High level algorithm definition and physical and mathematical optimisations. (MIPAS Level 2 Algorithm Theoretical Baseline Document)

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1 Lv Poduct Vaidation Dat: //0 Pag n. /35 chnica Not on: High v agoithm dfinition and physica and mathmatica optimisations (MIPAS Lv Agoithm hotica Basin Documnt Novmb 0 Issu 5 v. 0 (Compiant with ORM_ABC_PDS_V.0 and IPF V6.0 Divy of th study: Suppot to MIPAS Lv Poduct Vaidation upgad of th divy in th oigina study: "Dvopmnt of an Optimisd Agoithm fo Routin P, and VMR Rtiva fom MIPAS Limb Emission Spcta" Ppad by: Nam Institut B. Cai IFAC-CNR, Finz, Itay M. Caotti Univsity of Boogna, Itay S. Ccchini IFAC-CNR, Finz, Itay M. Höpfn KI, Kasuh, Gmany P. Raspoini IFAC-CNR, Finz, Itay M. Ridofi Univsity of Boogna, Itay L. Santui IFAC-CNR, Finz, Itay

2 Lv Poduct Vaidation Dat: //0 Pag n. /35 ABLE OF CONENS - INRODUCION 7. Changs fom Issu A to Issu 3 of th psnt documnt 8. Changs fom Issu 3 to Issu 4 of th psnt documnt 8. Changs fom Issu 4 to Issu 5 of th psnt documnt 9 - OBJECIVES OF HE ECHNICAL NOE CRIERIA FOR HE OPIMISAION HE MAHEMAICS OF HE RERIEVAL PROBLEM 4. Mathmatica convntions 4. hotica backgound 4.. h dict pobm 4.. h Gauss Nwton mthod h Lvnbg-Maquadt mthod Rviw of th possib convgnc citia 4..5 Us of xtna (a-pioi infomation in th invsion mod Us of th optima stimation, fo incusion of LOS ngining infomation (LEI in p, tiva 4..7 Covaianc matix and avaging kns of th tiva soution h goba fit anaysis High v mathmatics of th fowad mod h adiativ tansf 4.4. Convoution with th AILS Convoution with th FOV Instumnta continuum Summay of quid vaiabs Cacuation of th VCM of th masumnts Opations pfomd on th intfogam to obtain th apodisd spctum Computation of th VCM ativ to a sing micowindow Computation of th invs of th VCM Cacuation of th Jacobian matix K of th simuations Gnaisd invs Vaianc-covaianc matix of tangnt hights coctions SCIENIFIC ASPECS AND PHYSICAL OPIMISAIONS Choic btwn tiva of pofis at fixd vs and at tangnt atitud vs 5.. Rtiva at tangnt atitud and intpoation btwn tivd vaus Rtiva at fixd vs Discussion of th pobm Concusions 40

3 Lv Poduct Vaidation Dat: //0 Pag n. 3/35 5. Us of a-pioi infomation Accuacy impovmnt Systmatic os Hydostatic quiibium and LOS Engining infomation Latitud ffcts Angua spad Cimatoogica diffncs Eath mod and gavity Eath mod Gavity Ray-tacing and factiv indx Pssu-tmpatu dpndnc of factiv indx and ffcts on tangnt hights Mthod of ay tacing Lin shap moding Numica cacuation of th Voigt pofi Appoximation of th Voigt pofi by th Lontz function factos in th cas of CO and H O Lin-mixing Pssu shift Impmntation of Non-LE ffcts Sf boadning Continuum 5.. Instumnta continuum Na continuum Fa continuum Intpoation of th pofis in th fowad / tiva mod Intpoation of th tivd pofis to a us-dfind gid Optimizd agoithm fo constuction of initia guss pofis Gnation of initia guss continuum pofis Pofis guaization MAHEMAICAL OPIMISAIONS Radiativ ansf intga and us of Cutis-Godson man vaus Laying of th atmosph 7 6. Scant aw appoximation fo th cacuation of Cutis-Godson quantitis and dfinition of paths Squnc of th opations Intpoation of coss sctions fo diffnt gomtis Cacuation of spctum: xpoitation of sphica symmtis Us of intpoation fo th cacuation of Panck function Finit instumnt fid of viw. 8

4 Lv Poduct Vaidation Dat: //0 Pag n. 4/ Anaytica divativs Gna considations Divativ with spct to th voum mixing atio Divativ with spct to tmpatu Divativ with spct to th atmosphic continuum Divativ with spct to th tangnt pssu Indpndnc of tivd vaiabs Convgnc citia P-cacuation of in shaps Diffnt gids duing th coss-sction cacuation Coss-sction ook-up tabs Vaiab fquncy gids fo adiativ tansf computation 94 REFERENCES 96 APPENDIX A: MEMORANDUM ON DEERMINAION OF HE VCM OF ENGINEERING ANGEN HEIGHS IN MIPAS 99 APPENDIX B: EVALUAION OF RERIEVAL ERROR COMPONENS AND OAL ERROR BUDGE 09 APPENDIX C: GENERAION OF MW DAABASES AND OCCUPAION MARICES3 APPENDIX D: GENERAION OF LUS AND IRREGULAR FREQUENCY GRIDS (IG5 APPENDIX E: GENERAION OF MW-DEDICAED SPECRAL LINELISS 6 APPENDIX F: MIPAS OBSERVAION MODES 0 LIS OF HE MOS IMPORAN USED QUANIIES WIH HE RELAED SYMBOLS 4

5 Lv Poduct Vaidation Dat: //0 Pag n. 5/35 ACRONYMS LIS AILS ABD AMOS BA CKD CPU EC ECMWF ENVISA ESA FM FOV FR FS FWHM GN Apodisd Instumnt Lin Shap Agoithm hotica Basin Documnt Atmosphic ac MOcu Spctoscopy xpimnt Bitish Aospac Cough-Knizys-Davis Cnta Pocssing Unit Eo Consistncy Euopan Cnt fo Mdium-ang Wath Focasts ENVionmnt SAit Euopan Spac Agncy Fowad Mod Fid Of Viw Fu Rsoution masumnts acquid by MIPAS in th fist two yas of opations ( Foui ansfom Spctomt Fu Width at Haf Maximum Gauss Nwton (minimization mthod HIRAN HIgh-soution RANsmission mocua absoption databas HWHM IAPs IFOV IG ILS IMK LEI LM LOS LS LSF Haf Width at Haf Maximum Impmntd Atmosphic Pssus and mpatus, Instantanous Fid Of Viw Igua Gid Instumnt Lin Shap Institut fü Mtooogi und Kimafoschung LOS Engining Infomation Lvnbg-Maquadt (minimization mthod Lin Of Sight Low Statosph Last Squas Fit

6 Lv Poduct Vaidation Dat: //0 Pag n. 6/35 LE LU Loca hma Equiibium Look Up ab MIPAS Michson Intfomt fo Passiv Atmosphic Sounding MPD MW NESR NLSF NLE, Non-LE NR OFM OR ORM PDS POEM REC RFM RMS SVD A ULS VCM VMR ZFPD ZPD Maximum Path Diffnc spcta MicoWindow Nois Equivant Spcta Radianc Non-ina Last Squas Fit Non Loca hma Equiibium Na Ra im Optimizd Fowad Mod Optimizd Rsoution masumnts acquid by MIPAS fom Januay 005 onwad Optimisd Rtiva Mod Payoad Data Sgmnt Poa Obit Eath Mission Rsiduas and Eos Coation anaysis Rfnc Fowad Mod Root Man Squa Singua Vau Dcomposition angnt Atitud Upp oposph / Low Statosph Vaianc Covaianc Matix Voum Mixing Ratio Zo-Fid Path Diffnc Zo Path Diffnc

