Four-dimensional variational assimilation in the unstable subspace (4DVar-AUS) and the optimal subspace dimension

TL;DR

This paper introduces 4DVar-AUS, a method that leverages system instabilities by confining analysis increments to an optimal unstable subspace, improving data assimilation accuracy without needing adjoint models.

Contribution

The paper proposes a novel 4DVar-AUS method that exploits dynamical instabilities and identifies an optimal subspace dimension for improved analysis performance.

Findings

4DVar-AUS outperforms standard 4DVar across multiple assimilation windows.

Optimal subspace dimension aligns with the unstable and neutral manifold.

Method reduces analysis error without requiring adjoint integration.

Abstract

A key a priori information used in 4DVar is the knowledge of the system's evolution equations. In this paper we propose a method for taking full advantage of the knowledge of the system's dynamical instabilities in order to improve the quality of the analysis. We present an algorithm, four-dimensional variational assimilation in the unstable subspace (4DVar-AUS), that consists in confining in this subspace the increment of the control variable. The existence of an optimal subspace dimension for this confinement is hypothesized. Theoretical arguments in favor of the present approach are supported by numerical experiments in a simple perfect non-linear model scenario. It is found that the RMS analysis error is a function of the dimension N of the subspace where the analysis is confined and is minimum for N approximately equal to the dimension of the unstable and neutral manifold. For all assimilation windows, from 1 to 5 days, 4DVar-AUS performs better than standard 4DVar. In the presence of observational noise, the 4DVar solution, while being closer to the observations, if farther away from the truth. The implementation of 4DVar-AUS does not require the adjoint integration.