tellite etrievl Dt Assimiltion odgers C. D. Inverse Methods for Atmospheric ounding: Theor nd Prctice World cientific Pu. Co. Hckensck N.J. 2000 Chpter 3 nd Chpter 8 Dve uhl
Artist depiction of NAA terr tellite with MOPITT instrument http://www.spce.gc.c/sc/eng/pogee/2005/02_mopitt.sp Level 0 tellite instrument sensors unprocessed mesurements. Level 1 Unprocessed mesurements re converted into rdinces using instrument clirted lgorithms. In this step we hve instrument errors ut no etr informtion is dded in processing. Level 2 etrievl profiles of desired geophsicl quntities TqO 3 CO... re clculted using retrievl lgorithms sed upon tmospheric phsics. In this step priori informtion must e introduced to constrin the ill posed inverse prolem. This process introduces errors due to the phsics s well s from the priori.
The Prolem Assimilting geophsicl retrievls which we will refer to retrievls contining priori informtion introduces previousl known informtion not relted to the oservtion into our ssimiltion In some retrievls the priori informtion contined in retrievl mkes up the mjorit of the informtion -- In these cses ou re then ssimilting more to the priori thn to the oserved quntit The higher the percentge of informtion coming from the priori in the retrievl the greter the prolem Therefore we need to find new method to remove the priori informtion from the retrievl efore we ssimilte it
Orgniztion of Tlk 1. Development of the etrievl the odgers Approch 2. Error Anlsis of the etrievl 3. Discussion of new retrievl method which ccounts for this priori informtion
Development of etrievl oservtion i.e. rdinces f forwrd function true phsics stte vector true stte of the tmosphere i.e. TqO 3 CO forwrd function prmeters not included in the stte ut influence the mesurement i.e. spectrl line strengths error term errors not relted to the forwrd function prmeters i.e. detector noise
Development of etrievl f oservtion i.e. rdinces forwrd model error forwrd model n pproimtion of the true phsics forwrd function prmeters used in the model different due to the pproimtion of the phsics
Development of etrievl f etrievl estimte of the stte i.e. TqO 3 CO Inverse or etrievl method c estimted forwrd function prmeters Prmeters not in the forwrd function ut ffect retrievl for ll resonle inverse methods priori should e the onl prmeter
etrievl Theor Temp. Profile: etrievl: priori Profile: odgers 2000 igure 3.3 p.56
Newl Added lides I hve dded the net 16 slides to the presenttion which descrie the derivtion of odgers etrievl eqution I did not show these in clss due to time constrints
oservtion i.e. rdinces forwrd function true phsics Newl Added lides emote Mesurement stte vector i.e. TqO 3 CO f forwrd function prmeters not included in the stte ut influence the mesurement i.e. spectrl line strengths error term errors not relted to the forwrd function prmeters i.e. detector noise
Newl Added lides emote Mesurement forwrd model error f f forwrd function true phsics forwrd model n pproimtion of the true phsics forwrd function prmeters used in the model different due to the pproimtion of the phsics
Newl Added lides emote Mesurement oservtion i.e. rdinces f f forwrd model n pproimtion of the true phsics error term errors not relted to the forwrd function prmeters i.e. detector noise forwrd model error
Newl Added lides emote Mesurement f f priori estimte of the stte unrelted to ctul mesurement estimted forwrd function prmeters : ensitivit of orwrd model to stte : ensitivit of orwrd model to prmeters
Newl Added lides emote Mesurement f f
Newl Added lides emote Mesurement f f
etrievl estimte of the stte i.e. TqO 3 CO Newl Added lides etrievl Theor c Inverse or etrievl method Temp. Profile: Prmeters not in the forwrd function ut ffect retrievl for ll resonle inverse methods priori should e the onl prmeter c odgers 2000 igure 3.3 p.56 etrievl: priori Profile
Newl Added lides etrievl Theor c
forwrd model of priori nd est. forwrd funct. prm. Newl Added lides etrievl Theor c or n well ehved inverse method the method pplied to the priori should ield priori s the result : ensitivit of etrievl to mesurement
Newl Added lides etrievl Theor c
Newl Added lides etrievl Theor c
Newl Added lides etrievl Theor c
Newl Added lides etrievl Theor
Newl Added lides etrievl Theor : ll error terms 1. orwrd Model Prmeter 2. orwrd Model 3. Detector Noise
Newl Added lides etrievl Theor I A: Averging ernel the sensitivit of true stte to retrievl A / '
Newl Added lides etrievl Theor I A I A
Development of etrievl Through series of lineriztions nd n ssumption we rrive t: f : error terms 1. orwrd Model Prmeter 2. orwrd Model 3. Detector Noise ensitivit of the etrievl to mesurement ensitivit of orwrd model to prmeters
Development of etrievl A Averging ernel is the sensitivit of the retrievl to the true stte n n n n 1 1 11 L M O M L A 1 1 1 1 1 11 1 1 n n n n n n n n n M M M L M O M L M M ensitivit of the etrievl to mesurement ensitivit of the orwrd Model to the stte
Closer look t Averging ernel The ows of the ernel the retrievl t n level is n verge of the whole profile weighted this row. An idel inverse function would hve AI The re of the ernel ensitivit of retrievl to true profile Close to unit indictes high sensitivit The Columns of the ernel the response of the oserving sstem to δ-function disturnce t tht retrievl level. M 1 11 n n n M L n1 1 O L odgers 2000 igure 3.5 p.57 M n
