LECTURES ON INVERSE MODELING by Daniel J. Jacob, Harvard University January 2007

Transcription

1 LECURES ON INVERSE MODELING by Dnel J. Jcob, Hrvrd Unversty Jnury 007 FOREWORD INRODUCION.... BAYES HEOREM INVERSE PROBLEM FOR SCALARS VECOR-MARIX OOLS FOR INVERSE MODELING Error covrnce mtrces Gussn probblty dstrbuton functons for vectors Jcobn mtrx Model djont INVERSE PROBLEM FOR VECORS Anlytcl mxmum posteror (MAP) soluton Avergng kernel mtrx Peces of nformton n n observng system Exmple pplcton Sequentl updtng KALMAN FILER ( 3-D Vr ) ADJOIN APPROACH ( 4-D Vr ) OBSERVING SYSEM SIMULAION EXPERIMENS... 3

2 FOREWORD hese lectures on nverse modelng re prt of hlf-course, Models of Atmospherc rnsport nd Chemstry, tht I hve tught to grdute students t Hrvrd snce 00. Most of tht course focuses on the constructon of chemcl trnsport models, but I lso cover nverse modelng s t reltes to tmospherc chemstry pplctons: retrevng concentrtons from stellte rdnce mesurements, retrevng emssons from observed concentrtons, chemcl dt ssmlton. Orgnlly I dedcted just one lecture to nverse modelng, but ths hs grown to bout four lectures s my nterest nd experence n ths re hve grown. I frst educted myself n nverse modelng n 000 through redng course wth my former student Colette L. Held, n whch we worked through the book by Clve D. Rodgers, Inverse Methods for Atmospherc Soundng (World Scentfc, 000). I found t to be wonderfully rgorous nd cler, but dffcult.. My lectures re hevly nfluenced by ths book but try to be esly ccessble to begnnng grdute student wth bsc bckground n lner lgebr. I hve mde pont of usng Rodgers notton becuse I consder t model of clrty nd n order to encourge the nterested reder to go to hs book for further nformton nd nsghts. I would gretly pprecte feedbck from reders on ny topcs tht need to be clrfed, ny errors, nd ny generl ssues of content. Send me n eml t djcob@fs.hrvrd.edu. I would lke to thnk my students nd postdocs who hve worked on nverse modelng s prt of ther reserch nd educted me n the process: Colette L. Held, Emly Jn, Dyln B. Jones, Monk A. Kopcz. Pul I. Plmer. I would lke to extend specl thnks to Monk Kopcz for her comments on the djont modelng chpter. Dnel J. Jcob Jnury 007 Dnel J. Jcob, Lectures on Inverse Modelng,

3 1. INRODUCION Inverse modelng s forml pproch for estmtng the vrbles drvng the evoluton of system by tkng mesurements of the observble mnfesttons of tht system, nd usng our physcl understndng to relte these observtons to the drvng vrbles. We cll the vrbles tht we wsh to estmte the stte vrbles, nd ssemble them nto stte vector x. We smlrly ssemble the observtons nto n observton vector y. Our understndng of the reltonshp between x nd y s descrbed by physcl model F, clled the forwrd model: y = F( x, b ) + (1.1) where b s prmeter vector ncludng ll model vrbles tht we do not seek to optmze (we cll them model prmeters), nd s n error vector ncludng contrbutons from errors n the observtons, n the forwrd model, nd n the model prmeters. From nverson of equton (1.1), we cn obtn x gven y. In the presence of error ( 0 ), the best tht we cn cheve s sttstcl estmte, nd we need to wegh the resultng nformton gnst our pror ( pror) knowledge x of the stte vector before the observtons were mde. he optml soluton of x reflectng ths ensemble of constrnts s clled the posteror, the optml estmte, or the retrevl. We wll use these words nterchngebly. he choce of stte vector (.e., whch vrbles to nclude n x vs. n b) depends on wht vrbles we wsh to optmze, wht nformton s contned n the observtons, nd wht computtonl costs re ssocted wth the nverson. Inverse models hve three mjor pplctons n tmospherc chemstry: 1. Retrevl of tmospherc concentrtons from observed rdnces. Consder the problem of usng ndr spectr mesured from spce to retreve the vertcl profle of trce gs. he mesured rdnces t dfferent wvelengths represent the observton vector y, nd the trce gs concentrtons t dfferent vertcl levels represent the stte vector x. he forwrd model solves the rdtve trnsfer equton to clculte y s functon of x nd of ddtonl prmeters b ncludng surfce emssvty, tempertures, clouds, spectroscopc dt, etc. Inverson of ths forwrd model usng the observed spectr then provdes retrevl of x.. Optml estmton of surfce fluxes. Consder the problem of quntfyng surfce fluxes of gs on (lttude x longtude x tme) grd. he fluxes on tht grd represent the stte vector x. We mke observtons of tmospherc concentrtons of the gs from network of stes, representng n observton vector y. he forwrd model s chemcl trnsport model (CM) reltng x to y. he prmeter vector b ncludes meteorologcl vrbles nd ny chrcterstcs of the surfce flux (such s durnl vrblty) tht re smulted n the CM but not resolved n the stte vector. he nformton on x from the observtons s clled top-down constrnt on the surfce fluxes. A pror nformton on x bsed on our knowledge of the processes Dnel J. Jcob, Lectures on Inverse Modelng, 007.

