a in eq. (6.7).

108 18 6. Reconstruction Procedure discretisation agorithm a priori x a box φ biinear φ 1 ART SIRT BG, mean numb. boxes n, boxes j numb. nodes n, nodes r j iteration number k subtract BG composite scheme grid averaging constraints fitting peak nonnegativity x Figure 6.4: Choices for eements of the reconstruction procedure appied in part III. Taking the ensembe average and repeating the procedure for a parameters p 1,p 2,... P yieds the optima parameters as those for which Q(c I,ĉ I,p) E shows the optimum. In genera, measurement errors shoud be taken into account not ony if the parameter varied is the reguarisation parameter (iteration number) as in eqs. (5.14), so that finay the expression Q(c I,ĉ I,p) ǫ E (6.21) has to be optimised with respect to the parameters p.

109 7. Designing a Tomographic Experiment This chapter covers various aspects reevant for panning a tomographic measurement and its targets but not affecting the reconstruction procedure as such. Secs. 7.1, 7.2 dea with the setup of the path geometry and reated questions ike: How many ight paths are minimay necessary? or How much information wi be ost due to noise? etc. The incusion of additiona (point) measurements is discussed in sec. 7.3 depending on whether these are treated equay to the path measurements or regarded as a priori that might be overridden by the atter. The ast section examines how a tomographic reconstruction can be used for mode evauation (statisticay, see sec. 5.4) or even verification (deterministicay, e.g. sec. 5.3), inevitaby invoving both the reconstruction error and the mode uncertainty. It briefy addresses the probem of reating concentration vaues from point measurements to modeed vaues thus foowing up the discussion in secs. 2.3, Requirements for the setup number of ight paths c disc ĉ ĉ inv sec e.g., eqs. (5.8) maxima rec. error maxima disc. error box size V/ V grid dimension m n ĉ(r) c disc (r) V n number of ight paths size of spatia structures s V 1/(2 or 3) or s mesh size of ight paths e.g., eq. (7.1) m Figure 7.1: Estimation of the number of paths necessary for a maximay acceptabe reconstruction error or minimay resovabe spatia structures and vice versa, negecting the measurement error and assuming the even-determined case. Evident questions for a tomographic experiment are of the sort Given a hypothesis or mode assumption on the spatia distribution of a certain trace gas, what experimenta setup, i.e. essentiay what number of ight paths is needed for its verification or fasification?. This transates into a maximay acceptabe error of the reconstructed distribution or a minimum resoving power for spatia structures. Fig. 7.1 iustrates the concusion to the number of ight paths necessary to meet the requirements under the assumptions that no further (a priori) information is incuded and that the measurement error pays a minor roe. The mesh size of the ight paths in fig. 7.1 is understood as the typica, smaest area encosed by the crossing rays and, obviousy, it can vary significanty within the reconstruction area or for different directions. For DOAS measurements the compete geometry 19

110 11 7. Designing a Tomographic Experiment α α Figure 7.2: Iustration of the mesh size for a typica fan beam geometry generated by static emitting or receiving instruments. is generated by teescopes emitting (or receiving) ight for different anges α i (eevations in the case of vertica measurements). For exampe, assuming a number of n t static instruments and α i = α i+1 α i α to be more or ess the same for a instruments, then the size of the ight path meshes in the centre of the reconstruction area of size and the number m of the ight paths according to fig. 7.2 are reated roughy via 1 α O(1) α n t m 1, (7.1) 2 where the same number of rays and ange of beam spread α for a teescopes was assumed Light path geometry degrees of freedom and infuence of the a priori The preceding section reated the ight path geometry to the discretisation. Uness there is a continuous a priori c a (r) the roe of measurement error and a priori for the geometry can be discussed in the discrete vector space in terms of inear agebra. (i) Arguments based on the discrete forward mode ony: Without measurement error the arrangement of the ight paths for a given number of discretisation parameters n is as good as it can be if rank[a] = min(m, n). Components of x in the n rank[a]-dimensiona nuspace of A are given by information other than the measurement data. According to sec. 4.5, x can be represented in the basis of singuar vectors as x = j x j v j, where the v j with j = rank[a] + 1,...,n span the nuspace. Consequenty, one gets for the matrix V of the coumn vectors v j : If j >rank[a] { } { } jj = data, then x j is fixed competey by the. (7.2) 1 a priori or constraints V 2 How this picture changes in the presence of measurement errors has been impicity addressed in sec. 4.4 on the i-posedness of the tomographic reconstruction and in sec on the idea reconstruction grid. The former discussion suggests that the ight paths shoud be as inear independent as possibe. The atter discussion used the eigenvaues of A T A but it can be ed in the same way for AA T, showing

111 7.3. Incuding point measurements and profie information 111 that the geometry shoud be such that the individua paths have simiar ength and as itte overap in the boxes as possibe to ensure singuar vaues of simiar size. To quantify how much interdependency of the ight paths is acceptabe, it is hepfu to ook at the singuar vaue decomposition of A: d + ǫ = Ax = σ 1 u 1 v1 T x + + σ r u r vr T x, } {{ } r = rank[a]. (7.3) ǫ? As indicated, ony those modes of the decomposition add information which give a contribution above the noise eve. The singuar vectors are normaised to one so that this can be seen to be the case if σ j ǫ / v T j x ǫ / x, (7.4) where ǫ is the root mean square error, i.e. ǫ / m. 1 (ii) Arguments based on the inversion: Ony the case of a inear inversion by a matrix A inv is considered. In this case the resoution matrix R = A inv A, sec , can be used to study how for a given ight path geometry the origina concentration vaues x true j are mapped to the whoe reconstruction grid by x j = j R jj x true j. We note in passing that (½n R) jj = j >rank[a] V jj 2 for the generaised inverse, eqs. (4.37), so that for the idea reconstruction (R jj = 1) the sum over the nuspace indeed equas zero. The Bayesian approach in the specia form of the optima estimate gives bounds simiar to eq. (7.4) (see sec ): A mode j of the SVD of A increases the information content of the measurement, if for the singuar vaue σ j of à = S 1/2 ǫ ASa 1/2 hods σ j 1. (7.5) It was remarked at the beginning of sec. 4.7 that, even if not feasibe as a reconstruction method, the Bayesian method can be used as a diagnostic too for what-if scenarios, testing the outcome of an experiment for an assumed a priori state of the system. In this sense the optima estimate wi be used in sec to study the impact of measurement noise for different ight path geometries Incuding point measurements and profie information Panning a remote sensing experiment can invove point measurements in two ways The ocation of the point measurements is fixed how shoud the ight paths be chosen? The ight paths are fixed where shoud the point measurements be performed? A point measurement at the ocation r with resut c takes in the discrete framework the form c(r ) = [φ(r )] T x = c (with measurement error σ c ). (7.6) 1 A quite engthy cacuation without using the SVD for continuous kerne functions (and the over-determined case) in [Twomey, 1997, sec. 8.4] can be adopted to the discrete case and gives the same threshod.

112 Designing a Tomographic Experiment If there are p such measurements at ocations r ip, i p = 1,...p, their resuts can be joined in a vector c Êp and a matrix F Êp n as Fx = c with c = (c 1,...,c p ) T F ipj = φ j (r ip ). (7.7) For the box basis or if the ocation of the point measurement coincides with the grid nodes in the case of the bi-(tri)inear basis, the rows in the system become x ip = c ip. Concentration measurements aong a profie Π : r = r(s) Ê3, s Ê, can be considered as a series of point measurements c(r(s ip )) = c Π (s ip ), where the r(s ip ) ie aong the profie, and can thus in principe aso be expressed in the form of eq. (7.7). There are principay two ways to incorporate information from point or profie measurements into the tomographic reconstruction: (i) as additiona constraints ike in sec or (ii) as (additiona) a priori. (i) Point measurements as additiona constraints: The augmented system Ax = d defined in eq. (4.6a) can again be soved expicity. Assuming the under-determined case m+p < n with a priori x a and no further inequaity constraints, the east-norm soution is given by eq. (6.18). (ii) Point measurements as additiona a priori: The point measurement(s) wi in genera not provide an a priori for a components of x, so that further a priori x a is sti needed. The augmented a priori x a = F T (FF T ) 1 c + P F x a (7.8) draws rank[f] degrees of freedom from the point measurement(s) Fx = c and the remaining n-rank[f] from x a (appendix C). P F is the projector onto the nuspace of F. The east-squares east-norm soution of Ax = d without further constraints reads then as x = A T (AA T ) 1 ( d + P A F T (FF T ) 1 ) c + P F x a. (7.9) Case (i) treats remote sensing and point measurement in the same way. The (ad hoc) a priori x a contributes ony in the reduced n-rank[a] dimensiona nuspace N(A). In (ii) the a priori x a adds components in the intersection of N(A) and N(F): P A P F x a. 2 2 The difference is best iustrated by an exampe: Let there be«five boxes and one ight path through two boxes, a 1 1 second one through the other three such that A =. Let there further be a point measurement in the first box, i.e. F = (1 ), c = (c) T. Then for the augmented system 1 c 1 2 d 1 c (d c x a2 ) 1 x (i) = 1 B3 (d 2 + 2x a3 x a4 x a5 1 3 (d C 2 x a3 + 2x a4 x a5 ) A, whie x 2 (ii) = (d 1 c + x a2 ) B as for x (i) A 1 3 (d 2 x a3 x a4 + 2x a5 ) with augmented a priori x a = (c, x a2, x a3, x a4, x a5 ) T.

113 7.4. Aspects of mode evauation 113 Returning to the question of how both kinds of measurements shoud be set up, one finds that if they are meant to be compementary, ideay the ight paths or the ocation of the point measurement(s) shoud be set up such that the nuspaces of A and F according to eq. (7.7) have as itte in common as possibe, e.g. by choosing the row vectors of F as a inear combination of any p singuar vectors that span the nuspace of A. In practice, such a construction wi be hard to reaise and might be corrupted by noise for the reasons discussed in the preceding section. Therefore, an estimate of the degrees of freedom (or the information content) of different configurations of remote sensing and point measurement for an expected eve of noise as outined in section 7.2 can be hepfu. For the augmented system of case (i) the measurement error and a priori covariances are given by S ǫ = ( S ǫ L 2 S c ) and S a, where the system Fx = c was scaed with a typica ength L of the ight paths. The augmented a priori covariance for the case (ii) is given as (appendix C) S a = F T (FF T ) 1 S c (FF T ) 1 F + P F S a P F. (7.1) 7.4. Aspects of mode evauation Motivation for evauating the performance of dispersion or chemica transport modes on very sma spatia scaes was given in chap. 2 in the context of urban air poution monitoring. Evauation is understood here as the comparison of modeed concentrations with corresponding experimenta vaues. 3 According to sec. 2.3 this is more compicated than it sounds as both refer to fundamentay different quantities: For given meteoroogica conditions modes describe the mean fied given by eq. (2.3) pus, possiby, a more or ess accurate approximation of the turbuent part c : c mod (r) c(r,t) + c approx(r,t) for ū,t,p, etc. at time t, whereas a point measurement averaging over the period T = [t;t + t] yieds the vaue c exp (r, T ) c(r,t) T + c (r,t) T, (7.11) where c(r,t) T denotes the vaue of c(r,t) (assumed to be constant) during T. On the one hand, c mod and c exp wi differ substantiay for short averaging times c mod (r) c exp (r, T ) for t. On the other hand, for arge enough t c mod (r) c exp (r, T ) for t and stationary ū,t,p, etc. Because meteoroogica conditions are never stationary, a compromise must be found between averaging times ong enough to sufficienty reduce the randomness of the measured vaues and short enough to guarantee the same constant atmospheric parameters to be comparabe to the mode prediction. Averaging times of around 3 min are common but according to Schatzmann and Leit [22], who consider tracer measurements for a wind tunne mode of a street canyon for different averaging times, 3 Whereas vaidation woud mean a consistency check of the underying physica concepts and their transation into computer code.

114 Designing a Tomographic Experiment not aways sufficient to obtain representative vaues. Scaing the averaging times corresponding to the rea dimensions of the street canyon, the concusion on ambient measurements can easiy be drawn because in a wind tunne stationary boundary conditions for t can actuay be reaised. An even more pessimistic estimation is inferred from ong term time series of NO x concentrations measured in the origina street canyon where concentrations referring to the same boundary conditions scatter argey. It shoud be added in this context that it is common practice to convert a mean concentration c 1 for averaging time t 1 to another (usuay shorter) t 2 by a reation ( t 2 ) p c 2 = ( t 1 ) p c 1 with p typicay between.17 and.2 [Barratt, 22]. This procedure is entirey empirica, giving no information on the turbuent part for different times and it is furthermore controversia [Venkatram, 22]. The discussion so far concerned the representativeness of measured concentration vaues at a fixed pace with respect to time. Interpretation of measurement resuts at different paces r 1, r 2 and the same time interva T is simiary difficut, as the random and mean contributions in eq. (7.11) cannot be discriminated. One way to dea with the stochastic uncertainty in eq. (7.11) in order to compare it with the corresponding mode vaue is to use empirica or theoretica assumptions to reate the measured vaue c exp (r, T ) to the unknown mean c(r,t) or to derive bounds for the size of the random contribution c (r,t) T. For exampe, for urban poution where not ony the dispersion, but aso the emissions foow more or ess stochastic patterns, the experimenta concentrations are often assumed to be distributed og-normay around the (mode) mean (e.g., [Venkatram et a., 25], see aso [Seinfed, 1986, chap. 17]). The estimation of turbuent fuctuations has been shorty addressed in sec. 2.4 in terms of variances of the turbuent concentration fied within a pume. A more comprehensive approach is a decription by probabiity density functions [e.g., Yee and Chan, 1996; Ma et a., 25]. Whie these vaue-by-vaue comparisons are necessary when it comes to judging mode capabiity for individua concentrations or specia features of the distribution, for an evauation of the overa mode performance the stochastic character of the measured vaues can be reduced by considering statistic quaity measures ike the ones introduced in sec It is a bit difficut to specify the size of typica mode errors or vaues of statistic quaity measures because both heaviy depend on the kind of appication, the averaging times, atmospheric stabiity, the kind of species, the number of measurements and where they took pace etc. Aso the way and degree in which meteoroogica observations or immission measurements enter the mode prediction can be quite different (c.f. fig. 2.6 on page 34). In the context of roadway dispersion modeing Hed et a. [23] regard an agreement between a dispersion mode and measurement within 3% as perfect. For extensive cacuations with a CFD mode, see sec. 2.5, Tsai and Chen [24] find discrepancies between modeed and measured concentrations over an averaging period of h about 5 1% for CO, 2 5% for NO x and 1 15% for SO 2. Statistica evauation of dispersion modes on urban scaes, e.g. by means of concentration data from SF 6 dispersion experiments ike those mentioned in sec. 2.4, shows roughy the ranges isted in tabe 7.1 for statistica indices from tabe 5.1. The vaues shoud be taken with a grain of sat. Especiay NMSE and FB depend on the type of distribution. They are generay higher for sma scae pumes, higher for arger scae distributions (see aso sec ). Turning from the point measurements impicitey referred to ti now to tomographic remote sensing measurements, one expects the reconstructed fied to depend ess on the turbuent fied c (r,t). For

115 7.4. Aspects of mode evauation 115 Tabe 7.1: Ranges for some metrics from tabe 5.1 as found for dispersion and chemica transport modes on urban scaes [Seinfed and Pandis, 1998, sec. 23.8], [Hurey et a., 25; Brandt et a., 21; Moreira et a., 25; Venkatram et a., 24, 25] (In most cases for 1 h averages). NMSE.2 % IOA FB 5-15% r FA2 4-8 % PPA 15-2% the individua coumn densities, eq. (4.3), d(t) i T = ds c(r,t) T + ds c (r,t) T, i = 1,...,m, (7.12) LP i LP i the random part disappears for uniform fuctuations aong the ight path LP i if their ength scae is sma against the ength of LP i (c.f. sec. 2.4, especiay the exampe on p. 29). A numerica justification of this presumption coud be carried out with a Monte-Caro mode as the one provided by Backadar [1997]. Having mentioned this agreeabe property of path integrating measurements, scenarios for mode evauation ike the foowing ask for an estimation of the tota reconstruction error. (i) Given a modeed fied c mod (r) and a fied reconstructed from experiment ĉ(r), do both agree within the reconstruction error, i.e. c mod (r) ĉ(r)? < ĉ(r) (7.13) (ii) Given a presumed uncertainty of the mode, i.e. c min (r) c(r) c max (r), (7.14a) where c(r) is the unknown true distribution, coud a tomographic experiment further constrain the mode (or verify the assumed uncertainty)? That is c min (r)? < ĉ(r) + ĉ(r) and ĉ(r) + u ĉ(r)? < c max (r) (7.14b) in the sense of eq. (5.1). Whie probem (i) requires estimation of ĉ(r) independenty of the mode distribution, it is a consistent approach to (ii) to use the mode uncertainty for numerica error estimation as described in sec. 5.3, based on the ensembe E = {c I (r) c min (r) c I (r) c max (r)..., I = 1,...,N}, (7.15) where the dots refer to further specifications for c I (r) as described in appendix B. The bounds c min,

116 Designing a Tomographic Experiment c max coud be given by roughy known reative errors x of the mode, i.e. c min/max (r) = (1 ± x)c mod (r). (7.16) Aternativey, they coud be based on empirica experience (comparison with other modes etc) or be derived from statistica indices ike FA2. If, for exampe, FA2 1%, then one gets the bounds c min (r) =.5c mod (r), c max (r) = 2c mod (r). (7.17)

117 Part III. Numerica Resuts 117

118 8. 2-D Simuation Resuts for Gaussian Peaks This chapter presents simuation resuts for the 2-D reconstruction of trace gas distributions emerging from a point source by turbuent diffusion, which can be described by the Gaussian pume or puff mode (sec. 2.4). The case of emission puffs wi be studied in the tomographic experiment of the next chapter. To be precise, it is assumed that the 2-D concentration distribution on the section through the pume or puff is Gaussian. The orientation of this section that is, the reconstruction area is arbitrary, but for LP-DOAS experiments it is ikey to be horizonta. Superposition of severa Gaussians can be used to mimic more compex concentration fieds. It is assumed throughout that the trace gas species occurs ony as concentration puff, i.e. there is no natura or anthropogenic background. This might be unreaistic for most trace gases reevant for DOAS measurements, but at the very end of the chapter in sec. 8.5, it wi be shown that the preceding resuts hod for a smooth background, too, if subtracted before reconstruction. The simuation resuts fa into three parts on the parametrisation of the probem (sec. 8.2), the inversion agorithm (sec. 8.3) and the ight path geometry (sec. 8.4). A three parts empoy a set of test ensembes defined in sec The first two parts use a fixed test geometry which is more or ess the one reaised in the aforementioned experiment. The infuence of the parametrisation of the reconstruction resut in terms of grid dimension and grid combination schemes on different aspects of the reconstruction quaity is examined systematicay with respect to the extension of the puffs. The focus is on the smaer puffs of the test ensembes, for which box and biinear parametrisation are compared in detai with respect to discretisation and inversion errors. Finay, sampe reconstructions are shown. Anticipating the resuts of the second part (sec. 8.3) and in agreement with the iterature, which generay finds SIRT superior to ART, a reconstructions in sec. 8.2 use SIRT. The comparison of ART and SIRT in sec. 8.3 thus serves mainy to compare the infuence of the choice of the agorithm to the infuence of the parametrisation. Measurement noise and reguarisation are discussed ony for a specia case that wi be needed for the anaysis of the experiment in chap. 9. The third part of the simuation resuts starts with a short section (8.4.1) more of an outook iustrating the SVD as a diagnostic too. The main part consists of an anaysis of different geometries with the same ight path number by expicit numerica simuation for Gaussian distributions on the one hand (sec (i)), and agebraic and Bayesian concepts on the other (sec (ii)). The atter is used in sec in an exampe to optimay incude a point measurement in the sense of sec Finay, a simpe recipe is given to transfer simuation resuts for different numbers of ight paths (sec ), thus increasing the use of the specia cases studied here tremendousy. 118

119 8.1. Test distributions and ight path geometry Test distributions and ight path geometry = 1 a.u. ensembe σ x,y/ 2σ / 2σ / E E E E Light path geometry with three 9 -fans and 36 paths in tota. Variances and 2 σ-diameters of the peaks within the four ensembes. At 2 σ the exponentia has faen to 14% of its peak vaue. The area integra within 2 σ around the peak centre accounts for 95% of the tota integra. The ast coumn presents the ratio of the peak extension 2σ to the mesh size at the center of the geometry given by eq. (8.1). c(x,y) [a.u.] 1 c(x,y) [a.u.] /2 x [a.u.] /2 y [a.u.] /2 x [a.u.] /2 y [a.u.] Most narrow peak from E 1 with σ x = σ y =.3 Broadest peak from E 4 with σ x = σ y =.3 Tabe 8.1: Light path geometry used for the simuations in sections 8.2 and 8.3 and Gaussian concentration fieds. The reconstruction area is chosen as a square of side ength = 1 in arbitrary units (a.u.). The compete geometry of 36 ight paths is generated by three 9 -fans with 12 beams each, emitted by three teescopes sitting in the corners of the square (see tabe. 8.1). This number of optica paths corresponds to the one of the indoor experiment and it is high enough to guarantee a certain degree of reguarity required for a meaningfu comparison of different types of path geometries ater (sec ). The mesh size in the centre of the reconstruction area is according to eq. (7.1) 1 2 α n t m 1 = 1 π (8.1) The Gaussian test concentration distributions are ocated randomy within the reconstruction area. According to their peak width, they are divided into four ensembes E I, I = 1,...,4 : { ( ) E I = c I (x,y), I = 1,...,N c I (x,y) = C e 1 2 (x x ) 2 /σ 2 x +(y y)2 /σ 2 y, with random x,y,.1c max C C max, σ x,y as in tabe 8.1 }, (8.2)

120 D Simuation Resuts for Gaussian Peaks where C max = 1a.u. A random quantities are uniformy distributed and N is the number of random sampes. Taking the radius of the peak as 2σ, ensembe E 1 in tabe 8.1, with a smaest peak diameter simiar to the mesh size given by eq. (8.1), represents the ower imit of narrow peaks that are sti detectabe for the coverage with ight paths at hand. Here the aim is rather to ocate the peaks and to reconstruct tota amounts of concentrations, i.e., tota emissions, whereas for the broad peaks of ensembes E 3 and E 4 reconstruction of the actua distribution shoud become feasibe (see the ower panes in tabe 8.1). If the test concentration fied consists of more than one peak, maximum vaues for a Gaussians vary randomy between.1c max...c max. Maximay four peaks are considered at the same time, as reconstruction of any further, especiay sma peak becomes highy unreiabe for the given number of ight paths Parametrisation Referring to fig. 6.4 (p. 18), this section presents resuts for different choices within the discretisation procedure. As said, background concentration distributions in the sense of sec are assumed to be negigibe or to have been subtracted as proposed in sec (ii). A resuts in this section are for optimised iteration number k in the sense of sec The dependence on k wi be addressed in sec The nonnegativity constraint is aways impemented Grid dimension Severa quaity criteria from tabe 5.1 are cacuated within each ensembe depending on the grid dimension n. To concentrate on the important point, ony biinear basis functions and the simutaneous agorithm SIRT wi be considered. Furthermore, the grids are reguar with uniform spacing, so that a grid of dimension n has n x = n y = n nodes in each direction and grid spacing x = y = /( n 1). Nearness N EARN, representing the overa reconstruction error, the quaity of the reconstructed mean concentration in the form of the normaised mean bias N M B and the accuracy of reconstructed maxima of the highest peak PPA and of peak integras/tota emissions PIPA basicay cover the most reevant features one woud be interested in when trying to retrieve peak distributions. They are shown for one and four peaks of each ensembe in fig For ensembes E 3 and E 4 the mean a priori x a according to eq. (6.7) was chosen as iteration start x (), otherwise the a priori is zero. The most important points emerging from these curves are For narrow peaks (E 1 and E 2 ) a error measures in fig. 8.1 can be tremendousy reduced reative to the even-determined case n m = 36 by choosing discretisation grids that ead to highy under-determined systems of equations. For absoute maxima and tota emissions the improvement amounts to more than 5%. This hods aso for more (four) peaks. Here the ambiguity of individua quaity measures becomes apparent. Whie the overa reconstruction error for four peaks is aways smaer than for one peak (the reconstruction is ess accurate, as one might expect), the reative error of the mean concentration N M B behaves inversey. In the same way, the peak prediction accuracy is higher

121 8.2. Parametrisation 121 NEARN E I x/ peak 4 peaks E 1 E 2 E 3 E 4 NMB E I x/ n (a) Nearness x/ n (b) Normaised mean bias x/ PPA E I PIPA E I n (c) Error of the maximum peak prediction accuracy n (d) Error of the peak integra prediction accuracy Figure 8.1: Ensembe averages of different quaity criteria for one and four random peaks depending on the grid dimension. The number of ight paths is 36, the geometry is as in tab The biinear grid is reguar, so that dimension n means n 1 pixes in each direction. Aso shown is the grid spacing x. The a priori is zero for E 1, E 2 and the mean eq. (6.7) for E 3, E 4. SIRT was used (see sec. 8.3 for iteration numbers). The number of random sampes is N = 3. For the sake of cearness ony some standard deviations std EI ( ) are shown. (a) Overa reconstruction error NEARN. (b) Absoute error of the mean concentration NMB. A vaue of.25 means that the reconstructed spatia mean deviates by 25% from the rea one. (c) Absoute deviation of the reconstructed goba maximum from the rea one ( 1 PPA ). A vaue of.5 means deviation by 5%. (d) The same for the peak integra, i.e. a vaue of.1 means average over- or underestimation of tota emissions by 1%. for four peaks, whie the error of the reconstructed peak integra is ower than for one peak ony (suggesting that, with severa peaks, the argest accumuates concentration at the expense of the others). Whie the errors decrease with increasing n, the standard deviations std EI ( ) get arger, as one might expect because the degrees of freedom of the picture increase with growing n (visibe in the form of artefacts, ike bumps, in the reconstructed distribution). For broad peaks (E 3 and E 4 ) under- or even-determined discretisation grids ead to best resuts. For the broadest peaks from E 4 the overa reconstruction quaity rapidy deteriorates for higher dimensiona grids (resuting in arge artefacts).

122 D Simuation Resuts for Gaussian Peaks The optima grid dimension is not aways the same, but simiar for a quaity criteria. Soutions discretised by 1 1 pixes impy about 121/36 3 more unknown parameters than measured integras! How can the resuting continuous distributions for narrow peaks sti be better approximations of the rea peaks? The reasons are 1. For narrow peaks most of the grid nodes carry the concentration zero, and this is exacty the vaue provided by the a priori (in the inear case through eqs. (4.56)). 2. The discretisation error as defined in sec decreases with increasing grid dimension. This readiy expains the decrease in the overa reconstrcution error (N EARN) and aso the improved estimate of maximum concentration vaues (P P A), as the peak is spread over smaer pixes (mind that for 1 1 pixes the grid spacing x becomes comparabe to the peak diameter 2σ ). For the tota and peak area integras (NMB,PIPA) it shoud be noticed that these may increase if the origina concentration peak is spread over more pixes by the reconstruction whie the coumn densities are kept constant, as iustrated by the figure beow. c(x, y) c x x ĉ(x, y) LP c/ LP Figure 8.2: For the origina distribution (eft) the coumn densities are d 1 = d 2 = c x and the integra I over the compete reconstruction area is I = R dx dy c(x, y) = ( x) 2 c. For the reconstruction (right) d 1 = d 2 remain unchanged, but the area integra is now 5 3 I. 3. The nonnegativity constraint becomes active around zero so that the nuspace is reduced by the negative soutions. Putting points 1. and 2. together means that, when increasing the grid dimension, a possibe increase of the inversion error with a bad a priori being part of it remains smaer than the decrease of the discretisation error. This wi be further investigated in the next section. An iustration of point 3. wi be given in sec How can these findings be impemented to improve the reconstruction from rea experiments, i.e. when actua pume extensions are not known? First, the experimentaist is not as ignorant as a set of equations and there might we be information that does not enter the formuation of the tomographic reconstruction probem. For exampe, the dimension of an emission puff driven through the setup of the ight paths can roughy be inferred from the increase and decrease of coumn densities combined with wind speed. Or empirica dispersion coefficients might be usefu (sec. 2.4). A way to obtain information about the true peak extension within the reconstruction itsef is contained in tab. 8.3b, p. 129, which shows sampe reconstructions for each ensembe and various grid dimensions. The peak in tab. 8.3b, reconstructed from a Gaussian beonging to E 2, increases its height going from a grid with 8 8 pixes to 1 1 pixes, but decreases again for arger grid dimension whie artefacts appear. Both features together might be interpreted as the optima grid dimension being exceeded.

123 8.2. Parametrisation x inv + x reg / x id E1 E 2.4 Figure 8.3: Lower graph: Overa discretisation, inversion pus reguarisation and tota reconstruction errors for piecewise constant (box) and biinear parametrisation of four (narrow) concentration peaks from E 1 E 2 versus grid dimension. The grids are reguar and dimension n means n pixes for the box and ( n 1) ( n 1) pixes for the biinear basis. The under-determined soutions are practicay unreguarised, so that c reg(r). The a priori is zero as in fig. 8.1, again for SIRT and N = 3. Upper graph: The same for the reative discrete inversion error Basis functions c E1 E 2 biin. box ĉ ĉ inv + ĉ reg c disc n The piecewise constant and biinear basis functions are compared by expicity cacuating the discretisation error c disc (r) as proposed in sec The inversion error ĉ inv (r) immediatey foows as [φ(r)] T (x id x) and if the soution is reguarised we consider ĉ inv + ĉ reg = [φ(r)] T (x id x λ ) (It turns out that, especiay in the over-determined region, reguarisation can improve the soution even if there are no measurement errors. I postpone the discussion to sec. 8.3). Fig. 8.3 contains in the ower graph the mean overa discretisation error c disc (r) 2 and the inversion error of the reguarised soution ( ĉ inv + ĉ reg )(r) 2 for narrow peaks from ensembes E 1 pus E 2 for both types of basis functions. In fact, here ony the soutions with m n are reguarised and the reguarisation error hardy contributes for n m. As expected, the overa discretisation error, and therefore the tota reconstruction error, is far arger for the piecewise constant than it is for the piecewise biinear parametrisation. Furthermore, fig. 8.3 shows that: The squares of the overa discretisation and inversion error do indeed add up to the reconstruction error according to eq. (5.18). For the concentration distributions considered in fig. 8.3 the discretisation and inversion error are of simiar size for the optima grid dimension. The inversion error increases with grid dimension n. The decreasing tota reconstruction error is finay soey due to the faster decreasing discretisation error. In the under-determined regime the inversion error for the box basis is arger and faster growing than for the biinear basis. Concentrating on the under-determined cases n > m for the narrow peaks (the over-determined case appies to the broad peaks which are not part of these simuations), the ast point is most remarkabe because the discrete systems for both parametrisations are combined with the same a priori x a =

124 D Simuation Resuts for Gaussian Peaks reative error box biinear mean concentration 1 NMB 4% 5% tota emission 1 P IP A 15% 1% maximum concentration 1 P P A 4% 2% Tabe 8.2: Reative errors of concentration integras and maxima for a biinear grid of dimension n = 81. Distributions and a parameters are as in fig and it is not obvious why ĉ inv (r) shoud differ significanty. One has to be carefu though when carrying over concusions from continuous soutions toên and vice versa. For the norm this step takes the form ĉ 2 2 = x T Φx with Φ as in eqs. (5.5) yieding ĉ 2 = A x 2 for the box basis and reguar discretisation grids, where A is the area of a grid ce. The upper graph of fig. 8.3 shows that the reative norm of the discrete inversion error for the biinear basis is sti smaer for highy underdetermined systems of equations, but ess distinctive. The fact that the unreguarised east norm soution for the biinear parametrisation is better in terms of inversion errors has to be attributed to the a priori and/or constraints. For exampe, the biinear basis functions have overapping supports so that the matrix A has ess zero entries than the corresponding matrix for the box basis and the nonnegativity constraint might become more effective. So far error fieds c(r) have been considered, but with the piecewise constant basis representing box average concentrations (see eqs. (5.6)) one might ask whether the box representation enabes better reconstruction of spatia mean concentrations. In genera it does not, as iustrated by tabe 8.2 which gives the reative error of reconstructed mean concentrations (NMB) and tota emissions (PIPA) for an under-determined parametrisation by 8 8 pixes. The standard deviations of these errors within the test ensembes are of the size of the errors themseves, so that the difference in the mean concentration is not reay significant. Nevertheess the reconstruction errors of both mean concentration and tota emissions are persistenty arger for the box than for inear parametrisation for a under-determined grids. Recaing that for the cacuation of the peak integras the pixe size is taken into account (see expanation to eq. (5.2)), the fact that compared to the biinear basis the error of the peak integras for the box basis is reativey arger than the error of the mean concentration suggests the interpretation that the concentration is distributed on a far arger area than the peak or esewhere within the reconstruction area (artefacts). In both cases the inversion error woud become arger, agreeing with the curves in fig. 8.3 which show a arger inversion error for the box parametrisation in the under-determined case Grid transation As in the previous sections the effect of combining grids on the reconstruction is examined statisticay for quaity indices measuring the overa square error (nearness), the error of the spatia mean concentration (normaised mean bias N M B), the error of absoute maximum concentrations (peak prediction accuracy PPA) and the error of tota emissions PIPA). Fig. 8.5 shows averages within ensembes E 1 to E 4 for the composite scheme (sec (i)), the averaging scheme (sec (ii)) and the averaging scheme that uses the absoute maximum as additiona constraint (sec ) compared to the case of a singe grid. The origina reguar grid ( x = y) is shifted four times in each direction,

125 8.2. Parametrisation n = 5 5 n = 6 6 n = Figure 8.4: Condition number cond[a] = σ 1/σ rank[a], see eq. (4.27), for subgrids generated by shifting the origina biinear grid of dimension n ten times in x and y (c.f. fig. 6.2). Obviousy, the variation of cond[a] within the subgrids is ess distinct than the variation with n. cond[a] subgrid i.e. m = m x = m y = 4, resuting in N = (5( n 1) + 1) grid nodes in each direction for the fina composite grid of higher resoution, see eq. (6.5b). The number of shifts is not critica in that choosing m = 3 does not drasticay deteriorate the reconstruction resuts, whie shifting five or six times does not improve them significanty but increases cacuation time consideraby (according to eq. (6.5a) with m 2 ). Stricty speaking, different conditioning of the matrix A for each subgrid shoud be taken into account by adjusting the iteration number accordingy as iustrated in fig. 8.4, but at east for the averaging and composite scheme it affects the subgrid reconstructions in the same way and shoud not matter for a comparison. Therefore the same iteration number for a subgrids is chosen. First of a, genera concusions emerging from figs. 8.5a to 8.5d are: The performance of each reconstruction scheme heaviy depends on both the kind of concentration distribution (ensembe) and the quaity criterion used for evauation. The reative amount of change for different schemes (reative to the singe grid reconstruction) varies strongy within the quaity criteria. The composite scheme reduces the overa root mean square error for narrow peaks (E 1 ) by 8% whie the error of the reconstructed maximum concentration decreases by around 5%! The fact that, at the same time the accuracy of mean concentrations and emissions decreases by 3% and 25%, respectivey, can be understood as overestimation foowing from the construction of the scheme that combines maximum node vaues from different grids. It cannot be expected to conserve concentration integras. Originay proposed by Verkruysse and Todd [24] for better reconstruction of maximum concentration vaues in the context of indoor air contamination, the composite scheme thus works we for this purpose as ong as one is not interested in tota amounts of emitted gas. The restriction to narrow peaks is not addressed in a systematic manner in the aforementioned study. The authors use piecewise constant parametrisation combined with the MLEM approach (c.f. sec.4.9.1) and consider Gaussian peaks with σ/ , roughy corresponding to peaks from E 2 and E 3. For the geometry with 4 integration paths shown in fig. 3.5h and four peaks they obtain mean vaues for the overa error NEARN around.7 (singe grid),.5 (transated grids) and for the peak prediction error 1 PIPA around.35 (singe grid) and.2 (transated grid). Comparing these to the vaues for E 2 (to take the narrower peaks) in fig. 8.5a and 8.5c, it foows that aready the reconstruction with a singe biinear grid eads to better resuts than the grid transation method for the box basis, underining again the importance of optima parametrisation. The composite scheme does not work reiaby for peaks other than the narrow ones from ensembe E 1.

126 D Simuation Resuts for Gaussian Peaks NEARN E I singe grid composite average fixed peak NMB E I E 1 n = 1 1 E E E E 1 n = 1 1 E E E (a) Nearness (b) Normaised mean bias PPA E I PIPA E I E 1 n = 1 1 E E E (c) Error of the maximum peak prediction accuracy E 1 n = 1 1 E E E (d) Error of the peak integra prediction accuracy Figure 8.5: Ensembe averaged quaity indices for different methods of combining reconstructions from severa biinear grids, again for four peaks within each ensembe and a settings as in fig For each ensembe the optima grid in terms of NEARN was chosen. The grid dimension n refers either to the actua grid dimension (when reconstructing from a singe grid) or to the dimension of the origina grid used to generate the shifted grids. In the atter cases the grid was shifted four times in each direction, see fig The averaging method reduces the mean square error for a peaks and makes hardy any difference for the concentration integras as to be expected for a inear method. It does underestimate the peak maximum though as anticipated in sec , fig. 6.3, eading to ower peak prediction accuracy for the first three ensembes. For narrow peaks (E 1 ) the averaging scheme increases the peak error by 5%, taking us right to the modified approach where the maximum grid node vaue among a subgrids is impemented as a constraint for each subgrid reconstruction. This scheme does indeed give better estimates for the absoute maximum concentration for peaks from E 1 and E 2 but aso worse vaues for mean concentrations and tota emissions, especiay for very narrow peaks. This is not surprising because for these an enhanced peak concentration has reativey more impact on area integras (see aso fig. 8.2). To concude the statistic evauation, it can be said that simiar to the choice of the best reconstruction grid discussed before in sec , seection of the optima reconstruction scheme requires some idea about the extension of the peaks. The averaging scheme is safe in that it generay reduces the overa

127 8.2. Parametrisation 127 error, but at the expense of the peak reconstruction accuracy. For concentration peaks that ead to under-determined optima parametrisations the composite scheme and the averaging method with peak constraint offer aternatives for better maximum vaues, where the composite scheme shoud be preferred for very narrow peaks. Both schemes shoud not be used if one is interested in tota emissions. In fact, none of the mutipe grid schemes discussed here improves the quaity of reconstructed integration integras. Finay, tabes 8.3 and 8.4 iustrate the effect of the reconstruction schemes discussed. 1 Some characteristic features visibe in a reconstructions ike the peak reduction of the averaging scheme or the noisiness of the composite scheme foow directy from the construction of the method. Despite obvious shortcomings of individua schemes, it is hardy possibe to make out an optima method among a sampe reconstructions and it appears that the best quaitative picture arises by combining the information from different approaches. Tabe 8.4a is a good exampe where use of a singe grid (the 9 by 9 nodes grid in this case) can give a rather miseading picture. Another important aspect is contained in tabe 8.3b as mentioned earier. A four reconstruction schemes show a decrease of the maximum and a broadening of the peak for the very fine n = grid which can ony be interpreted as the a priori not working any more and the grid dimension being too arge. This behaviour might thus be considered as a criterion for the choice of the parametrisation. 1 3-D surface graphs instead of, for exampe, 2-D contour graphs are used for this purpose simpy because they visuaise structures more ceary.

128 D Simuation Resuts for Gaussian Peaks c(r) C max x origina ( E 1 ) y ĉ(r) C max C max C max y y x x x singe grid: n = 1 1 n = n = y C max C max C max y y x x x composite scheme: n = 1 1 n = n = y C max C max C max y y x x x averaging scheme: n = 1 1 n = n = y C max C max C max y y x x x fixed peak scheme: n = 1 1 n = n = y (a) c(r) from E 1 with σ x/ = σ y/ =.4 Tabe 8.3: Sampe reconstructions using a singe grid or one of the transation schemes, respectivey, for one Gaussian from each ensembe and various grid dimensions. A reconstructions use biinear parametrisation and SIRT. The grids are transated four times in each direction (m x = m y = 4).

129 8.2. Parametrisation 129 c(r) C max x origina ( E 2 ) y ĉ(r) C max C max C max y y x x x singe grid: n = 9 9 n = n = y C max C max C max y y x x x composite scheme: n = 9 9 n = n = y C max C max C max y y x x x averaging scheme: n = 9 9 n = n = y C max C max C max y y x x x fixed peak scheme: n = 9 9 n = n = y (b) c(r) from E 2 with σ x/ = σ y/ =.7 Tabe 8.3: Sampe reconstructions (continued).

130 D Simuation Resuts for Gaussian Peaks c(r) C max x origina ( E 3 ) y ĉ(r) C max C max C max C max y y y x x x x singe grid, n = 6 6 comp. scheme, n = 6 6 averag. scheme, n = 6 6 fix. peak scheme, n = 6 6 y (c) c(r) from E 3 with σ x/ = σ y/ =.15 Tabe 8.3: Sampe reconstructions (continued). Ony the even-determined grid is shown. c(r) C max x origina ( E 4 ) y ĉ(r) C max C max C max C max y y y x x x x singe grid, n = 5 5 comp. scheme, n = 5 5 averag. scheme, n = 5 5 fix. peak scheme, n = 5 5 y (d) c(r) from E 4 with σ x/ = σ y/ =.25 Tabe 8.3: Sampe reconstructions (continued). fig. 8.1a). Ony the under-detemined grid is shown (according to

131 8.2. Parametrisation 131 c(r) C max x origina two peaks y ĉ(r) C max C max C max C max y y y x x x x singe grid, n = 9 9 comp. scheme, n = 9 9 averag. scheme, n = 9 9 fix. peak scheme, n = 9 9 y C max C max C max C max y y y y x x x x singe grid, n = comp. scheme, n = averag. scheme, n = fix. peak scheme, n = (a) Two Gaussians, one from E 2 with σ/ = σ x/ = σ y/ =.6, C max = 1 a.u., the other from E 3 with σ/ =.13, C max =.7 a.u., respectivey. c(r) C max x origina four peaks y ĉ(r) C max C max C max C max y y y x x x x singe grid, n = 9 9 comp. scheme, n = 9 9 averag. scheme, n = 9 9 fix. peak scheme, n = 9 9 y (b) Four Gaussians from E 2 to E 4 with σ/ =.6,.8,.13,.2 and C max = 1,.5,.7,.2 a.u.. Tabe 8.4: Sampe reconstructions for severa peaks. Settings of the reconstruction as in tabe 8.3.

132 D Simuation Resuts for Gaussian Peaks 8.3. Agorithms: ART versus SIRT Soution of arge over-determined east squares probems by row acting methods ike ART and SIRT has been (and is sti) subject of extensive investigation in computerised tomography and there is itte need to go into much detai here. They have aso been appied to sma under-determined probems ike the one at hand in indoor gas tomography, but some of the respective studies [Drescher et a., 1996; Todd and Ramachandran, 1994a; Laeppe et a., 24] ack systematics and mathematica background necessary for a genera concusion. Therefore, some resuts comparing the performance of ART and SIRT aong the ines of the previous sections are presented, at the same time specifying detais of the iteration. At ast, the sensitivity to measurement noise is discussed Optima iteration number and reconstruction quaity σi [a.u.] n = 5 5 n = 6 6 n = 9 9 n = i Figure 8.6: Singuar vaues σ i of the matrix Ā beonging to the rescaed system eqs. (4.71) soved by ART and SIRT for reguar biinear grids of dimension n. The behaviour of the iteration numbers remains argey obscure in the aforementioned studies. According to sec we expect convergence ike x (k) x, where x is the (rescaed) east-squares minimum-norm soution, for error free data. The rate of convergence for the state vector components in the system of the singuar vectors is given by eq. (4.76) x (k) j x ( ) j x () j x ( ) j = (1 ω σ 2 j) k, j rank[ā]. Rescaed singuar vaues σ i for various grid dimensions are shown in fig. 8.6, whie the square of the right hand side was estimated in fig. 5.4a on page 91. Taking x () =, iteration numbers around 1 3 are necessary for singuar vaues of the order.1 to achieve reative deviations of the iterative soution from the idea x of about 1 2. For narrow peaks the discrepancy between a priori and idea soution wi in genera be smaer than for broader peaks, so that smaer iteration numbers suffice. Increasing the grid dimension wi both ower the arge singuar vaues and enhance the smaer ones, where the former eads to higher, the atter to smaer iteration numbers. It turns out, that for the peak distributions considered here the iteration number increases with n. This is shown in tab. 8.5 which aso confirms the predictions made before. The size of the iteration numbers for ART can immediatey be estimated from the construction of the iteration cyces of ART and SIRT eqs. (4.67,4.68): One projection step for SIRT corresponds to m (here 36) projections for ART so that one expects the iteration numbers of ART to be by a factor of the order 1/m smaer than those of SIRT.

133 8.3. Agorithms: ART versus SIRT 133 n E 1 E 2 E 3 E 4 ART SIRT ART SIRT ART SIRT ART SIRT a 4 (4) 1-15 a 4 (4) 15 5 (5) 3 15 (5) a 5 (5) 15-2 a 5 (7) 2 5 (8) 6 2 (6) a 6 (6) 3 7 (9) 4 1 (2 ) 6 3 (3) (9) 35 1 (15) (3 ) 4 4 (1 ) (15) (2) 8 2 (- -) (1 8) (2 5) 12 3 (- -) 5 5 (5) (22) 8 2 (- -) (- -) a Semi-convergent case Tabe 8.5: Approximate iteration numbers for which NEARN reaches convergence. The vaues are ensembe means for one Gaussian peak, in the case of SIRT numbers in brackets ( ) refer to four peaks. Underined figures indicate optima grids in terms of nearness. The case of convergence does not aways appy to the error free case. Semi-convergence as discussed in sec can occur even without substantia noise (a fact that has not been acknowedged in the studies mentioned in the introduction). For the peak distributions considered so far, it occurs ony for ART and over-determined systems as one woud expect but with additiona background or smooth distributions extending over the whoe reconstruction area, semi-convergence is dominant for sma dimensiona grids. With increasing grid dimension the iteration behaves convergent again. As far as cacuation time is concerned, ART is preferabe to SIRT. But, as reaised aso by Todd and Ramachandran [1994a] and Laeppe et a. [24], ART produces higher overa errors in terms of nearness, visibe in more noisy concentration maps with a tendency to artefacts. Here, I just want to point out two further aspects foowing from evauating the set of quaity indices used before for ART and SIRT, respectivey. Fig. 8.7 is anaogous to fig. 8.5, here for one peak ony, but the foowing concusions remain vaid for more peaks, too. 1. Combining compementary quaity measures adds information. For exampe, from the fact that ART shows a significanty higher mean bias than SIRT (fig. 8.7b) whie maxima and peak integras are simiar (figs. 8.7c,8.7d), it can be deduced that concentration is reconstructed outside the region of the peak, where originay there has not been any, i.e. aardvarks. This reiaby confirms what has been guessed from inspecting a imited set of reconstructions. 2. Again, the systematic approach reveas that the performance for a specific setting of the reconstruction in this case the agorithm depends on the distributions. For smooth distributions (E 4 ) ART and SIRT become very simiar and, indeed, there are studies that report better resuts for ART compared to SIRT. More important for the appications of this work is the observation that the improvement by the optima parametrisation (type of basis function, grid dimension) outweighs by far what can be gained by choosing one row acting method instead of another. This aso hods for the MLEM agorithm as the quantitative discussion on page 125 has shown Sensitivity to noise As found in the preceding section, there is no cear cut distinction between the error free convergent case and semi-convergence in the presence of noise. Foowing from the same discussion one expects

134 D Simuation Resuts for Gaussian Peaks NEARN E I SIRT ART E 1 E 2 E 3 E 4 NMB E I n (a) Nearness n (b) Normaised mean bias PPA E I PIPA E I n (c) Error of the maximum peak prediction accuracy n (d) Error of the peak integra prediction accuracy Figure 8.7: Ensembe averages of different quaity criteria for one random peak depending on the grid dimension for ART and SIRT. A settings as in fig Standard deviations are simiar for both agorithms and have been omitted for carity. the under-determined case to be ess susceptibe to noise than the over-determined one which is usuay deat with in the iterature. Here, I concentrate on the strongy under-determined case reevant for the measurements presented in the next chapter. Fig. 8.8 iustrates the mean impact of unbiased random noise on ART and SIRT for a sharp concentration peak with 2σ / =.5 ocated randomy within the reconstruction area. Ceary, random noise affects ART more than SIRT, which is somewhat pausibe because for the simutaneous iterative update corrections from each ight path are added up, thus canceing stochastic errors to some degree. Whie this observation remains generay true for other kinds of distributions, the very moderate effect of even arge errors is specia for the sharp peak distribution. Semi-convergence midy occurs for errors of about 2% and ony for ART. Nearness vaues increase for a eves of noise but the choice of the iteration number, i.e. the reguarisation parameter, is not critica and its vaues can more or ess be taken ike in the error free case. Fig. 8.9 shows how noise deteriorates the reconstruction of pume properties ike maximum concentrations and emissions for the same peak and SIRT. For random noise up to 1% a reaistic size for measurement noise the impact stays quite moderate. The contribution of noise to the pume integra error can be estimated by means of eqs. (5.13b) as

135 8.3. Agorithms: ART versus SIRT 135 Figure 8.8: Sensitivity of SIRT and ART to random noise versus iteration number for a narrow concentration peak ( 2σ/.5) ocated randomy in the reconstruction area. For integration paths with zero concentration an absoute error with variance of the size of the one with argest coumn density was assumed. NEARN k (ART) 1 SIRT no errors 1% 2% ART no errors 1% 2% 2 3 k (SIRT) foows (for the same iteration number the reguarisation part in eq. (5.13a) is the same as in the error free case). Assuming that the pume extends over n p grid nodes with the same perturbation x for each component, the eft hand side of eq. (5.13b) takes the form E [ x pert 2 2] np m( x) 2, where the transformation eqs. (4.71) was taken into account. The scaing factor L for σ ǫ = σ ǫ / L on the right hand side can according to sec be written as 2 /n φ, with being the characteristic ength of a ight path and n φ the number of basis functions (boxes) contributing to it. Assuming furthermore a constant concentration eve x within the pume of diameter, one can rewrite the absoute error with the hep of the reative data error r ǫ as σ ǫ = r ǫ d = r ǫ x. Putting the above LHS of eq. (5.13b) and the RHS in the form σ 2 ǫ ( 1 (1 ω σ 2 i ) k ) 2 ( 1 (1 ω σ 2 (rǫ σ i x)2 n i ) k ) 2 φ σ i i together eads to the foowing estimate for the error perturbation of the pume concentration x x r nφ ǫ n p m ( i i ) ( 1 (1 ω σ 2 i ) k ) 1/2 2. σ i The sum invoving the rescaed singuar vaues takes for a discretisation grids shown in fig. 8.6 a vaue around The number of grid points covered by a pume with /.2 on an grid is about 6. Combining this with n φ 1 and m = 36 resuts in a reative perturbation around x/x.2 for data noise of 1%, agreeing nicey with the number in fig. 8.9d. Two other aspects are worth mentioning. The first concerns the disproportionate sensitivity of the tota area integrated concentration (NMB) in fig. 8.9b. This is because the simuation puts an error on a zero coumn densities, too. Due to the nonnegativity constraint, this can ony ead to rising concentration vaues outside the pume whie the data residuas increase. Secondy, quaity measures and pume properties incorporated by them depend quite differenty on the reguarisation parameter. Whie the overa error is hardy affected by the iteration number, the other properties show ambivaent behaviour as indicated for the reative error of 2%. Stronger reguarisation, i.e. smaer iteration numbers, wi smooth the picture (see the discussion in sec ). This means smearing the peak and reducing the peak maximum, thus giving rise to higher errors of PIPA and PPA.

136 D Simuation Resuts for Gaussian Peaks 1.8 k = 2 k = 1 std E std ǫ 1.8 NEARN E.6.4 NMB E reative error reative error (a) Nearness (b) Normaised mean bias PPA E PIPA E reative error (c) Error of the maximum peak prediction accuracy reative error (d) Error of the peak integra prediction accuracy Figure 8.9: Impact of random noise on the overa reconstruction error, on the reconstrction mean concentration, maximum concentrations and tota emissions for the sharp concentration peak from fig The standard deviation std E refers to the variance within the reconstruction area, std ǫ to the ensembe averaged variance from the random error Light path geometry The discussion in the foowing is for biinear parametrisation and a singe reconstruction grid. Grid combination schemes are disregarded because a) for the arger cass of peaks considered shorty they woud not ead to significanty better resuts and b) for computationa ease. Appication of a grid combination scheme shoud not change the essentia points Singuar vaue decomposition of the geometry Scaar quantities ike the degrees of freedom (eq. (4.61)) and the information content (eq. (4.64)) of the measurement or the perturbation norm of the state vector (eqs. (5.12)) are governed by the singuar vaues σ i ony. But it can be instructive to ook at the singuar vectors u i (spanning the data space of the d) and v i (spanning the state space of the x), reating the eigenmodes to the ight paths and the reconstruction area, respectivey, too especiay, for very sma systems, see tabe 8.6. The singuar vectors beonging to the argest and smaest singuar vaues for the geometry used in

137 8.4. Light path geometry 137 grid v 5 x = A A (1, 1, 1, 1, 1) T Tabe 8.6: Under-determined probem with four ight (in red) paths and different parametrisation by five box basis functions (upper row). Whie the parametrisations are equay good in that they a ead to fu rank systems with simiar singuar vaues, the in this case 1-D nuspaces ook quite different (midde row) and eads to quite distinct retrievas (ast row, for the generaised inverse and a homogeneous fied). Mind that the nuspace does not contribute to box 1 for the 2 nd grid, to boxes 1 and 4 for the 3 rd and to box 3 for the 4 th grid, i.e. the retrieva in these boxes is exact. the previous sections and a reguar biinear grid of dimension n = 6 6 are shown in tabe 8.7. They show the osciating behaviour of the sma singuar vectors described as a genera feature in sec. 4.5(v) and iustrate how reguarisation smoothes the retrieva by cutting off sma singuar vaues and their vectors. The tabe aso contains an exampe of how the map from the singuar vectors u i to the ight paths can be used to identify certain modes, e.g. with sma singuar vaues. In this particuar exampe the reguar grid with n = 6 6 shows an exceptionay high condition number (see fig. 8.4 in which it corresponds to the subgrid with number ). The sma singuar vaue of the order 1 4 that is responsibe for it (see the figure for u 36 ) can ceary be attributed to some ight paths whose arrangement for the given grid eads to an accidenta inear dependency in the system matrix A. Different choice of the boxes containing these ight paths can improve the situation. Finay, tabe 8.8 indicates how the singuar vaue decomposition can hep with the construction of the grid for a given arrangement of ight paths, especiay if these cover the reconstruction area irreguary. Foowing the discussion in sec , the choice between different grids for two geometries shown can be made by opting for the one which gives more baanced singuar vaues. In both cases this choice give rise to a nuspace, more precisey the sum j >rank[a] V jj 2 according to sec. 7.2(i), that is more eveny distributed within the reconstruction area. This is indeed desireabe if the a priori is noncommitta. More quantitative arguments need specification of the a priori and the noise eve and wi be presented

138 D Simuation Resuts for Gaussian Peaks u 1 u 2 u 3 u 4 v 1 v 2 v 3 v 4 u 35 v 35 u 36 v 36 Tabe 8.7: Singuar vectors u i and v i spanning the space of coumn densities and state vectors, respectivey, for the argest four and smaest two singuar vaues of a reguar biinear grid of dimension n = 6 6 (see aso fig. 8.6). The components of u i are mapped to the ight paths whie those of v i are mapped to the grid nodes. Vaues in between the nodes are ineary interpoated. in the subsequent section.

139 8.4. Light path geometry 139 reguar geometry irreguar geometry 12 1 reguar grid irreguar grid Pj >rank[a] V 2 jj 12 1 reguar grid irreguar grid σi [a.u.] σi [a.u.] i singuar vaues i Tabe 8.8: Nuspaces and singuar vaues for two geometries and biinear reguar or irreguar grids, respectivey, of dimension n = 7 7. The vector P j >rank[a] V 2 jj is defined ony on the grid nodes and vaues in between are merey interpoated. The singuar vaues differ ony sighty and the better baanced sets of singuar vaues correspond to better baanced nuspaces.

140 D Simuation Resuts for Gaussian Peaks Comparison of different geometries (a) 2T9 (b) 3T9 (c) 4T9 Figure 8.1: Geometries with two (a,d), three (b,e) and four (c,f) teescopes and ight emitted in 9 (upper row)- and 18 (ower row)- fan beams. As before, the square area has a ength of = 1 a.u. In a cases m = 36. The geometry abeed 3T9 is the one considered so far. (d) 2T18 (e) 3T18 (f) 4T18 From a simpified theoretica point of view the question for the optima arrangement of a given number of integration paths has now been addressed on severa occasions, especiay in sec 7.2, and the answer basicay is: Diagonaise the matrix A as much as possibe, i.e. make the integration paths as inear independent as possibe. Light paths get more independent if emitted (or received) by a arger number of instruments or in wider anges. In practice, both strategies wi increase the mesh size according to eq. (7.1) and make the geometry more irreguar. For peak distributions the resoution in terms of the mesh size becomes crucia and it is by no means obvious that inear independency outweighs a arger mesh size. Then again the answer to this question might depend on the eve of noise, the impact of which is given argey by the singuar vaues, i.e. by the inear independency. The probem was investigated by means of the geometries depicted in fig. 8.1, a generated by m = 36 ight paths emitted (or received) in either 9 - or 18 -fan beams by between 2 and 4 teescopes. The ight paths were chosen to cover the reconstruction area more or ess reguary without caiming to present the idea soution (for exampe, with respect to the number of retro-refectors that woud be needed to reaise the ight paths in an actua LP-DOAS measurement). (i) First, the same statistica approach used earier for the evauation of different parametrisations and agorithms was chosen, this time for different geometries and random peaks taken from the union of the ensembes E 1 to E 3. These test distributions were chosen such that the mesh size shoud pay a roe and the same a priori (zero) coud be used for a peaks. Resuts for four random peaks reconstructed on a singe reguar biinear grid of dimension n = 7 7 are shown in fig Differences between the geometries are substantia (the reative standard deviation ies between 15% for the overa error NEARN and 5% for the error of the mean concentration), with a persistent trend showing that 1. increasing the number of emitting (or receiving) instruments generay enhances the reconstruction quaity, provided that the coverage with ight paths and their reguarity can be maintained. 2. For the imited number of ight paths considered here, arge 18 -fans ead to worse resuts.

141 8.4. Light path geometry 141 NEARN E1 E 2 E number of teescopes (a) Nearness NMB E1 E 2 E number of teescopes (b) Normaised mean bias 1 PPA E1 E 2 E number of teescopes (c) Error of the maximum peak prediction accuracy 1 PIPA E1 E 2 E number of teescopes (d) Error of the peak integra prediction accuracy Figure 8.11: Ensembe mean vaues of the overa error (a) and the errors of mean concentration (b), maximum concentrations (c) and tota emissions (d) for the geometries in fig. 8.1 and four peaks from ensembes E 1 to E 3. The reconstruction used a singe biinear grid with n = The performance of the configuration with two teescopes and 9 -fans (2T9 ) is exceptionay poor. A in a the curves agree remarkaby we with what one woud expect from just ooking at the geometries except maybe for the geometry caed 2T9. The ight path arrangement 3T9 used in the preceding sections represents a good average of the geometries examined here. To get an idea how the reconstruction quaity varies within the reconstruction area for the individua geometries the same quaity criteria as before were evauated for one peak on a fine grid of peak positions (x,y ) and mapped to the reconstruction area. The resuts are shown in the first three coumns of tabe 8.9 on page 142. Apart from the pattern of smaer nearness vaues if peak positions coincide with grid nodes, there are spots of arger reconstruction errors where the coverage with ight paths is coarse. The maps of the mean concentration and tota emissions show this pattern, too. Both get underestimated in gaps between ight paths and overestimated around the teescopes where mosty simiar ight paths from the same fan contribute. Taking the pots of mean concentration NMB and tota emissions PIPA together gives information about where the reconstructed trace gas is ocated: If the error of the mean concentration is distincty arger than for the concentration within the peak, the concentration is spread over a arger region (as is the case near the teescopes) or artefacts appear somewhere ese (which are here, for one peak, ess pronounced). The impact of random noise on the geometries was investigated for the very same peak distribution and a noise eve of 1%. Simiary to the findings in sec , the impact on the reconstruction quaity for this specia peak distributions is quite moderate. Furthermore, it turns out that the average aggravation is comparabe for a geometries. This is shown for the overa error NEARN in fig. 8.12, other quaity measures behave simiary. For individua geometries, the sensitivity to noise does vary though within the reconstruction area, in particuar for the specia cases 2T9, 2T18 and 4T18, see tab Whie the geometries 2T9 and 2T18 (not shown) are ess sensitive to noise in parts of the reconstruction area with poor coverage, the arrangement 4T18 shows higher sensitivity for these regions. An expanation for this opposed behaviour is that for the former two geometries the corresponding ight paths are highy inear dependent (amost parae at the top of

142 D Simuation Resuts for Gaussian Peaks 4T 18 4T 9 3T 18 3T 9 2T 18 2T 9 y x N EARN (x, y ) j N M B(x, y ) P IP A(x, y ) P j >rank[a] 2 Vjj Tabe 8.9: First to third coumn: Overa error (N EARN ), reconstructed mean concentration (N M B) and tota emissions (P IP A) as function of peak position for the geometries in fig The peak from ensembe E2 with σx / = σy / = 7 was reconstructed for 3 3 positions using a singe biinear grid with n = 9 9. Fourth coumn: Components j of the sum Vjj2 over the nuspace of A mapped to the nodes of the n = 9 9 grid. Vaues in between grid nodes are merey interpoated

143 8.4. Light path geometry NEARN 9 18 no error re. error number of teescopes Figure 8.12: Mean nearness for reconstruction of the peak as in tabe 8.9 without and with 1% noise. 2T9 4T18 Tabe 8.1: Noise sensitivity of the nearness, defined as (NEARN ǫ NEARN)NEARN 1 as a function of the peak position (x, y ) for reconstruction of the peak as in tabe 8.9 and seected geometries. 2T9 and in the centre of 2T18 ) whereas in the atter case of 4T18 they are totay independent. In the first case the parts of the reconstruction area with poor coverage of ight paths are reated with sma singuar vaues and are thus particuary affected by reguarisation, in the other case they are not. This is exacty what can be observed. (ii) The resuts for the comparison of various ight path arrangements have been obtained by expicit simuated reconstruction. Especiay the investigation of the reconstruction quaity within the area was restricted to a specific distribution. We are now going to show how the mosty agebraic arguments presented in sec. 7.2 can be used to arrive at simiar concusions, at east quaitativey, and start with an inspection of the nuspace according to eq. (7.2) ike before in tabe 8.8. Resuts, given in the fourth coumn of tabe 8.9 for a biinear grid of dimension n = 9 9, show indeed a correation between arge vaues of j >rank[a] V jj 2, i.e. arge nuspace, and high overa reconstruction error (mind again that the sum equas 1 R jj, where R is the resoution matrix of the generaised inverse). The correation is not very distinct and in the case of the geometry 3T9 hardy visibe. But one has to bear in mind that the first three coumns of the tabe refer to reconstruction which is a) noninear due to the nonnegativity constraint and b) for a specific distribution. The significance of both is nicey y x j (a) activity(x, y ) (b) average activity(j) (c) x x 2 / x 2 (x, y ) (d) P j >rank[a] V 2 jj Tabe 8.11: Activity of the nonnegativity constraint for reconstruction of the peak distribution as in tabe 8.9 and the geometry 3T9. (a) As function of the peak position (x, y ). A vaue of, e.g.,.9 at (x, y ) means that the constraint is active for 9% of the projections when the peak is at (x, y ). (b) Activity on node j averaged over a reconstructions. (c) Difference norm of state vectors x, x for reconstruction of the peak at (x, y ) with and without constraint. (d) Nuspace as in tab. 8.9.

144 D Simuation Resuts for Gaussian Peaks rank[a] d s σ a/σ ǫ = number of teescopes (a) Uniform a priori variance n p d s centre upper eft σ a/σ ǫ = number of teescopes (b) Nonuniform a priori variance with vaues σ a on n p = 3 3 nodes, zero ese H σ a/σ ǫ = number of teescopes (c) Uniform a priori variance Figure 8.13: Degrees of freedom d s and information H for the geometries of fig. 8.1 and different eves of noise. The biinear reconstruction grid has dimension n = 7 7. A covariance matrices are diagona. The area covered by the n p grid nodes in exampe (b) corresponds to about and is ocated either in the centre or the upper eft corner of the reconstruction area. iustrated by tabe 8.11 showing the activity of the nonnegativity constraint during the reconstruction of the peak in tabe 8.9 for the geometry 3T9. It is evident from the figure that the constraint is active amost aways and on a grid nodes with a tendency to fi up the nuspace. That is, the arger the nuspace the more active the constraint becomes, thus expaining at east party the difference in the patterns of overa reconstruction error and the nuspace. The strong noninearity of the inversion process foowing from tabe 8.11 is the reason why quantitative statements shoud rey on expicit numerica cacuations. Nevertheess, for a quaitative comparison of the geometries, we proceed with the inear approach and empoy the Bayesian method as reasoned in sec. 7.2(ii) to see how individua geometries perform for a given eve of noise and a supposed uncertainty of the true state. Using the degrees of freedom of the signa d s, defined by eq. (4.61a) in sec , as measures of performance means ooking at the diagona eements of the resoution matrix (or averaging kerne) R. We consider first a uniform a priori covariance S a = σa½n 2 corresponding to the case where the ocation of the peak within the reconstruction area is unknown and everywhere equay probaby and uncorreated measurement errors of the same size σ ǫ. In this case ony the ratio σ a /σ ǫ enters. The imit σ a /σ ǫ corresponds to the generaised inverse, for which d s = trace[r] = rank[a]. Fig. 8.13a shows how d s differ from the maximay possibe 36 for different eves of noise or better, for different degrees of ignorance about the true state vector. The parametrisation is the same as in fig. 8.11, i.e. a singe biinear grid of dimension n = 7 7. Writing the variances in terms of reative errors r ike σ a r a c p, σ ǫ r ǫ c p, with a representative peak concentration c p and peak diameter, gives the estimate σ a /σ ǫ r a /r ǫ 1. For the peaks considered in this section the ratio is thus of the order (O(.1)...O(1)) r a /r ǫ 1 (see tabe 8.1). The curves show substantia variation of d s for measurement with different geometries and a ranking very simiar to the one found from simuated reconstructions in fig. 8.11a. Subareas of the reconstruction area can be compared by choosing the a priori variance correspondingy. To examine the degrees of freedom of a pume measurement, e.g. in the centre, one woud assume a arge a priori variance within this subarea and a sma one outside. Fig. 8.13b iustrates this for a pume

145 8.4. Light path geometry 145 (a) d s (b) P j >rank[a] V 2 jj Tabe 8.12: (a) Degrees of freedom d s as function of the position of the point measurement for geometry 3T9 and the specia case that it ies on one of the nodes of the n = 9 9 biinear recontruction grid. A variances are diagona and σ a/σ ǫ = 1 for a measurements. (b) Nuspace as in tab Vaues between the grid nodes are merey interpoations. that extends over n p = 9 grid nodes either in the centre or the upper eft corner of the reconstruction area. The a priori covariance is again diagona with equa entries σa 2 for the nodes of interest and zero ese. Differences of up to one degree of freedom (out of nine) for the given eve of noise occur both with geometry and ocation of the pume. The figure suggests distincty better reconstruction resuts for the centre areas of the 9 -fan geometries, in agreement with fig Finay, fig. 8.13c is the counterpart of fig. 8.13a for the information content as defined by eq. (4.64). Both agree quaitativey (mind that the rea meaning of the information content ies in the number of states distinguishabe by the measurement and given by 2 H ), the same hods for the exampe with nonuniform a priori covariance (not shown). The important point foowing from comparing the geometries by means of (i) the overa reconstruction error on the one hand and (ii) the resoution matrix (averaging kerne) on the other is not that they differ in detais, but they agree so we. After a, evauation of the former in continuous state space takes fuy into account the shape of the distribution and its discretisation, whie the atter takes pace in discrete space, irrespective of the kind of distribution. This suggests that, in the end, it is the property of the inear independence that matters most, at east for the average of peaks taken into consideration here Point measurements The concusion in favour of the information concept at the very end of the preceding section justifies to use the degrees of freedom (or the information content) to anayse the optima pace of one or more additiona point measurements in a given geometry of remote sensing paths by just comparing their vaues for different positions. It was argued in sec. 7.3 that the measurement ocation shoud be such that the associated equation(s) ie in the nuspace of A. Considering one measurement and treating it as additiona constraint (case (ii) in sec. 7.3) gives vaues of d s as shown in fig for reconstruction on a n = 9 9 grid for the geometry abeed 3T9. A covariance matrices were again assumed to be diagona with equa entries and reative errors of the point and remote sensing measurement were taken to be of the same size. The measurement points were put on the nodes of the reconstruction

146 D Simuation Resuts for Gaussian Peaks grid for simpicity, but this shoud not restrict the concusion that emerges when comparing fig to the nuspace of 3T9 in tab The Bayesian approach indeed predicts a maximum increase of d s, if the point measurement takes pace where the nuspace associated with the remote sensing measurements is argest. According to the ast section, for noncommitta a priori this shoud hod equay for the reconstruction quaity Scaing with ight path number (a) 3T9 with m = 9 integration paths (b) 3T9 with m = 18 integration paths (c) 3T9 with m = 36 integration paths Figure 8.14: The same generic type of geometry with ight path numbers scaed by a factor f = 2. Numerica resuts from the beginning of this chapter referred to a specific geometry (3T9 ) with a fixed number of integration paths. The probem of transferring these numbers to arbitrary ight path configurations can be stated ike this: If, for a certain geometry g with m ight paths, simuations invoving test distributions c yied a vaue q for a quaity Q, ike the peak maximum precision, what wi the vaue q be for another geometry g consisting of m paths? I.e., Q(g(m),c) = q Q(g (m ),c) = q =? Such a concusion is easy, if the quaity Q obeys simpe scaing aws, for exampe with the meshsize. But sec , which deat with Q(g(m),c) for different geometries and fixed m, has shown that at east for the meshsize this is not the case. It was found that the generic type of the geometry, i.e. the inear independency of the ight paths, is more important than the meshsize for the same ight path number. Possibe scaing with quantities reated to the singuar vaue decomposition of the geometry was not investigated. Instead, I show how quaity measures scae with the ight path number m for the same type of geometry. The assumption goes ike this: Scaing m with a factor f, so that m = f m, gives for a more or ess reguar geometry a centre meshsize = f 1 (see eq. (7.1)). If the spatia dimensions of the concentration distributions are scaed by the same factor f 1, the quantity Q shoud stay invariant. For Gaussian distributions with variances σ this takes the form: Q ( g(f m),c(f 1 σ) ) = Q(g(m),c(σ)). (8.3) It was assumed here that Q has an appropriate normaisation, which is the case for a reative quaity measures in tabe 5.1 on page 97, except for the normaised mean square error NMSE. Taking both test distribution c and retrieva ĉ to be Gaussians, it is easy to see that the transformation σ f σ impies NMSE f 1 NMSE.

147 8.5. Background concentrations 147 2σ / m n/m NEARN NMSE IOA 1 NMB r FA2 1 PIPA One peak (E 1 ) 36 > (E 2 ) (E 3 ) (E 4 ) Four peaks (E 1 E 2 E 3 ) Tabe 8.13: Invariance of quaity indices Q, if ight path number and peak distribution are scaed according to eq. (8.3). The E 1, etc. refer to the ensembes from tabe 8.1 for m = 36. A vaues are ensembe averages for one or four random peaks and reconstruction on a singe, reguar biinear grid of dimension n. Some indices were taken into account ater for m = 9, 18, but simuations were not rerun for m = 36. These vaues are missing. To test the above scaing reation, we take again the geometry with three teescopes and 9 -fans 3T9 and a scaing factor f = 2 as shown in fig The test distributions arise from the origina ensembes E 1 to E 4 by scaing the variances with f, so that the ratio 2σ / is the same within each ensembe for the geometries with 9, 18 and 36 ight paths, respectivey. In other words, σ/ ranges in E 1 from.3 to.5 for m = 36 (see tabe 8.1), from.6 to.1 for m = 18 and so forth. Simuation resuts for reconstruction of one and four peaks on a singe biinear grid are given in tabe For comparison with typica vaues of quaity indices achieved by dispersion modes that have been presented in tabe 7.1, it contains some indices beyond the ones used so far. Apart from confirming the expected scaing behaviour for a measures except N M SE, the tabe exhibits arge variations in the sensitivity of different indices (for exampe, the index of agreement IOA seems to be rather insensitive). Whether the quaity of the reconstructed peak distribution can keep up with dispersion modes depends on the extension of the peaks reative to the meshsize (and the quaity index), as the comparison with tabe 7.1 shows. The invariance eq. (8.3) was found for other geometries, too, so that, in concusion, the recipe for appying simuation resuts for peak distributions from one geometry to a simiar geometry with a different number of ight paths simpy states: Suitaby normaised quaity indices for peak distributions with the same ratio 2σ / have equa vaues Background concentrations The simuations in this chapter have assumed that the concentration vaues go down to zero (or arbitrary sma vaues) far away from the peaks, i.e. that there is no background concentration in the sense of sec This precondition is of itte significance for broad peaks, as ong as the background is not too arge. But for narrow peaks, the a priori x a =, supported by the nonnegativity constraint, has been seen to be essentia. Instead of incorporating the background concentration into the a priori

148 D Simuation Resuts for Gaussian Peaks NEARN E I mean a priori subtract. b.g. E 1 E 2 E 3 E n (a) Nearness NMB E I n (b) Normaised mean bias PPA E I n (c) Error of the maximum peak prediction accuracy 1 PIPA E I n (d) Error of the peak integra prediction accuracy Figure 8.15: Same as fig. 8.1 for four random peaks and additiona random background concentration (see text), reconstructed either using a mean priori or by subtracting the background. x a, it was therefore suggested in sec (ii) to use it as a nonnegativity constraint after subtracting the associated path integras from the measurement data. Both methods are compared again for random peaks from each ensembe and an additiona nonuniform background distribution (see appendix B) with vaues varying randomy in the reconstruction area between 5% and 25% of the maximum peak vaue C max and maximum gradients of about C max /. 2 Fig shows the resuts for the specia case where the a priori is given by the mean concentration of a ight paths eq. (6.7), whie the vaue being subtracted is the minimum mean concentration eq. (6.14). Because the iteration number party serves as reguarisation parameter, it is given for competeness in tabe Except for the broadest peaks from ensembe E 4, the reconstruction errors obtained by subtracting the background are in amost a cases smaer than those using the mean a priori. Figs. 8.15a, 8.15b and 8.15d cannot be directy compared with the corresponding subfigures in fig. 8.1 for the case without background because, here, norm and integras contain aso the background. Detais on how much the reconstruction of peak properties is deteriorated by the presence of a background concentration fied wi depend on its variabiity and thus the quaity of the a priori (e.g., eqs. (6.1)). This has not been investigated systematicay, but, at east for the situation 2 For narrow peaks the contributions to the tota coumn density from the peak and the background are thus of simiar size. For broad peaks the contribution from the peak can be maximay 15 times that of the background.

