Continuum covariance propagation for understanding variance loss in advective systems

TL;DR

This paper investigates the causes of variance loss in covariance propagation for advective systems, revealing that hyperbolic PDE characteristics lead to discontinuities affecting covariance dynamics, which standard numerical methods fail to capture.

Contribution

It identifies the hyperbolic nature of covariance dynamics as a source of variance loss and proposes the need for local covariance propagation methods tailored to these discontinuities.

Findings

Variance loss and gain occur during covariance propagation even at full rank.

Discontinuities in covariance dynamics are caused by hyperbolicity near small correlation lengths.

Standard numerical methods inadequately capture covariance evolution near discontinuities.

Abstract

Motivated by the spurious variance loss encountered during covariance propagation in atmospheric and other large-scale data assimilation systems, we consider the problem for state dynamics governed by the continuity and related hyperbolic partial differential equations. This loss of variance is often attributed to reduced-rank representations of the covariance matrix, as in ensemble methods for example, or else to the use of dissipative numerical methods. Through a combination of analytical work and numerical experiments, we demonstrate that significant variance loss, as well as gain, typically occurs during covariance propagation, even at full rank. The cause of this unusual behavior is a discontinuous change in the continuum covariance dynamics as correlation lengths become small, for instance in the vicinity of sharp gradients in the velocity field. This discontinuity in the covariance dynamics arises from hyperbolicity: the diagonal of the kernel of the covariance operator is a characteristic surface for advective dynamics. Our numerical experiments demonstrate that standard numerical methods for evolving the state are not adequate for propagating the covariance, because they do not capture the discontinuity in the continuum covariance dynamics as correlations lengths tend to zero. Our analytical and numerical results demonstrate in the context of mass conservation that this leads to significant, spurious variance loss in regions of mass convergence and gain in regions of mass divergence. The results suggest that developing local covariance propagation methods designed specifically to capture covariance evolution near the diagonal may prove a useful alternative to current methods of covariance propagation.