On building the state error covariance from a state estimate

TL;DR

This paper proposes two algorithms to construct the state error covariance from a state estimate, improving data assimilation performance and complementing machine learning approaches in chaotic systems.

Contribution

It introduces novel algorithms for building state error covariance from estimates, enhancing data assimilation methods in chaotic models.

Findings

Algorithms yield good performance in toy data assimilation systems.

Methods are compatible with deep learning-based systems.

Potential to improve covariance estimation in chaotic systems.

Abstract

It was recently found with the aid of machine learning that for a variety of toy data assimilation systems with chaotic Lorenz-96 model it is possible to achieve a nearly-optimal data assimilation without carrying the state error covariance between cycles. This result does not look surprising on its own because not carrying covariance is the approach taken by standard 4D-Var, but it was found ``astonishing'' in the context of the machine learning-based system trained on the ensemble Kalman filter. This note proposes two algorithms for building the state error covariance from a state estimate that yield good performance and could be worked out by the deep learning-based system.