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

Shin'Ya Nakano

doi:10.5194/npg-28-93-2021

Abstract

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.

Highlights

The 4D ensemble variational method (4DEnVar; Lorenc, 2003; Liu et al, 2008) is a useful tool for practical data assimilation
We explore the conditions for achieving monotonic convergence to a local maximum of the objective function in a high-dimensional nonlinear context
The ensemble variational method is derived under the assumption that a linear approximation of a dynamical system model is valid over a range of uncertainty

Summary

Introduction

The 4D ensemble variational method (4DEnVar; Lorenc, 2003; Liu et al, 2008) is a useful tool for practical data assimilation. The 4DEnVar obtains the derivative of the objective function from the approximate Jacobian of a dynamical system model, which is estimated by using the ensemble of simulation results. The 4DEnVar algorithm is derived based on a lowdimensional linear approximation of the high-dimensional nonlinear system model. If the tangent linear of the system model is obtained, 4D variational data assimilation problems can be solved with the incremental formulation (Courtier et al, 1994), which can be regarded as an instance of the Gauss–Newton framework (Lawless et al, 2005). The present paper focuses on the iterative variational data assimilation algorithm for general uncertain problems in order to avoid the discussion on specific physical processes of geodynamo.

Ensemble-based method

Iterative algorithm

Rationale of the algorithm

Bayesian form

Experiments

Discussion and conclusions