7 Lv Poduct Vaidation Dat: //0 Pag n. 7/35 - Intoduction MIPAS (Michson Intfomt fo Passiv Atmosphic Sounding is an ESA dvopd instumnt opating on boad of th ENVISA satit aunchd on an a poa obit on Mach st, 00, as pat of th fist Poa Obit Eath Obsvation Mission pogam (POEM-. MIPAS masus th atmosphic imb-mission spctum in th midd infad ( cm -. Vtica distibution pofis of numous tac gass can b divd fom MIPAS obsvd spcta. Accoding to th cunt basin, atitud pofis of atmosphic pssu and tmpatu (p,, and of voum mixing atio (VMR of a fw high pioity spcis (O 3, H O, HNO 3, CH 4, N O, NO, F, F, N O 5 and CONO a outiny tivd fom MIPAS masumnts by th ESA Lv gound pocsso. Lv pocssing is a citica pat of th Payoad Data Sgmnt (PDS bcaus of both th ong computing tim quid and th nd fo a vaidatd agoithm capab of poducing accuat and iab suts. In th study "Dvopmnt of an Optimisd Agoithm fo Routin P, and VMR Rtivas fom MIPAS Limb Emission Spcta" a scintific cod fo na a tim (NR Lv anaysis was dvopd, suitab fo impmntation in ENVISA PDS and optimisd fo th quimnts of spd and accuacy. h suts of th study w usd by industy as an input fo th dvopmnt of th industia pototyp of th Lv cod. h fist masumnt was acquid by MIPAS on 4 Mach 00 and, stating fom Juy 00 nay continuous masumnts w acquid duing th fist two yas of satit opations. In this piod th masumnts w anaysd in NR and subsqunty off-in (OFL with th sam pocsso as th NR anaysis but using diffnt auxiiay data aowing to obtain mo accuat suts at th xpnss of an incasd computing tim. In th fist two yas of opation most of th masumnts w acquid in th nomina mod consisting of 7 swps p imb scan, with tangnt hights anging fom 6 to 68 km and stps of 3 km fom 6 to 4 km, of 5 km fom 4 to 5 km and of 8 km fom 5 to 68 km. Som masumnts w acquid in th so cad spcia mods howv ths masumnts w not pocssd by th ESA Lv pocsso. h masumnts ating to th fist two yas ( of MIPAS opations a oftn fd as Fu-Rsoution (FR masumnts. Du to pobms with th mio div of th intfomt, MIPAS masumnts w discontinud at th nd of Mach of 004. In Januay 005 MIPAS opations w sumd with a ducd maximum path diffnc (cosponding to a ow spcta soution of cm - instad of th oigina 0.05 cm - and with a fin vtica samping stp of th imb masumnts. hs masumnts acquid fom Januay 005 onwad a fd as Optimizd Rsoution (OR masumnts. h chang in th vtica samping stp impid a wosning in th conditioning of th tiva. o cop with this i-conditioning, a guaization schm was intoducd in th tiva mod. Stating fom Januay 005 sva nw spcia masumnt mods w pannd and a significant faction of masumnts was acquid in this configuation. Cunty ths masumnts a aso pocssd by th ESA Lv agoithm, th pocssing is howv imitd to atituds ow than appoximaty 75 km w Non-Loca hmodynamic Equiibium (NLE ffcts induc a sti accptab fowad mod o. Appndix F incuds a dtaid dsciption of th masumnt mods mpoyd duing th MIPAS mission. h psnt chnica Not povids th high v dfinition of th agoithm impmntd in th so cad Optimizd Rtiva Mod (ORM that is th scintific pototyp fo th MIPAS Lv gound pocsso opatd by ESA. his documnt incuds aso a discussion of th impmntd physica and mathmatica optimizations.

8 Lv Poduct Vaidation Dat: //0 Pag n. 8/35. Changs fom Issu A to Issu 3 of th psnt documnt h main changs consist in th intoduction of nw sctions gading th dsciption of itms that pviousy w ith dscibd in spas mmoandums and sma nots o not dscibd at a. Namy, th foowing nw sctions hav bn intoducd: Sct. 5.3: Intpoation of th tivd pofis to a us-dfind gid, Sct. 5.4: Optimizd agoithm fo constuction of initia guss pofis (incuding gnation of continuum pofis dscibd in sub-sct 5.4., Sct. 5.5: Pofis guaization Appndix A: Dtmination of th VCM of th ngining tangnt hights in MIPAS, Appndix B: Evauation of tiva o componnts and tota o budgt (incuds p o popagation appoach, Appndix C: Agoithm fo gnation of MW databass and occupation matics, Appndix D: Agoithm fo gnation of LUs and igua fquncy gids, Appndix E: Agoithm fo gnation MW-ddicatd spcta inists. Futhmo, Sct. 4.5 gading th cacuation of th VCM of th masumnts was stongy modifid in od to b consistnt with basin modifications. A nw sub-sction, 4.5.3, dscibing th mthod usd to cacuat th invs of th VCM of th masumnts was intoducd. his subsction pacs th od Sct. 6.3 (now movd. Additiona spas modifications w intoducd in od to mov obsot statmnts and mak th documnt in in with th cunt status of th study.. Changs fom Issu 3 to Issu 4 of th psnt documnt h main chang consists in th updat of th sction 5.5 dscibing th guaization adoptd, that was modifid as a consqunc of th chang of th obsvation scnaio aft Januay 005. Additiona spas modifications w intoducd in od to mov obsot statmnts and mak th documnt in in with th cunt status of th study. A notation chang concning adoptd symbos fo th usd vaiabs and paamts has bn pfomd. Som mismatchings hav bn coctd. Intoduction has bn updatd and modifid accoding to changs occud in instumnt masumnt mod. Appndix C has bn updatd; a figu poting a summay of th tota o fo th pofis tivd fom masumnts acquid aft Januay 005 has bn intoducd. Appndix F with th dsciption of th MIPAS obsvation mods has bn intoducd. A ist of th usd aconyms has bn addd togth with a ist of th main quantitis with th adoptd symbos.

9 Lv Poduct Vaidation Dat: //0 Pag n. 9/35. Changs fom Issu 4 to Issu 5 of th psnt documnt Documnt modifid in od to b compiant with ORM_ABC_PDS_V.0 and IPF V In paticua: Sction, Intoduction: adaptd fo compianc with th cunt study status. Sct nw infomation incudd about th Lvnbg-Maquadt mthod. Sct.s 4..4 and 6.8, nw convgnc citia incudd. Sct nw sction gading th cacuation of covaianc matix and avaging kns of th Lvnbg-Maquadt soution. Byond ths main modifications, th who documnt has bn visd to mov o updat outdatd sntncs. - Obctivs of th tchnica not Obctivs of th tchnica not a: povision of a summay of th quations impmntd in th Lv agoithm, dsciption of th considd options fo th optimisation of th cod, assssmnt of advantags and disadvantags of th individua options, povision of a ationa fo th choic of pfd option fo impmntation and idntification of a statgy fo vaidation of th choic itsf. Fo a futh sf-consistnt high-v dsciption of th agoithms impmntd in th Lv scintific cod of MIPAS, th ad shoud aso f to th paps of Ridofi t a. (000, Raspoini t a. (006 and Ccchini t a. (007.