4 determnng the fluxes (such s fuel combuston sttstcs, lnd type dt bses, etc) s clled bottom-up constrnt. Combnton of the top-down nd bottom-up constrnts provdes the optml estmte of x. Insted of the surfce fluxes themselves, we my wsh to optmze the drvng vrbles of surfce flux model; the pproch s exctly the sme but the dmenson of x my be gretly reduced. 3. Chemcl dt ssmlton. Consder the problem of constructng contnuous 3-D feld of concentrtons of trce gs over the globe on the bss of lmted mesurements of concentrtons t solted ponts nd scttered tmes. Such constructon my be useful to produce chemcl forecsts, to ssess the consstency of mesurements mde from dfferent pltforms, or to mprove estmtes of the concentrtons of non-mesured speces from mesurements of chemclly lnked speces. We defne the stte vector x(t) s the 3-D ensemble of grdded concentrtons t tme t, nd y s the vector of observtons vlble over the tme ntervl [t- t, t]. he forwrd model s CM ntlzed wth x(t- t) nd provdng forecst for tme t. hs forecst represents our pror nformton for x(t), to be corrected on the bss of the observtons over the tme ntervl [t- t, t]. Our stte vector here s n generl very lrge, whle the observton vector t ny gven tme s reltvely sprse. We refer to ths type of nverse model pplcton s dt ssmlton. Proper consderton of errors s crucl n nverse modelng. o pprecte ths, let us exmne wht hppens f we gnore errors. We lnerze the forwrd model y = F(x, b) round the pror estmte x tken s frst guess: y = F x b + K x x + Ο x x (, ) ( ) (( ) ) (1.) where K = y/ xs the Jcobn mtrx of the forwrd model wth elements k = y / x evluted t x = x. Let n nd m be the dmensons of x nd y, respectvely. j j In the bsence of error, m = n ndependent mesurements constrn x unquely. he Jcobn mtrx s then nxn mtrx of full rnk nd hence nvertble. We obtn for x: 1 x= x + K ( y F( x, b)) (1.3) If F s nonlner, the soluton (1.3) must be terted wth reclculton of the Jcobn round successve guesses for x untl stsfctory convergence s cheved. Now wht hppens f we mke ddtonl observtons, such tht m > n? In the bsence of error these observtons must necessrly be redundnt. However, we know from experence tht useful constrnts on n tmospherc system typclly requre very lrge number of mesurements, m >> n. hs reflects errors n the observtons nd n the forwrd model, descrbed by the error vector n equton (1.1). hus equton (1.3) s not pplcble n prctce; successful nverson requres dequte chrcterzton of the error nd consderton of pror nformton on x. he pror estmte hs ts own error: x = x+ (1.4) Dnel J. Jcob, Lectures on Inverse Modelng,

5 nd the nverse problem then nvolves weghtng the error sttstcs of nd to solve the optml estmton problem, wht s the best estmte of x gven y?. hs s done usng Byes theorem, presented below.. BAYES HEOREM Byes theorem provdes the generl foundton for nverse models. Consder pr of vectors x nd y. Let P(x), P(y), P(x,y) represent the correspondng probblty dstrbuton functons (pdfs), so tht the probblty of x beng n the rnge [x, x+dx] s P(x)dx, the probblty of y beng n the rnge [y, y+dy] s P(y)dy, nd the probblty of (x, y) beng n the rnge [x, x+dx, y, y+dy] s P(x,y)dxdy. Let P(y x) represent the pdf of y when x s ssgned to certn vlue. We cn wrte P(x,y)dxdy equvlently s P( x, y) dxdy = P( x) dxp( y x) dy (.1) or s P( x,y) dxdy = P( y) dyp( x y) dx (.) Elmntng P(x,y), we obtn Byes theorem: P( y x) P( x) P( x y) = P( y) (.3) hs theorem formlzes the nverse problem posed n chpter 1. Here P(x) s the pdf of the stte vector x before the mesurements re mde (tht s, the pror pdf). P(y x) s the pdf of the observton vector y gven the true vlue for x, whch the nstrument knows bout. P(x y) s the posteror pdf for the stte vector reflectng the nformton from the mesurements tht s, t s the pdf of x gven the mesurements y. he optml or mxmum posteror (MAP) soluton for x s gven by the mxmum of P(x y), tht s, the soluton to x P( x y) = 0 where x s the the grdent opertor n the stte vector spce. he probblty functon P(y) n the denomntor of (.3) s ndependent of x, nd we cn vew t merely s normlzng fctor to ensure tht P( x y) dx = 1. It plys no role n determnng the MAP soluton (snce t s ndependent of x) nd we gnore t n wht follows INVERSE PROBLEM FOR SCALARS Applcton of Byes theorem to obtn the MAP soluton s esest to frst understnd usng sclrs. Consder source relesng speces X to the tmosphere wth Dnel J. Jcob, Lectures on Inverse Modelng,

6 n emsson flux x. We hve n pror estmte x ± σ for the vlue of x, where σ s the error vrnce. For exmple, f X s emtted from power plnt, the pror nformton would be bsed on knowledge of the type nd mount of fuel beng burned n tht plnt, ny emsson control equpment, etc. We set up smplng ste to mesure the concentrton of X downwnd of the source. We mesure concentrton y ± σ where σ s the nstrument error vrnce. We then use CM to obtn reltonshp between x nd y s y = F( x) ± σ m (3.1) where σ m s the CM error vrnce. Let us ssume tht the CM reltonshp between x nd y s lner. Let us further ssume tht the nstrument nd CM errors re uncorrelted so tht the correspondng vrnces re ddtve. he mesured concentrton y s then relted to the true source x by y = kx± σ (3.) where the coeffcent k s obtned from the CM, nd σ s the observtonl error vrnce defned s the sum of the nstrumentl nd CM errors: σ = σ + σ (3.3) he observtonl error ncludes the forwrd model error, so t s not purely from observtons. hnk of t s the error n the observng system desgned to plce constrnts on the stte vector. After mkng the mesurement, we seek n mproved estmte ˆx of x tht optmlly ccommodtes the top-down constrnt from the mesurement nd the bottomup constrnt from the pror. We uses Byes theorem. Assumng Gussn error dstrbutons, we hve m 1 ( x x ) Px ( ) = exp[ ] (3.4) σ π σ 1 ( y kx) Py ( x) = exp[ ] σ π σ (3.5) Applyng Byes theorem (.3) nd gnorng the normlzng terms tht re ndependent of x, we obtn: ( x x ) ( y kx) Px ( y) exp[ ] (3.6) σ σ Fndng the mxmum vlue for P(x y) s equvlent to fndng the mnmum n the cost functon J(x): Dnel J. Jcob, Lectures on Inverse Modelng,

7 ( x x ) ( y kx) J( x) = + σ σ (3.7) whch s lest-squres cost functon weghted by the vrnce of the error n the ndvdul terms. It s clled χ cost functon, nd J(x) s formulted n equton (3.7) s clled the χ sttstc. he optml estmte to obtn nlytclly: ˆx s the soluton to J / x= 0, whch s strghtforwrd xˆ = x + g( y kx ) (3.8) where g s gn fctor gven by g = k kσ σ + σ (3.9) In (3.8), the second term on the rght-hnd sde represents the correcton to the pror on the bss of the mesurement y. he gn fctor s the senstvty of the retrevl to the observton: g = x / y. We see from (3.9) tht the gn fctor depends on the reltve mgntudes of σ nd σ /k. If σ << σ /k, then g 0 nd x x ; the mesurement s useless becuse the observtonl error s too lrge. If by contrst σ >> σ /k, then g 1/ k nd x y/ k ; the mesurement s so precse tht t constrns the soluton wthout recourse to the pror nformton. We cn lso express the retrevl soluton x. We hve ˆx n terms of ts proxmty to the true y = kx+ (3.10) where (wth vrnce σ ) s the observtonl error. Replcng n equton (3.8) we obtn x = x + (1 ) x + g (3.11) where = gk s n vergng kernel descrbng the reltve weghts of the pror x nd the true vlue x n contrbutng to the retrevl. he vergng kernel represents the senstvty of the retrevl to the true stte: = x/ x. he gn fctor s now ppled to the observtonl error n the thrd term on the rght hnd sde. We see from equton (3.9) tht the vergng kernel smply weghts the error vrnces σ nd (σ /k). In the lmt σ >> σ /k, then 1nd the pror does not contrbute to the soluton. However, our blty to pproch the true soluton s lmted by the thrd term g n equton (3.11) wth vrnce (g σ). Snce n the bove lmt g 1/ k, we obtn n fct x y/ k s derved prevously. he error on the retrevl s then defned by the observtonl error. Dnel J. Jcob, Lectures on Inverse Modelng,

