Information Content of Infrared Satellite Sounding Measurements with Respect to CO 2
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1 FEBRUARY 00 ENGELEN AND STEPHENS 7 Informaton Content of Infrared Satellte Soundng Measurements wth Respect to CO R. J. ENGELEN * AND G. L. STEPHENS Department of Atmospherc Scence, Colorado State Unversty, Fort Collns, Colorado (Manuscrpt receved 0 March 00, n fnal form 7 September 00) ABSTRACT Informaton theory s used to study the capabltes of the new-generaton satellte nfrared sounders [Atmospherc Infrared Sounder () and Infrared Atmospherc Soundng Interferometer (IASI)] for retrevng atmospherc carbon doxde (CO ) and for contrastng these new nstruments wth the current system of nfrared sounders [Televson and Infrared Observaton Satellte (TIROS) Operatonal Vertcal Sounder/Hgh-Resoluton Infrared Radaton Sounder (TOVS/)]. It s shown that nstruments lke and IASI wll be able to retreve column-averaged CO mxng ratos wth hgh enough accuracy (order of ppmv) to be useful for atmospherc CO nverson studes that try to estmate sources and snks of CO. On the other hand, the TOVS/ system s only able to retreve column-averaged CO mxng ratos wth an accuracy of the same order as the seasonal ampltude of atmospherc CO varatons (order of 0 ppmv). It s also shown that the constranng a pror covarance matrx has an mportant effect on what nformaton can be extracted from the observatons.. Introducton Atmospherc carbon doxde (CO ) has ncreased from about 80 to about 75 ppmv snce the begnnng of the ndustral era. The rate of combuston of fossl fuels s known from econometrc tabulatons (e.g., Andres et al. 996), and so the hstory of atmospherc CO can be used to nfer the ntegral of all other sources and snks n the earth system by subtractng ths anthropogenc emsson rate. Surface measurements from a sparse global network ndcate the bosphere and oceans have absorbed about one-half of the carbon emtted by fossl fuel consumpton durng the past 0 yr (e.g., Francey et al. 995). The geographc dstrbuton of these snks and the underlyng mechansms that control them, however, are too uncertan to predct how these processes wll change future emsson uptakes. A way to deduce the dstrbuton of sources and snks s to apply measurements of CO n an nverson model that takes nto account transport processes (e.g., Entng et al. 995; Gurney et al. 00). Untl now, the current flask network has been used for ths purpose, but the dstrbuton of these measurements s too sparse to provde the necessary coverage for these nverson studes (Engelen et al. 00). Al- * Current afflaton: European Centre for Medum-Range Weather Forecasts, Readng, Unted Kngdom. Correspondng author address: Rchard Engelen, ECMWF, Shnfeld Park, Readng RG 9AX, Unted Kngdom. E-mal: rchard.engelen@ecmwf.nt though the ndvdual flask measurements are of hgh precson (Masare and Tans 995), addtonal spatally resolved global maps of CO, wth a precson of about ppmv, offer the potental to mprove dramatcally our ablty to quantfy the sources and snks of CO and the dstrbuton of these fluxes (Rayner and O Bren 00). The study of Rayner and O Bren (00) ponts to the potental value of satellte-based remote sensng methods for studyng the carbon cycle. The recent study of Chédn et al. (00) demonstrates how the seasonal sgnatures of CO can be observed n Televson and Infrared Observaton Satellte Operatonal Vertcal Sounder/Hgh-Resoluton Infrared Radaton Sounder (TOVS/) measurements, and the study of Engelen et al. (00) hghlghts the potental of the future measurements of the Atmospherc Infrared Sounder (). The purpose of ths paper s to provde further analyss of the capabltes of the nextgeneraton sounders, such as and the Infrared Atmospherc Soundng Interferometer (IASI), and to contrast these capabltes wth those of the currentgeneraton TOVS/ nstruments. We attempt to quantfy CO nformaton content contaned n these types of nfrared measurement systems. The next secton outlnes the basc nformaton theory used to characterze these observng systems and llustrates the prncples wth use of a smple example. Ths theory s appled to and -lke observatons n secton, where t s shown that the observatons cannot be expected to resolve CO varaton below about 0 ppmv, whch barely resolves gross 00 Amercan Meteorologcal Socety