10 Lv Poduct Vaidation Dat: //0 Pag n. 0/ Citia fo th optimisation h cod must tak into account quimnts du to: chaactistics of input data, scintific quimnts of output data, coctnss of th atmosphic mod, coctnss of th instumnt mod, numica accuacy, obustnss in psnc of onous obsvationa data, ducd computing tim. h main difficuty is du to th ast quimnt, which, in psnc of th oths, imposs th sach fo physica and mathmatica optimisations in th impmntation of th cod. h psnt documnt povids a dsciption of th agoithm at a stag in which th cod has aady bn dvopd and thfo a choic among th possib options has aady bn mad. W must b awa, howv, that som choics hav bn mad on a puy thotica basis and thfo tsts pfomd with a data may suggst a diffnt appoach. h accptanc cition of th cod, and thfo ou choics, a basd on th combind dvopmnt of an Optimisd Rtiva Mod (ORM, an Optimisd Fowad Mod (OFM and a Rfnc Fowad Mod (RFM. Rtivas with th ORM fom spcta simuatd with th OFM and th RFM, with and without masumnt nois, aow th idntification of os du to:. masumnt o,. tiva o, 3. appoximations du to th optimisation. sts pfomd with diffnt computing accuacy and with diffnt pofi psntations dtmin, spctivy: 4. th numica accuacy 5. and th smoothing o. h accptanc cition quis th ORM to imit os. and 3. so that th ova o budgt incuding os fom. to 5. as w as systmatic o, is kpt bow th foowing quimnts: 3% o in tangnt pssu tiva, K o in tmpatu tiva, 5% o in VMR tivas, in th tangnt atitud ang 8-53 km. hs w th quimnts stabishd at th bginning of th study in absnc of spcific indications gading th utimat accuacy attainab fom MIPAS masumnts. Howv, tst tivas pfomd so fa hav shown that th abov quimnts cannot b mt in th who atitud ang xpod by th MIPAS scan and fo a th constitunts tivd by th ESA Lv pocsso.

11 Lv Poduct Vaidation Dat: //0 Pag n. /35 h pobm of assssing th utimat tiva accuacy attainab fom MIPAS masumnts has bn tackd in th famwok of th study fo th sction of optimizd spcta intvas (micowindows, MW fo MIPAS tivas (s Appndix C and Bnntt t a.,(999. An accptanc tst fo th cod, on th basis of actua tiva o has bn pfomd by using th compt OFM/ORM chain as w as th fnc spcta gnatd by th RFM. h statgy adoptd fo th opatd choics is th foowing: sinc an atitud o is dicty connctd to a pssu o which in tun cosponds aso to a VMR o, whnv th appoximation cosponds to an atitud o th appoximation is accptd if th o is ss than 0.5 km (cosponding to a % pssu o. Actuay, this is not a vy consvativ cition but it is sti satisfactoy bcaus it is appid ony fo th vauation of FOV and sf-boadning appoximations. If th appoximation dos not cospond to an atitud o, th appoximation is accptd on th basis of th adianc o. Random o must b sma than NESR (Nois Equivant Spcta Radianc, systmatic os must b sma than NESR dividd by th squa oot of th mutipicity of th ffct. If individua appoximations bhav as ith andom o systmatic os can ony b assssd by th fu tiva pocss. An ducatd compomis is mad by using an accptanc thshod qua to NESR/4. In Sct. 4 w summaiz th mathmatica invs pobm. Sct. 5 is ddicatd to th scintific aspcts that affct th atmosphic and th instumnt mod and to th cosponding physica optimisations. Sct. 6 is ddicatd to th choics atd to th impmntation of th cacuations in th computing softwa and to th cosponding mathmatica optimisations. 4 - h mathmatics of th tiva pobm 4. Mathmatica convntions h mathmatica convntions usd in th psnt tchnica not a hwith summaisd. h functions may hav th foowing attibuts: quaifis: quaifis a givn ony as subscipts (o as supscipt if subscipt is not possib and consist of a not that hps to distinguish th diffnt functions (.g. th Vaianc Covaianc matix S of diffnt quantitis o th sam function at diffnt vs of th cacuation (.g. th itation numb of a tivd quantity. Panthss a usd to spaat th quaifi fom th oth mathmatica opations that can b confusd with th quaifis (.g. to spaat quaifis fom tanspos o invsion opation. h vaiabs of th functions can appa ith as a subscipt o as agumnts. In od to povid a psntation consistnt with th convntion of matics and vctos, whnv possib, th vaiabs ativ to which th vaiabiity of th function is xpicity sampd within th cod a shown as a subscipt, whi vaiabs ativ to which a dpndnc ony xists impicity in th quations a shown as agumnts. Whn daing with matics and vctos, bod symbos a usd.

12 Lv Poduct Vaidation Dat: //0 Pag n. /35 h opation of convoution is indicatd with an astisk. 4. hotica backgound h pobm of tiving th atitud distibution of a physica o chmica quantity fom imbscanning obsvations of th atmosph, dops within th gna cass of pobms that qui th fitting of a thotica mod, that dscibs th bhaviou of a givn systm, to a st of avaiab obsvations of th systm itsf. h thotica mod dscibs th systm though a st of paamts so that th tiva pocdu consists in th sach of th st of vaus of th paamts that poduc th "bst" simuation of th obsvations. h most commony adoptd cition to accompish th obctiv is th minimisation of th function (gnay dfind as th summation of th o-wightd squad diffncs btwn obsvations and simuations with spct to th vau of th paamts. his cition is gnay fd as Last Squas Fit (LSF. Whn th thotica mod dos not dpnd inay on th unknown paamts th pobm, cad Non-ina Last Squas Fit (NLSF, cannot b sovd dicty by using a soution fomua, and an itativ pocdu must b usd. Sva mthods xist fo th NLSF, th on adoptd fo ou puposs is th Gauss Nwton (GN mthod modifid accoding to th Lvnbg-Maquadt cition (LM. In od to povid th famwok of th subsqunt discussion, th gna mathmatica fomuation of th pobm is hwith bify viwd. h mathmatica fomuation of th pobm is dscibd with fu dtais in Caotti and Cai, ( h dict pobm h signa I(ν,,z that achs th spctomt can b modd, by mans of th adiativ tansf quation (dscibd in Sct. 4.6, as a function S = S(b, x(z of th obsvationa paamts b and of th distibution pofi x(z of th atmosphic quantity which is to b tivd (z bing th atitud. Sinc th adiativ tansf dos not psnt a ina tansfomation, th pobm of diving th distibution x(z fom th obsvd vaus of S cannot b sovd though th anaytica invsion of th adiativ tansf quation. A ina tansfomation conncting S and x(z can b obtaind by opating a ayo xpansion of th adiativ tansf quation, aound an assumd pofi x z. In th hypothsis that x z is cos nough to th tu pofi to dop in a ina bhaviou of th function S, th ayo xpansion can b tuncatd to th fist tm to obtain: x z =xz S( b, x z S b, x z =S b, x z x z x z, z x z (4.. Not that th us of th intga is quid in th abov quation sinc th pofi x(z is h considd as a continuous function. Equation (4.. can b wittn as: wh: 0 b = K b S, x z x z dz (4..

