Behavior of the iterative ensemble-based variational method in nonlinear problems

Abstract

The behavior of the iterative ensemble-based data assimilation algorithm is discussed. The ensemble-based method for variational data assimilation problems, referred to as the 4D ensemble variational method (4DEnVar), is a useful tool for data assimilation problems. Although the 4DEnVar is derived based on a linear approximation, highly uncertain problems, in which system nonlinearity is significant, are solved by applying this method iteratively. However, the ensemble-based methods basically seek the solution within a lower-dimensional subspace spanned by the ensemble members. It is not necessarily trivial how high-dimensional problems can be solved with the ensemble-based algorithm which employs the lower-dimensional approximation based on the ensemble. In the present study, an ensemble-based iterative algorithm is reformulated to allow us to analyze its behavior in high-dimensional nonlinear problems. The conditions for monotonic convergence to a local maximum of the objective function are discussed in a high-dimensional context. It is shown that the ensemble-based algorithm can solve high-dimensional problems by distributing the ensemble in different subspace at each iteration. The findings as the results of the present study were also experimentally supported.