Inaccuracy of the variance evolution associated with discrete covariance propagation

Abstract Data assimilation methods often also employ the same discrete dynamical model used to evolve the state estimate in time to propagate an approximation of the state estimation error covariance matrix. Four‐dimensional variational methods, for instance, evolve the covariance matrix implicitly via discrete tangent linear dynamics. Ensemble methods, while not forming this matrix explicitly, approximate its evolution at low rank from the evolution of the ensemble members. Such approximate evolution schemes for the covariance matrix imply an approximate evolution of the estimation error variances along its diagonal. For states that satisfy the advection equation, the continuity equation, or related hyperbolic partial differential equations (PDEs), the estimation error variance itself satisfies a known PDE, so the accuracy of the various approximations to the variances implied by the discrete covariance propagation can be determined directly. Experiments conducted by the atmospheric chemical constituent data assimilation community have indicated that such approximate variance evolution can be highly inaccurate. Through careful analysis and simple numerical experiments, we show that such poor accuracy must be expected, due to the inherent nature of discrete covariance propagation, coupled with a special property of the continuum covariance dynamics for states governed by these types of hyperbolic PDE. The intuitive explanation for this inaccuracy is that discrete covariance propagation involves approximating diagonal elements of the covariance matrix with combinations of off‐diagonal elements, even when there is a discontinuity in the continuum covariance dynamics along the diagonal. Our analysis uncovers the resulting error terms that depend on the ratio of the grid spacing to the correlation length, and these terms become very large when correlation lengths begin to approach the grid scale, for instance, as gradients steepen near the diagonal. We show that inaccurate variance evolution can manifest itself as both spurious loss and gain of variance.