The Onsager--Machlup functional for data assimilation

TL;DR

This paper investigates the role of the Onsager--Machlup functional in data assimilation, clarifies how to incorporate prior distributions into cost functions, and proposes a new technique for estimating divergence terms in large systems.

Contribution

It provides numerical insights into the incorporation of the Onsager--Machlup functional in data assimilation and introduces a novel method for estimating divergence terms in high-dimensional models.

Findings

Divergence of the drift is crucial in weak-constraint 4D-Var.

Divergence is not necessary in MCMC with Euler scheme.

A new technique for estimating divergence and its derivatives is proposed.

Abstract

When taking the model error into account in data assimilation, one needs to evaluate the prior distribution represented by the Onsager--Machlup functional. Through numerical experiments, this study clarifies how the prior distribution should be incorporated into cost functions for discrete-time estimation problems. Consistent with previous theoretical studies, the divergence of the drift term is essential in weak-constraint 4D-Var (w4D-Var), but it is not nec essary in Markov chain Monte Carlo with the Euler scheme. Although the former property may cause difficulties when implementing w4D-Var in large systems, this paper proposes a new technique for estimating the divergence term and its derivative.