8 We cll (1-)x the smoothng error snce t lmts the blty of the retrevl to obtn solutons deprtng from the pror, nd we cll g. the retrevl error. We cn derve generl expresson for the retrevl error vrnce by strtng from equton (3.6) nd expressng t n terms of Gussn dstrbuton for the error n (x- x ). We thus obtn form n exp[ ( x x ) / σ ] where σ s the vrnce of the error n the posteror x. he clculton s lborous but strghtforwrd, nd yelds = + (3.1) ˆ σ σ σ / ( k ) Notce tht the posteror error s lwys less thn the pror nd observtonl errors, nd tends towrds one of the two n the lmtng cses tht we descrbed. Let us now consder stuton where our sngle mesurement y s not stsfctory n constrnng the soluton. We could n prncple remedte ths problem by mkng m mesurements y, ech ddng term to the cost functon (6.10). Assumng the sme observtonl error vrnce for ech mesurement: J( x) ( x x) ( y kx) σ σ = + m = 1 (3.13) We cn re-express J(x) s ( x x ) ( y kx) J( x) = + (3.14) σ σ / m Where < > denotes the men vlue nd σ / m s the vrnce of the error on the men of (y kx) (ths s the centrl lmt theorem). By ncresng m, we could thus pproch the true soluton: m xˆ < y > / k nd ˆ σ 0. However, ths works only f the observtonl error s (1) truly rndom, () uncorrelted between dfferent mesurements. Wth regrd to (1), t s crtcl to estblsh f there s ny systemtc error (lso clled bs) n the mesurement. In the presence of bs, no number of mesurements wll llow convergence to the true soluton; the bs wll be propgted through the gn fctor nd correspondngly ffect the retrevl. Accurte clbrton of the mesurng nstrument nd of the forwrd model s thus essentl. Wth regrd to (), nstrumentl errors (s from photon-countng) re often uncorrelted; however, forwrd model errors rrely re. For exmple, two successve mesurements t ste my smple the sme r mss nd thus be subject to the sme trnsport error n the CM used s forwrd model. It s thus mportnt to determne the error correlton between the dfferent mesurements. hs error correlton cn best be descrbed by ssemblng the mesurements nto vector nd constructng the observtonl error covrnce mtrx. Delng wth error correltons, nd lso delng wth multple sources, requres tht we swtch to vector-mtrx notton n our formulton of the nverse problem.. We do so n the next secton. Dnel J. Jcob, Lectures on Inverse Modelng,

9 Onr lst pont before we move on. We ssumed n the bove lner forwrd model y = F(x) = kx. Wht f the forwrd model s not lner? We cn stll clculte MAP soluton ˆx s the mnmum n the cost functon (3.7), where we replce kx by the nonlner form F(x). he error n ths MAP soluton s not Gussn though, so equton (3.1) would not pply. An lterntve s to lnerze the forwrd model round x s y k = to obtn n ntl guess x 1 of ˆx, nd then terte on the soluton by x x = x y reclcultng k =, nd so on. As we wll see, the ltter s the only prctcl x x = x 1 soluton s we move from sclr to vector spce. 4. VECOR-MARIX OOLS FOR INVERSE MODELING Let us now consder the problem of stte vector x of dmenson n wth pror vlue x for whch we seek n mproved estmte on the bss of n ensemble of observtons ssembled nto n observton vector y of dmenson m. he forwrd model s y = F( x) + (4.1) s n (1.1) but wthout the model prmeters to smplfy notton. Inverse nlyss requres defnton of error sttstcs nd pdfs for vectors, nd of the Jcobn mtrx for the forwrd model. he error sttstcs re provded by error covrnce mtrces, nd the pdfs re constructed n mnner tht ccounts for covrnce between vector elements. Numercl constructon of the Jcobn mtrx my be done usng ether the forwrd model or ts djont. We begn by descrbng these dfferent objects before proceedng to the soluton of the nverse problem n the followng sectons. 4.1 Error covrnce mtrces he error covrnce mtrx for vector s the nlog of the vrnce for sclr. Consder vector x = (x 1, x n ) of dmenson n. Its error covrnce mtrx S hs s dgonl elements the error vrnces of the ndvdul elements of x, nd s offdgonl elements the error covrnces between elements of x. Let us construct the pror error covrnce mtrx S of the stte vector for the nverson,.e., the nlog of σ n the sclr problem (secton 6.). he pror vlue for the stte vector s x nd the true vlue s x. he error vrnce vr (x - x, ) for element x s defned s the expected vlue of (x - x, ) when x s smpled over ts pror pdf P(x ). he error covrnce cov((x -x, ), (x j - x,j ))for the pr (x,, x j ) s defned s the expected vlue of the product (x x, )(x j x,j ) when x nd x j re smpled over ther respectve pror pdfs. he mtrx s thus constructed s: Dnel J. Jcob, Lectures on Inverse Modelng,

10 S vr( x1 x,1) cov( x1 x,1, xn x, n) = cov( x1 x,1, xn x, n) vr( xn x, n) (4.) We express t n compct mthemtcl form s S = E[( x x)( x x) ] where E s the expected vlue opertor returnng the expected vlue of the quntty,.e., ts verge vlue over lrge number of determntons. Constructng n ccurte error covrnce mtrx requres detled sttstcl nformton nd knowledge tht s often dffcult to obtn. Dependng on the problem, smple estmtes of errors my be suffcent. Exmple. If we ssume unform 50% error on the ndvdul elements of x wth no correlton between the errors on the dfferent elements, then the dgonl elements of S re 0.5x, nd the off-dgonl elements re ll zero. Let us now smlrly construct the observtonl error covrnce mtrx S of the error vector n the forwrd model (4.1) used to relte x to y: y = F( x) + (4.3) S s constructed s S vr( 1) cov( 1, n) = cov( 1, n) vr( n) (4.4) nd we cn express t n compct form s S = Ε[ ]. Included n re ll the sources of error tht would prevent the forwrd model from reproducng the observtons. hey cn be seprted nto nstrument errors ( ) nd forwrd model errors ( m ): = + m (4.5) hese errors re n generl uncorrelted so the correspondng error covrnce mtrces re ddtve: S =S +S m (4.6) he nstrument errors cn be determned from clbrton stndrds. he forwrd model errors re more dffcult to estmte. hey nclude errors n the model representtons of ll processes not correctble through djustment of the stte vector. Dnel J. Jcob, Lectures on Inverse Modelng,

