Radiative Transfer Models and their Adjoints. Paul van Delst
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1 Radiative rasfer Modes ad their Adjoits au va Dest
2 Overview Use of sateite radiaces i Data Assimiatio (DA) Radiative rasfer Mode (RM) compoets ad defiitios estig the RM compoets. Advatages/disadvatages
3 Use of sateite radiaces i DA Adjust the mode trajectory with data. Iterativey miimise the differece betwee a mode predictio ad data usig a cost/peaty fuctio e.g. ( X) ( X X b ) B ( X X b ) + ( Ym Y( X ) ( O + F) ( Ym Y( X ) J c J + X X b : Iput state vector ad backgroud estimate Y m Y(X): Measuremets ad forward mode B O F: Error covariaces of X b Y m ad Y(X) Iteratio step directio is determied from Y(X) iearised about X b ( X) Y( X ) + K( X )( X X ) Y here the K(X b ) are the Jacobias (K-Matrix) of the forward mode for the backgroud state X b K b k ( X ) Y X b k X X b b Lx Lx LxK Kx
4 Forward (FD) mode. he FD operator maps the iput state vector X to the mode predictio Y e.g. for predictor #: RM compoets ad defiitios () 3 + aget-iear (L) mode. Liearisatio of the forward mode about X b the L operator maps chages i the iput state vector X to chages i the mode predictio Y Or i matrix form: 3
5 Adjoit (AD) mode. he AD operator maps i the reverse directio where for a give perturbatio i the mode predictio Y the chage i the state vector X ca be determied. he AD operator is the traspose of the L operator. Usig the exampe for predictor # i matrix form RM compoets ad defiitios () Expadig this ito separate equatios: 3
6 RM compoets ad defiitios (3) K-Matrix (K) mode. Cosider a chae radiace vector R computed usig a sige surface temperature. For every chae ν R 5 B ( ) B( ν ) B ( ν ) R + FD L AD his is ot what you wat for DA/retrievas sice the sesitivity of each chae is accumated i the fia surface temperature adjoit variabe. Simpe soutio: B ( ν ) R So i the RM the K-matrix code simpy ivoves shiftig a the chae idepedet adjoit code iside the chae oop. hat s it. K
7 estig the RM compoets FD/L Start with the assumptio that the FD compoet is i good shape (e.g. vaidated with observatios radiosode matchups etc). L test agaist the forward mode. Ru the L mode with X iputs varyig from - X + X to give Y L. Ru the FD mode with X+X iputs ad differece from the ero perturbatio case to get the o-iear resut Y NL. Ispect Y L ad Y NL as a fuctio of X. L must be iear (d oh) for a X ad taget (d oh ) to the NL resut at X. Liearity of the L resut ca be checked by umerica differetiatio to give a costat for a X. Numerica differetiatio of NL resut shoud give same vaue as L at X. But accuracy of umerica derivative is a issue if the perturbatio resoutio is ow.
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10 Assume the FD mode iput vector X has K eemets ad the output vector has L eemets estig the RM compoets L/AD [ ] [ ] L K Y Y Y Y X X X X 3 3 Y X Ru the L mode j to K times with iput j i j i ; i savig the Y vector output each ru to give a LxK matrix L. X j i j i Y i savig the X vector output each ru to give a KxL matrix AD. he to withi umerica precisio Ru the AD mode j to L times with iput AD L
11 Iput: Output:
12 Iput: et Output:
13 Iput: Oo Output: Oo A
14 Advatages/Disadvatages Advatages Adjoit method produces Jacobias fuy cosistet with the forward mode. e defied set of rues for appyig method to code. Code tests are straightforward ad defiitive particuary for the L/AD test. Easy to icorporate mode chages improvemets additios etc. Good for sesitivity aayses. L used to ivestigate impact of sma disturbaces AD ca be used to ivestigate origi of the aomay. Disadvatages Compexity. Compared to fiite differeces (if oe ca ive with usig them) adjoit codig ca be a bit of a brai teaser. Very easy to produce code sower tha a sai i a straitjacket. Up frot code desig is a importat step. Have to be carefu whe vectorisig ad optimisig code.
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