Accuracy and Stability of The Continuous-Time 3DVAR Filter for The Navier-Stokes Equation

TL;DR

This paper analyzes the accuracy and stability of a continuous-time 3DVAR data assimilation filter applied to the Navier-Stokes equation, deriving an SPDE model and proving conditions for effective signal recovery.

Contribution

It introduces a continuous-time SPDE model for the 3DVAR filter on the Navier-Stokes equation and establishes stability and accuracy results in high-frequency observation regimes.

Findings

The filter can lock onto the true signal when enough low Fourier modes are observed.

Model uncertainty larger than data uncertainty (variance inflation) improves stability.

Numerical examples confirm theoretical stability and accuracy results.

Abstract

The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the Navier-Stokes equation. We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a damped-driven Navier-Stokes equation, with mean-reversion to the signal, and spatially-correlated time-white noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observations modes is small. Numerical examples are given to illustrate the theory.