149 8.6. Discussion 149 n E 1 E 2 E 3 E 4 subtract. b.g. mean a priori subtr. mean subtr. mean subtr. mean Tabe 8.14: Iteration numbers for fig The indicates optima vaues of NEARN for semi-convergence. Vaues without mean that NEARN has approximatey reached convergence for this iteration number. considered here, the peak maximum (fig. 8.15c) can be reconstructed with simiar accuracy as in the case without background (fig. 8.1c). With the background distribution modeed here being rather unspecific, it seems thus fair to say that 1. Subtracting a moderatey variabe background fied as formuated in eqs. (6.13), (6.14) rather than using a mean a priori in the east-squares minimum-norm scheme eads to significanty better resuts for narrow peaks (from ensembes E 1 to E 3 ), provided the soution is reguarised appropriatey. 2. For the differentia system that remains after subtraction of the background, the findings of the preceding sections appy, at east quaitativey. Detais depend on the actua background variabiity Discussion Some aspects concerning the comparison with existing studies are discussed here in brief, whie a further, concuding discussion is hed in chap. 11. Unike for routine appications of computerised tomography (CT), there is not a standard probem and no common way of evauating reconstruction methods in atmospheric science. For appications of CT in indoor gas dispersion, which come cosest to our kind of reconstruction probem, there are no pubications by different authors among a referred to in this thesis that use the same evauation method (Despite the fact that there is a arge set of standard metrics used in atmospheric modeing, see sec. 5.4). Furthermore, none of these indoor studies examines the parametrisation systematicay, neither in terms of the number of parameters, nor in terms of basis functions (Athough most of them dea with Gaussian peaks as we, where these issues matter). The possibiity of highy under-determined parametrisation is specia to narrow peaks on a negigibe or, as I have suggested, sufficienty smooth or known background. The fact that for this specia kind of distribution the representation by biinear basis functions not ony eads to smaer discretisation errors, but aso inversion errors, does not seem to be a reguarisation effect because, in genera, the conditioning of A for biinear functions is not better than for piecewise constant ones. On the other hand, for over-determined discrete inverse probems, e.g. in atmospheric profiing, the impact of the basis functions on the quaity of the retrieva has

150 D Simuation Resuts for Gaussian Peaks been attributed to different reguarisation behaviour (e.g., [Doicu et a., 24] and references therein). For either case, the choice of parametrisation of the discrete inverse tomographic probem certainy requires further attention. Most of the tomographic studies deaing with probems simiar to ours are concerned with agorithms, i.e. either with inversion agorithms for the same reconstruction principe (ike ART and SIRT) or with different principes atogether (ike the east-squares and the maximum entropy principes). In many cases, the comparison with other agorithms remains insufficient or unsystematic with respect to the kind of distribution (However, it was argued in sec that the kind of distribution does pay a roe, expaining some seeming contradictions in the iterature). For exampe, Drescher et a. [1996] finds better resuts for her method SBFM (see sec ) compared to ART on the grounds of ess noisy concentration maps, but does not compare it to SIRT which is known to give ess noisy resuts. Price et a. [21] introduce a method, basicay the Tikhonov approach with the third differences used for reguarisation, but compare it ony to SBFM using the data residua. The first method being over-determined and their discrete parametrisation being under-determined, there is hardy a point in comparing the data residua as argued in sec Furthermore, the authors do not show any sampe reconstruction with origina distribution. Doing this for Gaussian peaks suggests that the method is not ony smoothing, as intended by the authors, but oversmoothing in terms of sec Samanta and Todd [1999] have found that the MLEM principe (sec ) gives better resuts for the reconstruction of Gaussian peaks than the east-squares east-norm principe. This agorithm combined with box basis functions and a grid transation scheme was quantitativey compared to our approach on p. 125, in favour our method. This does obviousy not mean that the pure east-squares east-norm reconstruction principe is the utimate choice. In fact, for over-determined probems, experience from 1-D profie reconstruction suggests that it is not. But according to our findings, any (new) principe or inversion agorithm shoud be compared to others systematicay with respect to the kind of distribution and parametrisation. The same hods for the reguarisation approach. Light path geometries have been investigated by Todd and Bhattacharyya [1997]; Todd and Ramachandran [1994b] for indoor gas reconstruction, but apparenty inspired by X-ray tomographic setups they use an excessive number of 12 4 ight paths, combining fan beams and projection paths by additiona mirrors. By and arge, carrying out numerica simuations simiar to sec (i), the authors come to the same concusions, except for our geometries that are especiay irreguar. 3 Cehin, M., Computed tomography for gas sensing indoors using a modified ow third derivative method. Submitted to Atmospheric Environment.

151 9. 2-D Reconstruction of NO 2 Peaks from an Indoor Test Experiment The indoor experiment referred to in this chapter was carried out to study the performance of the mutibeam instrument, especiay deveoped for tomographic measurements [Mettendorf, 25; Pundt and Mettendorf, 25]. The experimenta conditions are highy artificia (see [Mettendorf et a., 26; Mettendorf, 25] for detais of the experiment and its compications), therefore it is not used here to discuss atmospheric measurements of trace gas pumes, but rather to iustrate the reconstruction procedure for an experimenta situation without further knowedge about the true concentration distribution. Ony one of the distributions measured and reconstructed is considered in sec. 9.2, as variations basicay are contained in the simuations of the previous chapter (see [Mettendorf et a., 26; Mettendorf, 25] for a discussion of a resuts). In particuar, the reconstruction error is estimated and re-estimated in sec. 9.3 in a consistent way from the reconstructed distributions and the simuations of chap. 8. Finay, sec. 9.4 examines the detectabiity of trace gas pumes in genera and for the argest point sources near Heideberg for iustrative purposes and as a case study for tomographic measurements currenty taking pace in Heideberg The experiment M 1 T 2 M 3 Figure 9.1: Light path geometry of the indoor experiment with 39 rays in tota. Beams, simutaneousy emitted by the teescopes (T), are redirected by mirrors (M) towards individua retro-refectors. The test distributions consist of poycarbonate cyinders with radius 1 m and NO 2 concentrations c 1 = (1.95 ±.15) 1 14 moec/cm 3, c 2 = (3.35 ±.1) 1 14 moec/cm 3. (This arrangement of the cyinders was abeed distribution no. 16 in [Mettendorf et a., 26; Mettendorf, 25].) y c 2 c 1 T 3 x T 1 M 2 15m 1m The experiment was carried out on a factory foor on an area of 15 1m 2 with three of the mutibeam instruments. Emission puffs were simuated by pacing one or two poycarbonate cyinders, each fied with NO 2 and a radius of 1m, at various positions in the area (fig. 9.1). The design of the ight path geometry was mainy dictated by instrumenta factors. Light beams eave the mutibeam teescope basicay in the same direction and have to be redirected by externa mirrors (one per beam) to the fina destination. To appear separatey on the mirrors the rays have to pass a 151

152 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment 2 experiment simuation sim.+ocation err. di [1 17 moec/cm 2 ] ight path i Figure 9.2: NO 2 coumn densities obtained from the experiment [Mettendorf et a., 26; Mettendorf, 25] and from integration over the known test distribution of fig The optica and integration path are twice the way from the teescope to the refector. The ocation error in the simuation takes into account an uncertainty of the cyinder position of about 5 cm. distance of severa metres. A further restriction is given by the fact that each mirror can be turned horizontay by maximay ±3, resuting in fans of maximay 12. These requirements ed to the arrangements of teescopes and mirrors as shown in fig. 9.1, i.e. three 9 -fans sitting in the corners of the test area. The number of ight paths that can be reaised at the same time is given by the number of rays emitted by each teescope (four in this case). To increase the number of ight paths in atmospheric measurements, the externa mirrors are successivey directed to different targets, so that the fina geometry used for reconstruction consists of more integration paths, yet not a referring to the same time (see the discussion in sec ). This scanning was carried out in the indoor experiment in three steps, each with 3 12 ight paths, so that the compete geometry comprises 36 paths. A beams emitted by the same teescope share the distance TM within the reconstruction area, which woud not be the case in rea atmospheric measurements or more precisey, this distance woud be competey negigibe compared to the remaining distance to the refector. Besides, it is unfavourabe for the tomographic reconstruction. Therefore, one optica path from the teescope to a retro-refector at M has been added for each teescope, thus removing the ambiguity introduced by the common path TM. The fina composite geometry with 39 ight paths is essentiay identica to the one abeed 3T9 in chap. 8. The three intermediate geometries with 12 ight paths (see [Mettendorf et a., 26; Mettendorf, 25] for detais) were set up under static conditions, i.e. the concentration distribution did not change. Contrary to the geometry with 12 ight paths in fig. 8.14, they were not intended to serve for reconstruction individuay, so ony the compete geometry of fig. 9.1 wi be considered in the foowing. Due to the short distances and the poycarbonate in the optica path, the DOAS measurement and its anaysis differ consideraby from an atmospheric measurement (see again [Mettendorf et a., 26; Mettendorf, 25] for detais in the foowing). In genera, coumn densities obtained from the DOAS anaysis agree we with expectations, as iustrated by fig. 9.2 for the concentration distribution with two puffs shown in fig In the simuations the finite beam diameter of around 6 3cm is negected and the error of the experimenta coumn densities is given by eq. (3.6), without taking into account any systematic errors, e.g. ike the one coming aong with an incorrect absorption cross section of NO 2. The average measurement error for a test distributions is about moec/cm 2 for beams passing through a cyinder and moec/cm 2 for beams that do not. For nonzero

153 9.2. Sampe reconstruction 153 ĉ(x, y) [1 14 moec/cm 3 ] a) n = 6 6 b) n = 7 7 c) n = 8 8 d) n = 9 9 I 1 = 2.9, I 2 = 5.3 I 1 = 5, I 2 = 3.5 I 1 = 3, I 2 = 6 I 1 = 6.2, I 2 = 7.1 e) n = 1 1 f) n = g) n = h) n = I 1 = 5.3, I 2 = 7 I 1 = 6.2, I 2 = 8.4 I 1 = 5.4, I 2 = 9.9 I 1 = 5.8, I 2 = 9.5 i) n = 11 11, aver. j) n = 11 11, comp. k) n = 13 13, aver. ) n = 13 13, comp. I 1 = 5.6, I 2 = 8.9 I 1 = 5.6, I 2 = 9.8 I 1 = 5.8, I 2 = 9. I 1 = 5.6, I 2 = 9.8 Tabe 9.1: Reconstruction of the coumn densities of fig. 9.2 using SIRT, x () =, k = 2, a singe biinear grid (a-h) and the averaging (i,k) or composite (j,) grid transation scheme. I 1 and I 2 are given by the integras over peaks 1 and 2: I i = (1 18 moec/cm) 1 R Peak i da ĉ(r). coumn densities in fig. 9.1 this amounts to reative errors between 3% and 1% Sampe reconstruction The reconstruction procedure is iustrated for the measurement shown in fig. 9.2, using the simuation resuts of the previous chapter but no information other than what is provided by the measured coumn densities themseves. The fact that most of the coumn densities have zero (or negigibe vaues) suggests the a priori x a = as iteration start x (). Starting the reconstruction with a singe reguar biinear grid of dimension n = 6 6 i.e. the over-determined case eads to the concentration map shown in tab. 9.1a 1. According to tabe 8.5 an iteration number of severa thousand steps shoud be chosen to assure convergence. Assuming the convergent and not the semi-convergent case which needs reguarisation is justified by the foowing. 1 Coour maps are a bit cearer than 2-D contour pots for the narrow peaks here.

154 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment The reconstructed map ceary shows two distinct peaks (and a hardy visibe, weak accumuation in the upper haf) which, on the grounds of the resuts of chap. 8, encourages to increase the grid dimension n. Even up to n = the reconstruction shows the same structure of two peaks and a very fat peak present in a maps except g, raising the question whether it genuiney beongs to the true distribution or whether it is an artefact of the reconstruction. The fact that no further artefacts appear when the grid dimension gets arger is an indicator for the a priori sti working and the peaks indeed being very narrow. But whie the area integras over the individua peaks for higher dimensiona grids give a fairy consistent picture, reconstruction of the peak maximum vaues ranging from 2.5 to moec/cm 3 for the higher peak is not concusive. To get a cearer picture of shape and maximum vaues, we note that in terms of the previous chapter the peaks beong to E 1, so that according to fig. 8.5 the averaging grid transation scheme is ikey to ead to a smaer overa error, whie the composite scheme shoud give a more precise vaue for the peak maxima. The corresponding concentration maps are shown in tab. 9.1i-. Again the peak integras, especiay of the ower peak, agree fairy we, but the maxima differ consideraby. Trusting that the resuts of fig. 8.5 appy here, too, the maxima of the composite scheme ought to be more accurate. Further evidence is provided by an expicit estimation of the reconstruction error as suggested in sec Discussion of the reconstruction error c(x,y) [1 14 moec/cm 3 ] c(x,y) [1 14 moec/cm 3 ] c(x,y) [1 14 moec/cm 3 ] x [m] y [m] x [m] y [m] 5 x [m] y [m] (a) (b) (c) Figure 9.3: Exampes of random fieds c I(x, y, ) E created by convouting Gaussian peaks with totay random fieds as described in the text. The numerica estimation of the reconstruction error is based on the foowing assumptions I. The true distribution consists of two peaks with negigibe background concentration. II. Taking the reconstructed peak maxima of peak 1 and 2, Ĉ, moec/cm 3, Ĉ,2 (3 6) 1 14 moec/cm 3 and the statistica scattering from fig. 8.5c, the true peak maxima are

155 9.3. Discussion of the reconstruction error 155 assumed to ie between moec/cm 3 C, moec/cm 3, moec/cm 3 C, moec/cm 3. (9.1) III. From the peak extensions reconstructed with the averaging and the composite grid transation scheme and from the meshsize of the geometry bounds for minimum and maximum 3σ-extension of the rea peaks, respectivey, are estimated as 1cm 3σ x,y 4cm. IV. The ocation (x,y ) of the peak centre reconstructed with the averaging and composite scheme is to ±5cm correct. For the case of the rea atmosphere, it is reasonabe to generate the ensembe E for the error estimation from Gaussians. Here the random peaks are created by convouting Gaussians with random fieds as described in appendix B such that the resuting fied c I (x,y) E fufis requirements I-IV and has at every point a maximum absoute gradient of c I 1 c Gaussian. The atter bound is not physicay motivated but intended to impose some variabiity on the otherwise smooth Gaussian. Sampes of this procedure are shown in fig Tab. 9.2 shows the mean error fieds ĉ(x,y,) (overestimation) and + ĉ(x,y,) (underestimation) as defined by eq. (5.14b) for reconstruction on a singe 1 1 pixe grid and the averaging and composite scheme, respectivey. The maps ook very simiar, with overstimation smeared around the peaks and underestimation concentrated mainy on the maximum of the higher peak. According to sec. 5.3 the contribution from the measurement error is cacuated by adding random errors with variances given by the meaurement errors to the actuay measured coumn densities and taking the standard deviation std ǫ [ĉ] around the reconstructed mean.. We notice first that the very weak concentration peak in the reconstructions from experimenta data found in the previous section is very ikey an artefact, caused either by the imperfect inversion (tab. 9.2a-c) or by errors of the measurement data (tab. 9.2i-k). Second, cacuation of the area integras of ± ĉ, i.e. the mean errors, shows smaer vaues for the averaging scheme, foowed by the singe grid reconstruction. The difference between averaging and composite scheme amounts to 7%. Reconstruction of the peaks on a singe grid is significanty more susceptibe to measurement noise. Finay, mean errors for reconstruction on a pixe grid are throughout arger than the corresponding ones for the 1 1 pixe grid, without changing the ranking of the methods. Disregarding the resuts of the composite scheme on grounds of arger reconstruction errors fits neaty into the picture obtained from the reconstructed maps in tab. 9.1, where maximum vaues of the singe grid and the averaging scheme generay are in much better agreement compared to the composite method. To get a cearer picture of the reconstructed distribution and its error, fig. 9.4 shows concentration profies aong the x-axis through the centre of the true peaks. For the cross section considered here, the reconstruction errors ± ĉ (ong dashed ine) are argest at the peak maxima, which for the Gaussian error ensembe from above tend to be underestimated, especiay for singe grid reconstruction of

156 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment ĉ(x, y) [1 14 moec/cm 3 ] E (a) singe grid (b) averaging scheme (c) composite scheme I = 7.1 ( 7.5) I = 6.8 ( 7.3) I = 7.3 ( 8.) + E (e) singe grid (f) averaging scheme (g) composite scheme I = 5.6 (5.9) I = 5.4 (5.7) I = 5.8 (6.3) std ǫ[ĉ(x, y)] (i) singe grid (j) averaging scheme (k) composite scheme I = 2.6 (3.1) I = 1.8 (2.3) I = 1.9 (2.) Tabe 9.2: Mean error fieds ĉ, indicating overestimation, (a-c) and +ĉ, indicating underestimation, (df) for reconstruction of the random distributions from the ensembe E described in the text and iustrated in fig. 9.3 (N = 2 sampes) on a 1 1 pixe grid. The impact of the random measurement error (i-k) is cacuated for the experimenta coumn densities using the estimated measurement error. I is given by the integra over the entire reconstruction area A: I = (1 18 moec/cm) R 1 da ĉ(r). A Vaues in brackets ( ) refer to a pixe grid. the peak 2 (fig. 9.4d). Whie the impact of the measurement error (dotted ine) is indeed sma, the standard deviations of ± ĉ are of simiar size as the errors themseves. Taking the standard deviations into account as they suggested in sec. 5.3 gives in the case of the averaging scheme for the peaks with { } 1.9 Ĉ,1/2 = 1 14 moec/cm 3, 3.1 { } 5.6 da ĉ = 1 18 moec/cm, (9.2) 8.9 Peak,1/2

157 9.3. Discussion of the reconstruction error 157 c(x) [1 14 moec/cm 3 ] c ĉ + ± ĉĉ... ± std E [ ± ĉ]... ± std ǫ[ĉ] c(x) [1 14 moec/cm 3 ] c ĉ + ± ĉĉ... ± std E [ ± ĉ]... ± std ǫ[ĉ] c(x) [1 14 moec/cm 3 ] x [cm] 8 (a) peak 1, averaging scheme 9 c ĉ + ± ĉĉ... ± std E [ ± ĉ]... ± std ǫ[ĉ] 1 c(x) [1 14 moec/cm 3 ] x [cm] 11 (b) peak 2, averaging scheme 12 c ĉ + ± ĉĉ... ± std E [ ± ĉ]... ± std ǫ[ĉ] x [cm] (c) peak 1, singe grid x [cm] (d) peak 2, singe grid Figure 9.4: Profies aong the x-axis through the centres of the peaks in tab. 9.1 reconstructed on a 1 1 grid and successive addition of the reconstruction errors ±ĉ, their standard deviations std E[ ±ĉ] and the propagated measurement error std ǫ[ĉ]. Aso shown the true cyinder distribution (the uncertainty of about ±5 cm in the cyinder position is not taken into account). the foowing bounds on the peak maximum vaues and the peak integras (tota emissions): { } { } 1.5 (1.8) 1 14 moec/cm (2.2) C,1/ moec/cm 3, 2.5 (2.9) 5. (4.) { } { } 3.1 (4.3) (6.8) moec/cm da c 1 18 moec/cm, (9.3) 5.3 (7.1) 16.3 (11.9) Peak,1/2 where the numbers ( ) in brackets refer to ĉ ± ± ĉ without standard deviations std E [ ± ĉ]. The wide ranges, particuary for the second peak with ower and upper bounds for the maxima and integras differing by factors two and three, respectivey, are quite dissatisfying. They are to a certain extent caused by the very pessimistic bounds on the random ensembe eqs. (9.1). The ranges in eqs. (9.1) can be narrowed down, for exampe, by assuming instead of I-IV on page 154 that I. In agreement with the error bounds cacuated from I-IV, the distribution reconstructed on a 1 1 pixe grid by the averaging scheme (tab. 9.1i and fig. 9.4a,b) is within ±5% of its vaues correct.

158 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment c(x,y) [1 14 moec/cm 3 ] 5 x [m] y [m] c(x,y) [1 14 moec/cm 3 ] 5 x [m] y [m] c(x) [1 14 moec/cm 3 ] c ĉ + ±ĉĉ... ± std E [ ±ĉ] (a) random distributions x [cm] (b) profie aong the x-axis through the centre of peak 2 Figure 9.5: Exampes of random concentration fieds generated from the fied reconstructed on a 1 1 pixe grid using the averaging scheme under the assumptions I and II, p. 157, (a) and reconstruction errors for peak 2 (b). The measurement error is not incuded in the figure. II. The peak centre is reconstructed with an accuracy of ±5cm as before. Generating again random distributions agreeing with I and II (see fig. 9.5a) eads to mean reconstruction errors ± ĉ and standard deviations shown for peak 2 in the profie fig. 9.5b. Notaby the bound on the underestimation + ĉ and as expected its standard deviation std E [ + ĉ] are consideraby reduced. The new bounds on the reconstructed peaks are now (incuding the measurement error as in eq. (9.3)) { } { } 1.5 (1.8) 1 14 moec/cm (2.1) C,1/ moec/cm 3, 2.6 (2.8) 4.2 (3.6) { } { } 3.7 (4.8) (6.4) moec/cm da c 1 18 moec/cm (9.4) 5.9 (8.1) 11.6 (1.5) Peak,1/2 with vaues in brackets again for the case without the standard deviation of ± ĉ. Under the assumptions made, the estimated error of the peak maxima in eqs. (9.2) are thus maximay 2%(1%) for peak 1 and 35%(15%) for peak 2, whereas those of the integras are 3%(15%) at most in both cases. Finay, comparing the reconstructed fieds and their error estimates to the known cyinder distributions with { } 1.95 ±.15 C,1/2 = 1 14 moec/cm 3, 3.35 ±.1 { } 6.1 ±.5 da c = 1 18 moec/cm, 1.5 ±.3 Peak,1/2 one finds that On the one hand, the true peak maxima and tota emissions ie not ony within the arger error bounds that take the scattering of the reconstruction errors ± ĉ into account, but aso within the ranges based on the mean errors ± ĉ E (and the measurement error) ony.

