Model error and sequential data assimilation. A deterministic formulation

TL;DR

This paper introduces a deterministic formulation of the extended Kalman filter that models model errors as a universal quadratic evolution law, improving accuracy over traditional white noise assumptions in data assimilation.

Contribution

It proposes a new deterministic approach to model error evolution in the extended Kalman filter, based on recent findings of universal behavior, and demonstrates its effectiveness in a Lorenz system.

Findings

Deterministic error evolution accurately approximates short-term error growth.

The quadratic law for error covariance improves filter accuracy.

Correlation with initial errors is less significant in short-term dynamics.

Abstract

Data assimilation schemes are confronted with the presence of model errors arising from the imperfect description of atmospheric dynamics. These errors are usually modeled on the basis of simple assumptions such as bias, white noise, first order Markov process. In the present work, a formulation of the sequential extended Kalman filter is proposed, based on recent findings on the universal deterministic behavior of model errors in deep contrast with previous approaches (Nicolis, 2004). This new scheme is applied in the context of a spatially distributed system proposed by Lorenz (1996). It is found that (i) for short times, the estimation error is accurately approximated by an evolution law in which the variance of the model error (assumed to be a deterministic process) evolves according to a quadratic law, in agreement with the theory. Moreover, the correlation with the initial condition error appears to play a secondary role in the short time dynamics of the estimation error covariance. (ii) The deterministic description of the model error evolution, incorporated into the classical extended Kalman filter equations, reveals that substantial improvements of the filter accuracy can be gained as compared with the classical white noise assumption. The universal, short time, quadratic law for the evolution of the model error covariance matrix seems very promising for modeling estimation error dynamics in sequential data assimilation.