11 Exmple. Consder the problem of nvertng CO surfce fluxes for ndvdul contnents on the bss of n ensemble of worldwde CO tmospherc observtons t surfce stes, nd usng n Eulern CM s the forwrd model. Contrbutons to the forwrd model error wll nclude: 1. he model trnsport error;. Sptl nd temporl smoothng ntrnsc to the model (grd resoluton, tme step), preventng t from resolvng fne-scle vrblty n the observtons ths s clled the representton error; 3. Error n the dstrbuton of pror CO surfce fluxes on subcontnentl scles tht s not corrected by djustment of the stte vector on contnentl scles ths s clled the ggregton error. he error covrnce mtrx s n generl complcted object to nterpret. Egendecomposton cn be useful tool to dentfy the domnnt error ptterns. In the he error covrnce mtrx S for vector x hs full rnk, snce otherwse would mply tht n element s perfectly known. It s lso symmetrc snce the covrnce opertor s symmetrc. It therefore hs orthonorml egenvectors e wth egenvlues λ. Egennlyss of S s useful to dgnose the domnnt error ptterns. S cn be spectrlly decomposed long ts egenvectors s S= λee (4.7) Consder now the bse for x defned by the egenvectors. In tht bse, egenvector e hs vlue of 1 for ts th element nd vlue of zero for ll ts other elements, so tht the error covrnce mtrx s dgonl mtrx of egenvlues: λ1 0 S = (4.8) 0 λ n he egenvlue λ thus represents the error vrnce ssocted wth the orthonorml error pttern e. By egenvlue decomposton of S nd rnkng of egenvlues, one cn dentfy the domnnt error ptterns nd the correspondng vrnces. 4. Gussn probblty dstrbuton functons for vectors Applcton of Byes theorem (chpter ) requres formulton of probblty dstrbuton functons for vectors. We derve here the generl Gussn pdf for vector x of dmenson n wth expected vlue <x> nd error covrnce mtrx S. If the errors on the ndvdul elements of x were uncorrelted (.e., f S were dgonl), then the pdf of the vector would smply be the product of the pdfs for the ndvdul elements. hs smple soluton cn be cheved by trnsformng x to the bss of egenvectors e of S. Let Dnel J. Jcob, Lectures on Inverse Modelng,

12 us ssemble the egenvectors s the columns of mtrx E; then z = E (x - <x>) s the trnsformed vlue of x - <x> n the egenvector bss. he pdf of z s then 1 z P( z) = exp[ ] (4.9) λ 1/ ( πλ) he determnnt of mtrx s the product of ts egenvlues: S = λ (4.10) Further defnng Λs the dgonl mtrx of egenvlues, we cn rewrte (4.9) s: P 1 1 = z (4.11) ( π ) S 1 ( z) exp[ Λ z] n / 1/ nd replce z: 1 1 P( ) = exp[ ] n / 1/ ( π ) -1 x (x-<x>) EΛ E(x-<x>) S (4.1) he spectrl decomposton (4.7) of S cn be expressed n terms of E s S=EΛE Snce S s symmetrc mtrx, E = E -1, nd therefore -1-1 S =EΛ E resultng n the generl pdf expresson for vector x: 1 1 P( ) = exp[ ] n / 1/ ( π ) -1 x (x- < x >) S (x- < x >) S (4.13) (4.14) (4.15) 4.3 Jcobn mtrx he Jcobn mtrx s lnerzton of the forwrd model tht enbles pplcton of mtrx lgebr to the nverse problem. It represents the senstvty of the observton vrbles y to the stte vrbles x, ssembled n mtrx form: y K = xf = x (4.16) Dnel J. Jcob, Lectures on Inverse Modelng,

13 wth ndvdul elements k = y / x. If the forwrd model s lner, then K does not j j depend on x nd fully descrbes the forwrd model for the purpose of the nverson. If the forwrd model s not lner, then K s functon of x nd represents lnerzton of the forwrd model round x. It needs to be clculted ntlly for the pror vlue x, representng the ntl guess for x, nd then re-clculted s needed for updted vlues of x durng tertve convergence to the soluton. Dependng on the degree of non-lnerty, K my not need to be re-clculted t ech terton. Constructon of the Jcobn mtrx my be done nlytclly f the forwrd model s smple, s n 0-D chemcl model where the evoluton of concentrtons s determned by locl recton rtes. If the forwrd model s complcted, such s 3-D CM, then the Jcobn must be constructed numerclly. he stndrd pproch, nd the best to use f the dmenson of the stte vector s less thn tht of the observton vector (n < m), s to buld the Jcobn mtrx column by column by successvely perturbng the ndvdul elements x of the stte vector by smll ncrements x, nd pplyng the forwrd model to obtn the resultng perturbton y. If the observtons re sprse or the stte vector s lrge so tht n > m, then more effectve wy to construct the Jcobn s through the model djont, s descrbed below. 4.4 Model djont he djont of model s the trnspose of ts Jcobn mtrx. It turns out to be very useful n nverse modelng pplctons where observed concentrtons re used to constrn stte vector of sources or concentrtons t prevous tmes. We wll dscuss ths n chpter 7. It cn lso be n effcent tool for numercl constructon of the correspondng Jcobn mtrx. Consder CM dscretzed over tme steps [t 0, t, t n ], nd let y n represent the vector of concentrtons t tme t n,. We wsh to determne ts senstvty to the stte vector x t tme t 0 (generlzton to tme-nvrnt stte vector or to stte vector elements for dfferent tmes wll be shown lter to be mmedte). he correspondng Jcobn mtrx s Κ = y n/x. By the chn rule, yn yn yn-1 y1 y0 K = =... x y y y x n-1 n- 0 (4.17) where the RHS s product of mtrces. As dscussed n secton 4.3, the stndrd wy to construct the Jcobn numerclly s by successvely perturbng the ndvdul elements of x nd pplyng the CM to obtn the resultng perturbton y. Another wy s to tke the trnspose of the Jcobn mtrx,.e., the djont of the CM: yn yn-1 y y 1 0 y0 y y... 1 n-1 y n... yn-1 yn- y0 x x y0 yn- yn-1 K = = (4.18) where we hve mde use of the property tht the trnspose of product of mtrces s equl to the product of the trnsposed mtrces n reverse order. he djont model descrbed by (4.18) offers n lternte wy of constructng the Jcobn mtrx by Dnel J. Jcob, Lectures on Inverse Modelng,

