Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model

TL;DR

This paper analyzes the 3DVAR data assimilation method applied to the Lorenz '63 model, demonstrating how variance inflation stabilizes the filter and improves accuracy in partial, noisy, and sequential data scenarios.

Contribution

It provides a theoretical analysis of the 3DVAR method in the Lorenz '63 model, highlighting the role of variance inflation in stabilization and accuracy enhancement.

Findings

Variance inflation stabilizes the 3DVAR filter.

The analysis applies to partial and noisy data.

Both discrete and continuous data streams are considered.

Abstract

The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for high-dimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an on-line fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz '63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy.