Richard Menard;
Martin Deshaies-Jacques
Environment and Climate Change Canada
The 8th EnKF Data Assimilation Workshop
Diagnostic of analysis error covariance by cross-validation
Talk:
Menard_EnKF_DA_Workshop_2018_v4.ppt
We present a general theory of estimation and optimization of the analysis error covariance without relying on forecast. The method is based on cross-validation that is by partitioning the original observation data set into a training set, to create the analysis, and an independent (or passive) set, used to evaluate the analysis . We have also developed a geometric interpretation of the method based on Hilbert space representation of random variables. A number of statistical estimation formulas of analysis error covariance are derived that can be used in either passive or active observation spaces. In particular we use the variance of passive observation–minus-analysis residuals and show that the true analysis error variance can be estimated, without relying on the optimality assumption. This approach is used to obtain near optimal analyses that are then used to evaluate the analysis error using several different methods at active and passive observation sites. We compare the estimates according to the method of Hollingsworth-Lönnberg (1989), Desroziers et al., a new diagnostic we developed, and the perceived analysis error computed from the analysis scheme, to conclude that, as long as the analysis is near optimal, all estimates agree within a certain error margin. Except for to the Desroziers et al. diagnostics, all other diagnostics of analysis error covariance are, by construction, symmetric matrices, and in practice also positive definite. We also examine how these diagnostics are valid when active observation errors are correlated with background errors. Applications are presented in the context of surface analysis and to obtain the localization correlation length.
We present a general theory of estimation and optimization of the analysis error covariance without relying on forecast. The method is based on cross-validation that is by partitioning the original observation data set into a training set, to create the analysis, and an independent (or passive) set, used to evaluate the analysis . We have also developed a geometric interpretation of the method based on Hilbert space representation of random variables. A number of statistical estimation formulas of analysis error covariance are derived that can be used in either passive or active observation spaces. In particular we use the variance of passive observation–minus-analysis residuals and show that the true analysis error variance can be estimated, without relying on the optimality assumption. This approach is used to obtain near optimal analyses that are then used to evaluate the analysis error using several different methods at active and passive observation sites. We compare the estimates according to the method of Hollingsworth-Lönnberg (1989), Desroziers et al., a new diagnostic we developed, and the perceived analysis error computed from the analysis scheme, to conclude that, as long as the analysis is near optimal, all estimates agree within a certain error margin. Except for to the Desroziers et al. diagnostics, all other diagnostics of analysis error covariance are, by construction, symmetric matrices, and in practice also positive definite. We also examine how these diagnostics are valid when active observation errors are correlated with background errors. Applications are presented in the context of surface analysis and to obtain the localization correlation length.
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