NUMERICAL METHOD FOR UNCERTAINTY ASSESSMENT OF THE TOPOGRAPHIC PARAMETERS DERIVED FROM DEMS

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1 NUMERICAL METHOD FOR UNCERTAINTY ASSESSMENT OF THE TOPOGRAPHIC PARAMETERS DERIVED FROM DEMS B. Bajat a, S.Rath b, D. Blagojevic a a Uiversity of Belgrae, Faculty of Civil Egieerig, Departmet of Geoesy, Serbia & Moteegro. [email protected] b Hamburg Uiversity of Techology, River & Coastal Egieerig, D-1073 Hamburg. ABSTRACT Topography has a great ifluece o may characteristics which are coecte to the istict area o Earth such as hyrology, microclimate characteristics, vegetatio cover properties, geological structures, mieral eposits, plat characteristic etc. Digital terrai moels represet segmets of spatial ata bases relate to presetatio of terrai features a laforms. Square gri elevatio moels (DEMs) have emerge as the most wiely use structure urig the past ecae because of their simplicity a simple computer implemetatio. They have become a importat segmet of Topographic Iformatio Systems (TIS), storig atural a artificial lascape i forms of igital moels. This ki of ata structure is especially suitable for morphometric terrai evaluatio a aalysis which is very importat i moelig Earth surface a atmospheric processes or evirometal moelig applicatios. Treatig topographic parameters like slope, aspects a terrai curvature as the first a seco erivatives or terrai surface, they ca easily be calculate i such ata structures. DEMs are use as proxies for the actual terrai surface a etermie as a gatherig of the measure terrai ata a iterpolatio techiques. Accorig to this statemet, the quality of DEMs are characterie by the ucertaity beig referre to ata moel-base ucertaity a ata base ucertaity. Accorig to the regular gri structure of poits, ucertaities i DEMs are mostly treate through the height accuracy of the poits. Well kow methos to assess the ucertaity of erive parameters, such as Mote Carlo stochastic simulatio techiques, are alreay use for these purposes. This metho has both goo a iferior characteristics. Prouctio of umerous realiatios of DEMs for particular terrai a subsequet statistical aalysis may be time cosumig a limite with available computer memory. Simple umerical solutio of a variace propagatio techique is presete i this paper as a possible alterative for ucertaity assessmet of erive topographic parameters. Both ucorrelate a correlate ucertaity scearios are take ito accout. Obtaie results are cofirme with Mote Carlo approach with high level of agreemet. Keywors: DEM, ucertaity, topographic parameters. 1 INTRODUCTION Gatherig, maiteace a visualiatio of relief ata are mostly use GIS (Geographical Iformatio System) fuctios. This segmet of GIS applicatios is well kow as Digital Terrai Moel (DTM). Durig the last ecae this has become coveiet tool for represetig terrai surface, ot oly i GIS applicatios, but i computer assiste software for projectig, as well. DTM bases are orgaie i ifferet maers, like Triagular Irregular Networks (TIN) or i regular lattice (GRID). Term DEM is relate to terrai height atabases with regular gri structure or altitue matrix, a the term Digital Terrai Moels (DTM) is mostly relate to TIN structures. DEMs approach for ata base structures is suitable for Natioal Heights Data Bases which cover the area of the whole coutry area as a part of atio-wie spatial ata bases available from the atioal mappig agecies. This ki of ata bases are very useful i various applicatio areas i GIS, like ecological stuies, 3D urba mappig, evirometal moitorig, lascape plaig, geological aalysis, civil egieerig, flooplai aalysis, risk maps etc. They are also very suitable i complex aalysis, especially i superimposig other ki of ata, like satellite images or remotely sesig iformatio. The ew a improve methos of a aalysis allow users to process eve more complex applicatio tasks with