14 pplyng successve perturbtons y to the ndvdul elements of y n nd successvely ( ) pplyng the opertors y / y, -1 n n-1 ( y / y ) n-1 n- ll the wy to ( y x ) obtn the senstvty y/ x whch s row of the Jcobn mtrx. hus we construct the Jcobn mtrx row by row, nsted of column by column s prevously. hs pproch s more economcl f dm(y) < dm(x), becuse s we wll see lter the cost of runnng the djont model s comprble to tht of runnng the forwrd model. A sngle run of the djont model from t n to t 0 cn determne the senstvty of y n to stte vector elements t dfferent tmes, or to tme-nvrnt stte vector n the ltter cse, one sums the senstvtes to the stte vector for the successve steps bck n tme. Constructon of the djont model K requres lnerzton of the forwrd model (CM) to express t s product of mtrces tht we cn then trnspose. Consder the mtrces Z = y / y descrbng the evoluton of the CM over ndvdul tme steps [t -1, t ], nd the mtrx Z = / 0 y0 x, such tht 0 / to K = Z o Z 1...Z... Z n (4.19) o determne the ndvdul mtrces Z we need to decompose the CM n terms of ts ndvdul lnerzed opertors. Consder CM wth successve pplcton of opertors for emssons (E), chemstry (C), convecton (C o ), nd dvecton (A) over model tme step [t -1, t ]. Ech opertor updtes the concentrton over the tme step, nd ths updte cn be descrbed by mtrx representng the lnerzed opertor. For exmple, the updte from dvecton cn be wrtten y = A y 1where A s the lnerzed dvecton opertor. hus we cn wrte Z s product of mtrces representng the lnerzed opertors: Z = A C C E (4.0) nd ts trnspose s then o, Z = E C C A o, (4.1) Constructon of Z = / 0 y0 x s done n exctly the sme wy except tht the lnerzton of the relevnt CM opertor s done wth respect to x rther thn to y. In the common pplcton where we re nterested n the senstvty of concentrtons to emssons, the lnerzton wth respect to x s done n the emsson opertor E. hs s strghtforwrd to do, s dscussed below. Constructon of the djont of forwrd model thus nvolves two steps. he frst s the constructon of ngent Lner Model (LM) tht lnerzes the ndvdul opertors of the forwrd model. he second s the trnsposton of these lner opertors. Constructon of the LM from non-lner opertors s n rduous tsk nd commercl softwre pckges re vlble for ths purpose. Dnel J. Jcob, Lectures on Inverse Modelng,

15 For smple lner opertors, however, constructon of the djont model my be strghtforwrd. Consder for exmple lner dvecton opertor n whch the trnsport of mss from grdbox j to grdbox k over tme step [t -1, t ] s descrbed by the mtrx coeffcent jk such tht yk, / yj, 1 = jk. In the trnsposed opertor, ths coeffcent pples to the reverse flow from grdbox k to grdbox j, tht s, y / y =. We thus see tht reversng the wnds s ll we need to do to obtn j, k, 1 jk the trnspose of lner dvecton opertor. hs mkes sense n terms of the generl djont model phlosophy of propgtng senstvtes bck n tme. Even f the dvecton scheme s wekly non-lner, usng reverse wnds my be n cceptble pproxmton for the ske of ese n buldng the djont. he vldty of the pproxmton cn be tested s wll be dscussed n chpter 8. As nother exmple of smple djont constructon, consder lner chemcl opertor consstng of locl frst-order loss. he mtrx for tht opertor s dgonl snce there re no nterctons between speces or grdboxes. rnsposton does not chnge the opertor, whch s then clled self-djont; we cn pply the orgnl opertor n the djont model. Agn, ths mkes sense; f speces decys wth certn tme constnt, then the senstvty of concentrtons to condtons bckwrd n tme wll decy wth tht sme tme constnt. he emsson opertor s smlrly self-djont when expressed n terms of the senstvty of concentrtons to emssons (.e., for constructon of the mtrx Z = / 0 y0 x ). It s null mtrx when expressed n terms of the senstvty to concentrtons for the prevous tme step (.e., for constructon of the mtrx = y / y ). Z INVERSE PROBLEM FOR VECORS he vector-mtrx tools presented n chpter 4 llow us to pply Byes s theorem to obtn n optml estmte of stte vector x (dm n) on the bss of the observton vector y (dm m), the pror nformton x, the forwrd model F, nd the error covrnce mtrces S nd S (the reder s encourged to return to chpter 3 s needed for smple pplcton of Byes theorem to the sclr nverson problem, whch helps develop ntuton for the mterl presented n the present chpter). We need to lnerze the forwrd model, f t s not lredy, n order to use mtrx lgebr.. hs s done by ylor expnson bout the pror vlue s descrbed by (1.), where K = y/ x= x F(x)s the Jcobn mtrx. If the forwrd model s lner, K s nvrnt wth x. If t s not, then K must be clculted ntlly for x = x nd re-clculted tertvely s the nverson progresses. In ths chpter we ssume tht the forwrd model s lner or hs been lnerzed so tht y=kx+ where s the observtonl error vector prevously ntroduced n chpter 1. (5.1) Dnel J. Jcob, Lectures on Inverse Modelng,