2 7 JOURNAL OF APPLIED METEOROLOGY VOLUME seasonal swngs n CO, whereas and IASI are expected to resolve mean tropospherc CO mxng ratos below about ppmv.. Informaton content a. Theory To calculate the nformaton content of nfrared satellte radances wth respect to atmospherc CO concentratons, we followed the descrpton by Rodgers (000), whch s based on the defnton by Shannon and Weaver (99). The nformaton content of a set of observatons s defned by the change n the logarthm (base ) of the number of dstnct possble states of the system beng measured. If we defne the possble states of a system wth a probablty dstrbuton functon P and the entropy of ths system as S(P), the nformaton content of a set of observatons s the change n entropy H S(P ) S(P ), () where P represents our knowledge before the observatons are made and P s our knowledge after the observatons are made. If we assume Gaussan probablty dstrbutons, we can defne the entropy as S(P) ln S, () where S s the covarance matrx that descrbes our knowledge of the system. The entropy can be seen as the logarthm of the volume of state space occuped by the probablty densty functon defned by the covarance matrx. The nformaton content s then H ln S ln S ln SS, () wth S beng the pror covarance and S beng our posteror covarance. It therefore represents the reducton n volume of the pror probablty functon by makng the observatons. Ths means that of all the possble atmospherc profles wthn the atmospherc profle space defned by the a pror covarance matrx, a total of H profles can actually be dstngushed by the observatons. Another useful measure of nformaton s the degrees of freedom for sgnal. Although the total degrees of freedom of a set of observatons s equal to the number of observatons, only a selecton of these total degrees of freedom s ndependent and sgnfcant wth respect to the measurement nose. The degrees of freedom for sgnal are therefore defned as the number of ndependent peces of nformaton n a measurement that can be observed above the nose of the observatons. Usng the precedng covarance defntons, we can wrte the degrees of freedom for sgnal as the trace of the same matrx product as s used n the defnton of the Shannon nformaton content (S ): S TABLE. Retreval error, degrees of freedom for sgnal, and nformaton content for a smple measurement. y (ppmv) a (ppmv) x (ppmv) d s H Small measurement error d tr(ss ). Large measurement error s () b. A smple flask measurement example A smple example can show the use of the earlerdefned measures of nformaton n an observaton. Assume we want to retreve the true value of the CO concentraton x from a drect flask observaton y gven some a pror guess of x. If the observaton ncludes an error, we have the followng relaton between x and y: y x. (5) We can then calculate the retreval error gven the measurement error and the a pror error from x ( a y ), (6) where standard devatons are used to represent the errors. The nformaton content and degrees of freedom for sgnal can then be calculated from () and (), respectvely. Table shows the results for two cases: () a small-measurement-error case (measurement error s 0.5 ppmv and a pror error s ppmv) and () a largemeasurement-error case (measurement error s ppmv and a pror error s 0.5 ppmv). Although the retreval error x s the same for both retrevals, the degrees of freedom and the nformaton content are very dfferent. In the case wth small measurement error, the degrees of freedom for sgnal s almost ; n the case wth large measurement error, the degrees of freedom for sgnal s almost zero. As expected, the nformaton content of the low nose measurement s much larger than the nformaton content of the hgh nose measurement. The low observatonal nose retreval can dstngush 6 values wthn the a pror varance of ppmv, whch n ths scalar case s equal to the sgnalto-nose rato defned by a / y. In other words, although the retreval error does not dstngush between the lowand hgh-nose cases, the degrees of freedom and the nformaton content dfferentate between the two cases and dentfy the better measurement system. c. A more general example In a more realstc envronment, such as the retreval of CO concentratons from nfrared satellte observatons (Engelen et al. 00) that we are consderng n ths paper, a lnear relaton between the retreval var-