2 combiatios of geometrical, topological a thematic aspects usig hybri ata, i.e. raster ata, as well as, vector ata [1] DTM cocept ha starte as a sta aloe applicatio itee for ifferet egieerig tasks. Nowaays, it is a part of the spatial ata bases which store the atural a artificial lascape i the form of igital moels. DEM are compulsory compoet of oe categories of spatial iformatio systems kow as Topographic Iformatio System (TIS) []. Geomorphometry, efie as sciece which treats the geometry of the lascape with quatitative parameters, attemptig to escribe the form of la surface, represets oe of the isciplies where DEMs proucts are completely implemete. Slope, aspect, profile (vertical) a cotour (horiotal) curvature (covexity) are basic attributes of la surface at the vertex or its close eighborhoo, a they form a coheret system for terrai aalysis a escriptio, easily calculate with such arrage atabases of terrai heights[3]. DEM ata is subject to errors such as ay other spatial ata source. DEM users must keep i mi that results obtaie by aalysis a processig of such ata epe o the quality of this ata. Quality assessmet of obtaie results iicates their reliability a suitability for particular applicatios. Sice a ataset is prouce rather for various tha just for oe specific applicatio, the quality of ataset ca be assesse oly by kowig the ata quality elemets, as well as by the ata quality overview elemet [4]. DEM QUALITY ASSESSMENT Terrai heights i DEM ata bases are cosiere as poit features i which heights compoets are store as attributes. This cocept is well kow as.5 GIS ata moel. Attribute accuracy is the mai elemet of quality of such spatial atabases. Accuracy ca be efie as closeess of agreemet betwee the test result a the accepte referece or true value, where referece values are ata sources of higher accuracy. However, a true value or locatio is a luxury ot ofte available i spatial ata. I the case of spatial ata it refers to ata of larger scale a resolutio or usage of istrumet of higher accuracy or more recet measuremets. All those imply that efiitio of accuracy is relative a for that reaso term accuracy is substitute with ucertaity. May wors with similar cootatio are use to express term ucertaity, like reliability, cofiece, accuracy, error etc. This term is especially suitable for reportig the assertiveess of variables where true values are ukow. Uer these circumstaces it is ot feasible to calculate exactly the error of the appraisal. A useful step i assessig the ucertaity is to cosier the factors by which the error is ifluece [5]. Two DEM-relate ucertaity categories are commoly recogie. The primary is ata moel-base ucertaity, resultig from iffereces betwee the form of ata moel a actual elevatio surface. The seco is ata-base ucertaity, referrig to iffereces betwee the elevatio of locatio specifie i the ata set a actual elevatio at that locatio [6]. DEM altitues are give as iterval ata, a root mea square error (RMSE) or staar eviatio coul be use as a ucertaity metrics. The RMSE calculate from resiuals betwee moels heights a grou truth poits is commoly use measure of the accuracy for the DEMs proucts [7]. Aother powerful measure is the staar eviatio of resiuals, obtaie from iscrepacies betwee moel heights a terrai heights at the particular locatios. where: i i 1 σ, 1 _ i - resiual betwee height i DEM a true height i DEM - t, - umber of cotrol poits. _ i (1)

3 Accuracy evaluatios ofte uses fiel survey ata base o the Global Positioig System (GPS) because of high accuracy. Performig cotrol measuremets is a staar proceure for evaluatig the quality of DEMs proucts. It ca also be performe by DEMs users if they are ot acquaite with lieage of ata. Locatios of the cotrol poits have to be raomly chose [8]. Developig aequate GIS methos a techiques which ca take ucertaity of spatial ata ito accout urig their processig are crucial for errors halig i subsequet aalysis [9]. The problem of error propagatio through GIS operatios ca be uertake by usig statistical theory [10]. This aalytical aalysis ca be achieve whe algebraic relatioships betwee iput ata a output results are simple, e.g. slope calculatios i geomorphometry aalysis (equatio ). where: σ slope - staar eviatio of a slope, - gri spacig, r- correlatio betwee elevatio errors. σ slope ( 1 r) σ () Due to sufficietly complex aalyses beig performe i may GIS applicatios this aalytical approach is usually impossible. Istea, ucertaity must be propagate by simulatios a aalye from the results, kow as Mote Carlo approach [11]. This is more geeral approach where error is moele stochastically through umerous simulatios of probable realiatios of DEM by proucig a istributio of results which may be assesse statistically. Two issues are importat for this approach, the magitue of error of the iput ata as well as the spatial structure of error. The spatial structure of error may be represete by semivariograms or covariograms. Proucig umerous realiatios of DEMs for a particular terrai a subsequet statistical aalysis may be time cosumig a limite by the available computer memory. 3 NUMERICAL SOLUTION FOR ERROR PROPAGATION A alterative of aalytical approach of DEMs ucertaity assessmet which avois complicate formula erivatios is offere i this sectio. Value of primary geomorphological parameters for particular gri cell is calculate as fuctio of heights of surrouig cells: where: y- value of a particular geomorphological parameter. i - terrai height i surrouig gri cell. yf( 1,,... ) (3) - umber of surrouig cells, 4 for rook s case a 9 for quee s case (fig. 1.)

4 Figure 1. Chess rules for rook s a quee s motio Sice formula (3) is ofte oliear, the lieariatio is give by: Partial erivatives: f f f y y f ai i 0 are calculate with aproximate values of the surrouig cells. Numerical proceure for calculatio gives us the followig equatio: a i (,,... +,... ) (,,... ) f 1 i f 1 where is a small arbitrary icremet. Equatio (4) ca be preset as: or i matrix form: y y0 + a1 1+ a a (7) [ ] [ ] y A Z + y, A a a a, Z (8) T T T The variace of y ca be etermie by rules of error propagatio law: (4) (5) (6) σ A C A σ A C A T T y y (9) where σ y is variace of particular geomorphological parameter, C is covariace matrix of error cells heights: where meas that, σ ij σ ji. σ ii is variace a C σ11 σ1 σ 1 σ1 σ σ (10) σ1 σ σ σ ij is covariace of heights errors. The matrix C is also symmetric, which Whe heights errors are ucorelate, matrix C has the followig structure: C σ σ 0 (11) 0 0 σ

5 I the case that variaces of heights errors of surrouig cells are equal with σ value, the: C σ σ I where I represets the uit matrix. For give coitioes i equatios (11) a (1) the variace of geomorphological parameters will be: σ a σ + a σ a σ y 1 11 ( a1 a... a ) σ σ y It is ecessary to etermie o iagoal terms of matrix C for corellate heights errors. Uer the assumptio that corellatio betwee heights errors becomes weaker with loger istace, oe of possible moels for correlatio is expoetial fuctio: where ij - istace betwee i a j cells D correlatio istace. ij D ij σ e (1) (13) σ (14) Figure. Flow chart for umerical solutio for ucertaity assessmet of topographical parameters Presete proceure is suggeste as practical solutio with regar to time a computig resources for ucertaity assesmet of calculate topographical parameters. Results obtaie with this approach have bee verifyie by comparig them with the results obtaie from Mote Carlo stochastic simulatios. 4 FORMULAS FOR CALCULATION OF TOPOGRAPHICAL PARAMETERS It is straightforwar to obtai primary geomorphological parameters i every gri cell accorig to the above metioe scheme by calculatig partial erivatives with metho of fiite iffereces: x y xx yy xy ( ) ( ) 8 ( ) ( ) x y + + xy Slope S shows the rate of chage of height of the terrai surface with istace: (15) S + + x y x y (16) Slope cotrols ruoff a soil loss, thickess of soil horios, some plat characteristics.

6 Aspect A is the orietatio of the lie of steepest escet, which is measure i clockwise irectio startig from orth: o y o x A 180 arcta + 90 x x I associatio with slope, aspect cotrols isolatio a evapotraspiratio, it iflueces thickess of soil horios a some plat properties, also. Horiotal curvature K c is the curvature i the horiotal plae of cotour lie. Vertical curvature K p is the curvature i the vertical plae of a flow lie. They represet the rate of chage of a first eriviate such as slope or aspect i particular irectios. Tagetial curvature K t is curvature i a iclie plae perpeicular to both the irectio of flow a the surface. Tagetial curvature represets horiotal curavture multiplie by the sie of the slope agle, a it is more appropriate tha horiotal curvature for stuig flow iveregece a covergece whe terrai slope is too small[1]: K K p c xxx + xy x y + yy y P 3 P Q xx y xy x y yy x x xxy xy x y + yy x Q P P + Kt P Q Horiotal a vertical curvatures are the etermiig local factors of the yamics of overla a itrasoil water, they ifluece soil moisture, ph, thickess of soil horios, orgaic matter, plat cover istributio[13]. 5 CASE STUDY DEM was prouce by igitiig cotour layers from two ajacet sheets of the topographic maps of scale 1:5000, with cotour iterval of 5m, with total area of 13.5 square kilometers for the research purpose. Test locatio is the resort area Zlatibor i south wester Serbia with miimum height of 850m a maximum height of 1174m. This area is hilly plateau, with the exceptio of the west a orth-west part with greater slopes of terrai. Digitie polylies with height attributes were broke ito vertices a such big amout of obtaie poits was reuce by usig the Douglas-Peucker algorithm for polylie simplificatio to poits. A DEM was prouce i two steps: A iitial TIN was prouce usig a Delauay triagulatio, beig subsequetly coverte ito regular gri with 10m resolutio (350 rows by 400 colums). + y (17) (18)