16 5.1 Anlytcl mxmum posteror (MAP) soluton Folllowng the generl pdf formulton for vectors (secton 4.), the pdfs from Byes theorem n chpter re gven by x = x x S x x + c -1 ln P( ) ( ) ( ) 1 (5.) 1 ln P( y x) = ( y Kx) S ( y Kx) +c -1 1 ln P( x y) = ( x x ) S ( x x ) + ( y Kx) S ( y Kx) + c 3 (5.3) (5.4) where c 1, c, c 3 re constnts. he mxmum posteror (MAP) soluton s the vlue of x tht yelds the mxmum of P(x y), or equvlently the mnmum of the sclr-vlued cost functon J(x): -1 1 J ( x) = ( x x ) S ( x x ) + ( y Kx) S ( y Kx) (5.5) o fnd ths mnmum, we solve for xj ( x) = 0 : x = S x x + K S Kx-y = xj ( ) ( ) ( ) (5.6) he soluton s strghtforwrd nd cn be expressed n compct form s xˆ = x + G( y Kx ) (5.7) wth G gven by G = S K ( KS K + S 1 ) (5.8) or equvlently by: G = ( K S K+ S ) K S (5.9) (the second form s more expedtous to compute f m > n). G s the gn mtrx nd descrbes the senstvty of the retrevl to the observtons,.e., G = xˆ / y. he error covrnce mtrx Ŝ of ˆx cn be clculted s n chpter 3 for the sclr problem by rerrgng the rght-hnd sde of (5.4) to be of the form ˆ 1 ( x xˆ) S ( x xˆ). he lgebr s strghtforwrd nd yelds ˆ 1 S= ( K S 1 K+ S 1 ) (5.10) Note the smlrty of our equtons for chpter 3. J ( x), xˆ, G, Sˆ to those derved for sclrs n Dnel J. Jcob, Lectures on Inverse Modelng,

17 We ponted out n the sclr problem the dnger of over-nterpretng the pprent reducton n error vrnce on the stte vector from σ to ˆ σ s result of ccumultng lrge number of observtons. he sme concern pples here. he reducton n error from S to Ŝ ssumes tht the observtonl error s truly rndom nd tht error covrnces n the observtons re fully ccounted for. hese requrements re often not stsfed, n whch cse Ŝ wll underestmte the ctul posteror error. An often more relstc wy of ssessng the error n ˆx s through n ensemble of nverse clcultons wth vrous perturbtons to model prmeters, observtonl vlues, nd covrnce error estmtes wthn ther expected uncertntes. 5. Avergng kernel mtrx A useful wy to express the blty of n observtonl system to constrn the true vlue of the stte vector s wth the vergng kernel mtrx A= x/ ˆ x, representng the senstvty of the MAP soluton ˆx to the true stte x. A s the product of the gn mtrx G = xˆ / y nd the Jcobn mtrx K = y/ x: A=GK (5.11) Replcng (5.11) nd (5.1) nto (5.7) we obtn n lternte form of the MAP soluton: xˆ = Ax+ ( I A) x + G (5.1) n where I n s the dentty mtrx of dmenson n. Note the smlrty to (3.11) n the sclr problem. A s weghtng fctor for the reltve contrbutons to the retrevl from the true stte vs. the pror estmte. Ax represents the contrbuton of the true stte to the soluton, (I n A)x represents the contrbuton from the pror, nd G represents the contrbuton from the rndom observtonl error mpped onto stte spce by the gn mtrx G. A perfect observtonl system would hve A = I n. We cll (I n A)x the smoothng error (becuse t smoothes the soluton towrds the pror) nd G the retrevl error. Equton (5.1) lso provdes n lternte expresson for the MAP error covrnce mtrx Ŝ : ˆ S= E[( x-x)(x-x ˆ ˆ) ] = E(( I A)( x x )( x-x ) ( I -A) ) + E[ G G] = ( I -A)S (I -A) +GS G n n n n (5.13) from whch we see tht Ŝ cn be decomposed nto the sum of smoothng error covrnce mtrx ( I - A)S (I - A) nd retrevl error covrnce mtrx GS G. mtrx s n n Algebrc mnpulton yelds n lternte form of the vergng kernel 1 A= In ˆ SS (5.14) Dnel J. Jcob, Lectures on Inverse Modelng,

18 whch shows how the reltve reducton n error enbled by the observtonl system reltve to the pror provdes mproved knowledge of the stte vector. he vergng kernel mtrx s very useful thng to know bout n observton system, nd s essentl for testng or comprng two dfferent observton systems used to determne x. In the cse of n nlytcl MAP soluton nvolvng explct clculton of the Jcobn nd gn mtrces, s descrbed bove, A comes out of the soluton nlytclly ether by (5.11) or (5.14) (whchever form s most convenent to compute). Other pproches to the nverse problem, nvolvng for exmple neurl networks tht ft x to y emprclly on the bss of pror correltons, or the djont pproch descrbed n chpter 7 tht solves numerclly for xj ( x) = 0, do not provde vergng kernel mtrces s prt of ther solutons. An vergng kernel mtrx cn stll be constructed numerclly column by column by (1) tkng smll perturbtons x to ndvdul elements of the stte vector, () pplyng the forwrd model to obtn the resultng perturbton y, (3) pplyng observtonl error to y, nd (4) pplyng the retrevl to y + to obtn xˆ. 5.3 Peces of nformton n n observng system A concept relted to the verge kernel mtrx s the number of peces of nformton n n observng system towrds constrnng n n-dmensonl stte vector. he number of peces of nformton s often clled the number of degrees of freedom for sgnl (DOFS) wth notton d s. It cn be determned s the reducton n the normlzed error on x due to the mesurement. We express the normlzed error on x pror to the mesurement wth the χ cost functon J (x ): -1 ( x) ( x x) S ( x x) J = (5.15) whch hs n expected vlue of n, representng the number of peces of nformton to be obtned for perfect knowledge of the system. After the mesurement hs been mde, the vlue of ths cost functon becomes J xˆ x xˆ S x xˆ) -1 ( ) ( ) = ( (5.16) he DOFS s gven by the dfference n the expected vlues of J (x ) nd J ( ˆx ): d = E[( x x ) S ( x x )] E[( x xˆ) S ( x xˆ)] = n E[( x xˆ) S ( x xˆ)] (5.17) s he quntty notton: -1 ( x xˆ) S ( x xˆ) s sclr nd s thus equvlent to ts trce n mtrx ( x xˆ) S ( x xˆ) = tr(( x xˆ) S ( x xˆ)) = tr(( x xˆ)( x xˆ) S ) (5.18) where the lst equlty explots the commuttvty of the trce opertor: tr(ab) = tr(ba). We thus obtn for the posteror vlue of J : Dnel J. Jcob, Lectures on Inverse Modelng,

19 -1 E[( x xˆ) S ( x xˆ)] = [tr(( x xˆ)( x xˆ) S )] = tr( SS ) -1-1 E ˆ (5.19) so tht ds ˆ -1 tr( ) tr( ˆ -1 = n SS = I SS ) = tr( A) n (5.0) he number of peces of nformton n n observng system (or degrees of freedom for sgnl) s gven by the trce of the vergng kernel mtrx. 5.4 Exmple pplcton he fgure below (from Jcob et l., J. Geophys. Res. 003) shows n ts left pnel typcl vergng kernel mtrx for retrevl of vertcl profles of crbon monoxde (CO) mxng rtos from the MOPI stellte nstrument. hs nstrument mkes ndr mesurements of IR terrestrl emsson round the 4.6 µm CO bsorpton bnd. he rdnces mesured t dfferent wvelengths represent the observton vector for the nverse problem, nd the CO mxng rtos t n = 7 dfferent vertcl levels from the surfce to 150 hp represent the stte vector. he vergng kernel mtrx s represented n the fgure row by row,.e., ech lne wth symbol represents one row of the vergng kernel mtrx for the level ndcted by the symbol, nd descrbes the senstvty of the retrevl t tht level to the true CO mxng rtos t dfferent lttudes. Dnel J. Jcob, Lectures on Inverse Modelng,