3 FEBRUARY 00 ENGELEN AND STEPHENS 75 ables x and the observatons y can be constructed such that y Kx, (7) where K s the weghtng functon matrx. Usng a constrant n the form of an a pror covarance matrx S a, we can wrte the retreval error covarance matrx as T Sx (S a KSy K), (8) where S y s the measurement error covarance matrx. Because the nformaton content and the degrees of freedom for sgnal are defned wth respect to the a pror covarance matrx and the measurement covarance matrx, we scale the weghtng functon matrx accordngly: / K S KS / y a. (9) As shown n Rodgers (000), the sngular vectors of ths scaled weghtng functon matrx K represent ndependent vertcal profles that can be measured wth the set of observatons. The respectve sngular values are a drect measure of the sgnal-to-nose rato takng nto account our pror knowledge of the atmospherc state. Thus, sngular vectors wth a sngular value larger than contan usable nformaton about the atmospherc state, and sngular vectors wth a sngular value smaller than are domnated by the measurement nose. Rodgers (000) also shows that we can use these sngular values to calculate the degrees of freedom for sgnal and the nformaton content d s (0) ( ) H ln( ). () Therefore, by calculatng the sngular values of K we have three measures of nformaton for a certan set of observatons consderng our pror knowledge: () the number of ndependent measurements made to better than measurement error, () the degrees of freedom for each ndependent measurement and the total degrees of freedom for the set of observatons, and () the Shannon nformaton content of each ndependent measurement and the total Shannon nformaton content for the set of observatons. In the next secton, we wll also consder the estmated error n the retreved columnaveraged volume mxng rato, whch can be calculated from the retreval error covarance matrx S x as follows (Rodgers and Connor 00). T g Sg, () where g s an operator that converts the level volume mxng ratos to a column-averaged mxng rato: p g, () p where p s the pressure thckness of layer.. Applcaton to and observng systems Usng the above-descrbed theory, we calculated the nformaton content wth respect to atmospherc CO for observatons by TOVS/ and. We used a broadband Malkmus radatve transfer model to calculate the weghtng functons (Engelen et al. 00). For, we used a spectral resoluton of cm for the band between 500 and 500 cm ; for we used the spectral channels senstve to CO defned by the half-wdth of the nstrumental response functons, whch s about 5 cm for the longwave channels and about 5 cm for the shortwave channels. The channels used were channels 7 and 5 7 (Smth et al. 979). The measurement covarance matrx for both nstruments was specfed as a dagonal matrx wth standard devatons of 0.5 K on the dagonal elements. Ths error ncludes uncertantes n the temperature profle, whch acts n ths smple setup as an nput for the radatve transfer model. These temperatures could come from the Advanced Mcrowave Soundng Unt or a weather forecast model. The above assumptons for the measurement covarance matrx are optmstc, especally because errors n the assumed temperature profles wll ntroduce correlatons. Also, the value of 0.5 K s small. Therefore, we wll also use a value of.0 K n one of the experments. The a pror covarance matrx has, n our frst example, dagonal elements of 6 (ppmv) (standard devaton of ppmv) and off-dagonal elements specfed as follows: S j a exp( z z j /H), () where the lnear-scale heght H s set to 5 km and where the mnmum vertcal correlaton s set to 0.5. The lowest km, whch represents the boundary layer, was decoupled from the rest of the atmosphere by settng the correlatons to zero. Ths covarance matrx setup, ncludng the uncertanty estmate of ppmv, was based on hourly output of CO profles from a GCM smulaton (S. Dennng, Colorado State Unversty, 00, personal communcaton). Although the