A unified framework for the analysis of accuracy and stability of a class of approximate Gaussian filters for the Navier–Stokes equations

Abstract Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. In these cases, nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation (DA) techniques. A number of DA algorithms used commonly in practice, such as the Ensemble Kalman Filter (EnKF) or Ensemble Square Root Kalman Filter (EnSRKF), suffer from serious drawbacks such as catastrophic filter divergence and filter instability. The analysis of stability and accuracy of these DA schemes has thus far focused either on finite-dimensional dynamics, or on complete observations in case of infinite-dimensional dissipative systems. We develop a unified framework for the analysis of several well-known and empirically efficient DA techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals , between the approximate-Bayesian and control-theoretic approaches to DA. We consider the ‘model’ dynamics governed by the two-dimensional incompressible Navier–Stokes equations (NSEs) and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time DA techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced NSEs. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the L 2 norm and of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the DA algorithms and covariance inflation and localization which are employed in practice for improved filter performance.