20 Consder n ths fgure the retrevl of the CO mxng rto t 700 hp (nverted trngles). We see tht the retreved vlue s ctully senstve to CO t ll lttudes,.e., t s not possble from the retrevl to nrrowly dentfy the CO mxng rto t 700 hp (or t ny other specfc lttude). he temperture contrst between vertcl levels s not suffcent. We retreve nsted brod CO column weghted towrd the mddle troposphere ( hp). In fct, the retrevl t 700 hp s more senstve to the CO mxng rto t 500 hp thn t 700 hp. Physclly, ths mens tht certn mxng rto of CO t 500 hp wll gve spectrl response smlr to lrger mxng rto t 700 hp, becuse 500 hp hs greter temperture contrst wth the surfce. We lso notce n the vergng kernel mtrx some negtve vlues, for exmple the retrevl t 700 hp s negtvely dependent on CO n the strtosphere t 150 hp. hs s not physcl result but s llowed by the MAP sttstcl ft; t s reflected n the posteror error covrnce mtrx Ŝ by negtve correlton between the retreved vlues t 700 nd 150 hp. he trce of ths prtculr vergng kernel mtrx s 1., so tht MOPI provdes 1. peces of nformton on the vertcl profle. One my thus expect good nformton on some vertclly weghted column but not on grdents. Vsul nspecton of the left pnel shows tht the retrevls t the surfce, 850, 700, nd 500 hp ll gve the sme CO column weghted towrds the mddle troposphere, wheres the retrevls t 350, 50, nd 150 hp ll gve smlr wek sgnl n the upper troposphere nd lower strtosphere. he lrgest pece of nformton n MOPI s thus the CO column weghted towrd the mddle troposphere, wth smller pece of nformton weghted towrd hgher lttudes. he rght pnel of the Fgure shows the pplcton of the vergng kernel mtrx to the vldton of the MOPI nstrument wth n underpss rcrft vertcl profle extendng from the surfce to 00 hp. he CO mesurements from rcrft (thn sold lne) hve hgh ccurcy nd cn be vewed s defnng the true profle. Applyng the vergng kernel mtrx to ths true profle (bnned by the MOPI retrevl levels) yelds the dshed lne; ths s wht MOPI would see f ts cpblty were s dvertsed by the error nlyss tht led to the vergng kernel mtrx. We see tht the vertcl structure s lrgely lost, s would be expected (the vertcl grdent s mnly from the pror). hs rcrft profle processed wth the vergng kernel mtrx cn be then compred to the MOPI retrevl, shown by the thck sold lne. he two hve smlr vertcl grdents but ths merely reflects the pror nformton. More nstructve s tht the columns re smlr, wth 6% postve bs for MOPI. hs bs represents systemtc error n the retrevl tht s not ccounted for n the error nlyss Sequentl updtng Anlytcl dervton of the MAP soluton usng (5.7) or equvlently (5.1) requres constructon of the Jcobn mtrx K = y/ x nd of the gn mtrx G. Numercl constructon of K requres number n of forwrd model clcultons, to be possbly terted f the forwrd model s nonlner. he ssocted computtonl costs Dnel J. Jcob, Lectures on Inverse Modelng,

21 cn be tremendous nd lmt the mngeble sze of the stte vector. As we wll see n chpter 7, ths dffculty cn be ddressed wth n djont pproch to the nverse problem. he sze m of the observtonl vector s lso of concern becuse of the ssocted mtrx multplctons nvolved n the constructon of G. Lmttons on m cn however be crcumvented wthn the frmework of the nlytcl soluton by usng sequentl updtng. In ths pproch, the observton vector s prttoned nto smller pckets of observtons tht re successvely ngested nto the nverse nlyss. he MAP soluton ( ˆx,Sˆ ) obtned fter processng of one pcket s then used s pror for the next pcket, nd so on. he fnl soluton s exctly the sme s f the entre observton vector were ngested t once. he only lmtton s tht observtons n dfferent pckets must be tken to be uncorrelted,.e., S for the ensemble of observtons must be vewed s block dgonl mtrx where the blocks re the ndvdul pckets. 6. KALMAN FILER ( 3-D Vr ) So fr we hve used nverse nlyss of observtons to constrn fxed vlue of the stte vector. In fct, we my wnt to use observtons dstrbuted n tme to constrn stte vector evolvng wth tme, subject to some pror knowledge of ths stte vector nd ts evoluton wth tme, nd not solvng for ll tmes t once (whch would quckly mke the problem computtonlly ntrctble). A strghtforwrd wy to do ths, commonly clled the Klmn flter or 3-D Vr, s to terte n tme the nlytcl MAP soluton presented n secton D here refers to the use of observtons t gven tme step to constrn the stte vector t tht tme step, subject to pror nformton from pror or posteror tmes. 4-D Vr, descrbed n the next secton, refers to the use observtons t gven tme to constrn the stte vector over rnge of tmes. he Klmn flter cn be run ether forwrd or bckwrd. We descrbe the forwrd flter frst.. Consder n ensemble of observtons collected t dscrete tme steps over n ntervl [t 0, t n ]. let y be the ensemble of observtons collected t tme t, nd x the correspondng vlue of the stte vector. Strtng from pror knowledge (x, S ) t tme t 0, we use the observtons y 0 nd the nlytcl MAP soluton descrbed n secton 5.1 to derve best estmtes ( x,s ˆ ˆ 0 0 ) for tht tme. We then progress forwrd n tme, usng pror knowledge of the tme evoluton of x expressed by lner (or lnerzed) evoluton opertor M wth error M : x =Mx -1 + M (6.1) In the smplest cse, we could ssume persstence for x, n whch cse M would be the dentty mtrx. Consder tme t -1 for whch we hve the MAP soluton ( x ˆ ˆ -1,S -1 ). We use (6.1) to derve n pror vlue for x s Dnel J. Jcob, Lectures on Inverse Modelng,