13 Lv Poduct Vaidation Dat: //0 Pag n. 3/35 ( b=s b, x z -S b, x z (4..3 S S b, x z K( b, x (4..4 x z xz xz x z x z =x z. (4..5 Equation (4.. is an intga quation that psnts a ina tansfomation of th unknown Δ x (z ading to th obsvations Δ S (b by way of th kn K( b, x. 4.. h Gauss Nwton mthod In th cas of pactica cacuations, th mathmatica ntitis dfind in Sct. 4.. a psntd by disct vaus. Actuay, w wi da with a finit numb (M of obsvations and a finit numb (N of vaus to psnt, in a vcto x(z, th atitud distibution of th unknown quantitis (ths N vaus wi b dnotd as "paamts" fom now on. As a consqunc th intga opato of Eq. (4.. bcoms a summation and th quation itsf can b xpssd in matix notation as: In quation (4..6: Δ S = K Δ x (4..6 Δ S is a coumn vcto of dimnsion M (namy Mx. h nty m of Δ S is th diffnc btwn obsvation and th cosponding simuation cacuatd using th assumd pofi x z (Eq K is a matix (usuay dnotd as Jacobian matix having M ows and N coumns. h nty k i of K is th divativ of obsvation i with spct to mnt of paamt aay x (Eq Δ x is a coumn vcto of dimnsion N (namy Nx. h nty (Δ x i of Δ x is th coction ndd to th assumd vau of paamt x z in od to obtain its coct vau x z. h goa of th tiva is th dtmination of this vcto. h pobm is thfo that of th sach fo a "soution matix" G (having N ows and M coumns that, mutipid by vcto Δ S povids Δ x. If th vcto Δ S is chaactisd by th vaianc-covaianc matix S m (squa of dimnsion M, th function which must b minimisd is dfind as: and matix G is qua to: χ =Δ S (S m - Δ S (4..7 G ( K S K K S. (4..8 m m

14 Lv Poduct Vaidation Dat: //0 Pag n. 4/35 h supscipt dnots th tanspos and th supscipt - dnots th invs of th matix, if th invs of S m dos not xist, its gnaisd invs must b usd instad (s Kaman (976 and Sct If th unknown quantitis a suitaby chosn, matix ( K Sm K is not singua, thought it might b i-conditiond. If th a minimum of th function is found and S m is a coct stimat of th os, th quantity dfind by quation (4..7 has an xpctation vau qua to (M - N and a standad dviation qua to M N. h vau of th quantity povids thfo a good stimat of th quaity of M N th fit. Vaus of which dviat too much fom unity indicat th psnc of incoct M N assumptions in th tiva. h unknown vcto Δ x is thn computd as: Δ x = G Δ S (4..9 and th nw stimat of th paamts as: x z =x z ( z (4..0 x h os associatd with th soution to th invsion pocdu can b chaactisd by th vaianc-covaianc matix (S x of x(z givn by: - S = GS G = K S K (4.. - x m m Matix S x pmits to stimat how th xpimnta andom os map into th unctainty of th vaus of th tivd paamts. Actuay, th squa oot of th diagona mnts of S x masus th oot man squa (.m.s. o of th cosponding paamt. h off-diagona mnt s i of matix S x, nomaisd to th squa oot of th poduct of th two diagona mnts s ii and s, povids th coation cofficint btwn paamts i and. If th hypothsis of inaity mad in Sct. 4.. about th bhaviou of function S is satisfid, Eq. (4..0 povids th sut of th tiva pocss. If th hypothsis is not satisfid, th minimum of th function has not bn achd but ony a stp has bn don towad th minimum and th vcto x(z computd by Eq. (4..0 psnts a btt stimat of th paamts with spct to x z. In this cas th who pocdu must b itatd stating fom th nw stimat of th paamts which is usd to poduc a nw matix K. Convgnc citia a thfo ndd in od to stabish whn th minimum of th function has bn appoachd nough to stop th itations h Lvnbg-Maquadt mthod h Lvnbg-Maquadt (LM mthod intoducs a modification to th pocdu dscibd in th pvious sub-sction. his modification pmits to achiv th convgnc aso in th cas of stongy non-ina pobms. h LM mthod consists in modifying matix ( K S K bfo using it in m

15 Lv Poduct Vaidation Dat: //0 Pag n. 5/35 (4..8 fo th cacuation of G. h modification consists in ampifying th diagona mnts of matix A ( K S K accoding to: m ( ( m ii ( ( m ii M K S K K S K (4.. wh M is a positiv scaa with th ffct of damping th nom of th coction vcto Δ x, hnc ducing th isk of pocting th paamts vcto fa away fom th oca inaity gion. h modification (4.. aso otats th coction vcto Δ x, fom th GN diction towads th diction of, hnc incasing th chanc of obtaining a sma with th updatd paamts vcto. h agoithm pocds as foows:. cacuat th function and matix A fo th initia vaus of th paamts,. st M to a initia "sma" vau (.g and modify A accoding to Eq. (4.., 3. cacuat th nw stimat of th paamts fo th cunt choic of M using quation (4..9, 4. cacuat th nw vau of using quation (4..7, 5. if cacuatd at stp 4 is gat than that cacuatd at stp, thn incas M by an appopiat facto (.g. 0 and pat fom stp 3 (mico itation, 6. if cacuatd at stp 4 is sma than that cacuatd at stp, thn dcas M by an appopiat facto (.g. 0, adopt th nw st of paamts to comput a nw matix A and pocd to stp 3 (maco itation. h (maco itations a stoppd whn a p-dfind convgnc cition is fufid. An advantag of using th LM mthod is that th cacuation of th Jacobian matix can b avoidd in th micoitations. Fo th dvopmnt of th ORM cod, howv, sinc most opationa tivas do not to da with a stongy non-ina pobm and sinc th cacuation of th Jacobian matix is fast whn pfomd within th fowad mod, simutanousy with th cacuation of th imb-adiancs, th ORM is optimizd fo a Gauss Nwton oop (maco-itation, i.. th Jacobian matix is computd aso in th mico-itations oops. As a sid ffct th LM modification (4.. impovs th conditioning of matix A and intoducs a guaizing ffct that is mosty ost duing th itations, whnv sufficint infomation on th tivd paamts in psnt in th obsvations. his fatu pmits to avoid th isk of intoducing biass in th soution. Mo dtais on th guaizing ffct of th LM mthod can b found in Doicu t a. (00. h bhaviou of th LM mthod is citicay viwd and compad to th ikhonov guaization with constant stngth in Ridofi t a. (0. Fo a dp undstanding of th guaizing LM mthod w sti commnd Ridofi t a. (0 and spciay a th ptinnt fncs citd thin Rviw of th possib convgnc citia W viw h sva conditions which can b considd fo th dfinition of a convgnc cition.. h ativ vaiation of th function obtaind in th psnt itation with spct to th pvious itation is ss than a fixd thshod t i..:

