Retrieval Theory Mar 23, 2008 Vijay Natraj

The Inverse Modeling Problem Optimize values of an ensemble of variables (state vector x ) using observations: a priori estimate x a +  a Measurement vector y Forward model y = F(x) +  “MAP solution” “optimal estimate” “retrieval” Bayes’ theorem

Applications for Atmospheric Concentration Retrieve atmospheric concentrations (x) from observed atmospheric radiances (y) using a radiative transfer (RT) model as forward model Invert sources (x) from observed atmospheric concentrations (y) using a chemical transport model (CTM) as forward model Construct a continuous field of concentrations (x) by assimilation of sparse observations (y) using a forecast model (initial-value CTM) as forward model

Optimal Estimation Forward problem typically not linear No analytical solution to express state vector in terms of measurement vector Approximate solution by linearizing forward model about reference state x 0 K: weighting function (Jacobian) matrix K describes measurement sensitivity to state.

Optimal Estimation Causes of non-unique solutions  m > n (more measurements than unknowns)  Amplification of measurement and/or model noise  Poor sensitivity of measured radiances to one or more state vector elements (ill-posed problem) Need to use additional constraints to select acceptable solution (e.g., a priori)

Bayes’ Theorem a priori pdfobservation pdf normalizing factor (unimportant) a posteriori pdf Maximum a posteriori (MAP) solution for x given y is defined by solve for P(x,y)dxdy  Bayes’ theorem

Gaussian PDFs Scalar x Vector where S a is the a priori error covariance matrix describing error statistics on ( x-x a ) In log space: Similarly:

Maximum A Posteriori (MAP) Solution Bayes’ theorem: MAP solution: minimize cost function J : Solve for Analytical solution: with gain matrix bottom-up constraint top-down constraint

Averaging Kernel A describes the sensitivity of retrieval to true state and hence the smoothing of the solution: smoothing error retrieval error MAP retrieval gives A as part of the retrieval: Sensitivity of retrieval to measurement

Degrees of Freedom Number of unknowns that can be independently retrieved from measurement DFS = n: measurement completely defines state DFS = 0: no information in the measurement