159 9.4. Some aspects of atmospheric measurements of emission pumes 159 On the other hand, the profies in figs. 9.4 and 9.5 ceary show that the reconstructed distributions do not agree with the true fieds within their error bounds. Both observations refect fundamenta probems of the error estimation for the inverse probem aready discussed quaitativey in the introductory section 5.1. The scattering of the error fieds ± ĉ(r), i.e. the standard deviations std E ([ ± ĉ(r)], is to a arge extent given by the variabiity of the test distributions used for the statistica estimate. As pointed out in sec. 5.3, an ensembe of smooth concentration fieds and one containing highy fuctuating fieds can, in principe, give rise to the same mean error fieds ± ĉ(r) E, but their standard deviations std E [ ± ĉ(r)] wi be quite different. The variabiity of the random distributions above is mainy controed by the choice of the maximum gradient. The fact that the ower and upper error bounds seem to be overestimated means that the arbitrary set variabiity of the ensembe is arger than the variabiity of the true concentration fied. The inconsistency of the spatia distributions seen in the peak profies is not due to an ensembe of test functions that is chosen too sma in terms of ower, upper bounds and variabiity, but it has to be attributed to the generic form of the ensembe: continuous Gaussian peaks. Choosing discrete step functions as generating distributions woud have increased the reconstruction error such that the rea (cyinder) and the reconstructed distributions are compatibe. However, in the atmosphere there woud be itte evidence for such an artificia choice if turbuent dispersion pays a roe. Athough the two issues are reated, the probem of the unknown variabiity is ikey to pay a ess important roe the onger the averaging time of the measurement is and thus the smoother the atmospheric distribution becomes Some aspects of atmospheric measurements of emission pumes The resuts from the indoor experiment cannot be transferred to atmospheric measurements because apart from the quite different experimenta conditions (see [Mettendorf et a., 26; Mettendorf, 25] for detais) the experiment assumed static puffs during the measurement cyce, which is rather unreaistic even for short times (see sec ), and the ratio of background to pume NO 2 concentrations woud not correspond to atmospheric measurements. Therefore, a reaistic setup, simiar to measurements currenty taking pace over the centre of Heideberg [Pöher, 26] is considered in the foowing instead. The area is assumed to be a square of size = 2 2km 2 and the number of ight paths generated by three mutibeam instruments is taken to be 18, corresponding to the geometry in fig. 8.14b, p. 146, so that the simuation resuts of the previous chapter can at east approximatey be adopted by downscaing as described in sec In particuar, the smaest peak that can sti be reconstructed corresponds to the ower imit of ensembe E 2 in tab. 8.1, p. 119, with a 2σ diameter given by 2σ / =.2, i.e. here, min = 4m. From a remote sensing point of view, the question whether ocay enhanced concentrations can be measured purey depends on the detection imit of the trace gas species. The emission puff with mean concentration c on top of the background c BG depicted in fig. 9.6a can be detected, if the puff s

160 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment c BG c T9 3T9 c = c c = 2 c c = 1 c L σi/.6.4 c + c BG d BG d = d + d BG (a) Pume with concentration c on a background c BG. c + c d d BG = d + d (b) Pume concentration c and residua c after subtracting the background i (c) Singuar vaues for the geometries 2T9, 3T9 with m = 18 ight paths (figs.8.1, 8.14) scaed by the ength of the square reconstruction area. The threshods indicate the singuar vaues above the detection eve for a pume concentration c according to eq. (9.5). 15 Figure 9.6: Detection of trace gas pumes. contribution to the tota coumn density is above the detection imit δ for the coumn densities d d BG > δ or c > δ 2, where is the diameter of the puff. Detection imits of the mutibeam instrument for a ight path ength of 5km are isted in tab. 9.3 for some trace gases (see aso tab. 2.2, p. 22, for typica mixing ratios of these trace gases.) From the inversion point of view, actua vaues of detection imits and measurement noise become important for the information content of the experiment and for the reguarisation procedure. Whie the atter has been argued in sec. 4.4 and demonstrated in secs , 9.3 to be ess critica for the under-determined reconstruction of peaks than it woud be for arge over-determined probems, it is instructive to see how many degrees of freedom of the measurement system do in fact contribute for a given eve of noise. According to sec. 7.2(i) it is given by the singuar vaues with σ j ǫ/ x 2, where ǫ is the root mean square error of the measurements. Writing simiary to the approximations made on page 135 x 2 n p c for a pume with mean concentration enhancement c extending over n p grid nodes and taking ǫ as the detection imit δ, the inequaity becomes Here, d σ j / 2 np δ d or σ j / 2 np c c /. (9.5) is the coumn density of the pume ike in fig 9.6a and c the concentration minimay detectabe for a ight path ength of 2. For a narrow peak, e.g. 2σ / =.3 or 2σ = 6m, O 3 NO 2 SO 2 HONO CH 2 O detection imit [ppb] for 5 km Tabe 9.3: Detection imits of the mutibeam instrument for a ight path of ength 2L = 5 km [Mettendorf, 25].

161 9.4. Some aspects of atmospheric measurements of emission pumes 161 tab suggests n/m = 2, corresponding to a 5 by 5 pixe grid and thus n p something ike three or four. If the mean concentration c of the pume is around the detection imit c, ony singuar vaues σ j /.3 give contributions above the detection imit (fig. 9.6c, dashed ine), for c 2c the threshod is.15 (dotted ine) and so forth. The detection imit of NO 2, SO 2 and CH 2 O for the ight path ength 2 = 2 6m is c =.8,.3 and 1.25ppb, respectivey, so that for reaistic poution eves (see tab. 2.2) practicay a modes contribute (in the case of geometry 3T9 ). If the tota measurement error, incuding noise, systematic errors and errors arising from changes of the concentration fied during the measurement, is moderate, the decisive factors for the reconstruction of trace gas peaks are the discretisation and inversion errors. For negigibe background concentration the reative discretisation and inversion errors do not depend on absoute peak concentrations and the detectabiity of a pume can be inferred from the simuations in the previous chapter. The situation, where a pume with mean concentration enhancement c is reconstructed by subtracting the supposed coumn densities of a moderatey variabe background concentration c BG (r) is shown in fig. 9.6b. The coumn densities after subtraction are up to a residua concentration fied c(r) given by those of the pume d d BG = d + d = 2 c + 2L c, where c is the average concentration of the residua fied on the whoe reconstruction area. On the one hand, this procedure stricty speaking demands d d. On the other hand, the presumaby erratic character of the residua suggests to treat its coumn densities as noise, and the simuations in sec have shown that reconstruction of a (singe) pume sti makes sense for error eves beow 1%. Taking into account that there are further measurement errors, one gets the rough bound or d d 1%, (9.6a) c c.1 /L.1 /. (9.6b) For exampe, for a oca enhancement of NO 2 around 1ppb with / =.3 as above and c BG 1ppb, the inequaity reads which seems to be a reaistic requirement. c.3ppb 1 3 c BG, Finay, the up to now hypothetica dimensions of the pume ought to be put in context to reaistic dispersion of trace gas in the atmosphere. To this end, we consider a point source at distance x away from the optica path, which is perpendicuar to the wind direction for simpicity (see fig. 9.7a). The wind direction is assumed to be constant. The horizonta dimensions of the pume are given by the horizonta dispersion coefficients σ y. For the Pasqui-Gifford parametrisation (appendix A) and the most frequent neutra stabiity cass D, the 2σ-diameter at downwind distance, e.g., 5 km is 2σ 18m. Assuming a wind speed of ū = 5m/s for this cass, the trave time for this distance

162 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment z q ū y h L/2 +L/2 z x x d(x, z, h) [1 3 µg/m 2 ] q = 1 3 µg/s ū = 1 m/s A = E = D B F h = 5 m 1 m 15 m (a) Specia geometry with the optica path of ength 2L at height z parae to the y-axis downwind distance x [km] (b) Coumn density d(x, z = 2 m, h) for an inert tracer according to eq. (9.7) for different stack heights and the urban Briggs parametrisation of the dispersion coefficients. The orientation of the optica path of ength 4 km is as in figure a. Figure 9.7: Measurement of a coumn density in a continuous pume emitted by a point source. is 16min. For stabiity casses A (extremey instabe) and F (extremey stabe) the corresponding vaues are 2σ 188m and 54m, but at ower and higher wind speeds, respectivey. Whie the pume passes the reconstruction area of ength = 2km (in 7min at ū = 5m/s), the extension 2σ increases to 253 m (A), 146m (D) and 73m (F). In terms of the ensembes of chap. 8, the peaks woud thus beong to E 2 to E 4. Turning to concentration eves in the pume, we take the most simpe case of a continuous point source and, negecting refection at the mixing ayer, the coumn density d measured aong the ight path for the geometry in fig. 9.7a can simpy be cacuated by integrating eq. (2.5a) aong y d(x,z,h) = 2 π [ q exp ( ū σ z (z ) h)2 2σz 2 + exp ( (z + ) ] h)2 erf( 2σz 2 L ) 2 2σ y, (9.7) with σ y, σ z evauated at x and erf being the error function. Just as an exampe, we consider a cement works which happens to be ocated at about 5 km distance south-east of Heideberg and study the impact on coumn densities measured in Heideberg for the scenario of fig. 9.7a. 2 The European poutant emission register (EPER) 3 states mean annua emissions of q SOx kg/a = 12g/s, q NOx kg/a = 16g/s for this pant. The actua direct emission of SO 2 can be written as q SO2 = EF SO 2 f t q SOx, EF SOx } {{ } r where EF denotes emission factors and r is the ratio of directy emitted SO 2 to SO x. The factor f t is a 2 This wind vector does not represent the predominant wind direction. 3

163 9.4. Some aspects of atmospheric measurements of emission pumes 163 dimensioness factor accounting for deviations from the annua mean and the corresponding expression hods for NO 2. Furthermore, the pume undergoes chemica transformation which in the case of NO 2 can be characterised by the downwind NO x concentration and the Leighton ratio L = [NO]/[NO 2 ] (c.f. sec. 2.2) and for SO 2 takes the simpe form of an exponentia decay by exp( kx /ū) with a vaue of k = 4, s 1 suggested in [ISC-3, 1995] for urban environment. For x = 5km and a wind speed of 2m/s, the exponentia takes a vaue of.99, so that the SO 2 pume can be treated as inert. Incuding the downwind chemica decay by the factor D(x ), we write d D (x,z,h) = D(x )d(x,z,h) = D(x )r f t d (x,z,h). (9.8) Except for the most instabe conditions (A), the coumn density at x = 5km does not depend significanty on the effective stack height h, as shown in fig. 9.7b, where d(x,z,h) is cacuated for different stack heights for a ength 2L = 4km of the ight path at height z = 2m, using the Briggs urban parametrisation of the dispersion coefficients. 4 Vaues of the coumn densities d (x = 5km,z = 2m,h = 1m) without the correction factors D, r and f in eq. (9.8) and corresponding average concentrations c aong the ight path amount to SO x NO x d c d c stab. cass ū [1 3 µg/m 2 ] [µg/m 2 ] [ppb] [1 3 µg/m 2 ] [µg/m 2 ] [ppb] [m/s] A/B D E/F 2 Taking the factors D and r into account wi reduce these concentrations further, whie f t might we be much arger than one. Average concentration vaues aong a ight path above the centre of Heideberg obtained by Rippe [25] during measurements in December 24/January 25 range from around.5ppb to 14ppb for SO 2 and from 2ppb to maximay 45ppb for NO 2. Whether the pume coud be detected as such depends apart from the meteoroogica conditions, the activity of the cement works and the chemica deveopment of the pume on the actua emission factors and on the concentration eves in Heideberg itsef. For a pume with / 1m/2km =.2 (stabiity cass D, c.f. p. 161) eqs. (9.6) require c/ c.5. Taking the corrected mean concentration c of SO 2, for exampe, as a tenth of c above, i.e. c.2ppb, this means c 1 3 something unachievabe. The same anaysis can be carried out for a arge coa fired power station, the GKM, Mannheim, ocated 15km north-west of Heideberg. The EPER specifies q SOx kg/a = 7g/s, q NOx kg/a = 12g/s. The ratio r = EF NO2 /EF NOx takes vaues beow 1% (see sec. 2.2), whie according to the UK emission database 5 EF SO2 /EF NO2 2.7 for coa. Using the Pasqui-Gifford parametrisation (for rura environment) and ignoring again the mixing ayer, the uncorrected coumn densities d and average concentrations c are 4 We use the units [µg/m 2 ] common in air poution monitoring. 5

164 D Reconstruction of NO 2 Peaks from an Indoor Test Experiment SO x NO x d c d c stab. cass ū [1 3 µg/m 2 ] [µg/m 2 ] [ppb] [1 3 µg/m 2 ] [µg/m 2 ] [ppb] [m/s] B D F 2 where first vaues refer to an effective stack height of 2m, the second to 1m. With the more reaistic higher effective stack height and having the vaues of the factors r in mind, these concentrations can for highy above-average activity of the power station be a significant contribution to Heideberg s poution eve, but in view of the pume extension 2σ 8m(F) 6m(A), it woud rather appear as an enhanced background than a distinct pume. The two exampes discussed represent the most important point sources for NO 2 and SO 2 in the vicinity of Heideberg. Therefore, it can be concuded that (narrow) pumes with concentration maxima distincty above the city eve have to originate from within the area of Heideberg.

165 1. 2-D Reconstruction of Mode Trace Gas Distributions above a Street Canyon This ast chapter of numerica resuts focuses on tomographic reconstruction of trace gas distributions in an urban environment. For ack of experimenta data resuts from an eaborate mode system, utimatey designed to cacuate concentrations within and in the vicinity of a highy pouted city street canyon, wi be empoyed to simuate measurements with a moderate number of ight paths. The reason for choosing these mode distributions is simpy that they cover a horizonta area arge enough and with sufficient spatia resoution to represent a reaistic state of the atmosphere on the scae of our tomographic measurement. We are neither concerned with the particuarities of the mode, nor with the specific meteoroogica situation or the emission scenario, but ony with the spatia patterns of the distributions. The description of the mode situation in sec. 1.1 is therefore kept rather brief, before turning to the exempary reconstruction of NO 2 in sec The reconstruction procedure turns out to be far ess fexibe than the one for peak distributions, instead reguarisation becomes crucia. This becomes even more evident in the error cacuations of the subsequent sec. 1.3, which examines the possibiity to verify the mode NO 2 distribution by means of the tomographic setup. The evauation procedure foows the suggestion of sec. 7.4 on mode evauation Mode system, set-up and resuts Mode resuts used for the simuations of this chapter were obtained by the mode system M-SYS, which was briefy referred to in sec 2.5 as a too for the evauation of ambient air quaity according to the EU framework directive 96/62/EC (see sec. 2.1). The trace gas distributions presented shorty were cacuated by D. Grawe 1 for the street canyon Göttinger Straße in Hanover in the framework of the VALIUM project, which was carried out especiay for the deveopment and vaidation of these toos [Schatzmann et a., 26]. Motivation for the deveopment of M-SYS was the assumption that contributions from a reevant scaes have to be modeed propery to predict air poutant concentrations on the street scae. Athough ony the innermost mode domain wi be considered here, some characteristics of the modes shoud be mentioned for the sake of competeness. More detais and references can be found in [Trukenmüer et a., 24]. Two modes are used to cacuate transport and chemistry of poutants on the mesoscae: the non-hydrostatic Mesoscae Transport and Stream Mode METRAS [Schünzen et a., 1996] and the 1 Grawe, D., persona communication. 165

166 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon Figure 1.1: Non-uniform mode grid for the microscae street canyon cacuations. The area is m 2, resoved by 15 m near the boundaries and 1.5 m at the centre, where Göttinger Straße heads towards the roundabout in the North. Aso shown the traffic emissions for the case study used here. They are highest for Göttinger Straße and the roundabout (The picture is miseading because of the uneven grid) [IER, University of Stuttgart. Courtesy of D. Grawe, now with the Division of Environmenta Heath and Risk Management, University of Birmingham]. Mesoscae Chemistry Transport Mode MECTM [Lenz et a., 2]. For the case at hand they are appied on three eves with horizonta domains of 2 2km 2 (resoution 16km), km 2 (4km) and 112 2km 2 (1km), respectivey, to assess air quaity on a regiona scae and provide boundary conditions for one way nesting of the modes. A domains are centred on Hanover- Brunswick, where the street canyon of Göttinger Straße is ocated. Boundary conditions for the outermost mode are given by interpoated observations in the case of meteoroogica parameters and are party based on cimatoogies in the case of background concentrations (see [Trukenmüer et a., 24] for detais). The innermost mesoscae mode incudes major point and area sources (such as motorways and cities etc.) that contribute to the background concentrations of the microscae environment of the street canyon. The atter is defined horizontay by a 1 1km 2 area with the street canyon at its centre and verticay up to a height of about 4m. Fow fieds are cacuated on a non-uniform grid using the obstace-resoving microscae mode MITRAS [Schünzen et a., 23]. The grid spacing varies horizontay from 15m at the atera boundaries to 1.5m at the centre of the street canyon, see fig The vertica grid consists of 5 ayers with spacing 1.5m at the bottom and 3m at the top. Concentrations within the street canyon environment are computed by the microscae modification of MECTM, caed MICTM 2. Both modes are based on the gas phase chemica mechanism RADM2 [Stockwe et a., 199] and cacuate concentrations of 59 species. Benzene (an important poutant subject to EU reguation, see aso tab. 2.1, p. 21) is not modeed individuay because emission inventories are acking. The obstace-resoving mode MITRAS was evauated using wind tunne data. The comparison shows adequate agreement for the fow fieds in genera, but near the edges of buidings and at roof top resuts depend on the turbuence cosure scheme chosen [Schünzen et a., 23]. The system M- SYS was compared to experimenta observations of background concentrations of SO 2, NO 2 and O 3 on a regiona scae with very good resuts for SO 2 and NO 2, but ess accurate predictions for O 3 [Trukenmüer et a., 24]. The microscae performance of the mode system was evauated by intensive fied measurements within the street canyon during the VALIUM project and incuded point and remote sensing measurements of important traces gases and the tracer SF 6 [Schäfer et a., 25]. Comparing 3min averages of point measurements taken at 2m above ground and the predictions by 2 Grawe, D.: Verknüpfung von Modeen und Messungen zur Konzentrationsvorhersage, PhD-thesis, Fachbereich Geowissenschaften, University of Hamburg (in German), in preparation, 26.

167 1.1. Mode system, set-up and resuts 167 (a) Buiding height (street canyon at x ) (b) Horizonta wind vector 3 m above ground (c) Horizonta wind veocity 3 m above ground (d) Vertica wind veocity 3 m above ground Figure 1.2: Buidings and wind fied at 3 m above ground for the microscae modes. The street canyon is at x [Courtesy of D. Grawe, now with the Division of Environmenta Heath and Risk Management, University of Birmingham]. M-SYS shows mutua agreement we within a factor of 2. 3 The trace gas distributions considered in the foowing were obtained from M-SYS under the assumption that therma infuences can be negected on the microscae, which eads to a stationary soution for the street canyon. For the 3min run of the mode system with 1min time steps it was furthermore assumed that background emissions in the mesoscae modes do not change and that the ony emissions reevant for the microscae modes originate from traffic. As the highest buidings within the microscae mode domain reach 3 m, see fig. 1.2a, the optica paths of the tomographic remote sensing measurement are taken to ie in a pane at around 3m above ground. The associated wind fied is shown in figs. 1.2b-d. Measuring coumn densities at 3m is admittedy somewhat high (for exampe, concentrations of NO 2 are about ten times ower than at ground). But there are severa reasons why a tomographic measurement at this height yet is interesting, especiay from a modeer s point of view. 3 Grawe, D., persona communication.

168 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon c mod [ppb] y [m] -5 x [m] 5 c mod [ppb] y [m] -5 x [m] c mod [ppb] y [m] -5 x [m] 5 (a) NO 2 at t = (b) O 3 at t = (c) HONO at t = c mod [ppb] y [m] -5 x [m] 5 c mod [ppb] y [m] -5 x [m] 5 c mod [ppb] y [m] -5 x [m] 5 (d) CH 2 O at t = (e) SO 2 at t = and t = 3 min Figure 1.3: M-SYS mode distributions of seected trace gases on the section at 3 m above ground. The street canyon is at x. Time steps of the mode run are 1 min. First, it gives information about the background contribution to the street canyon poution eve. Second, as mentioned before, at east for the microscae transport mode MITRAS empoyed here, vertica momentum fuxes way above roof top can depend significanty on the turbuence cosure [Schünzen et a., 23, especiay fig. 2]. And third, reated to this topic, it has been observed in the fied measurements mentioned above that tracers were upifted from the street canyon, over the roofs and down into the backyards. This effect occurs in mode simuations, too [Schünzen et a., 23] and it woud be interesting to test the quantitative agreement between measurement and mode. Finay, fig. 1.3 shows exampes of 2-D mode distributions for the height 3m. 4 A species show concentration peaks (or sinks in the case of O 3 ) rising at the street canyon and extending pume ike aong the wind direction to the right hand side of the street canyon. They are caused by the buiding structure 5 and not very high at 3m, for exampe maximay 3ppb for NO 2. For some trace gases, ike NO 2, the background concentration fied on the eft hand side of the street canyon is very smooth 4 Again 3-D contour pots are chosen to make the spatia variabiity cearer. 5 Grawe, D., persona communication.

169 1.2. Sampe reconstruction for NO y [m] x [m] 25 (a) c mod (x, y) [ppb] (NO 2 at t = ) y [m] x [m] (b) c id (x, y) [ppb] y [m] x x [m] (c) ĉ(x, y) [ppb] Figure 1.4: 2-D NO 2 distributions 3 m above ground. (a) M-SYS mode as in fig. 1.3a. (b) Idea parametrisation on a reguar biinear grid of dimension n = 4 4. (c) Distribution reconstructed on a grid with n = 4 4 and mean a priori. The iteration number is k = 5 and vaues of the quaity indices are: NEARN =.57, NMSE = , IOA =.89, NMB = 2 1 3, FB = 4 1 4, r 2 =.67 and FA2 = 1%. compared to the peak structure, others ike O 3 show more background variabiity. The concentration fieds shown in fig. 1.3 change ony sighty during the 3min interva. The most significant change occurs for SO 2 due to a change in the background Sampe reconstruction for NO 2 We consider the reconstruction of NO 2 from measurements with three teescopes and a moderate number of 18 optica paths, arranged in 9 -fans as in fig. 8.14b on page 146. The meshsize in the centre is about 2 25m, the smaest peak structures have a size of about s 3m. It is thus pure chance, if such a peak is hit by a ight path ike the one in the centre of fig. 1.4a. But even then a peak with 3m and concentration enhancement c 3ppb woud give rise to a coumn density 18ppb, which is far beow the detection imit of the coumn density δ = 1ppb m according to tab. 9.3 (The minima peak extension for an enhanced NO 2 concentration of 3ppb woud be 17m). The NO 2 distribution with a smooth background to the eft of the street canyon offers a good opportunity to iustrate different estimates of the background concentration c BG according to sec (i). Foowing eq. (6.11), the over-determined case of a singe, reguar 3 by 3 pixe grid with n = 4 4 eads to c BG = 8.64ppb for eq. (6.11), coming very cose to the vaues ppb in figs. 1.3a, 1.4a. Vaues for the under-determined cases eqs. (6.12) and a reguar grid with n = 5 5 are c BG = 8.77, 9.12, 4.56ppb for eqs. (6.12a), (6.12b), (6.12c). The ast case does not reproduce the physica background, the first two ead to amost identica reconstructions ĉ.