uncertanty at ndvdual levels s ppmv, the uncertanty n the column-averaged mxng rato s. ppmv [usng ()] because of the correlatons between the levels. Fgure shows the sngular vectors and ther correspondng sngular values for both and wth an a pror standard devaton of ppmv. Only sngular values that are larger than are sgnfcant wth respect to the measurement error. has no sgnfcant vectors; has two sgnfcant vectors. The frst two sngular vectors represent broad vertcal patterns wthout much vertcal resoluton. The thrd sngular vector adds some vertcal resoluton but s not sgnfcant. The degrees of freedom and the Shannon nformaton content for ths setup are shown n Table. The total degrees of freedom for are only 0.. The total Shannon nformaton content s 0.8, whch
4 76 JOURNAL OF APPLIED METEOROLOGY VOLUME FIG.. Frst four sngular vectors wth ther correspondng sngular values for (left) and (rght). Measurement error was set to 0.5 K, and a pror error was set to ppmv. means that only 0.8. dfferent atmospherc states can be detected. Ths means that wthn the a pror uncertanty of ppmv less than two dfferent atmospherc states can be dstngushed. For, however, the total degrees of freedom s.6 and the total Shannon nformaton content s., whch translates nto fve dstngushable atmospherc states wthn the a pror uncertanty. The retreval error of the column-averaged mxng rato s.8 ppmv for and. ppmv for. There s some CO sgnal n the radances, but t s clearly not enough to observe atmospherc CO concentratons at better than ppmv. On the other hand, radances observed by wll be capable of provdng sgnfcant atmospherc CO nformaton, especally total column values, as s shown by the frst two sngular vectors. What s also nterestng to note s that the sngular vectors for and are very smlar. It s apparent that there s not a great dfference n what structures both nstruments can observe; the dfference s n the sgnal-to-nose rato reflected by the nformaton content. If we ncrease our a pror uncertanty to 0 ppmv (wth a column-averaged uncertanty of 7.6 ppmv), whch s close to the seasonal ampltude of atmospherc CO concentratons, the radances have a clearer sgnal, as shown n Table. The sngular vectors are TABLE. Sngular values and the contrbuton of each sngular vector to the degrees of freedom and nformaton content for and wth a pror CO errors of ppmv and observaton errors of 0.5 K. The total degrees of freedom and the total nformaton content are shown n the bottom row. the same as n Fg., but the sngular values (and therefore the degrees of freedom) and the Shannon nformaton content have changed. now has one sgnfcant sngular vector, and the degrees of freedom have ncreased to almost. The total Shannon nformaton content s now 0.78, whch represents almost two dfferent atmospherc states. Ths means that s able to estmate a column-averaged CO concentraton as represented by the frst sngular vector when our pror knowledge s on the order of 0 ppmv. stll has two sgnfcant sngular vectors. The total number of atmospherc states that can be detected by has ncreased to. The column-averaged retreval errors are now 5. ppmv for and. ppmv for. Ths result shows that can sgnfcantly mprove over the a pror estmate but does not reach an uncertanty that would be small enough to have a sgnfcant effect n CO nversons. Our estmate of the effect of errors n the assumed temperature profle on the forward radatve transfer model s conservatve. We assumed an error of 0.5 K, but t could easly be as large as K. Table shows the retreval statstcs for an a pror error of ppmv and a total measurement and forward model error of K for both and. As before, does not have any sgnfcant sngular values and ts degrees of freedom dropped to 0.06 wth.0 dstngushable atmospherc state. The retreval error of the column-av- TABLE. The same as n Table but wth a pror CO errors of 0 ppmv and observaton errors of 0.5 K. d s H d s H d s H d s H