16 Lv Poduct Vaidation Dat: //0 Pag n. 6/35 ( x ( x it ( x it it t (4..4 wh it is th cunt itation indx.. h maximum coction to b appid to th paamts fo th nxt itation is bow a fixd thshod t i..: Max ( x it ( x ( x it it t (4..5 diffnt thshods can b vntuay usd fo th diffnt typs of paamts (, p, and VMR. h absout vaiations of th paamts can aso b considd instad of th ativ vaiations, whnv an absout accuacy quimnt is psnt fo a paamt (as fo th cas of tmpatu. Non-tagt paamts such as continuum and instumnta offst paamts shoud not b incudd in this chck. 3. Sinc th xpssion (4..5 is singua whnv a paamt is qua to zo, an atnativ fomua which can b considd is: it it - it it x x S x x x N it t 3 (4..5bis H psnts th nomaizd chi-squad tsting th compatibiity of x it with x it- within th o dscibd by th covaianc matix S x it. h quantity f oughy psnts th avag distanc btwn of x it and x it- masud as a faction of th o ba S. Unss a sconday minimum of th cost function has bn appoachd, f masus aso th convgnc o. his considation can b usd to st th thshod t 3 on th basis of th maximum accptab convgnc o. Fo xamp, if w qui th convgnc o to b sma than /0 of th o du to masumnt nois, thn w shoud sct t 3 / h ason that discouagd using (4..5bis sinc th vy bginning of th ORM dvopmnt, is that S x it dos not ay psnt th nois o of th soution whn th tiva is fa fom th convgnc. h xpinc gaind in tivas fom a data howv showd that th intitation changs of S x a usuay magina. 4. h diffnc btwn th a ( is ss than a fixd thshod t 3 : LIN it x and th chi-squa computd in th ina appoximation it it ( x LIN ( x it ( x t 4 (4..6

17 Lv Poduct Vaidation Dat: //0 Pag n. 7/35 wh LIN is computd using th xpssion: KGΔ S KG S m Δ S ( h itation indx has achd a maximum aowd vau (t 4 : it t 4 (4..8 h choic of th most appopiat ogica combination of th abov conditions (which povids th convgnc cition is discussd in th sction of mathmatica optimisations (s Sct Us of xtna (a-pioi infomation in th invsion mod Whn som a-pioi infomation on th tivd paamts is avaiab fom soucs xtna to th MIPAS intfomt, th quaity of tivd paamts can b impovd by incuding this infomation in th tiva pocss. Assuming th a-pioi infomation as consisting of both an stimat x A of th vcto of th tivd paamts and of th vaianc covaianc matix S A atd to x A, th combination of th tivd vcto with th xtnay povidd vcto x A can b mad, aft th convgnc has bn achd, by using th fomua of th wightd avag: x x S S S GΔ S x (4..9 o x A x S A A Intoducing th xpicit xpssions of G and S x givn spctivy by quations (4..8 and (4.., quation (4..9 bcoms: A m A m S x A o x K S K S K S Δ S x S x (4..0 his is th so cad optima stimation fomua (s Rodgs (976 and Rodgs (000. Eq. (4..0 can b usd aso at ach tiva itation stp, in pac of q. (4..9, fo diving th nw stimat of th unknowns (at th sam tim aso th obsvations x A a incudd in th vcto of th obsvations fo computing th. Whn quation (4..0 is usd in th itations of th tiva, th a-pioi stimat of th tivd paamts povids infomation on th unknown quantitis aso at th atituds wh th masumnts may contain ony poo infomation. In this cas th tiva pocss is mo stab (s aso Sct. 5.. Howv, whn using quation (4..0 in th tiva itations, th xtna infomation and th tiva infomation a mixd duing th minimisation pocss and thfo thy cannot b individuay accssd at any tim. his pvnts to asiy stimat th coction and th bias intoducd by th a-pioi infomation on th tivd quantitis. h dcision on whth to us quation (4..0 duing th tiva itations o to us (4..9 duing th tiva and (4..0 aft th convgnc has bn achd chify dpnds on th typ of a-pioi infomation w a daing with. In th cass in which th usd a-pioi infomation is xpctd not to poais th suts of th tiva (.g. in th cass in which indpndnt a-pioi stimats a avaiab fo diffnt tivas, quation (4..0 can b pofitaby usd duing th tiva itations.

18 Lv Poduct Vaidation Dat: //0 Pag n. 8/35 Futh advantags and disadvantags of th us of a-pioi infomation a dscibd as a scintific aspct in Sct Us of th optima stimation, fo incusion of LOS ngining infomation (LEI in p, tiva Engining LOS data a updatd at ach scan and thfo constitut an ffctiv and indpndnt souc of infomation which can b outiny usd in p, tivas and dos not bias th tivd pofis. In this cas it is ay woth to us fomua (4..0 at ach and itation stp and t th LOS infomation to hp th convgnc of th tiva. In this cas th a-pioi infomation dos not povid dicty an stimat of th unknowns of th tiva, but a masumnt of a quantity that is atd to th unknowns by way of th hydostatic quiibium aw. h ngining infomation on th pointing consists of a vcto z containing th diffncs btwn tangnt atituds of th swps of th cunt scan and of a Vaianc-Covaianc Matix (VCM V z atd to th vcto z. h componnts of th vcto z a dfind as: wh z z z Nsw z z (4.. Nsw z Nsw N sw is th numb of swps of th considd scan. If w dfin th vcto Δ S as: Δ S =z- z tg (4.. wh z tg is th vcto of th diffncs btwn th tangnt atituds at th cunt itation; instad of quation (4..6 w hav a coup of quations dfining th tiva pobm: Δ S =K Δ x Δ S,L =K L Δ x,l (4..3 wh th matix K L is th acobian that inks th diffncs btwn tangnt atituds with th vcto of th unknowns. his matix has to b -computd at ach tiva itation (as matix K; th cip fo th cacuation of this matix is givn in Sct h function to b minimisd bcoms: Δ S Δ Δ S Δ (4..4 S m S S z S and th vcto Δ x,l which minimiss this is givn by: x,l m L z L m S L z S Δ K S K K S K K S Δ K S Δ (4..5

19 Lv Poduct Vaidation Dat: //0 Pag n. 9/35 hfo, if w dfin matics A, B, and B L as: A K SmK K LSz K L B B K Sm K S L L z (4..6 quation (4..5 bcoms: Δ A BΔ B Δ (4..7 x,l S L S In th ina gim, this quation povids th soution of th tiva pobm. At ach tiva itation th tiva pogam has to comput matics K, K L, A, B and B L, thn, sinc LM agoithm is usd, matix A has to b modifid accodingy to quation (4..3 and aftwads usd in quation (4..7 in od to div ŷ. In this appoach, th quation which dfins th ina chi-squa LIN is: LIN S x m S x S L x z S L x Δ KΔ S Δ KΔ Δ K Δ S Δ K Δ (4..8 this is th quation to b usd instad of quation ( Cacuation of th acobian matix K of th ngining tangnt atituds (A Lt s xpicity wit th scond componnt of quation (4..3: ΔzΔz tg K A Δ x (4..9 It is ca fom this ation that th componnt i, of K A is: z i K A( i, with i=,..., N sw x wh I top is th tota numb of fittd paamts in th cunt tiva. - and =,..., I top (4..30 Now, bing x p th vcto of th unknowns of p, tiva, it is composd as foows: h fist N sw mnts psnt th tangnt pssus, h mnts fom N sw+ up to * N sw psnt th tangnt tmpatus, h mnts fom * N sw + up to I top psnt atmosphic continuum and instumnta offst paamts. Sinc ngining tangnt atituds do not dpnd on continuum and offst paamts K A (i,=0 fo i=,.., N sw- and =* N sw +,..., I top. On th oth hand th ngining tangnt atituds a connctd with tangnt pssus and tangnt tmpatus though hydostatic quiibium aw.

