Linear Form of the Radiative Transfer Equation Revisited. Bormin Huang

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1 Lnear Form of the Radate Tranfer Equaton Reted Bormn Huang Cooerate Inttute for Meteorologcal Satellte Stude, Sace Scence and Engneerng Center Unerty of Wconn Madon The 5 th Internatonal TOVS Study Conference (ITSC-5) Maratea, Italy, 4- October 26

2 Outlne Smth lnear form (Aled( Otc, 99) ). t dual form. The lnear form wth exact analytcal Jacoban. the tangent-lnear and adjont model. The lnear form wth exact analytcal Jacoban. the lnear form wth nexact analytcal Jacoban. Why ue the forward model wth exact analytcal Jacoban for hycal retreal and data amlaton? Summary Future work

3 Bll Smth Acheement n Phycal Retreal Bll ha been oneerng the hycal retreal of temerature and aborbng conttuent rofle from the radance ectra nce the late 6. H mot recent lnear form of the RTE wa ublhed n the landmark aer n Aled Otc (99). Th monochromatcally aroxmate lnear form and t arant hae been tll ued n the hycal retreal [e.g. Ma et al., 2, L et al., 2]. H work nred Huang et al. (Aled Otc, 22) to uccefully dere the lnear form wth exact analytcal Jacoban for the wdely-ued McMlln- Flemng-Eyre-Woolf tye of forward model (e.g. RTTOV, RTIASI). In th talk I wll roe that there ext the dual rereentaton of Smth lnear form (99), and how ome mathematcally nteretng outcome dered from th dual form.

4 Smth Lnear Form (99). t Dual Form ob S R = B ( ) τ ( ) B ( ) dτ ( ) = τ τ R B ( ) ( ) B ( ) d ( ) P δr B δτ δb τ = ( ) ( ) + ( ) ( ) δ R R R ob δ B ( ) B ( ) B ( ) δτ τ τ ( ) ( ) ( ) Smth Lnear Form It Dual Form B( ) d[ δτ( ) ] δb( ) dτ( ) δb ( ) ( ) ( ) [ ( )] dτ B dδτ δr δb τ B δτ = ( ) ( ) + ( ) ( )

5 δ R = B( ) δ τ( ) + δb( ) τ( ) δr β ( ) τ ( ) δt + = = δb( d ) τ( ) B ( ) d[ δτ( ) ] β ( ) τ ( ) δt( ) dln τ ( ) dt( ) β( ) τ ( ) δu( ) dln τ ( ) du ( ) δ R = B + δ ( ) τ( ) B( ) δτ( ) B ( ) [ ( )] ( ) ( ) dδτ δb dτ B ( T ( )) δb ( ) = δt( ) β ( ) δt( ) T ( ) d δτ τ δu du ( ) ( ) ( ) ( ) = = τ dt = d τ dτ ( ) τ ( ) d ln τ ( ) δ R β ( ) τ ( ) δ T + = = ln τ ( ) U ( ) q ( ) d g = τ ( ) τ ( ) dln ( ) dt( ) ( ) ln ( ), du ( ) du ( ) dln τ ( ) du ( ) ( ) = ln τ ( ) du ( ) du ( ) du d du ( ) β ( ) τ ( ) δ T( ) dln τ ( ) du o ( ) dt ( ) β ( ) τ ( ) ( dln τ ( ) δ U ) du ( )

6 δr β ( ) τ ( ) δt + = = β ( ) τ ( ) δt ( ) dln τ ( ) dt( ) β( ) τ ( ) δu( ) dln τ ( ) du ( ) δr β ( ) τ ( ) δ T + = = du( ) β ( ) τ ( ) δ T ( ) dln τ ( ) du o ( ) dt ( ) β ( ) τ ( ) ( dln τ ( ) δu ) du ( ) The effecte temerature rofle of the th aborbng ga: dt( ) δt( ) δt( ) δu( ) du ( ) The effecte temerature rofle of the th aborbng ga: du dt δt( ) δt( ) δu( ) du du ( ) ( ) ( ) ( ) Fnal Lnear Form δt δt = δr β ( ) τ ( ) β ( ) ( ) τ ( ) dln τ ( )

7 dt( ) δt( ) δt( ) δu( ) du ( ) du dt δt( ) δt( ) δu( ) du du ( ) ( ) ( ) ( ) du ( ) = + [ ] U ( ) U ( ) T( ) T( ) dt( ) du ( ) ( ) ( ) ( ) dt ( ) T T + U du ( ) = U ( ) + T ( ) T ( ) dt ( ) A ecal cae: T T ( ) = ( ) du ( ) = + [ ] U ( ) U ( ) T( ) T( ) dt( ) The general cae: T T { ( ) } ( ) ( ) du dt ( ) U U T T d T T dt d ( ) = ( ) ( ) ( ) + ( ) ( ) ( ) The retreal qualty of aborbng ga rofle deend on the qualty of temerature frt gue!

