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Retrieval Theory Mar 23, 2008 Vijay Natraj
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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
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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
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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.
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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)
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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
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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:
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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
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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
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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
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