5 FEBRUARY 00 ENGELEN AND STEPHENS 77 TABLE. The same as n Table but wth a pror CO errors of ppmv and observaton errors of K. TABLE 5. The same as n Table but for a dagonal a pror covarance matrx wthout any vertcal correlatons. d s H d s H d s H d s H eraged volume mxng rato s.0 ppmv. The retrevals also degrade, but there s stll one sgnfcant sngular value. The total degrees of freedom s now 0.95, and the number of dstngushable atmospherc states s.. The retreval error of the column-averaged volume mxng rato s.8 ppmv. For the case wth a 0-ppmv a pror error specfcaton, smlar degradaton results are obtaned. Although our specfcaton of the a pror covarance matrx s an estmate based on model output, we now show the role of well-specfed vertcal correlatons n the covarance matrx. Fgure and Table 5 show the results of the frst experment, but now wth a dagonal a pror covarance matrx that contans no vertcal correlatons at all. Because the levels are completely uncorrelated, errors at dfferent levels start to compensate when we calculate the column-averaged uncertanty. Ths column-averaged uncertanty of the a pror dagonal covarance matrx here specfed s 0.98 ppmv. Both and show sngular vectors wth more vertcal structure, but the sngular values have decreased sgnfcantly. does not have any sgnfcant sngular vectors at all, and the number of sgnfcant sngular vectors for s also reduced to zero. The number of dstngushable atmospherc states s now.0 for and. for. The column-averaged uncertanty for s 0.97 ppmv, whch s bascally equal to the a pror uncertanty. The column-averaged uncertanty for s 0.87 ppmv. So, although the nondagonal a pror covarance matrx seems to constran the retreval more than the dagonal a pror covarance matrx, the amount of nformaton that can be retreved from the observatons s actually hgher for the nondagonal matrx. The reason for ths result s that the weghtng functons see only large-scale structure. The nondagonal covarance matrx constrans the smallscale structure but has a larger varance at the larger scale, therefore allowng retreval of more nformaton about the large-scale structure than does the pure dagonal covarance matrx. All analyses n ths secton have been carred out for ndvdual profles. However, to reduce the retreval error, spatal and temporal averagng could be appled so as to render CO dstrbutons on spatal and temporal scales that are useful for current nverson studes. Most recent CO nverson studes have used monthly mean observatons and a transport model grd on the order of 5 0 (e.g., Gurney et al. 00; Kamnsk et al. 00; Rödenbeck et al. 00). Most areas wth sgnfcant cloudness would allow the averagng of at least several satellte observatons on these space and tme scales. However, although averagng wll reduce the random component of the observaton error, any systematc errors n the retreved values wll reman n the averaged product. These systematc errors arse from bases n the a pror estmates, bases n the radatve transfer modelng, and bases n the temperature feld. Furthermore, most of these bases are spatally heterogeneous, whch FIG.. Same as n Fg., but now wth a dagonal a pror covarance matrx (no vertcal correlatons).
6 78 JOURNAL OF APPLIED METEOROLOGY VOLUME wll create errors n the horzontal gradents of the averaged CO felds. Therefore, a proper characterzaton of especally these systematc errors s crucal for a correct nterpretaton of the results.. Summary One of the mportant advantages of the hgher-resoluton sounders planned n the Natonal Polar Orbter Envronmental Satellte System (NPOESS) and European Meteorologcal Operatonal Polar-Orbtng Satellte (METOP) era s the possblty for extractng nontradtonal nformaton from the measurements they provde. It s wthn ths context that we descrbe the extent to whch CO may be drectly retreved from radance data or data from smlar emsson-based spectrometer systems usng nformaton content theory. The analyses presented n ths paper are also contrasted wth the same analyses appled to TOVS/. We have shown that the retreval of CO column concentratons from hgh-spectral-resoluton nfrared sounders looks promsng, confrmng the earler results of Engelen et al. (00). These retrevals potentally offer hgh enough accuracy to be useful for CO nverson studes that seek to estmate snks and sources, although the nformaton s concentrated manly n the free troposphere and only very broad structures can be observed. Ths result suggests that weghted