20 Lv Poduct Vaidation Dat: //0 Pag n. 0/35 h tansfomation which ads to z stating fom P, is dfind by th hydostatic quiibium: z i i i Pi n P I i fo i =,..., N sw (4..3 wh P and indicat, as usua, pssu and tmpatu and i is qua to: M i g0 ( z, (4..3 R wh g 0 is th accation of gavity at th man atitud of th ay z z i z / and atitud s ; M is th ai mass and R th gas constant. If th atituds a masud in km and in Kvin, w gt M/R = h acobian matix J associatd with th tansfomation (4..3 is a ( N sw-; N sw matix containing th divativs: i zi J( i, P J ( i, z i sw N fo i =,..., N fo i =,..., N sw sw and and =,..., N = N sw sw +,..., N sw (4..33 and thfo, diving quations (4..3 w obtain: J i i i, Pi ( i J ( i, P n i sw N i Pi i Pi fo i =,... SW N i fo i =,... Nsw - and =,... N sw Nsw - and = N sw +,... N sw (4..34 wh th function is dfind as: if [ag] = RUE ag ( if [ag] = FALSE Considing that th oigina vcto of th unknowns of p, tiva contains aso continuum and offst paamts, matix K A can b obtaind by xtnding matix J with as many coumns as quid to ach th dimnsion ( N I. As afomntiond, ths xta coumns contain ony sw -; top zos du to th fact that th tangnt atituds do not dpnd on continuum and offst paamts. Fo what concns th vaianc-covaianc matix S z of MIPAS tangnt hights quid fo th impmntation of th quations xpaind in Sct. 4..6, this matix is divd using a simp agoithm basd on MIPAS pointing spcifications. his agoithm is dscibd in Appndix A.

21 Lv Poduct Vaidation Dat: //0 Pag n. / Covaianc matix and avaging kns of th tiva soution h covaianc matix (VCM and th avaging kns (AKs, Rodgs, 000 a diagnostic toos commony usd to chaactiz th soution of th tiva. In paticua th VCM w cacuat dscibs th mapping of th masumnt nois o onto th soution, whi th AKs dscib th spons of th systm (instumnt and tiva agoithm to infinitsima vaiations in th tu atmosphic stat, hnc chaactizs th vtica soution of th tivd pofis. h diffnt agoithms a impmntd in th ORM to cacuat VCM and AKs of th LM soution. h th mthods psnt diffnt vs of sophistication and a sctab via a switch. Mthod : VCM and AKs of th LM soution, in th GN appoximation. If matix ( K Sm K of Eq. (4.. is w-conditiond (fo th invsion invovd in Eq. (4..8 and if th itativ pocss convgs within th machin numica pcision, thn th LM soution coincids with th GN soution, thfo its VCM ( S x and AK ( A x a cacuatd as (s Rodgs (000: S ( K S K (4..36 x A x m I (4..37 wh I is th idntity matix of dimnsion qua to th numb of mnts in th stat vcto x. Mthod : VCM and AKs of th LM soution, in th sing-itation appoximation. If matix ( K Sm K of Eq. (4.. is i-conditiond and / o th tiva itations a stoppd by som physicay maningfu cition bfo th xact numica convgnc is achd, thn th xpssions (4..36 and (4..37 may b a ough appoximation, as th LM and th GN soutions do diff. In this cas th LM damping tm must b takn into account. h LM soution x LM at th ast itation can b wittn as: xlm xi K Sm K MD K Sm y f( x i (4..38 wh x i is th stat vcto stimat at th scond-ast itation, y th obsvations vcto with VCM S m, f th fowad mod and K its Jacobian vauatd at x i. W aso intoducd D diag K S m K, wh th symbo diag[...] indicats a diagona matix with diagona mnts qua to thos of th matix potd within th squad backts [...]. If w assum x i to b indpndnt of y (sing itation appoximation, fo th VCM and AKs of th LM soution w asiy gt: m M m m M m M m S K S K D K S K K S K D (4..39 x A K S K D K S K (4..40 x Mthod 3: VCM and AKs of th LM soution, taking into account th who minimization path. h imiting appoximations of mthods and iustatd abov can b avoidd with a mathmatica tick. W stat by witing th gnic fom of th itativ Eq.(4..38 as: xi+ xi Ki Sm Ki idi Ki Sm y f( xi xi Gi y f( x i (4..4

22 Lv Poduct Vaidation Dat: //0 Pag n. /35 h w xpicity addd a subscipt i to a quantitis dpnding on th itation count i, and w intoducd th gain matix: G K S K D K S (4..4 i i m i i i i m If w intoduc th itation-dpndnt matix i as: i k x y i k (4..43 and w assum th tiva is stoppd (by som maningfu cition at itation i, thn fomay, th VCM and th AK of th LM soution can b wittn as: S S (4..44 x A y x x y x y x x K (4..45 Matics i can b cacuatd as th divativ of Eq. (4..4 with spct to y. Ngcting th divativs of K i with spct to x i (hypothsis aady xpoitd in th Gauss-Ntwon appoach itsf, and consqunty with spct to y, w gt: G I K (4..46 i i i i i Raanging Eq. (4..46 and considing that th initia guss x 0 dos not dpnd on th obsvations y, w obtain th foowing cusiv fomua fo th matics i : with (4..47 Equation (4..47 fo i=0,,, - dtmins. his matix is thn usd in Eq.s (4..44 and (4..45 to povid th VCM and th AK of th soution x. Eqs. (4..44 and (4..45 show that both th VCM and th AK dpnd on which, in tun, as shown by Eq. (4..47, dpnds on th path in th paamt spac foowd by th minimization pocdu, fom th initia guss to th soution. Not that, if an itation stp is don with i = 0 (Gauss-Nwton itation fom Eq. (4..4 w gt G i K i =I and fom Eq. (4..47 it foows that is indpndnt of th stps pfomd bfo th considd itation. hfo, w can say that a Gauss-Ntwon itation sts th mmoy of th path foowd bfo that itation. his ast mthod 3 was fist intoducd in Ccchini and Ridofi (00, it dos not us hypothss such as w-conditiond invsion, xact numica convgnc o sing-itation tiva, thfo in gna it is fa mo accuat than th mo usua mthods and dscibd ai. h ativ accuacy of mthods and 3 is citicay viwd and tstd in Ccchini and Ridofi (00.

