Using the (iterative) ensemble Kalman smoother to estimate the time correlation in model error

Numerical weather prediction systems contain model errors related to missing and simplified physical processes, and limited model resolution. While it has been widely recognized that these model errors need to be included in the data assimilation formulation, providing prior estimates of their spatio-temporal characteristics is a hard problem. We follow a systematic path to estimate parameters in the model error formulation, specifically related to time-correlated model errors. This problem is more difficult than the standard parameter estimation problem because the model error parameters are only visible through the random model error realisations. By concentrating on linear and nonlinear low-dimensional systems, we are able to highlight the many aspects of this problem, using state augmentation in an ensemble Kalman smoother (EnKS) and its iterative variant (IEnKS). It is not possible to estimate the model error parameters in one assimilation window because enough information has to be gathered to see the parameters through the random errors, even when every time step is observed. If only one parameter is estimated in a linear one-dimensional system the EnKS works well, but when we try to estimate two parameters the method fails. An IEnKS is able to find the correct parameter values for the linear system. For the highly nonlinear logistic map the IEnKS can get stuck in local minima, but with careful tuning of the step length in the iterations and careful transformation of the solution space the correct parameter values can be found. The main conclusion is that estimating model error parameters –even in low-dimensional systems– is a difficult problem, but via careful reformulation of the problem practical solutions can be found