8 Matlab examle

9 The exact analytcal Jacoban aroach. the tangent-lnear & adjont aroach. Same numercal recon! 2. The exact analytcal Jacoban aroach eeral tme fater!! Examle: T T ( A B)( ) A Td ( d) = ( ) e d V T C T + Comute dv δv = δtd dtd Source: Paul an Delt

10 Why Ue Exact Analytcal Jacoban n Retreal and Data Amlaton? Phycal Retreal (D-Var): Cot( X ) = R ob R( X ) or ob Cot( X ) = R R( X ) + λ X X WP Data Amlaton (3D/4D-Var): Cot( X,...) = Atmoherc Dynamc Term( X,...) + R ob R( X ) ~ deal choce for multertral ounder (e.g. HIRS, GOES, MODIS) ~ an underdetermned roblem (wth reect to a tycal forward model) retreal Cot( X,...) = Atmoherc Dynamc Term( X,...) + X X ~ deal choce for hyerctral ounder (e.g. AIRS, IASI, GIFTS) ~an oerdetermned roblem (wth reect to a tycal forward model) Atmoherc dynamc term bacally goerned by the aer-stoke equaton, whch ha no analytcal Jacoban. Thu, t tangent-lnear/adjont model are needed for data amlaton. The exact analytcal Jacoban for the wdely-ued McMlln-Flemng-Eyre-Woolf tye of radate tranfer/forward model (e.g. RTTOV, RTIASI) are derable (Huang et al., Aled Otc, 22).

11 Summary. The clacal deraton of the lnear form of the RTE by Smth et al. (99) reewed. It dual form dered. 2. The orgnal lnear form aear to be a ecal cae of t dual form when the temerature frt gue haen to be the true temerature rofle. 3. Lnear form wth nexact analytc Jacoban make retreal reult unrelable! 4. The exact analytc Jacoban mlementaton an effcent alternate to the tangent-lnear/adjont model for hyerectral retreal and data amlaton roblem wth the wdely-ued McMlln-Flemng-Eyre-Woolf tye of forward model (e.g. RTTOV, RTIASI).

12 Future Work The Remote Senng GEOME (Geometrcal( Exloraton of onlnear Otmzaton n Meaurement Enronment) Project: Unelng the Radance Hyerace for Quantfyng Geohycal Retreal

13 The remote enng GEOME roject am to ole the followng long-tandng fundamental roblem n ae remote enng Gen a enor ecfcaton, t forward model and exact Jacoban, an, to quantfy the nformaton content of each channel,.e., the bet exected contrbuton from each channel to the retreal of temerature and aborbng gae at each reure leel, the nformaton content of a enor,.e., the bet exected retreal accuracy that et the tattcal lmt for all retreal algorthm, the error of a fat model n term of the degradaton of the bet exected retreal accuracy, a comared to t LBL counterart, the mact of enor noe leel on the bet exected retreal accuracy, the frt gue tolerance a afety meaure beyond whch no retreal algorthm can reach the bet exected retreal, and the retreal effcency a robutne meaure for any retreal algorthm, a comared to the bet exected retreal. Alcaton: Loy comreon retreal mact tude: to conclude the retreal degradaton (due to loy comreon) by the bet exected retreal accuracy that et the lmt for all oble retreal algorthm. Otmal channel electon for retreal wth artal channel: to relee the comutatonal burden n retreal and data amlaton a wth the mnmum retreal degradaton for hyerectral ounder (e.g. AIRS, IASI, GIFTS). Future enor degn & trade-off tudy for rk reducton: to ae the nformaton content (the bet exected retreal accuracy) of a enor degned wth arou ectral range, ILS reoluton, and noe leel. Dfferent lng ece hae dfferent genome. So do dfferent t enor!