22 x =M xˆ, -1 (6.) nd the ssocted pror error covrnce mtrx s [ ˆˆ ] [ ] ˆ S = E M M + E = MS M + S, M M -1 M (6.3) where ˆ s the error on xˆ -1 nd SM s the error covrnce mtrx for the evoluton opertor. We thus obtn tme-dependent MAP solutons for x over the ntervl [t 1, t n ]. A problem wth the forwrd flter for some pplctons s tht the observtons t tme t do not constrn the stte vector t pror tmes. An lterntve s to use bckwrd flter n whch we strt from pror knowledge (x, S ) t tme t n. he observtons y n t tme t n re used to obtn MAP soluton for x n, nd we then progress bckwrd n tme followng the sme procedure s wth the forwrd flter. he only dfference s tht we must use n evoluton opertor M runnng bckwrd: x =M x + M (6.4) 1 In the bckwrd flter, observtons t tme t re not llowed to constrn the stte vector t posteror tmes. It s possble to run the Klmn flter for the sme observtonl dt set frst forwrd, then bckwrd, to llow observtons to constrn the stte vector n both temporl drectons. 7. ADJOIN APPROACH ( 4-D Vr ) When very lrge number of observtons s vlble, s from stelltes, we would lke to use ths nformton to constrn very lrge stte vector feturng hgh sptl nd temporl resoluton, commensurte wth the detl n the observtons nd lmted solely by the resoluton of the forwrd model. hs s mprctcl n the nlytcl solutons to the nverse problem descrbed bove, s these requre explct constructon of the Jcobn mtrx s well s multplctons of mtrces hvng the dmenson of the stte vector. he djont pproch ddresses ths dffculty through the pplcton of the djont model. he djont pproch, lke the nlytcl pproch, seeks to mnmze the cost functon J(x) gven by (5.5), but t does so numerclly rther thn nlytclly. Strtng from the ntl guess x, t computes the cost functon grdent x J ( x ) tertvely n combnton wth steepest-descent numercl lgorthm to fnd mn(j(x)). Stndrd steepest-descent lgorthms re descrbed n textbooks on numercl methods; populr one s the BFGS lgorthm. he fgure below llustrtes how steepest-descent lgorthm ppled to successve guesses x, x, x, x pproches mn(j(x)). Dnel J. Jcob, Lectures on Inverse Modelng,

23 x x x x Mnmum of cost functon he mn tsk n the djont pproch s the effcent computton of x J ( x) t ech terton of the steepest-descent lgorthm. x J ( x ) s gven by (equton (5.6)): = xj ( x) S ( x x) xf S ( F(x)-y) (7.1) where F(x) s the forwrd model (not necessrly lner), nd = K s the model -1 djont (secton 4.4) ppled here to the vector S (F(x) y) whch represents the weghted error n the blty of our guess for x to mtch the observtons. he -1 components of S (F(x) y) re clled the djont forcngs. Implementton of the djont pproch s s follows. We strt from the pror x s ntl guess nd mke one pss of the forwrd model through the perod [t 0, t n ] of the observtonl record of nterest. Observtons my be scttered over tht perod. -1 We collect the correspondng vlues S (y-f(x )) of the djont forcngs nd clculte x J ( x ) followng (7.1): x F = 1 xj ( x) K S ( F(x )-y) (7.) hs clculton s done by pplyng the djont model to the djont forcngs s descrbed n secton 4.4. We strt from the djont forcngs t tme t n nd then work bckwrd n tme, pckng up ddtonl djont forcngs long the wy, untl we rech t 0. he vrbles through whch the djont forcngs re propgted bckwrd n tme wth the djont model re clled the djont vrbles. An mportnt spect of ths clculton s tht the Jcobn mtrx s never explctly constructed. he vlue of J ( x ) computed x Dnel J. Jcob, Lectures on Inverse Modelng, 007.

24 from (7.) s pssed to the steepest-descent lgorthm, whch mke n updted guess x. We then reclculte J ( x ) for tht updted guess, x = xj ( x) S ( x x) K S ( F(x)-y) (7.3) pss the result to the steepest-descent lgorthm whch mkes n updted guess x, nd so on. Ech terton thus nvolves one pss through the forwrd model over [t o, t n ] followed by one pss of the djont model over [t n, t 0 ]. Forwrd nd djont models typclly hve comprble computtonl requrements, nd the requrements for the djont model re nsenstve to the dmenson of x. Incresng the dmenson of x stll entls some penlty, however, s t generlly ncreses the number of tertons requred for convergence. he correctness of the cost functon grdents x J ( x ) produced by the nverse model cn be checked wth smple fnte-dfference test nd ths s stndrd procedure. o do ths test, pply the forwrd model to x, clculte the cost functon J(x ), nd repet for smll perturbton x + x. he resultng fnte-dfference pproxmton J( x + x) J( x) xj ( x) x (7.4) cn then be compred to the vlue obtned wth the djont model. 8. OBSERVING SYSEM SIMULAION EXPERIMENS After hvng desgned n observton system nd relted nverse model to optmze the estmte of stte vector x, t s good de to test ths mchnery wth n observng system smulton experment (OSSE), both to test the vlue of the observng system nd lso to test tht the nverse model s workng properly. he OSSE conssts of exmnng the blty of synthetclly generted observtons wth the forwrd model (pseudo-observtons) to retreve vlues of x consstent wth expecttons. In n OSSE, we strt by selectng some resonble but rbtrry vlue of x s the true vlue for the purpose of the test. We then generte pseudo-observtons y followng equton (4.3) by pplyng the forwrd model to x nd ddng rndom nose consstent wth our knowledge of S. We then sk the queston, how well cn our observng system retreve x gven y? o ths end we strt from n pror estmte x constructed by pply rndom nose to x n mnner consstent wth the pror error covrnce mtrx S. From ths pror nd usng the pseudo-observtons y, we then clculte MAP soluton ˆx, compre t to the true vlue x to ssess the mert of the observton system, nd determne whether the two re consstent wthn the posteror error covrnce mtrx Ŝ s test of the nverse model. OSSEs re routnely conducted n the desgn phse of stellte msson to test whether the msson cn stsfctorly ddress ts scentfc objectves. In tht cse, Dnel J. Jcob, Lectures on Inverse Modelng,

25 the focus s on whether the MAP soluton provdes sgnfcntly mproved estmte of x reltve to x. he vergng kernel mtrx A s useful dgnostc. It wll typclly provde n overoptmstc ssessment of the cpblty of the observng system, s dscussed n chpter 5, becuse the errors used to generte the pseudo-dt re rndom. But f the observng system fls tht test then t should clerly be redesgned. Dnel J. Jcob, Lectures on Inverse Modelng,