170 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon c(r) ĉ u c mod < c(r) ĉ u c min < ĉ u ĉ u ĉ c mod ĉ < c mod ĉ c max ĉ < c max c min ĉ (a) Evauation of the mode r (b) Evauation of the mode bounds r Figure 1.5: Consistency of the experimenta retrieva ĉ with the mode c mod and its bounds c min/max, understood as 1σ check. Upper and ower bounds of the reconstructed distribution are given by ĉ /u = ĉ + /u ĉ. (a) Case (i). (b) Case (ii). Simuating error free measurements of the mode NO 2 distribution with the above optica geometry of 18 ight paths and reconstruction using SIRT with different grid dimension, a priori and reconstruction schemes reveas that 1. The over-determined parametrisation gives smaer overa errors and better statistica indices than the under-determined one. 2. Optima iteration numbers, especiay in the under-determined case, ie ong before convergence. 3. The optima choice for the a priori is the mean eq. (6.7). 4. Neither any of the the grid-shifting schemes, nor subtracting or fitting the background eads to better resuts than reconstruction with a singe n = 4 4 grid and mean a priori. 5. Except when subtracting the background, the nonnegativity constraint is not active. Measurement errors have been negected everywhere for the moment. Point 4. is not too surprising as, even after idea remova of the backgrund, the distribution does not represent a genuine peak distribution ike in chap. 8. Points 3. and 5. are pausibe. However, the second observation seems to be a sign of a crucia difference to the reconstruction of peaks. The fact that the under-determined soutions have to be strongy reguarised suggests that the eastsquares east-norm soution and thus the east-norm principe might not onger be the optima seection criterion for this kind of distribution. The reconstruction resut on an over-determined n = 4 4 grid shown in fig. 1.4c does not bear much resembance to the origina distribution above the street canyon, but comparison with the idea parametrisation of the mode distribution by the same grid (according to sec ) in fig. 1.4b makes cear that a fied of this kind has to be expected from the given spatia resoution Discussion of mode evauation We examine the two questions raised in the context of mode evauation in sec. 7.4:

171 1.3. Discussion of mode evauation 171 Figure 1.6: Optima estimate using x id from fig. 1.4b as a priori with reative variances of 5% and 1% noise on the coumn densities obtained from the mode distribution in fig. 1.4a. (a) The state vector. (b) A posteriori variances. Vaues in between the grid nodes are merey interpoated y [m] x [m] (a) ˆx [ppb] y [m] x [m] (b) σˆxi [ppb] (i) Given a distribution ĉ reconstructed from a tomographic experiment and a mode prediction c mod for the same situation, do they agree within the recontruction error? (ii) If not, do they agree within the uncertainty of the mode? The first question is iustrated by fig. 1.5a with the reconstruction error given by its ower and upper bounds ĉ and ĉ u, the second by fig. 1.5b with the mode uncertainty given by the bounds c min and c max. We take the NO 2 distribution from figs. 1.3a,1.4a as mode prediction c mod and assume its error according to the specification for the street canyon cacuations given above on page 167 to be a factor of two. That is, we take it for granted in the foowing that the vaues of the true concentration fied c ie between c min (r) =.5 c mod (r) c(r) 2 c mod (r) = c max (r), (1.1) (Especiay the upper bound might seem pessimistic, but we reca that buiding effects can change concentration vaues to a high degree (see sec. 2.5) and that the mode treatment of fows around buidings is far from trivia. See the discussion above.) Because we do not have experimenta coumn densities, measurements wi be simuated for true concentration fieds c that are obtained by randomy varying the mode within its uncertainty. Case studies for such simuated measurements wi be presented in sec Before, the reconstruction error has to be quantified. The next section wi show that an estimation based on eq. (1.1) aone eads to reconstruction errors that are too arge to aow any rea evauation of the mode. Therefore, further assumptions wi be added to the above inequaity bounds in sec Estimation of the reconstruction error using the optima estimate To get a first idea about the predictive power of the tomographic measurement under the given circumstances, we empoy the optima estimate, which makes sense because the nonnegativity constraint is not active (5. above). The idea discretisation in fig. 1.4b is used as a priori x a and the data error covariance is assumed to be of the form S ǫ = σǫ½n 2 with σ ǫ given by a reative measurement error of 1%. The asymmetric bounds in eq. (1.1) cannot be represented by a Gaussian a priori probabiity density. Instead, we use the covariance S a = diag(σa 2 i ) with σ ai =.5 x ai, that is a ±5% 1σ error. The state vector ˆx and the diagona eements of its covariance matrix Sˆx are shown in fig. 1.6.

172 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon Ceary, the 1σ uncertainty of the a posteriori ies within the bounds of eq. (1.1). But aso evident its minimum vaue of about 2ppb is amost as high as the peak structures, and in fact a structues, of the origina mode distribution. And the optima estimate does not even incude the discretisation error arising from the finite representation of the continuous fied. To be more precise, the optima estimate provides a measure for the reconstruction error using an ensembe of distributions that are Gaussian distributed around the a priori, with a spatia variabiity given by the grid spacing. However, the variabiity of the mode distribution is much higher Estimation of the reconstruction error using a random ensembe We now narrow down the true distribution further by assuming additionay to eq. (1.1) that not ony its concentration vaues c but aso its gradients c are confined by the mode. Generating random concentration fieds as described in appendix B, it turns out that it is possibe to constrain c such that the background behaviour to the upwind eft hand side of the street canyon, the increase of concentration downwind to its right and the appearance of the origina peaks can be reproduced with a variabiity of the concentration vaues that sufficienty expoits eq. (1.1). Peak centres appear at the origina sites. To account for mode uncertainties of exact positions an arbitrary shift of the whoe distribution in the pane by maximay 1% (±1m) is aowed. To be precise, we now assume that the true distribution c for the atmospheric state modeed by c mod ies within the bounds: I..5 c mod (x,y) c(x,y) 2 c mod (x,y). II. c(x,y) is maximay f max c mod (x,y) in the sense of eqs. (B.3), (B.4) with f max = 5. III. c(x,y) c(x + x,y + y ) with x, y 1m. The exact vaue f max = 5 for the factor constraining the gradient is arbitrary. For exampe f max = 1 resuts in simiar, yet very noisy concentration fieds. Furthermore, as discussed at the end of sec. 9.3, the choice of maximum gradients affects the bounds of the reconstruction error. Here, f max is chosen such that the variabiity of the mode distribution is reproduced and that the bounds given by I are expoited. The ensembe E of distributions c I, I = 1,...N, for estimation of the reconstruction error is given by random fieds satisfying I-III. Sampes are shown in fig. 1.7a to 1.7f, whie figure 1.7c represents a distribution that was obtained by a shift of more than 1 m in the direction of negative y. (The 2-D contour pots in bottom pane wi be referred to ater.) The standard deviation around the ensembe mean c I (x,y,) E is shown in fig. 1.8a. Indeed, it exhibits higher variabity in the peak region to the right of the street canyon and the ensembe created from I-III appears to be a sensibe random variation of c mod, reproducing a smooth background to the eft of the street canyon and decreasing peak concentrations towards the boundary right from it. 6 6 Mind that c I (x, y, ) E c mod (x, y). In genera, for a random number c uniformy distributed between c max and c min c = 1 2 (cmax + c min) std[c] = (cmax c min). In particuar for c max = 2c mod, c min = 1 2 c mod: c = 5 4 c mod and std[c] = 3 4 c mod.

173 1.3. Discussion of mode evauation 173 c I [ppb] y [m] -5 x [m] c I [ppb] y [m] -5 x [m] 5 c I [ppb] y [m] -5 x [m] 5 (a) c 1 (x, y) (b) c 2 (x, y) (c) c(x, y) c I [ppb] y [m] -5 x [m] 5 c I [ppb] y [m] -5 x [m] 5 c I [ppb] y [m] -5 x [m] 5 (d) (e) (f) 5 25 y [m] y [m] y [m] x [m] (g) c 1 (x, y) [ppb] x [m] (h) c 2 (x, y) [ppb] x [m] (i) c(x, y) [ppb] 25 5 Figure 1.7: Random concentration fieds generated from the mode NO 2 distribution (figs. 1.3a,1.4a) according to points I-III, except c(x, y) in fig. 1.7c, which vioates III in that y > 1 m. The bottom pane shows 2-D contour pots of c 1, c 2 and c for ater reference. Figures 1.8b and 1.8c show ower and upper bounds of the reconstruction error fied ĉ(x, y) cacuated for E according to eq. (5.14d) u/ ĉ(x,y) = ± ĉ(x,y) E ± std E [ ± ĉ(x,y)]. (1.2) Measurement noise was not taken into account. It foows that

174 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon y [m] x [m] 25 (a) std E [c I (x, y)] [ppb] 5 y [m] x [m] (b) ĉ(x, y) [ppb] without noise y [m] x [m] (c) uĉ(x, y) [ppb] without noise Figure 1.8: Random ensembe E generated from I-III (a) Standard deviation around the mean of E. (b) Lower and (c) upper bounds of the reconstruction error without noise, i.e. ĉ = c disc + ĉ inv + ĉ reg and /u understood as u/ = ± E ± std E[ ±]. If the true background shows a variabiity simiar to the mode (it is in fact higher for the distributions from E, see fig. 1.7), it can be reconstructed with a precision of about ±.5ppb. Reconstruction of the peak structure is hardy possibe, especiay not for the peak at y 5m. If the ocation of the peaks within the reconstruction area is competey unknown, the resuting reconstruction error of at east ±2ppb wi exceed the structures in c mod. Simuations where III is reeased to a shift over the entire area yied bounds /u ĉ between ±1.5 and ±4ppb. Even without measurement error, this is aready of the order of what was obtained from the optima estimate, see fig. 1.6b. Additiona information ike II or III has to be provided to enabe discrimination of the structures of the unkwown true distribution within the reconstruction error of the above tomographic setup Case studies Simuated measurements are now studied for the true 2-D NO 2 concentration being c 1, c 2 or c from fig. 1.7 and the mode prediction aways being c mod from figs. 1.3a,1.4a. The fieds c 1 and c 2 represent the case of true distributions consistent with the mode bounds. The fied c in fig. 1.7c serves as an exampe of the inconsistent case. (As discussed in sec. 7.4, to be correct this inconsistent case requires an independent estimate of the reconstruction error. With no such estimate avaiabe, we wi use the above estimate of the reconstruction error.) Reconstructions with and without measurement errors are treated seperatey in the foowing. Measurement noise can be negected If the experimenta error pays ony a minor roe in the tomographic measurement, the parameters of the reconstruction are ike in fig. 1.4c. 2-D maps reconstructed from simuated measurements of c 1, c 2 and c are shown in the top pane of tab. 1.1 on page 175. Ad (i): To find out whether these reconstructed fieds ĉ 1, ĉ 2 and ĉ are consistent with the mode distribution c mod, differences according to fig. 1.5a are formed, indicating inconsistency whenever

175 1.3. Discussion of mode evauation y [m] y [m] x [m] x [m] x [m] ĉ 1 (x, y) ĉ 2 (x, y) ĉ(x, y) y [m] y [m] y [m] x [m] x [m] x [m] ĉ 1u (x, y) c mod (x, y) ĉ 2u (x, y) c mod (x, y) ĉ u(x, y) c mod (x, y) y [m] y [m] x [m] x [m] x [m] c mod (x, y) ĉ 1 (x, y) c mod (x, y) ĉ 2 (x, y) c mod (x, y) ĉ (x, y) y [m] y [m] y [m] y [m] x [m] x [m] x [m] ĉ 1u (x, y) c min (x, y) ĉ 2u (x, y) c min (x, y) ĉ u(x, y) c min (x, y) y [m] (a) true distribution c 1 [ppb] (b) true distribution c 2 [ppb] (c) true distribution c [ppb] Tabe 1.1: Reconstruction of a simuated, error free measurement (1 st pane from the top, parameters as in fig. 1.4c), consistency with the mode c mod (2 nd & 3 rd pane) and with the mode bounds (bottom pane) according to fig. 1.5 for the distributions c 1, c 2 and c from fig Red and yeow (and green) indicate that the mode overestimates, bue (and green) that it underestimates the true distribution.

176 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon y [m] x [m] 25 (a) c mod (x, y) c(x, y) [ppb] y [m] x [m] (b) c max(x, y) c(x, y) [ppb] y [m] x [m] (c) c(x, y) c min (x, y) [ppb] Figure 1.9: Evauation of (a) the mode c mod and (b,c) the mode 1σ bounds by infinitey many idea point measurements of c covering the whoe reconstruction area. In (a) these measurements woud find a pattern of over-and underestimations above the street canyon (dashed ine) and in the vicinity from the right. they become negative. Negative vaues of the fied ĉ u c mod in the second pane from the top show where mode concentrations are too high to agree with the measurements. In the same way, c mod ĉ < means that mode vaues are too ow. Within the 1σ bounds of the reconstruction error defined by eq. (1.2) the mode quite correcty woud agree with none of the true distributions estimated from the measurement. The discrepancies identified from the error anaysis agree we with what one woud expect from comparing the origina distributions in figs. 1.7g-1.7i with c mod in fig. 1.4a. But the absoute numerica vaues of the differences beow 1 ppb are rather sma, except for the case of c 1 where the background concentration is consideraby overestimated by the mode c mod. Idea point measurements of the rea distribution immediatey aow to compare true and mode concentrations and thus mode verification at the ocations of the measurement. But a sufficient number of these sampes at the right paces is crucia in order to get a consistent picture of the mode performance, as iustrated by fig. 1.9a where the concentration fied c representing the true distribution is compared to the mode prediction. Whie point sampes in the eft haf of the picture and near the right boundary woud be representative of a arger area round the pace where they were taken, the interesting region above the street canyon requires quite a few measurements to get the pattern of over- and underestimation. This is exacty the probem of the representativeness of point measurements in an area with compex buidings addressed by Schünzen et a. [23] (c.f. sec. 2.1). Ad (ii): We define the ower and upper mode bounds c min, c max of the ensembe defined by I-III through the 1σ bounds c max/min = c I (x,y,) E ± std E [c I (x,y,)] (1.3) with std E [c I ] as in fig. 1.8a. It turns out that the distributions c 1 and c 2, which are in E, vioate the 1σ bounds, whie the fied c / E is consistent with them (tab. 1.1, bottom pane). The seeming inconsistency of the former, especiay c 1, in the region without peaks is due to the fact that their background concentrations take vaues at the ower end of admissibe vaues (see aso the footnote on page 172). The differences between the fieds c mod and c appear mainy in the area of the peaks and cannot be resoved by the tomographic setup, see aso fig. 1.9.

177 1.3. Discussion of mode evauation y [m] 9.6 y [m] 9.6 y [m] x [m] (a) ĉ(x, y) [ppb], k = x [m] 25 (b) ĉ(x, y) [ppb], k = x [m] (c) ĉ(x, y) [ppb], k = y [m] y [m] y [m] x [m] (d) std ǫ[ĉ(x, y)] [ppb], k = x [m] (e) std ǫ[ĉ(x, y)] [ppb], k = x [m] (f) std ǫ[ĉ(x, y)] [ppb], k = Figure 1.1: Reguarised soutions for a simuated measurement of c mod (a-c) and their standard deviation for a reative error of 1% on the synthetic coumn densities (d-f) for different vaues of the iteration number. A other parameters are as in fig Subfigure (c) is the same as fig. 1.4c. Measurement noise has to be taken into account The impact of random noise on the reconstruction, i.e. the perturbation error in terms of sec , can be estimated aong the ines of sec Equation (5.13b) takes again the form E [ x pert 2 ] 2 nm( x) 2 = i ( 1 (1 σ 2 i ) k σ i ) 2, where x represents the mean perturbation error of the state vector components, k is the iteration number and, as usua, n, m are the number of grid points and ight paths, respectivey. Introducing the mean reative error r ǫ of the coumn densities, the spatia mean concentration c and a typica number n φ of basis functions contributing to a ight path ( 3 5 in this case) gives x r ǫ c nφ nm i ( 1 (1 σ 2 i ) k σ i ) 2. For reative errors of about 1%, c 9ppb and n φ = 4 one gets x 1, 1.4, 3.2ppb for k = 25, 5, 5.

178 D Reconstruction of Mode Trace Gas Distributions above a Street Canyon y [m] x [m] (a) ĉ mod (x, y) [ppb] y [m] x [m] 4.5 (b) ĉ mod u (x, y) [ppb] y [m] x [m] (c) ĉ 1u (x, y) c mod (x, y) [ppb] Figure 1.11: Tota reconstruction error incuding noise. (a,b) Upper and ower bounds ĉ u/ = ±ĉ E ± std E[ ±ĉ]±std ǫ(ĉ) of the reconstruction error for the distribution reconstructed from a simuated measurement of c mod. (c) Consistency of c mod and the reconstruction from a measurement of c 1 (same as in tab. 1.1(a), but now with noise). The iteration number is k = One finds that the mean perturbation is substantia, its numerica vaue simiar to the noiseess part of the reconstruction error (fig. 1.8) and it strongy depends on the choice of the reguarisation parameter k. For the mode distribution c mod this is further pointed out in fig. 1.1, which shows reguarised soutions for a simuated measurement (a-c) and their standard deviations for 1% noise (d-e) for different degrees of reguarisation. Ceary, the iteration number in this case has more impact on the perturbation than on the east-squares soution. Not ony absoute vaues of the standard deviation, but aso the spatia patterns vary with k (for exampe, maximum vaues shift from the interior to the boundaries of the reconstruction area with growing iteration number). The weaker reguarised case with k = 5 is very simiar to the stochastic reguarisation by the optima estimate above (see fig. 1.6, p. 171), agreeing with the fact that the a priori is not very we constrained. Adding the perturbation error for the coumn densities from c mod to the reconstruction error without noise (c.f. fig. 1.8) according to eq. (5.14e) eads to the bounds of the tota reconstruction error shown in fig. 1.11a and 1.11b. These now have become so arge that mode evauation is hardy possibe within the bounds given by I-III, and among the distributions considered in the noisefree case ony distribution c 1 with very ow background concentrations can safey be distinguished from the mode c mod (fig. 1.11c). Athough evauation of the specific mode distribution here with ow absoute concentration variations in the reconstruction area was found to be hard, if not impossibe for reaistic measurement errors, it appears fair to concude that estimating the bounds of the reconstruction error from an ensembe of suitabe random disributions is a consistent numerica scheme not ony hepfu to narrow down the uncertainty of the tomographic retrieva, but aso a precise too for the evauation of modes.

179 11. Concusion and Outook Concusions This thesis theoreticay investigates the possibiity to retrieve 2-D distributions of trace gas concentrations from active DOAS measurements aong a number of m ight paths that is reaistic at the current experimenta state (m 1 4). The conventiona aspects of the retrieva method can be summarised as foows: The approach discretises the inverse probem of finding the concentration fied for measured coumn densities by parametrising the unknown distribution by a imited number of n oca basis functions. The resuting inear discrete inverse probem is repaced by a east-squares east-norm probem for the discrete state vector, which is soved iterativey by the simutaneous iterative or agebraic reconstruction technique (SIRT and ART), commony used in computerised tomography. Reasons for this approach were given in sections 4.2 and 6.1. The choice of the iterative soution was justified in section by its fexibiity with respect to additiona (in)equaity constraints. It aso impies a specific reguarisation behaviour depending on the iteration number. The important nove contributions to tomography with a ow number of integration paths are: A systematic investigation of the parametrisation with respect to the shape of the concentration fied, incuding the number of parameters, the kind of basis function (piecewise constant and inear) and new schemes that take into account severa reconstruction grids. Whie the focus is usuay put on the inversion agorithm, it was shown here for peak distributions, which represent an important cass of atmospheric concentration fieds, that the parametrisation pays an equay, if not more important roe. A second aspect that becomes vita for tomography with ow spatia resoution, but which is mosty ignored, is the treatment of the compete reconstruction error, incuding the discretisation. The detaied approach of this thesis has reveaed that the very common piecewise constant (box) functions shoud be avoided whenever possibe even for the reconstruction of spatia mean vaues. A systematic anaysis of ight path geometries using aternative arguments has pointed out not ony their tremendous impact on the quaity of the reconstruction, but aso given insight into the reasons, incuding the roe of measurement errors. Furthermore, it was suggested how numerica resuts can, to a certain degree, be generaised for arbitrary numbers of ight paths. Besides, it was attempted throughout to present simiar and aternative concepts from different discipines in a common context in order to put them into perspective (e.g., sections 4.2, 4.8.1, 4.8.3, 4.9) and to keep the discussion open for future deveopments (e.g., sections and 4.7). In more detai, the findings of this thesis can be stated in the foowing way. The systematic approach to simuated tomographic measurements of 2-D Gaussian distributions ( peak distributions ) with respect to their extension and parametrisation in chapter 8 shows that 179

180 Concusion and Outook 1. for narrow emission peaks the reconstruction quaity in terms of root mean square error, bias and tota emissions can be tremendousy improved by choosing parametrisations that ead to under-determined east-squares probems (m < n), provided that underying background concentrations are known, negigibe or smooth enough to be subtracted as proposed in section (a quantification of smooth enough was given in section 9.4) in other words, that a zero concentration is a good a priori (after subtraction). At the same time, this means that the east-norm seection criterion and the nonnegativity constraint becomes more, the perturbation error ess important (sections 4.4, and tab. 8.11). 2. for the under- or even-determined soutions of 1. the biinear parametrisation not ony gives rise to smaer discretisation errors (which is evident) but aso to smaer inversion errors (which is not evident). This hods aso for bias and tota emissions, i.e. for mean vaues of the concentration fied. 3. the east-norm soution appears to be appropriate, at east no evidence against it was found. 4. the effectiveness of any of the schemes combining shifted grids by taking either the average, the maximum node vaues or the average whie keeping the absoute maximum fixed strongy depends on the distribution and the feature of the reconstruction one is most interested in. The averaging scheme is safe in that it generay reduces the root mean square error of the reconstruction. 5. from an experimenta point of view, the tempora resoution of the measurement is as important as the spatia resoution if the concentration peaks represent emission puffs subject to wind transport (sec ). The discussion of the fan beam geometries in section 8.4 reveas that 6. increasing the number of emitting and/or receiving systems whie keeping the number of ight paths fixed generay eads to better reconstruction resuts (provided that the geometry does not get too irreguar, producing arge gaps). 7. for the sma numbers of ight paths considered in this thesis, making the fans wider in genera deteriorates the reconstruction quaity. 8. for the peak distributions according to 1. the sensitivity to measurement errors does not vary much with the geometry, but can vary within the recontruction area. 9. numerica resuts of simuations for a certain geometry can be carried forward to geometries with different numbers of ight paths, but of a simiar type, by scaing the distributions with the mesh size of the geometry (section 8.4.4). Furthermore, the discussion presents the singuar vaue decomposition as a usefu diagnostic too for the agebraic properties of the measurement system, independenty of the distributions (see aso section 9.4, fig. 9.6), and the optima estimate as a simpe method to anayse different physica scenarios without having to perform engthy simuations (which in this case eads to very simiar resuts, c.f. section 8.4.2).