7 Figure 3. DEM of test area Zlatibor A GPS survey of cotrol poits has bee carrie out, usig fast static a real time kiematics (RTK). A seve parameters similarity trasformatio was use to brig GPS cooriates ito official referece system. Ellipsoial GPS heights were trasforme ito orthometric system by polyomial moel of seco egree. The achieve accuracy of heights is estimate to be better tha 3 cm. Calculate σ for ata set was approximately 1.5m, a obtaie result is i accorace with the expecte accuracy of the cartographic source ata use for DEM prouctio [14]. Estimate correlatio istace of errors was D110m. The estimate ucertaities (staar eviatio) of topographic parameters obtaie by stochastic simulatios σ sim are compare with values σ a of staar eviatios obtaie aalytically by propose algorithm. Both correlate a ucorrelate heights errors were take ito cosieratio. Mote Carlo simulatios were carrie out by aig geerate error fiels to iitial DEM [15]. Both ucorrelate (worst case sceario) a correlate error fiels have bee geerate. For each sceario 5 simulatios were mae, as it was ecessary to yiel a statistically useful result at the 0.05 sigificace level [16]. The compliace of staar eviatios was checke for each gri cell by testig of hypothesis: H 0 : estimate values of staar eviatios are similar σ a σ sim H 1 : estimate values of staar eviatios are ot similar σ a σ sim Aalytically obtaie value σ a has f egrees of freeom while simulate estimate has f 1 4 egrees of freeom, with umber of simulatios 5. Depeig o test statistics, followig critical values (for 99% probability) were use: 0.99 (,4) 0.1, F F0.99 H0 F F H (4, ) 1.79 Table 1. Acceptace of Fisher s test of similarity of simulate a aalytically obtaie ucertaity estimates for each topographic parameter F_σ S F_σ A F_σ kc F_σ kp F_σ kt Ucorrelate error fiels 89.7% 61.9% 69.8% 97.7% 96.7% Correlate error fiels 84.8% 71.8% 73.0% 90.9% 85.9% Lege: F_σ i - Fisher's test for i th topografical parameter, σ S - staar eviatio of a slope, σ A - staar eviatio of a aspect, σ kc - staar eviatio of cotour curvature, σ kp - staar eviatio of profile curvature, σ kt - staar eviatio of tageetial curvature.

8 The acceptace of Fisher s test (acceptace of hypothesis H 0 ) is give i percetages which show the rate betwee the umber of cell where hypothesis H 0 is accepte i regar to whole umber of gri cells. The agreemet of these estimates varies from oe parameter to the other, with a view that this agreemet is somewhat lower for aspects, but which is quite expecte cosierig specifies of those aspects closer to values of 0 o or 360 o respectively. 7 REMARKS This aalysis ca ot give a aswer which oe of both iscussse methos for ucertaity estimatio gives more realible results. Nevertheless it cofirms that simple umerical solutios for aalitical metho gives similar results as stochastic simulatios. This proceure coul be suggeste as simple a ecoomic regarig to software a time resources for a uceraity aalyses i geomorphometry. REFERENCES 1 Glemser, M., Klei, U., Fritsch, D., Stru G., 000: Complex Aalysis Methos i Hybri GIS Usig Ucertai Data. I: GIS - Geo-Iformatiossysteme, No., pp , Wichma Verlag, Heielberg. Kraus K.,1995: From igital elevatio moel to topographic iformatio system. I: Week, D. Fritsch a D. Hubbie (e.): 45th.Photogram metric week, pp Stuttgart 3 Evas I.S, 1998: What o Terrai Statistic Rally Mea I: Lae S.N., Richars K.S. (e): Laform Moitorig Moellig a Aalysis, pp Joh Wiley &Sos, Loo 4 ISO/TC11, 001: ISO Geographic iformatio Quality priciples 5 Isaaks E., Srivastava R., 1989: A Itrouctio to Applie Geostatistics. Oxfor Uiversity Press, pp New York, NY 6 Shortrige A., 001: Characteriig Ucertaity i Digital Elevatio Moels., I: Husaker C.T., Goochil M.F., Friel M.A., Case T.J. (e.): Spatial Ucertaity i Ecology,pp Spriger- Verlag New York. 7 Caruso V.M., 1987: Staars for igital elevatio moels. Proceesigs of the ACSM/ASPRS Aual Covetio, pp Baltimore. 8 Li Z. 1991: Effects of Check Poits o the Reliability of DTM Accuracy Estimates Obtaie from Experimetal Tests. PERS, 57, (10),pp Opeshaw S., 1994: Learig to live with errors i spatial atabases, I: Goochil M.F, Gopal S. (e.): The accuracy of spatial atabases, pp Taylor&Fracis, Loo 10 Burrough P.A., 1986: Priciples of Geographical Iformatio Systems for La Resources Assessmet, Oxfor Uiversity Press, New York, 194 pp. 11 Goochil M.F., 1995: Attribute accuracy. I: Guptill S.C.& Morriso J.L (e): Elemets of spatial ata quality, pp Elsevier. 1 Gallat J.C., Wilso J.P., 000: Primary Topographic Attributes. I: Gallat J.C., Wilso J.P. (e): Terrai Aalysis-Priciples a Applicatios,pp Joh Wiley&Sos, New York. 13 Florisky I.V., Semiar: Digital Terrai Moels: Theory a Practice 14 Merchat, D.C.,1987: Spatial accuracy specificatio for large scale topographic maps, PERS 53, pp Fisher P., 1991: First Experimets i Viewshe Ucertaity: The Accuracy of the Viewshe Area. PERS 57 (10), pp Hope A.C.A., 1968: A simplifie Mote Carlo sigificace test proceure. Joural of the Royal Statistical Society B, 30, pp