free tropospherc column retreval s feasble. On the other hand, the current TOVS/ nstrument s only able to detect sgnals comparable to the seasonal ampltude of atmospherc CO. Both nstruments could beneft from spatal and temporal averagng of the ndvdual retrevals, but any systematc error would be retaned n the averaged CO values, whch could lead to errors n the spatal gradents of the CO felds. The use of the exstng -yr record of observatons s probably of lmted use for nverson studes, but the data mght be very useful to get a long-term observatonal vew of the seasonal atmospherc CO varablty. Specfyng a correct a pror covarance matrx s very mportant to obtan the most nformaton from the observatons. Small-scale vertcal structures should be constraned more strongly than large-scale vertcal structures because the weghtng functons are generally broad and the retreval can therefore only retreve nformaton about the large-scale structures. Plans exst to augment the data obtaned by the hghresoluton sounders wth data from spectrometers desgned to measure the spectrally reflected sunlght at ultrafne resoluton n specfc CO absorpton bands. The CO measurement approach usng these measurements s descrbed n O Bren and Rayner (00) and employs radance measurements n two carefully selected absorpton bands located n the near-nfrared regon of the solar spectrum. The complementary nature of these observatons and the extent to whch they add nformaton on boundary layer CO are currently under nvestgaton. Acknowledgments. The work descrbed n ths paper was supported by NASA Contract NCC5-6 and NOAA Contract NA7RJ8. REFERENCES Andres, R. J., G. Marland, I. Fung, and E. Matthews, 996: A dstrbuton of carbon doxde emssons from fossl fuel consumpton and cement manufacture, Global Bogeochem. Cycles, 0, 9 0. Chédn, A., A. Hollngsworth, N. A. Scott, S. Serrar, C. Crevoser, and R. Armante, 00: Annual and seasonal varatons of atmospherc CO, N O, and CO concentratons retreved from NOAA/TOVS satellte observatons. Geophys. Res. Lett., 9, 69, do:0.09/00gl008. Engelen, R. J., A. S. Dennng, K. R. Gurney, and G. L. Stephens, 00: Global observatons of the carbon budget.. Expected satellte capabltes for emsson spectroscopy n the EOS and NPOESS eras. J. Geophys. Res., 06, ,,, and TransCom Modelers, 00: On error estmaton n atmospherc CO nversons. J. Geophys. Res., 07, 65, do:0.09/00jd0095. Entng, I., C. Trudnger, and R. Francey, 995: A synthess nverson of the concentraton and dc of atmospherc CO. Tellus, 7B, 5 5. Francey, R., P. Tans, C. Allson, I. Entng, J. Whte, and M. Troler, 995: Changes n the oceanc and terrestral carbon uptake snce 98. Nature, 7, 6 0. Gurney, K. R., and Coauthors, 00: Towards robust regonal estmates of CO sources and snks usng atmospherc transport models. Nature, 5, Kamnsk, T., W. Knorr, P. J. Rayner, and M. Hemann, 00: Assmlatng atmospherc data nto a terrestral bosphere model: A case study of the seasonal cycle. Global Bogeochem. Cycles, 6, 066, do:0.09/00gb006. Masare, K., and P. Tans, 995: Extenson and ntegraton of atmospherc carbon doxde data nto a globally consstent measurement record. J. Geophys. Res., 00, O Bren, D. M., and P. J. Rayner, 00: Global observatons of the carbon budget. : CO concentratons from dfferental absorpton of reflected sunlght n the.6 m band of CO. J. Geophys. Res., 07, 5, do:0.09/00jd Rayner, P. J., and D. M. O Bren, 00: The utlty of remotely sensed CO concentraton data n surface source nversons. Geophys. Res. Lett., 8, Rödenbeck, C., S. Houwelng, M. Gloor, and M. Hemann, 00: CO flux hstory nferred from atmospherc data usng a global nverson of atmospherc transport. Atmos. Chem. Phys. Dscuss.,, Rodgers, C. D., 000: Inverse Methods for Atmospherc Soundng: Theory and Practce. World Scentfc, 8 pp., and B. J. Connor, 00: Intercomparson of remote soundng nstruments. J. Geophys. Res., 08, 6, do:0.09/ 00JD0099. Shannon, C. E., and W. Weaver, 99: The Mathematcal Theory of Communcaton. Unversty of Illnos Press, pp. Smth, W. L., H. M. Woolf, C. M. Hayden, D. Q. Wark, and L. M. McMlln, 979: The TIROS-N Operatonal Vertcal Sounder. Bull. Amer. Meteor. Soc., 60,
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