# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.06663