Development of etrievl A More Useful orm for error nlsis A I A
Error Anlsis A I A We sutrct the true profile since we wnt to know how close to the retrievl is to the truth. This is useful to us since trditionll the ssimiltion compred the model generted profile to the retrieved profile directl. Temp. Profile: etrievl: odgers 2000 igure 3.3 p.56
Error Anlsis A I A I A moothing orwrd Model Prmeters orwrd Model etrievl Noise s f m Wht we need to investigte for ech term: 1. The men error 2. The error covrince
Men moothing Error s A I Becuse the true stte is not normll known we cnnot estimte the ctul smoothing error Wht we require is description of the sttistics of the error which must e clculted from the men nd covrince over some pproprite ensemle of sttes if s A I This would occur if the priori is equl to the men vlue of n ensemle of sttes for ll profiles We will ssume this term to e smll Note: this ssumption cn fil depending upon the priori 0
moothing Error Covrince Where is the priori error covrince mtri we re sometimes given or we cn clculte it with other informtion } { } { Τ Τ Τ s s s s E E I A I A Epected Vlue Opertor We will NOT ssume this term to e smll
Men Model Prmeter Error f The error in the retrievl due to errors in the forwrd model prmeters trightforwrd to evlute: If the forwrd model prmeters hve een estimted properl nd the model is liner s fr s the re concerned The sensitivit of the retrievl to the mesurement derive lgericll or perturtion from the inverse method The sensitivit of the forwrd model to the forwrd model prmeters. derive lgericll or perturtion from the forwrd model If it is good retrievl lgorithm then the errors will e unised nd the ssumed model prmeter error will e smll We will ssume this term to e smll Note however if the stellite instrument chnges or if the instrument is not full chrcterized this ssumption is wrong.
Model Prmeter Error Covrince } { } { Τ Τ Τ Τ E E f f f f Where is the forwrd model prmeter error covrince mtri ver hrd to determine Note gin if the stellite instrument chnges or if the instrument is not full chrcterized this ssumption is wrong. We will ssume this term to e smll
Men orwrd Model Error nd Covrince ememer is the difference etween the forwrd function nd the forwrd model Modeling error cn e hrd to evlute ecuse it requires model for the forwrd function which includes the correct phsics If the correct phsics is known nd cn e modeled ccurtel the forwrd model then evluting modeling error is strightforwrd. We will ssume this term to e smll Note if forwrd function is not known in detil or so horrendousl comple tht no proper model is fesile this ssumption m e wrong. Note if the instrumenttion chnges chrcter in orit this ssumption will e wrong
Men etrievl Noise nd Covrince Men Mesurement noise is: Assumed rndom Assumed unised Assumed uncorrelted etween chnnels m { Τ } Τ E m m We will NOT ssume either term to e smll
etrievl Error Overview umming up the men etrievl Error ssumed onl contriution the retrievl noise term: umming up the etrievl Error Covrince ssumed onl contriutions from the smoothing nd the retrievl noise terms s I A moothing Prmeters Model Noise 0 0 s s 0 0 f f 0 0 0 0 m m s m Τ Τ I A I A
etrievl Error Overview Oviousl there re mn ssumptions regrding the errors from stellite retrievls using odger s pproch This is wh creful correlted mesurements using independent instruments like lloons irorne dtsets or high precision ccurte stellite profiles for stellite vlidtions must e performed throughout the life of the stellite to insure tht these ssumptions remin vlid
etrievl Dt Assimiltion X X W W f [ H X ]... In the pst the ove reltionship ws used for the dt ssimiltion the clculted retrievl H X the model ckground converted into the retrievl spce using the oservtion opertor which did NOT ccount for the dded priori informtion in the retrievl
New Assimiltion of etrievl Method In the pst: X X W [ H X ] Tking into ccount: A I A Modif the retrievl I A A Modif the H-opertor H X AH X [ H X ] X X W Now we hve ccounted for priori informtion in the retrievl
New Assimiltion of etrievl Method Error Anlsis Previousl: sutrct true stte vector from oth sides X X W H X A [ ] I A Now: sutrct verging kernel*true stte vector from oth sides X X W [ AH X ] A A A
New Error Anlsis No longer smoothing term in the error nlsis ut ll other terms the sme s efore so: A Τ A Prmeters Model Noise 0 0 f f 0 0 0 0 m m
Clculting New etrievl Error Covrince New etrievl Error Covrince: m iven the old etrievl Error Covrince: s m Comining: A I s A I o the New etrievl Error Covrince is the difference etween the Old etrievl Error Covrince nd the contriution of the priori error covrince. Τ
Artist depiction of NAA terr tellite with MOPITT instrument http://www.spce.gc.c/sc/eng/pogee/2005/02_mopitt.sp Conclusions Incorportion of stellite retrievls into dt ssimiltion is not lws stright forwrd. Creful considertion to the quntities ssimilted s well s to the errors of these quntities must e tken ome of the issues with the ssimiltion of retrievls in the pst hve een due to the inclusion of the priori informtion in these retrievls. These issues m e delt with through removl of priori dt from the retrievl within the ssimiltion sstem I onl presented one method -- there re others
Newl Added lide Other Assimiltion Methods [ ] H H H A I X A X X W X X hown Method 1: [ ] H H H X A X A I X W X X Method 2: Method 3: [ ] H H H X A X X W X X
Thnk ou!