23 Lv Poduct Vaidation Dat: //0 Pag n. 3/ h goba fit anaysis In th goba-fit intoducd by Caotti, (988, th who atitud pofi is tivd fom simutanous anaysis of a th sctd imb-scanning masumnts. h tiva is basd on th ast-squas cition and ooks fo a soution pofi that has a numb p of dgs of fdom sma than o qua to th numb of th obsvd data points. In pactic th pofi is tivd at p disct atituds and at intmdiat atituds an intpoatd vau is usd. In this appoach, th vcto Δ S that appas in Eq. (4..9 is th diffnc btwn a th sctd obsvations and th cosponding simuations (a th spcta intvas and a th imb-scanning masumnts a incudd in this vcto, vntuay aso a-pioi infomation can b incudd. h unknown vcto Δ x may contain a diffnt vaiab dpnding on th tiva w a pfoming, in gna it is, howv, an atitud dpndnt distibution which is sampd at a numb of disct atituds as w as som spctoscopic and instumnta paamts (.g. atmosphic continuum. h us of th LM mthod fo th minimisation of th function quis th computation of th quantitis that appa in th quations (4..8 and (4..9, namy: simuations fo a th imb-scanning masumnts and a th sctd micowindows, th vaianc covaianc matix S y of th obsvations, th Jacobian matix K h simuations of th obsvd spcta a pfomd using th fowad mod dscibd in Sct h vaianc covaianc matix atd to th apodisd spcta data (obsvations is divd stating fom nois vs, apodisation function and zo fiing infomation, using th agoithm dscibd in Sct h Jacobian matix containing th divativs of th simuatd spcta with spct to th unknown paamts is computd as dscibd in Sct High v mathmatics of th fowad mod h task of th fowad mod is th simuation of th spcta masud by th instumnt in th cas of known atmosphic composition. hfo, this mod consists of:. th simuation of th adiativ tansf though th Eath s atmosph fo an optima instumnt with infinitsima fid of viw (FOV, infinit spcta soution and no distotions of th inshap.. th convoution of this spctum with th apodisd instumnt in shap (AILS to obtain th apodisd spctum which incuds in shap distotions. 3. th convoution of ths spcta with th FOV of th instumnt. Not that whi stp. povids a mod of th atmosphic signa, stps. and 3. simuat instumnta ffcts. Not a th instumnta ffcts a howv simuatd in th fowad mod, sinc th tiva is pfomd fom caibatd spcta, instumnt sponsivity and phas os a coctd in v b pocssing. h AILS which incuds th ffcts of finit soution, instumnt in-shap distotions and apodization is aso povidd by Lv b pocssing.

24 Lv Poduct Vaidation Dat: //0 Pag n. 4/ h adiativ tansf In od to obtain th spcta S, z (i.. th intnsity as a function of th wavnumb fo th ( g diffnt imb gomtis (dnotd by th tangnt atitud z g of th obsvation g th foowing intga fo th adiativ tansf has to b cacuatd: Wh: S(, z B(, ( s d (, s (4.4. g b s g g = wavnumb z g = tangnt atitud of th optica path g s g = co-odinat aong th in of sight (LOS bonging to th optica path with th tangnt atitud z g S(,z g = spcta intnsity (s g = tmpatu aong th Lin of Sight B(, = souc function (,s g = tansmission btwn th point s g on th LOS and th obsv ocatd at s 0. his quantity dpnds on th atmosphic composition, pssu and tmpatu though th co-odinat s. b = indicato fo th fathst point that contibuts to th signa Und th assumption of oca thmodynamic (LE quiibium B(, is th Panck function: 3 hc B(, (4.4. hc xp K B with h = Panck s constant c = vocity of th ight K B = Botzmann s constant h tansmission can b xpssd as a function of s g : s g (, sg xp k(, s' ( s' ds' (4.4.3 s 0 with p( sg ( sg = numb dnsity of th ai K ( s p(s g = pssu B g and th wightd absoption coss sction:

25 Lv Poduct Vaidation Dat: //0 Pag n. 5/35 ms N VMR k(, s k (, s x ( s (4.4.4 g m m g m wh N ms = numb of diffnt mocua spcis that absob in th spcta gion und considation VMR x s = voum mixing atio (VMR of th spcis m at th point s g m ( g k m (,s g = absoption coss sctions of th spcis m g In th tiva mod th atmosphic continuum mission is takn into account as an additiona spcis with VMR = and th cosponding coss sction is fittd as a function of atitud and micowindow (s Sct Fo th continuum cacuation in th sf standing fowad mod th coss sctions a takn fom a ook up tab and th a VMR of th continuum spcis is usd (s Sct Equation (4.4. can now b wittn as: S(, s g = b sg s0 b sg s0 B(, ( s B(, ( s g g d (, s ds k(, s g g g ds ( s g g (, s g ds g (4.4.5 In od to dtmin th intga (4.4.5 two basic stps a ncssay: th ay tacing, i.. th dtmination of th optica path s g and, consqunty, th tmpatu (s g, th pssu p(s g and th voum mixing atio x VMR m(s g aong th LOS and th cacuation of th absoption coss sctions k m (,s g Ray tacing h in of sight in th atmosph is givn by th viwing diction of th instumnt and th distibution of th factiv indx in th atmosph. h factiv indx n(p(s g,(s g is dtmind as a function of pssu and tmpatu by th atmosphic mod (s sction 5.5. Absoption coss sction cacuation h absoption coss sction of on mocua spcis m as a function of tmpatu and pssu is givn by th foowing sum ov a ins of th spcis: k m ins A (,, p Lm, ( Am, ( m,,, p (4.4.6 wh L m, ( = in stngth of in of spcis m

26 Lv Poduct Vaidation Dat: //0 Pag n. 6/35 m, A m = cnta wavnumb of in of spcis m A, (- m,,,p = in pofi (in-shap h in stngth is cacuatd by th fomua: L m, ( L m, ( 0 Q Q m m hce" xp ( 0 K B ( hce" xp K B0 m, m, hc, xp K B hc, xp K B0 m m (4.4.7 with L m, ( 0 = in stngth at fnc tmpatu 0 Q m ( = tota intna patition function E m, = ow stat ngy of th tansition V h basic in shap is th Voigt function A,, - th convoution of th Dopp A D m, ( m, m, ( m, p L, and th Lontz pofi A,, : A m, ( m, p V m, ( m, m, m, m, m, p D L,, p A (, A (,, (4.4.8 h Dopp pofi is givn by th fomua n (, D m A m, ( m,, xp n (4.4.9 D D m, m, with th haf width at haf maximum (HWHM of th in: wh B D K m, m, n (4.4.0 M c m M m = mocua mass of spcis m h Lontz function is: A L m, and th Lontz HWHM: L m, ( m,,, p (4.4. ( L m, m, L m, 0 m, L p 0 m, 0 (4.4. p with : L 0, m = Lontz haf width at fnc tmpatu 0 and fnc pssu p 0

27 Lv Poduct Vaidation Dat: //0 Pag n. 7/35 m, = cofficint of tmpatu dpndnc of th haf width Using th substitutions: and (4.4.3 x m, n y m, n L m, D m, m, D m, (4.4.4 th Voigt function can b wittn as: with: A n m,,, p K( xm,, ym (4.4.5 D V m, (, m, K( x y t m, m,, ym, ( xm, t ym, dt ( Convoution with th AILS In od to tak into account th finit spcta soution of th instumnt distotion of th in-shap by th instumnt th apodisation of th obsvd spcta, th spctum S(,z g is convoutd with AILS(, giving: S A (, z S(, z AILS( (4.4.6 g g AILS( is th Apodisd Instumnt Lin Shap that is obtaind by convouting th masud ILS with th apodisation function usd fo th apodisation of th obsvd spcta Convoution with th FOV h FOV of an instumnt is an angua distibution. In cas of satit masumnts, ik MIPAS, a (nay ina ationship xists btwn viwing ang and tangnt atitud, thfo th FOV can b psntd using a imb-scanning-ang-invaiant atitud distibution. FOV(z g,z dscibs th finit FOV of MIPAS as a function of th atitud z. In th cas of MIPAS, FOV(z g,z is psntd by a picwis ina cuv tabuatd in th input fis. Fo th simuation of th spctum affctd by th finit FOV (S FA (,z g th foowing convoution is cacuatd: S FA (, z S (, z FOV ( z, z (4.4.7 g A g

28 Lv Poduct Vaidation Dat: //0 Pag n. 8/ Instumnta continuum Fo th simuation of th instumnta continuum an additiona (micowindow dpndnt and swp indpndnt tm is addd to S, z. his tm is fittd in th tiva pogam. F ( g Summay of quid vaiabs Fo th atmosphic mod: pssu aong th in of sight g p(s g tmpatu aong th in of sight g (s g voum mixing atio aong th in of sight g x VMR (s g Fo th ay tacing: atitud and viwing diction of th instumnt o tangnt atitud (in cas of homognousy ayd and sphica atmosph z g Fo th coss sction cacuation: cnta wavnumb of tansition of spcis m m, fnc in stngth of tansition of spcis m L m, ( 0 ow stat ngy of tansition of spcis m E m, tota intna patition function of spcis m Q m ( mocua mass of spcis m M m fnc Lontz haf width of tansition of spcis m L0 m, cofficint of tmpatu dpndnc of th haf width m, Fo th AILS convoution: apodisd instumnt in shap AILS( Fo th FOV convoution: fid of viw function FOV(z g,z

29 Lv Poduct Vaidation Dat: //0 Pag n. 9/ Cacuation of th VCM of th masumnts h vaianc covaianc matix (VCM of th siduas S m, usd in Eq. (4..4, is in pincip givn by th combination of th VCM of th obsvations S y and th VCM of th fowad mod S FM : S m S S (4.5. y FM Howv, sinc: th ampitud of th fowad mod o is not accuaty known; coations btwn fowad mod os a difficut to quantify in th tiva agoithm w choos to us S m = S y. h ntity of th fowad mod os wi b vauatd by anayzing th bhaviou of th - tst fo th diffnt micowindows, at th diffnt atituds. In paticua, th obtaind - tst wi b compad with its xpctd vau as dtmind on th basis of th tota o vauatd by th so cad Rsiduas and Eo Coation (= REC" anaysis (s Piccoo t a. (00. Hwith w dscib how th VCM of th obsvations S y can b divd. Evn if th points of th intfogams masud by MIPAS a sampd indpndnty of ach oth (no coation btwn th masumnts, th spcta data a affctd by coation. h coation aiss fom th data pocssing pfomd on th intfogam (.g. apodisation. Fo this ason th nois vs povidd by Lv B pocssing do not fuy chaactis th masumnt os and th computation of a compt VCM S y of th spctum S( is ndd. In Sction 4.5. w dscib th opations pfomd on th intfogam in od to obtain th apodisd spctum. On th basis of ths opations, in Sct w dscib how th vaianc covaianc matix S y of th obsvations can b divd. Finay, in Sct th pocdu usd to invt S y is dscibd Opations pfomd on th intfogam to obtain th apodisd spctum h standad MIPAS intfogam is a doub-sidd intfogam obtaind with a nomina maximum optica path diffnc (MPD of +/- 0 cm in th FR masumnts and +/- 8 cm in th OR masumnts. h apodisd spctum S ˆ ( is obtaind by subsqunty pfoming th foowing opations on th intfogam:. Zo-fiing Duing Lv B pocssing, in od to xpoit th fast Foui ansfom agoithm, th numb of points of th intfogam is mad qua to a pow of by xtnding th intfogam with zos fom th MPD to th Zo-Fid Path Diffnc (ZFPD. h masud intfogam is thfo qua to an intfogam with maximum path diffnc ZFPD, mutipid by a boxca function (d dfind as:

30 Lv Poduct Vaidation Dat: //0 Pag n. 30/35 MPD d 0 whn d [ MPD, MPD] whn d [ MPD, MPD] (MPD = Max. Path Diffnc (4.5.. Foui ansfomation (F Lt us ca S th spctum obtaind fom th masud (and zo-fid intfogam NAHR ( and S th spctum that woud hav bn obtaind fom th intfogam with maximum HR ( MPD path diffnc ZFPD. S, S and F[ ( d] a a givn in th samping NAHR ( HR ( gid. ZFPD Sinc th F of th poduct of two functions is qua to th convoution of th F s of th two functions, w obtain: MPD S ( S ( F[ ( d]. (4.5.3 NAHR HR 3. R-samping at th fixd gid Sinc a p-dfind and constant gid is quid by th ORM fo its optimisations, th spctum - is -sampd at a fixd gid 0.05 cm, with D qua to 0 cm D ( cm, with D qua to 8 cm fo th OR masumnts. h pfomd D opation can b wittn as: - d ZFPD S * F, (4.5.4 NA S NAHR wh ( S NA is cacuatd at th fixd gid. D In this cas th opation (4.5.4 is not a cassica convoution among quantitis that a dfind on th sam gid (.g. Eq. (4.5.3, but is a -samping pocss which changs th gid spacing fom of S NAHR to of S NA. ZFPD D his opation dos not intoduc coation btwn th spcta points ony if MPD D. If MPD D th sut of (4.5.4 is qua to th F of a ±0 cm (±8 cm fo OR intfogam. If MPD Dth sut of (4.5.4 is qua to th zo-fiing to 0 cm (8 cm fo OR path diffnc of an intfogam with path diffnc MPD, thfo th spcta points a coatd to ach oth. h NESR vaus givn in th Lv B poduct a computd aft this -samping stp.

31 Lv Poduct Vaidation Dat: //0 Pag n. 3/35 4. Apodisation h apodisd spctum S ˆ ( is obtaind by convouting th spctum S with th apodisation function ap (, sampd at. D S ˆ * (4.5.5 S NA ap NA( 4.5. Computation of th VCM ativ to a sing micowindow In th cas of MIPAS data th micowindows a usuay w spaatd and it is asonab to assum that th spcta data points bonging to diffnt micowindows a uncoatd. As a consqunc, th vaianc covaianc matix of th spctum S y is a bock-diagona matix with as many bocks as many micowindows a pocssd, and th dimnsion of ach bock is qua to th numb of spcta points in th cosponding micowindow. W assum that diffnt points of th micowindow a chaactisd by th sam o, but diffnt micowindows can hav diffnt os. In this sction w div th ationship that appis to ach bock and fo simpicity with S y w f to a sing bock ath than th fu VCM of th obsvations. h coation btwn diffnt spcta points of th micowindow is du to th apodisation pocss and to th zo-fiing that is psnt in th cas of MPD < D. If MPD D, ony th apodisation is a caus of coation and th VCM S y of th apodisd spctum ˆ ( S, (S NA is a diagona matix sinc S can b computd fom th VCM S NA associatd with NA th spcta points of S a uncoatd and fom th Jacobian J of th tansfomation (4.5.5: NA S y JSNAJ SNAJJ (4.5.6 In Eq. (4.5.6 th od of th opations has bn changd bcaus S NA is a diagona matix and a th diagona mnts a qua. h diagona vaus a qua to (NESR, wh NESR is th quantity cacuatd aft opation 3 of Sct h cacuation of matix J is staightfowad. Fom th xpicit xpssion of th convoution (4.5.5: S ˆ( ( ( (4.5.7 i S NA it foows that th nty i,k of matix J is qua to: ap i J ( (4.5.8 i, k ap i k and th vaianc covaianc matix S y can b computd as: y NESR, i, k k, NESR i ap i k ap k k S J J ( (. (4.5.9