Javier Amezcua;
Peter Jan van Leeuwen
University of Reading
The 8th EnKF Data Assimilation Workshop
Ensemble smoothers in the presence of model error. Exact and ensemble solutions.
Talk:
Amezcua_8EnKf18.pdf
Variational data-assimilation methods based on ensembles have gained great popularity in the last decade, and are seen by many as the way forward in e.g. numerical weather prediction. Errors in the model equations are increasingly being included. This is typically done, however, in rather ad-hoc approximate ways, and without understanding how these approximations affect the solution and how they interfere with approximations inherent in the finite-size ensembles.
We show the first systematic evaluation of the influence of the approximations to model errors in the solution to the data-assimilation problem, and their interference with approximations from the finite-size ensembles in the linear data assimilation problem. Since most ensemble-based variational methods are solved iteratively through a set of successive linearised problems, the present work has direct relevance for non-linear problems.
We derive the full solution from Bayes Theorem and investigating its characteristics, aided by analytical solutions in the scalar case. Then we focus on approximations in the temporal characteristics of the model errors, generating exact solutions for quite general temporal auto-correlations, for infinite and finite ensemble sizes, both for the analysis mean and covariance approximations.
Numerous new results are derived. For instance, we find that for additive model errors model errors that are exponentially auto-correlated over time, an incorrect correlation time scale can have substantial negative effects on the solutions, and an over estimation of the correlation time scale is worse than an underestimation. Furthermore, the resulting space-time Kalman gain can be written as the true gain multiplied by two factors, a linear term that contains the errors due to both time-correlation errors and finite ensemble effects, and a nonlinear term related to the inverse part of the gain, with the linear term dominating. By assuming that both errors are relatively small, we are able to disentangle the contributions from the different approximations to the data-assimilation solution. We find that the analysis mean is mainly affected by the time-correlation errors, while the analysis covariance is affected by these errors and an in-breeding term. This latter term is similar to in-breeding terms found for ensemble Kalman filters in the past, but is now derived for a smoother in the presence of auto-correlated model error.
Variational data-assimilation methods based on ensembles have gained great popularity in the last decade, and are seen by many as the way forward in e.g. numerical weather prediction. Errors in the model equations are increasingly being included. This is typically done, however, in rather ad-hoc approximate ways, and without understanding how these approximations affect the solution and how they interfere with approximations inherent in the finite-size ensembles.
We show the first systematic evaluation of the influence of the approximations to model errors in the solution to the data-assimilation problem, and their interference with approximations from the finite-size ensembles in the linear data assimilation problem. Since most ensemble-based variational methods are solved iteratively through a set of successive linearised problems, the present work has direct relevance for non-linear problems.
We derive the full solution from Bayes Theorem and investigating its characteristics, aided by analytical solutions in the scalar case. Then we focus on approximations in the temporal characteristics of the model errors, generating exact solutions for quite general temporal auto-correlations, for infinite and finite ensemble sizes, both for the analysis mean and covariance approximations.
Numerous new results are derived. For instance, we find that for additive model errors model errors that are exponentially auto-correlated over time, an incorrect correlation time scale can have substantial negative effects on the solutions, and an over estimation of the correlation time scale is worse than an underestimation. Furthermore, the resulting space-time Kalman gain can be written as the true gain multiplied by two factors, a linear term that contains the errors due to both time-correlation errors and finite ensemble effects, and a nonlinear term related to the inverse part of the gain, with the linear term dominating. By assuming that both errors are relatively small, we are able to disentangle the contributions from the different approximations to the data-assimilation solution. We find that the analysis mean is mainly affected by the time-correlation errors, while the analysis covariance is affected by these errors and an in-breeding term. This latter term is similar to in-breeding terms found for ensemble Kalman filters in the past, but is now derived for a smoother in the presence of auto-correlated model error.
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