181 11.1. Concusions 181 Taking the indoor experiment (chapter 9) as an exampe of a rea word measurement of emission puffs where the true concentration distribution is not known proves that ad 1.&3. for peak distributions it is possibe to optimise the discretisation grid in a consistent way without knowing the true width of the peaks. Comparing reconstructions from different grid combination approaches and from a singe grid can ead to a better picture of the true concentration fied (because the schemes are sensitive to different features of the peak) and thus to the a posteriori preference of a certain scheme. 1. the numerica approach to estimate the compete reconstruction error fied from reconstructions of admissibe (random) distributions eads to sensibe upper and ower bounds of this fied and aows to identify artefacts in the origina reconstruction. It is important to get the variabiity of these admissibe distributions right in order to neither over- nor underestimate the reconstruction error. Finay, appying the same reconstruction schemes to highy resoved mode distributions ( smooth distributions 1 ) above a city street canyon (chapter 1) suggests that 11. if the mode distributions refect the true variabiity of trace gas concentrations in the environment of compex buidings, their tomographic reconstruction is a rather chaenging task. 12. contrary to the peak distributions, for the on a arger scae smooth mode distribution the over-determined soution achieves minima overa reconstruction errors and neither grid combination schemes nor subtraction of the background improves the reconstruction. The severe semi-convergence of the under-determined soution suggests that the east-norm seection criterion might not be appropriate for this kind of distribution, but comparison with aternative reconstruction principes for reaistic atmospheric distributions is necessary to cear this point. 13. data errors have strong impact now and, therefore, reguarisation becomes crucia. The infuence of the basis functions on individua contributions to the reconstruction error in the i-posed overdetermined case has not been examined, but it is known, for exampe from atmospheric profie retrieving [e.g., Doicu et a., 24], that the choice of the basis functions affects reguarisation (c.f (iv)). Therefore, from an experimenta point of view the i-posedness of the probem demands minima data errors. 14. contrary to a point measurement that within its errors can directy be compared to a mode prediction at the measurement site, the interpretation of a tomographicay reconstructed fied is not possibe without information on the true variabiity of the atmospheric trace gas. The numerica scheme proposed here, essentiay parametrising the a priori concentrations by functiona bounds and bounds for the gradients, eads to distinct error patterns consistent with the true distributions. Tomographic measurements with ow spatia resoution can be used to verify specific mode predictions if ony the uncertainty of the mode can be narrowed down sufficienty. 1 Which here means they cannot be represented by narrow peaks on a moderatey smooth background

182 Concusion and Outook Outook The resuts of this work wi be appied to the measurements presenty taking pace in Heideberg that were briefy addressed in sec Moreover, current and future technica deveopment of active DOAS instruments, making these smaer, cheaper and easier to hande, wi reduce the expense to set up tomographic experiments with eventuay increasing numbers of ight paths. With simiar progress of the passive technique and a resuts of this thesis equay hoding for any remote sensing method with we defined ight paths, future tomographic MAX-DOAS measurements proposed by Frins et a. [26] (c.f. sec. 3.4) woud represent an extremey versatie appication. Tomographic DOAS measurements from sateite simiar to the suggestion by Feming [1982] (sec. 3.3) are a further exampe that aows straightforward use of our resuts. In principe, everything said hods for IR and LIDAR remote sensing as we. Given this perspective, certain aspects of the reconstruction procedure shoud be reconsidered, where we ony refer to the discrete approach using basis functions with oca support. For appications with reguar ight path geometries ike sateite measurements possiby air craft measurements c.f. fig. 3.5, transform methods might become feasibe (sec ). Methods ike the expicit fit of Gaussian peaks to the measurement data (SBFM, c.f ) require further investigation, especiay with respect to the stabity of their soution (but distributions ike those in fig. 1.3 do not ook very promising for this specia approach). Basis functions Whie it was expicity shown in this thesis that inear parametrisation is superior to piecewise constant parametrisation in every respect for a reguar reconstruction area, the impementation of the biinear discretisation given by eq. (4.13) is not very fexibe when it comes to parametrising concentration fieds on an irreguar area with ony few basis functions. Different schemes, ike trianguar discretisation, might be more appropriate. Another approach worth ooking at is the parametrisation by higher order poynoms. This does make the probem noninear, but B-spines, for exampe (c.f. sec ), have exceent approximation and, as reported by [e.g., Doicu et a., 24], reguarisation properties. Furthermore, they are very fexibe with respect to their knots. [Baussard et a., 24] use a sma number of adaptivey chosen B-spines to parametrise their inverse probem, giving rise to the expectation of sma discretisation and inversion errors. Reconstruction principe In image reconstruction the east-norm principe is frequenty repaced by a maximum-entropy principe, or atogether by a maximum ikeihood approach (sec ). Whether these woud give better resuts for atmospheric reconstruction probems remains to be shown. By the same token, the iterative reguarisation of the east-norm soution adopted here from image reconstruction shoud be compared with aternative methods commony used for atmospheric inverse probems. This woud be, in particuar, the Tikhonov method, which if augmented with a nonnegativity constraint is essentiay the constrained optimisation principe eq. (4.51). This formuation of the reconstruction probem woud become especiay attractive if the reguarisation parameter δ in eq. (4.51) for a given reguarisation matrix D coud be quantitativey derived from atmospheric parameters. For exampe, if D is given by the first differences from maximum gradients of the concentration fied. Fehmers [1996] empoyed

183 11.2. Outook 183 this method for tomographic reconstruction of the ionosphere (see aso fig. 3.5a, p. 44). 2 Estimation of the reconstruction error The Monte Caro estimation of the reconstruction error proposed in this thesis is simpe and fexibe, but aso time consuming and has the disadvantage of taking into account aso distributions that do not agree with the measurements. A scheme that is consistent in this respect woud be highy desirabe. Using the a posteriori covariance of the discrete optima estimate is ony possibe if the a priori can be formuated in terms of Gaussian probabiity densities and the discretisation error can be negected (provided that the east-squares approach has been chosen as reconstruction principe). 2 Computer code for an agorithm soving this and more genera constrained optimisation probems is avaiabe, e.g., from the Numerica Agorithms Group [

184 Part IV. Appendices 184

185 A. Atmospheric Stabiity Casses and Dispersion Coefficients Stabiity casses wind speed day m [m/s] incom. soar radiation (insoation) thiny overcast 3/8 strong moderate sight or 4/8 ow coud coud cover cover < 2 A A B B F F 2 3 A B B C E F 3 5 B B C C D E 5 6 C C D D D D > 6 C D D D D Tabe A.1: Atmospheric stabiity categories based on wind speed and insoation (see text). On the basis of experimenta observations (the Nebraska Prairie Grass Project) Pasqui [1961] proposed to describe different states of the atmosphere by means of six categories A to F derived from readiy avaiabe meteoroogica observabes. Tabe A.1 defines them in terms of wind speed and insoation [e.g., Barratt, 22]. Estimation of the insoation from soar eevation and sky cover can be found in [Barratt, 22, tabe 3.9]. Tabe A.2 reates the casses to some meteoroogica parameters. Pasqui cassification phenomena freq. of mean wind boundary ayer σ α stabiity occur. [%] speed [m/s] depth [m] A extremey strong therma unstabe instabiity, bright sun A B B moderatey transitiona period unstabe moderate mixing B C C sighty unstabe transitiona periods, sight mixing C D D neutra strong winds, overcast day/night transitions E sighty stabe transitiona periods, night-time mod. winds F mod. stabe cear night time skies, very im. vert. mixing F G G extremey stabe Tabe A.2: Seected meteoroogica parameters reated to stabiity categories [Barratt, 22, tabes 3.7,13]. The parameter σ α was defined by eqs. (2.2) as σ α = σ v/ū. Vaues are taken from [Backadar, 1997, tabe 1.1]. Estimates of T L for various atmospheric stabiities have been given by severa authors. Those proposed by 185

186 186 A. Atmospheric Stabiity Casses and Dispersion Coefficients Draxer are given in the foowing tabe (see Backadar [1997] and references therein): T L surface source eevated source [s] stabe unstabe stabe unstabe atera vertica Dispersion coefficients There are numerous parametrisations of the pume dispersion coefficients defined in sec. 2.4 [Seinfed and Pandis, 1998]. Tabes A.3, A.4 give functionas forms known as Pasqui-Gifford and Briggs curves, respectivey [Barratt, 22]. Samping times are around 1 min. Stabiity cass x[m] σ x(x) = σ y(x)[m] σ z(x)[m] A x x og 1 σ z = og 1 x (og 1 x) 2 B x x og 1 σ z = og 1 x +.18 (og 1 x) 2 C x x.91 D x.9.93 x og 1 σ z = og 1 x +.61 (og 1 x) 2 E x x og 1 σ z = og 1 x +.7 (og 1 x) 2 F x.9.57 x og 1 σ z = og 1 x (og 1 x) 2 Tabe A.3: Pasqui-Gifford parametrisation of the dispersion coefficients. Stabiity cass σ x(x) = σ y(x)[m] σ z(x) [m] Open country A.22 x ( x).5.2 x B.16 x(- -).5.12 x C.11 x(- -).5.8 x( x).5 D.8 x(- -).5.6 x( x).5 E.6 x(- -).5.3 x( x) 1 F.4 x(- -).5.16 x( x) 1 Urban A B.32 x ( x).5.24 x( x) +.5 C.22 x(- -).5.2 x D.16 x(- -).5.14 x( x).5 E F.11 x(- -).5.8 x( x).5 Tabe A.4: Briggs parametrisation of the dispersion coefficients for rura and urban conditions.

187 B. Generation of Random Test Distributions We consider the case that the 2-D random scaar fieds c I(r), I = 1,..., N, are generated from a given function c(r) in the foowing way. The c I(r) are defined on a finite number of grid nodes r J, J = 1,..., n x n y such that 1. for the function vaues on the nodes c min[c(r J)] c I(r J) c max[c(r J)], i.e. ower and upper bounds on each node are functions of c s vaue on it. For exampe, f min c(r J) c I(r J) f max c(r J), (B.1) (B.2a) ike a factor of 2 uncertainty, or c(r J) [c(r J)] c I(r J) c(r J) + [c(r J)], (B.2b) ike a ±5% uncertainty. Depending on the spacing of the nodes r J, this constraint on its own can resut in much arger gradients than the ones of the origina fied c, and for very fine grids it eads to unphysicay fuctuating vaues. Therefore, it is necessary to restrict the gradients and possiby higher derivatives of c I. Here, we restrict ourseves to the first differences and a constraint of the specia form 2. for the first differences f c(r J) min/max ci(rj) f c(r J) max/min x x x for c(r J) x (B.3) and the same for y. To generate variabiity where the origina fied is very smooth, e.g. c, this sign preserving reation is reaxed to f min and the same for y. c(rj) ci(rj) f c(rj) max x x x if c(rj) some treshod, x (B.4) We further consider two different impementations of the constraints. (i) Direct Monte-Caro generation of the c I(r J) such that one of the equations (B.2) and eq,. (B.3), (B.4) are satisfied, starting on a certain grid node. The gradients of c have to be computed beforehand. (ii) Convoution c I(r) = r(r) c(r) 187

188 188 B. Generation of Random Test Distributions with a random fied r constrained by r min r(r J) r max, (B.5) r max x and the same for y, (B.6) such that eq. (B.2a) with f min = f max and eq. (B.3) are fufied. This impies The fied r is produced as in (i). r min/max = f min/max, r max = f max f max c(r J) max( r(rj), r(rj) ). x y Both methods have been tested and, for the right choice of parameters, ead to simiar resuts. In any case, the mean and standard deviation of a c I shoud be checked for bias and actua variabiity within the above bounds. Uness stated differenty, a random numbers are uniformy distributed.

189 C. Auxiiary Cacuations Proof of equation (5.8a): Estimate of the discretisation error It is with c disc(r) = c(r) [φ(r)] T Φ 1 c Z c j = dv φ j(r)c(r) Ω j Z Φ jj = dv φ j(r)[φ j (r)] T Ω j and Ω j the support of basis function φ j. Expand c j around a point ρ = (x ρ, y ρ) in Ω j: Z c j c(ρ) dv φ j(r) Ω j + c Z x (ρ) dv φ j(r)(x x ρ) + (x y) Ω j Z c 2 x (ρ) dv φ j(r)(x x ρ) 2 + (x y) 2 Ω j Z + 2 c x y (ρ) dv φ j(r)(x x ρ)(y y ρ). Ω j Taking ρ as the grid node r j in the case of the biinear basis and as the box centre in the case of the box basis, inear terms in x and y vanish for pixes entirey in the reconstruction area. The integra in the first contribution is Z Ω j dv φ j(r) = x i y i = A i, where A i is the area of the rectange formed by four grid nodes and in the case of the biinear basis the grid is assumed to be reguar in both x-and y-direction merey for simpicity. Simiary, for the quadratic term 8 Z < 1 dv φ j(r)(x x j) 2 12 = Aj( xj)2 box, Ω j : 1 6 Aj( xj)2 reguar biinear and the same for y. The vector c gets thus 8 < A j c(r j) c j : A j c(r j) ( x j) 2 2 c x 2 + ( y j) 2 2 c (r y j) 2 ( x j) 2 2 c x 2 + ( y j) 2 2 c y 2 (r j) box, reguar biinear which aows to estimate c disc(r) if Φ 1 is known expicitey. For the box basis Φ 1 ij = A 1 i δ ij gives the desired expression. The cacuation of Φ 1 in the biinear case is more invoved and not considered here. 189

190 19 C. Auxiiary Cacuations Proof of equations (6.11) & (6.12): Fitting the background In the over-determined case the east-squares probem of eq. (6.1b) min d By 2, with B = (A ) y = (x, is given by c BG) T y = (B T B) 1 B T d A T A A T = (A T ) T T! 1 A T d T d!. The inverse I = ` J j j T j in the ast expression can be computed from giving Inserting these expressions into yieds equation (6.11). A T A A T (A T ) T T! j = `A T ( T T )A 1 A T, I =½, J = T `A T ( T T )A 1, j = ( T ) T A`A T ( T T )A A T. x = JA T d + ( T d)j, c BG = j T A T d + ( T d)j The under-determined case min y y T Hy, d = B y contains the norms x 2 2, y 2 2 and P j (x j c BG) 2 by the choices H = C A, H = C A and H = n 1, C A respectivey. The norma equations of the second kind for the above minimum-norm principe Hy = 1 2 BT γ, d = By cannot be soved straightforwardy if H is singuar, which is the case for the ast two matrices H. Writing H =! H h h T h

191 191 with non-singuar H, the norma equations take the form with soution where Hx + hc BG = 1 2 AT γ, h T x + hc BG = 1 2 T γ, Ax + c BG = d c BG = N 1 v T (AH 1 A T ) 1 d, n x = H 1 A T N 1 o H 1 A T (AH 1 A T ) 1 vv T H 1 hh T H 1 A T + H 1 h T (AH 1 A T ) 1 d, N = v T (AH 1 A T ) 1 v h T H 1 h + h, v = AH 1 h and γ = 2(AH 1 A T ) 1`(AH 1 h ) c BG + d. Equations (6.12) foow for the specia choices of H, h and h above. The sum x + c BG(1,..., 1) T = x + c BG e can be written as x + c BG e = H 1 A T (AH 1 A T ) 1 d + ½ H 1 A T (AH 1 A T ) 1 A x a where Ae = was used and x a = c BG H 1 (He h) can be interpreted as a priori (c.f., e.g. eq. (4.42)). Particuary, x a = c BG e for the matrices H above. Proof of equations (6.17) & (6.18): Additiona constraints In the over-determined case (m + f n), the east-squares soution of Ax = d, with A =! d d =, in the form of eq. (6.17) immediatey foows from c A F! and x = (A T A) 1 A T d by the matrix mutipication in bocks: A T A = A T F T A F! = A T A + F T F. The under-determined case with a priori x a has the soution x = x a + A T (AA T ) 1 (d Ax a),

192 192 C. Auxiiary Cacuations where the cacuation of the inverse (AA T ) 1 = I = ` I 1 I 3 I T 3 I 2 is carried out as on page 19 and yieds I 1 = (AP FA T ) 1, I 2 = (FP AF T ) 1, I 3 = (AP FA T ) 1 AF T (FF T ) T = (AA T ) 1 AF T (FP AF T ) 1, with P A defined by P A =½ A T (AA T ) 1 A, see eq. (4.55), and the same for P F. For A T (AA T ) 1 = ( A T I 1 + F T I T 3 which competes the proof. A T I 3 + F T I 2 ) one gets thus A T I 1 + F T I T 3 = P FA T (AP FA T ) 1, A T I 3 + F T I 2 = P AF T (FP AF T ) 1, Proof of equations (7.8)-(7.1): Point measurements as additiona a priori The augmented a priori x a is constructed such that f = rank[f] of its degrees of freedom are determined by the point measurements, expressed as Fx = c, and the remaining n f degrees from the ad hoc a priori x a. This is in fact a pure east-norm probem with soution according to eq. (4.56c) x a = F T (FF T ) 1 c + P Fx a. (C.1) As it provides an instructive approach to the east-norm probem, we derive eq. (C.1) expicity by using the singuar vaue decomposition F = UΣV T and write fx x aj = V jj x a + j n X j =1 j =f+1 V jj x a j. (C.2) Inserting this into Fx a = UΣx a = c gives the unique soution x a j = Σ 1 j (U T c) j for j = 1,..., f by projecting onto the orthogona compement of the nuspace. The remaining components are set to the components of x a in the system of the singuar vectors v Inserting these equations back into eq. (C.1) gives x a j = (V T x a) j for j = p + 1,..., n. x aj = (Σ U T c) j + Σ n j =f+1v jj V T j kx ak, where the matrix product in the second sum is the projector onto the nuspace of F and Σ = ` Σ 1 r as in eq. (4.36). Using the singuar vaue decomposition, the first expression can be seen to be (F T (FF T ) 1 c) j, thus estabishing eq. (7.8). Equation (7.9) foows immediatey from eq. (4.36). The augmented covariance matrix S a can ike any matrix inên n be decomposed in the foowing way S a = P F S ap F + P F S ap F + P F S ap F + P F S ap F,

193 193 where P F is again the projector onto the nuspace of F and PF the projector onto its orthogona compement. The form eq. (7.1) foows from 1. c = Fx a PF S apf = F T (FF T ) 1 S c(ff T ) 1 F, with S c = Eˆ(c c)(c c). T 2. No correation between the physica and unphysica subspaces reated to the point measurements and the ad hoc a priori. 3. P F S ap F = P FS ap F.

194 D. Software & Numerics CWinApp CDocManager CDocTempate 1 CDocument casses for agorithms, fieds, parametrisation, data, geometry evauation etc. CMainFrame... 1 CChidFrame 1 CView 1... n 2-D/3-D view casses for grid, ight paths, fieds Figure D.1: The MFC Singe Document/View architecture (simpified). Dashed boxes indicate code that remains argey untouched, round boxes contain the actua C++ casses for cacuations and dispay. Most of the cacuations were carried out within a programme based on code originay deveoped by T. Laeppe for reconstruction and dispay of experimenta or simuated data [Laeppe et a., 24]. The code is written in C++ and based on the Microsoft Foundation Casses (MFC), using the Singe Document/View architecture. MFC is a Microsoft C++ ibrary providing, e.g., casses and methods for windows as they are known from the operating software with the same name. The doc/view architecture is a specia way to manage the storage of data (in the document), its dispay in one or mutipe windows (views), the user interaction and the coordination and update of the views. Singe doc/view means that there can be ony one document in the appication, but sti mutipe views of it (In fact, this architecture has been decared obsoete by now). The virtua agorithms represent ony part of the compete code that is automaticay created by Visua Studio s (the Microsoft programming interface) appication wizard. Actua graphic dispay of the data uses the freeware OpenGL casses. Whie the particuar architecture aows convenient visuaisation, the whoe code seems heaviy overoaded for purey scientific purposes and is sometimes hard to modify without some knowedge about MFC. Furthermore, it reies on Visua Studio, which is ony commerciay avaiabe. Practicay a origina casses have been modified for the simuations of this thesis. The agorithm for the singuar vaue decomposition (as we as vector and matrix casses) were adopted from the TNT/JAMA ibrary. 1 Any other specia routines have been taken from [Press et a., 1992]. A coour pots in this thesis have been generated by a programme deveoped by B.C. Song which essentiay does the same job as the earier version by Laeppe, but without using Microsoft s or any other commercia ibrary. Contour pots have been created using the freeware gnupot. 1 Pozo, R., Mathematica and Computationa Sciences Division, Nationa Institute of Standards and Technoogy, 194

195 References Andersen, A. H., and A. C. Kak, Simutaneous Agebraic Reconstruction Technique (SART): A superior impementation of the ART agorithm, Utrason. Imag., 6, 81 94, Austen, J. R., S. J. Franke, C. H. Liu, and K. C. Yeh, Appication of computerized tomography techniques to ionospheric research, Proceedings of the URSI and COSPAR Internationa Beacon Sateite Symposium on Radio Beacon Contribution to the Study of Ionization and Dynamics of the Ionosphere and to Corrections to Geodesy and Technica Workshop, part 1, 23 35, Backus, G. E., and J. F. Gibert, Uniqueness in the inversion of inaccurate gross earth data, Phi. Trans. R. Soc. Lond., 266, , 197. Barad, M. L. (Ed.), Project Prairie Grass, A Fied Program in Diffusion, vo. I and II of Geophysica Research Papers, No. 59, Air Force Cambridge Research Center, 1958, Report AFCRC-TR Barratt, R., Atmospheric dispersion modeing. An introduction to practica appications, Business and Environment Practitioner Series, Earthscan Pubications Ltd., London (UK), 22. Bäumer, D., B. Voge, and F. Fieder, A new parameterisation of motorway-induced turbuence and its appication in a numerica mode, Atmos. Environ., 39(31), , 25. Baussard, A., E. L. Mier, and D. Préme, Adaptive B-spine scheme for soving an inverse scattering probem, Inverse Probems, 2(2), , 24. Beotti, C., F. Cuccoi, L. Facheris, and O. Vasei, An appication of tomographic reconstruction of atmospheric CO 2 over a vocanic site based on open-path IR aser measurements, IEEE Trans. Geosci. Remote Sensing, 41(11), , 23. Beniston, J. P., M. Wof, M. Beniston-Rebetez, H. J. Kösch, P. Rairoux, and L. Wöste, Use of Lidar measurements and numerica modes in air poution research, J. Geophys. Res., 95(D7), , 199. Berkowicz, R., O. Herte, N. N. Sørensen, and J. A. Michesen, in Modeing Air Poution from Traffic in Urban Areas. Fow and Dispersion Through Obstaces, edited by R. J. Perkins and S. E. Becher, pp , 1997, Carendon Press, Oxford, Björck, Å., Numerica methods for east squares probems, SIAM, Phiadephia, Backadar, A. K., Turbuence and diffusion in the atmosphere, Springer, Band, V. V., J. P. Guarco, and T. V. Ederedge, Observations of NO 2 formation in two arge natura gas fired boiers, Proceedings of the 2 Internationa Joint Power Generation Conference, 2. Bobrowski, N., G. Hönninger, B. Gae, and U. Patt, Detection of bromine monoxide in a vocanic pume, Nature, 423, , 23. Brandt, J., J. H. Christensen, L. M. Frohn, F. Pamgren, R. Berkowicz, and Z. Zatev, Operationa air poution forecasts from European to oca scae, Atmos. Environ., 35(Supp. 1), S91 S98,

196 196 References Briggs, G. A., Diffusion estimation for sma emissions, ARL Report ATDL 16, NOAA, Britter, R. E., and S. R. Hanna, Fow and dispersion in urban areas, Ann. Rev. Fuid Mech., 35, , 23. Byer, R. L., and L. A. Shepp, Two-dimensiona remote air poution monitoring via tomography, Opt. Lett., 4, 75 77, Byrne, C., Likeihood maximization for ist-mode emission tomographic image reconstruction, IEEE Trans. Medica Imaging, 2(1), 13 12, 21. Byrne, C., A unified treatment of some iterative agorithms in signa processing and image reconstruction, Inverse Probems, 2, 13 12, 24. Cavetti, D., G. Landi, L. Reiche, and F. Sgaari, Nonnegativity and iterative methods for i-posed probems, Inverse Probems, 2, , 24. Carin, B. P., and T. A. Louis, Bayes and empirica Bayes methods for data anaysis, Chapman & Ha, Roca Raton, Censor, Y., and G. T. Herman, On some optimization techniques in image reconstruction from projections, App. Numer. Math., 3, , Corsmeier, U., M. Koher, B. Voge, H. Voge, and F. Fieder, BAB II: A project to evauate the accuracy of rea-word traffic emissions for a motorway, Atmos. Environ., 39, , 25a. Corsmeier, U., D. Imhof, M. Koher, J. Kühwein, R. Kurtenbach, M. Petrea, E. Rosenbohm, B. Voge, and U. Vogt, Comparison of measured and mode-cacuated rea-word traffic emissions, Atmos. Environ., 39, , 25b. Dempster, A., N. Laird, and D. Rubin, Maximum ikeihood from incompete data via the EM agorithm, J. Roy. Statist. Soc., 39, 1 38, Doicu, A., F. Schreier, and M. Hess, An iterative reguarization method with B-spine approximation for atmospheric temperature and concentration retrievas., Environmenta Modeing and Software, 2, , 24. Drescher, A. C., A. J. Gadgi, P. N. Price, and W. W. Nazaroff, Nove approach for tomographic reconstruction of gas concentration distributions in air: Use of smooth basis functions and simuated anneaing, Atmos. Environ., 3(6), , Drescher, A. C., D. Y. Park, A. J. Gadgi, S. P. Levine, and W. W. Nazaroff, Stationary and time-dependent tracer gas concentration profies using open path FTIR remote sensing and SBFM computed tomography, Atmos. Environ., 31, , Ducaux, O, E. Frejafon, H. Schmidt, A. Thomasson, D. Mondeain, J. Yu, C. Guiaumond, C. Pue, F. Savoie, P. Ritter, J. P. Boch, and J. P. Wof, 3D-air quaity mode evauation using the Lidar technique, Atmos. Environ., 36, , 22. Fehmers, G. C., Tomography of the ionosphere, Ph.D. dissertation, Technische Universiteit Eindhoven, The Netherands, Fehmers, G. C., An agorithm for quadratic optimization with one quadratic constraint and bounds on the variabes, Inverse Probems, 14, , 1998.

197 References 197 Fesser, J. A., Penaized weighted east-squares image reconstruction for positron emission tomography, IEEE Trans. Medica Imaging, 13(2), 29 3, Finayson-Pitts, B., and J. N. Pitts, Chemistry of the Upper and Lower Atmosphere: Theory, Experiments, and Appications, Academic Press, New York, 2. Fischer, M. L., P. N. Price, T. L. Thatcher, C. A. Schwabe, M. J. Craig, E. E. Wood, R. G. Sextro, and A. J. Gadgi, Rapid measurements and mapping of tracer gas concentrations in a arge indoor space, Atmos. Environ., 35, , 21. Fisher, B., J. Kukkonen, M. Piringer, M. W. Rotach, and M. Schatzmann, Meteoroogy appied to urban air poution probems: Concepts from COST 715, Atmos. Chem. Phys., 6, , 26. Feming, H. E., Sateite remote sensing by the technique of computed tomography, J. App. Meteoro., 21, , Frins, E., N. Bobrowski, U. Patt, and T. Wagner, Tomographic mutiaxis-differentia optica absorption spectroscopy observations of Sun-iuminated targets: a technique providing we-defined absorption paths in the boundary ayer, App. Opt., 45, , 26, accepted. Geyer, A., B. Aicke, S. Konrad, J. Stutz, and U. Patt, Chemistry and oxidation capacity of the nitrate radica in the continenta boundary ayer near Berin, J. Geophys. Res., 16(D8), , 21. Gifford, F. A., Use of routine meteoroogica observations for estimating atmospheric dispersion, Nuc. Saf., 2, 47 51, Gibert, P. F. C., Iterative methods for the three-dimensiona reconstruction of an object from projections, J. Theoret. Bio., 36, , Giui, D., L. Facheris, and S. Tanei, Microwave tomographic inversion technique based on stochastic approach for rainfa fieds monitoring, IEEE Trans. Geosci. Remote Sensing, 37(5), , Goub, C. H., and C. F. van Loan, Matrix computations, John Hopkins University Press, Batimore, Gordon, R., R. Bender, and G. T. Herman, Agebraic reconstruction techniques (ART) for three-dimensiona microscopy and X-ray photography, J. Theoret. Bio., 29, , 197. Groetsch, C. W., Inverse probems in the mathematica sciences, Vieweg, Braunschweig, Gu, S. F., and G. J. Danie, Image reconstruction from incompete and noisy data, Nature, 272, , Hadamard, J., Sur es probèmes aux dérivées partiees et eur signification physique, pp , 192, Princeton University Buetin, 192. Hak, C., I. Pundt, S. Trick, C. Kern, U. Patt, J. Dommen, C. Ordóñez, A. S. H. Prévôt, W. Junkermann, C. Astorga-Loréns, B. R. Larsen, J. Meqvist, A. Strandberg, Y. Yu, B. Gae, J. Keffmann, J. Lörzer, G. O. Braathen, and R. Vokamer, Intercomparison of four different in-situ techniques for ambient formadehyde measurements in urban air, Atmos. Chem. Phys., 5, , 25. Hanna, S. R., Confidence imit for air quaity modes as estimated by bootstrap and jacknife resamping methods, Atmos. Environ., 23, , Hanna, S. R., and E. M. Insey, Time series anayses of concentration and wind fuctuations, Boundary-Layer Meteoro., 47, , 1989.

198 198 References Hanna, S. R., R. Britter, and P. Franzese, A baseine urban dispersion mode evauated with Sat Lake City and Los Angees tracer data, Atmos. Environ., 37, , 23. Hansen, P. C., Rank-deficient and discrete i-posed probems, Siam, Phiadephia, Hart, A., B.-C. Song, and I. Pundt, Reconstructing 2d concentration peaks from Long Path DOAStomographic measurements: Parametrisation and geometry within a discrete approach, Atmos. Chem. Phys., 6, , 26. Hashmonay, R. A., M. G. Yost, and C.-F. Wu, Computed tomography of air poutants using radia scanning path-integrated optica remote sensing, Atmos. Environ., 33, , Hecke, A., A. Richter, T. Tarsu, F. Wittrock, C. Hak, I. Pundt, W. Junkermann, and J. P. Burrows, MAX- DOAS measurements of formadehyde in the Po-vaey, Atmos. Chem. Phys., 5, , 25. Hed, T., D. P. Y. Chang, and D. A. Niemeier, UCD 21: an improved mode to simuate poutant dispersion from roadways, Atmos. Environ., 37, , 23. Herman, A., and S. W. Rowand, Three methods for reconstructing pictures from X-rays: A comparative study, Comput. Graph. Image Proc., 1, , Herman, G. T., A. Lent, and S. W. Rowand, ART: Mathematics and appications. A report on the mathematica foundations and on the appicabiity to rea data of the agebraic reconstruction techniques, J. Theor. Bio., 42, 1 32, Heue, K.-P., Airborne Muti Axis DOAS instrument and measurements of two dimensiona tropospheric trace gas distributions, Ph.D. dissertation, Institut für Umwetphysik, Universität Heideberg, Germany, 25. Heue, K.-P., A. Richter, M. Bruns, J. P. Burrows, C. v. Friedeburg, U. Patt, I. Pundt, P. Wang, and T. Wagner, Vaidation of SCIAMACHY tropospheric NO 2-coumns with AMAXDOAS measurements, Atmos. Chem. Phys., 5, , 25. Hönninger, G., C. v. Friedeburg, and U. Patt, Muti Axis Differentia Absorption spectroscopy (max-doas), Atmos. Chem. Phys., 4, , 24a. Hönninger, G., H. Leser, O. Sebastián, and U. Patt, Ground-based measurements of haogen oxides at the Hudson Bay by Active Long Path Doas and Passive MAX-DOAS, Geophys. Res. Lett., 31(L4111), 24b. Hönninger, G., N. Bobrowski, E. R. Paenque, R. Torrez, and U. Patt, Reactive bromine and sufur emissions at Saar de Uyuni, Boivia, J. Geophys. Res., 31(L411), 24c. Hurey, P., W. L. Physick, A. K. Luhar, and M. Edwards, The air poution mode (TAPM) version 3. part 2: Summary of some verification studies., CSIRO Atmospheric Research Technica Papers No. 72, CSIRO, 25. ISC-3, User s guide for Industria Source Compex (ISC 3) Dispersion Modes, vo. 2 (Description of mode agorithms), EPA 454/B 95 3a, USEPA, Jiang, M., and G. Wang, Convergence studies on iterative agorithms for image reconstruction, IEEE Trans. Med. Imaging, 22(5), , 23. Kaczmarz, K., Angenäherte Aufösung von Systemen inearer Geichungen, Bu. Acad. Poon. Sci., A35, , Kak, A. C., and M. Saney, Principes of computerized tomographic imaging, Siam, Phiadephia, 21.

199 References 199 Kessemeier, J. K., and M. Staudt, Biogenic voatie organic compounds (VOC): An overview on emission, physioogy and ecoogy, J. Atmos. Chem., 33(1), 23 88, Kesser, Ch., W. Brücher, M. Memmesheimer, M. Kerschgens, and A. Ebe, Simuation of air poution with nested modes in North Rhine-Westphaia, Atmos. Environ., 35(Supp. 1), S3 S12, 21. Koher, M., U. Corsmeier, U. Vogt, and B. Voge, Estimation of gaseous rea-word traffic emissions downstream a motorway, Atmos. Environ., 39, , 25. Kösch, H. J., Probing the Atmosphere: Air Poution Studies by LIDAR, Ph.D. dissertation, Freie Universität Berin, Germany, 199. Kösch, H. J., P. Rairoux, J. P. Wof, and L. Wöste, Simutaneous NO and NO 2 DIAL measurements using BBO crystas, App. Opt., 28(11), , Kunitsyn, V. E., E. S. Andreeva, A. Y. Popov, and O. G. Razinkov, Methods and agorithms of ray radiotomography for ionospheric research, Annaes Geophysicae, 13(12), , Laeppe, T., V. Knab, K. U. Mettendorf, and I. Pundt, Longpath DOAS tomography on a motorway exhaust pume: Numerica studies and appication to data from the BAB II campaign, Atmos. Chem. Phys., 4, , 24. Leahy, R. M., and C. L. Byrne, Recent deveopments in iterative image reconstruction for PET and SPECT., IEEE Trans. Med. Imaging, 19, , 21. Lenz, C.-J., F. Müer, and K. H. Schünzen, The sensitivity of mesoscae chemistry transport mode resuts to boundary vaues, Env. Monitoring and Assessment, 65, , 2. Linz, P., Uncertainty in the soution of inear operator equations, BIT, 24, 92 11, Ma, Y., Z. Boybeyi, S. Hanna, and K. Chayantrakom, Pume dispersion anaysis on surface concentration and its fuctuations from MVP fied experiment, Atmos. Environ., 39, , 25. McEroy, J. L., and F. Pooer, The St. Louis dispersion study, Report AP 53, U.S. Pubic Heath Service, Nationa Air Poution Contro Administration, Mettendorf, K. U., Aufbau und Einsatz eines Mutibeam Instrumentes zur DOAS-tomographischen Messung zweidimensionaer Konzentrationsverteiungen, Ph.D. dissertation, Institut für Umwetphysik, Universität Heideberg, Germany, 25. Mettendorf, K. U., A. Hart, and I. Pundt, An indoor test campaign of the tomography ong path differentia absorption spectroscopy, J. Environ. Monit., 8, , 26. Minerbo, G., MENT: A maximum entropy agorithm for reconstructing a source from projection data, Comput. Graphics Image Process., 1, 48 68, Moreira, D. M., J. C. Carvaho, and T. Tirabassi, Pume dispersion simuation in ow wind conditions in the stabe and convective boundary ayers, Atmos. Environ., 39, , 25. Mueer, K., and R. Yage, Rapid 3D cone-beam reconstruction with the Simutaneous Agebraic Reconstruction Technique (SART) using 2D texture mapping hardware, IEEE Trans. Med. Imag., 19, , 2. Munk, W., P. Worcester, and C. Wunsch, Ocean acoustic tomography, Cambridge Univ. Pr., Cambridge UK, 1995.

200 2 References Natterer, F., The mathematics of computerized tomography, Siam, Phiadephia, 21. Neophytou, M. K., D. A. Goussis, E. Mastorakos, and R. E. Britter, The conceptua deveopment of a simpe scae-adaptive reacting poutant dispersion mode, Atmos. Environ., 39, , 25. Noet, G. (Ed.), Seismic Tomography with appications in goba seismoogy and exporation geophysics, D. Reide Pubishing Company, Dordrecht, Ocese, L. E., and B. M. Tosei, Deveopment of a mode for reactive emissions from industria stacks, Environmenta Modeing & Software, 2, , 25. Pasqui, F., The estimation of the dispersion of windborne materia, Metero. Mag., 9, 33 49, Perner, D., D. H. Ehhat, H. W. Pätz, U. Patt, E. P. Röth, and A. Voz, OH-radicas in the ower troposphere, Geophys. Res. Lett., 3, , Peters, C., S. Pecht, J. Stutz, K. Hebestreit, G. Hönninger, K. G. Heumann, A. Schwarz, J. Winterik, and U Patt, Reactive and organic haogen species in three different European coasta environments, Atmos. Chem. Phys., 5, , 25. Petritoi, A., R. Ravegnani, G. Giovanei, D. Bortoi, U. Bonaf, I. Kostadinov, and A. Ouanovsky, Off-axis measurements of atmospheric trace gases by use of an airborne utravioet-visibe spectrometer, App. Opt., 41(27), , 22. Phiip, N. P., Poutant tomography using integrated concentration data from non-intersecting optica paths, Atmos. Environ., 33, , Pieke, R. A., W. R. Cotton, R. L. Wako, C. J. Tremback, W. A. Lyons, L. D. Grasso, M. E. Nichos, M. D. Moran, D. A. Wesey, T. J. Lee, and J. H. Copeand, A comprehensive meteoroogica modeing system RAMS, Meteor. Atmos. Phys., 49, 69 91, Patt, U., Dry deposition of SO 2, Atmos. Environ., 12, , Patt, U., Air monitoring by Differentia Optica Absorption Spectroscopy (DOAS), in Air Monitoring by Spectroscopic Techniques, vo. 127 of Chemica Anaysis Series, edited by M. W. Sigrist, 1994, John Wiey & Sons, Inc., New York, Patt, U., D. Perner, G. W. Harris, A. M. Winer, and J. N. Pitts, Detection of NO 3 in the pouted troposphere by differentia optica absorption, Geophys. Res. Lett., 7, 89 92, 198. Pöher, D., Tomographic two-dimensiona Long Path DOAS measurements of trace gas distributions above the city of Heideberg, Phd progress report, University of Heideberg, 26. Press, W. H., B. P. Fannery, and S. A. Teukosky, Numerica recipes in C: The art of scientific computing, Cambridge Univ. Pr., Cambridge UK, Price, P. N., M. L. Fischer, A. J. Gadgi, and R. G Sextro, An agorithm for rea-time tomography of gas concentrations using prior information about spatia derivatives, Atmos. Environ., 35, , 21. Puen, J., J. P. Boris, T. Young, G. Patnaik, and J. Isein, A comparison of contaminant pume statistics from a Gaussian puff and urban CFD mode for two arge cities, Atmos. Environ., 39, , 25. Pundt, I., and K. U. Mettendorf, Mutibeam ong-path differentia optica absorption spectroscopy instrument: A device for simutaneous measurements aong mutipe ight paths, App. Opt., 44(23), , 25.

201 References 21 Pundt, I., K. U. Mettendorf, T. Laeppe, V. Knab, P. Xie, J. Lösch, C. von Friedeburg, U. Patt, and T. Wagner, Measurements of trace gas distributions using Long-path DOAS-Tomography during the motorway campaign BAB II: experimenta setup and resuts for NO 2, Atmos. Environ., 39(5), , 25. Reis, M. L., and N. C. Roberty, Maximum entropy agorithms for image reconstruction from projections, Inverse Probems, 8, , Ridde, A., D. Carruthers, A. Sharpe, C. McHugh, and J. Stocker, Comparisons between FLUENT and ADMS for atmospheric dispersion modeing, Atmos. Environ., 38, , 24. Rippe, B., Vorarbeiten für Langpfad DOAS Tomographische Messungen über der Stadt Heideberg, Master s thesis, Institut für Umwetphysik, Universität Heideberg, Germany, 25. Rodgers, C. D., Inverse methods for atmospheric sounding: Theory and practice, Word Scientific, London, 2. Samanta, A., and L. A. Todd, Mapping chemicas in air using an environmenta CAT scanning system: Evauation of agorithms, Atmos. Environ., 32, , Sauer, K., and C. Bouman, A oca update strategy for iterative reconstruction from projections, IEEE Trans. on Signa Processing, 41(2), , Scaes, J. A., and A. Gersztenkorn, Robust methods in inverse theory, Inverse Probems, 4, , Scaes, J. A., and R. Snieder, To Bayes or not to Bayes, Geophysics, 62, , Schatzmann, M., and B. Leit, Vaidation and appication of obstace-resoving urban dispersion modes, Atmos. Environ., 36, , 22. Schatzmann, M., W. Bächin, S. Emeis, J. Kühwein, B. Leit, W. J. Müer, K. Schäfer, and H. Schünzen, Deveopment and vaidation of toos for the impementation of European air quaity poicy in Germany (Project VALIUM), Atmos. Chem. Phys., 6, , 26. Schünzen, K. H., K. Bigake, C. Lüpkes, U. Niemeier, and K. v. Sazen, Concept and reaization of the mesoscae transport- and fuid-mode METRAS, METRAS Techn. Rep. 5, Meteoroogica Institute, University of Hamburg, Schünzen, K. H., D. Hinneburg, O. Knoth, M. Lambrecht, B. Leit, S. Lopez, C. Lüpkes, H. Panskus, E. Renner, M. Schatzmann, T. Schoenemeyer, S. Trepte, and R. Woke, Fow and transport in the obstace ayer First resuts of the microscae mode MITRAS, J. Atmos. Chem., 44, , 23. Schäfer, K., S. Emeis, H. Hoffmann, C. Jahn, W. J. Müer, B. Heits, D. Haase, W.-D. Drunkenmöe, W. Bächin, H. Schünzen, B. Leit, F. Pascheke, and M. Schatzmann, Fied measurements within a quarter of a city incuding a street canyon to produce a vaidation data set, Int. J. Environ. Po., 25, , 25. Seinfed, J. H., Atmospheric chemistry and physics of air poution, Wiey and Sons, Seinfed, J. H., and S. N. Pandis, Atmospheric chemistry and physics, Wiey and Sons, Shepp, L. S., and Y. Vardi, Maximum ikeihood reconstruction for emission tomography, IEEE Trans. Med. Imaging, MI 1, , Shirey, T. R., W. H. Brune, X. Ren, J. Mao, R. Lesher, B. Cardenas, R. Vokamer, L. T. Moina, M. J. Moina, B. Lamb, E. Veasco, T. Jobson, and M. Aexander, Atmospheric oxidation in the Mexico City Metropoitan Area (MCMA) during Apri 23, Atmos. Chem. Phys., 6, , 26.

202 22 References Song, C.H., G. Chen, S.R. Hanna, J. Crawford, and J.J. Davis, Dispersion and chemica evoution of ship pumes in the marine boundary ayer: Investigation of O 3/NO y/ho x chemistry, J. Geophys. Res., 18(D4), 4143, 23. Souhac, L., C. Pue, O. Ducaux, and R. J. Perkins, Simuations of atmospheric poution in Greater Lyon an exampe of the use of nested modes, Atmos. Environ., 37, , 23. Stern, R., and R.J. Yamartino, Deveopment and first evauation of micro-cagrid: A 3-D urban-canopy-scae photochemica mode, Atmos. Environ., 35(Supp. 1), S149 S165, 21. Stockwe, W. R., P. Middeton, J. S. Chang, and X. Tang, The second generation regiona acid deposition mode chemica mechanism for regiona air quaity modeing, J. Geophys. Res., 95(D1), , 199. Stu, R. B., An Introduction to Boundary Layer Meteoroogy, Kuwer Acad. Pub., Dordrecht, Boston, London, Stutz, J., and U. Patt, Numerica anaysis and error estimation of differentia optica absorption spectroscopy measurements with east squares methods, App. Opt., 35, , Stutz, J., and U. Patt, A new generation of DOAS instruments, in Instrument Deveopment for Atmospheric Research and Monitoring, edited by J. Bösenberg et a., pp , 1997, Springer, Berin, Heideberg, Stutz, J., B. Aicke, R. Ackermann, A. Geyer, A. White, and E. Wiiams, Vertica profies of NO 3, N 2O 5, O 3, and NO x in the nocturna boundary ayer: 1. observations during the Texas Air Quaity Study 2, J. Geophys. Res., 19(D1236), 1 14, 24. Sykes, R. I., and R. S. Gabruk, A second-order cosure mode for the effect of averaging time on turbuent pume dispersion, J. App. Met., 36, , Tanabe, K., Projection method for soving a singuar system of inear equations and its appications, Numer. Math., 17, , Tarantoa, A., Inverse Probem Theory, Esevier, New York, Tarantoa, A., Inverse probem theory and mode parameter estimation, SIAM, Phiadephia, 25. Tarantoa, A., and B. Vaette, Inverse probems: Quest for information, J. Geophys., 5, , Tikhonov, A. N., Soution of incorrecty formuated probems and the reguarization method, Soviet Math. Dok., 4, , Todd, L., and R. Bhattacharyya, Tomographic reconstruction of air poutants: Evauation of measurement geometries, App. Opt., 36(3), , Todd, L., and D. Leith, Remote sensing and computed tomography in industria hygiene, Am. Ind. Hyg. Assoc. J., 51(4), , 199. Todd, L., and G. Ramachandran, Evauation of agorithms for tomographic reconstruction of chemica concentrations in indoor air, Am. Ind. Hyg. Assoc. J., 55(5), , 1994a. Todd, L., and G. Ramachandran, Evauation of optica source-detector configurations for tomographic reconstruction of chemica concentrations in indoor air, Am. Ind. Hyg. Assoc. J., 55(12), , 1994b.

203 References 23 Trampert, J., and J. J. Lévêque, Simutaneous reconstruction technique: Physica interpretation based on the generaized east squares soution, J. Geophys. Res., 95(B8), , 199. Trukenmüer, A., D. Grawe, and K. H. Schünzen, A mode system for the assessment of ambient air quaity conforming to EC directives, Meteoro. Zeitschrift, 13(5), , 24. Tsai, M. Y., and K. S. Chen, Measurements and three-dimensiona modeing of air poutant dispersion in an urban street canyon, Atmos. Environ., 38, , 24. Twomey, S., On the numerica soution of Fredhom equations of the first kind by the inversion of the inear system produced by quadrature, J. Assoc. Comput. Mach., 1, 97 11, Twomey, S., Introduction to the mathematics of inversion in remote sensing, Dover Pubication Inc., New York, Van der Suis, A., and H. A. van der Vorst, Numerica soution of arge sparse inear agebraic systems arising from tomographic probems, in Seismic Tomography, edited by G. Noet, chapter 3, pp , Reide, Hingham, Mass., Vardi, Y., L. A. Shepp, and L. Kaufman, A statistica mode for positron emission tomography, J. Amer. Statist. Assoc., 8, 8 37, Vardouakis, S., N. Gonzaez-Fesca, B. E. A. Fisher, and K. Periceous, Spatia variabiity of air poution in the vicinity of a permanent monitoring station in centra Paris, Atmos. Environ., 39, , 25. Veite, H., B. Kromer, M. Mößner, and U. Patt, New techniques for measurements of atmospheric vertica trace gas profies using DOAS, Environmenta Science and Poution Research, specia issue 4, 17 26, 22. Venkatram, A., The expected deviation of observed concentrations from predicted ensembe means, Atmos. Environ., 13, , Venkatram, A., The uncertainty in estimating dispersion in the convective boundary ayer, Atmos. Environ., 13, 37 31, Venkatram, A., Accounting for averaging times in air poution modeing, Atmos. Environ., 36, , 22. Venkatram, A., V. Isakov, J. Yuan, and D. Pankratz, Modeing dispersion at distances of meters from urban sources, Atmos. Environ., 38, , 24. Venkatram, A., V. Isakov, and J. Pankratz D.and Yuan, Reating pume spread to meteoroogy in urban areas, Atmos. Environ., 39, , 25. Verkruysse, W., and L. A. Todd, Improved method grid transation for mapping environmenta poutants using a two-dimensiona CAT scanning system, Atmos. Environ., 38, , 24. Vokamer, R., A DOAS study on the oxidation mechanism of aromatic hydrocarbons under simuated atmospheric conditions, Ph.D. dissertation, Institut für Umwetphysik, Universität Heideberg, Germany, 21. Voz-Thomas, A., H. Geiss, A. Hofzumahaus, and K. H. Becker, Introduction to specia section: Photochemistry experiment in BERLIOZ, J. Geophys. Res., 18(D4), 8252, 23. Von Gasow, R., M. G. Lawrence, R. Sander, and P. J. Crutzen, Modeing the chemica effects of ship exhaust in the coud-free marine boundary ayer, Atmos. Chem. Phys., 3, , 23.

204 24 References Von Gasow, R., R. Von Kuhmann, M. G. Lawrence, U. Patt, and P. J. Crutzen, Impact of reactive bromine chemistry in the troposphere, Atmos. Chem. Phys., 4, , 24. Wagner, T., B. Dix, C. v. Friedeburg, U. Frieß, S. Sanghavi, R. Sinreich, and U. Patt, MAX-DOAS O 4 measurements: A new technique to derive information on atmospheric aerosos Principes and information content, J. Geophys. Res., 19(D225), 8252, 24. Wagner, T., J. Burrows, T. Deutschmann, B. Dix, F. Hendrick, C. von Friedeburg, U. Frieß, K.-P. Heue, H. Irie, H. Iwabuchi, J. Keer, C. McLinden, H. Oetjen, E. Paazzi, A. Petritoi, U. Patt, P. Postyyakov, J. Pukite, A. Richter, M. van Roozendae, A. Rozanov, V. Rozanov, R. Sinreich, S. Sanghavi, and F. Wittrock, Comparison of box-air-mass-factors and radiances for MAX-DOAS-geometries cacuated from different UV/visibe radiative transfer modes, Atmos. Chem. Phys. Discuss., 6, , 26. Wayne, R. P., Chemistry of Atmospheres: An Introduction to the Chemistry of the Atmospheres of Earth, the Panets, and their Sateites, Oxford University Press, 3rd edition, 2. Weinbrecht, S., S. Raasch, A. Ziemann, K. Arnod, and A. Raabe, Comparison of arge-eddy simuation data with spatiay averaged measurements obtained by acoustic tomography presuppositions and first resuts, Bound. Layer Met., 111, , 24. Wenig, M., B. Jähne, and U. Patt, Operator representation as a new differentia absorption spectroscopy formaism, App. Opt., 44(16), , 25. Wimott, C. J., On the vaidation of modes, Phys. Geog., 2, , Wofe, D. C., and R. L. Byer, Mode studies of aser absorption computed tomography for remote air poution measurement, App. Opt., 21(7), , Yamartino, R.J., J.S. Scire, G.R. Carmichae, and Y.S. Chang, The CALGRID mesoscae photochemica grid mode I: Mode formuation, Atmos. Environ., 26A, , Yee, E., and R. Chan, A simpe mode for the probabiity density function of concentration fuctuations in atmospheric pumes, Atmos. Environ., 31(7), , 1996.

205 Bedanken möchte ich mich herzich bei Herrn Prof. Patt für die abschießende Betreuung dieser Arbeit, Herrn Prof. Jähne für die zweite Begutachtung und insbesondere bei Caudia Hak, Kaus-Peter Heue und Denis Pöher für die sorgfätige Korrektur ihrer ersten Versionen. Mein besonderer Dank geht auch an David Grawe für die freundiche Bereitsteung seiner Modedaten, ae Koegen des IUP, von denen ich in fachichen Diskussionen ernen konnte, sowie an ae, die für Spaß und Kurzwei gesorgt habe und nicht zuetzt an Frau Cos für ihre stete, unerschütteriche Hifsbereitschaft. Above a I woud ike to thank Bing-Chao Song who not ony taught me things about computers I did not dream of, gave me insight into a Chinese way of thinking (both with incredibe patience) and made my time at the IUP a ot more fun, but who is aso one of the finest persons I have ever met.