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Retrieval Theory

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 + e a. “MAP solution” “optimal estimate” “retrieval”. Bayes’ theorem. Forward model y = F(x) + e.

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Retrieval Theory

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  1. Retrieval Theory Mar 23, 2008 Vijay Natraj

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

  3. 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

  4. 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 x0 • K: weighting function (Jacobian) matrix • K describes measurement sensitivity to state.

  5. 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)

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

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

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

  9. 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

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

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