An evaluation of methods for normalizing diffusion‐based covariance operators in variational data assimilation

Abstract

AbstractDeveloping effective ways to model and cycle the background‐error covariance matrix is an active area of research in data assimilation. An important aspect of this problem when using a filter to model the background‐error correlations is the computation of normalization factors to ensure that the diagonal elements of the modelled correlation matrix are all equal to one. Updating the parameters of a flow‐dependent correlation model on each assimilation cycle requires updating the normalization factors, which is costly using traditional methods such as randomization. In this article, we discuss the normalization problem within the context of a diffusion filter‐based covariance model used for background‐error modelling in a variational data assimilation system for the global ocean. We evaluate various methods for estimating normalization factors when the diffusion tensor of the correlation model is derived from an ensemble of ocean states. Our results show that estimates produced using inexpensive methods derived from analytical considerations of the diffusion equation can have significant errors, especially near boundaries. Estimates obtained using randomization with a small sample size (∼100) are more accurate in a globally averaged sense but are noisy and can have unacceptably large errors locally. Next, we focus on the specific problem of accounting for flow‐dependent correlation parameters in the vertical component of the diffusion operator only, which is especially important near the surface for the assimilation of sea surface temperature observations. Remarkably accurate estimates are obtained by approximating the normalization matrix as a separable product of two normalization matrices: one computed using randomization with the horizontal diffusion operator only and the other computed using randomization with the vertical diffusion operator only. If the parameters of the horizontal component of the diffusion operator are static, then only the normalization factors of the flow‐dependent vertical component need to be recomputed on each cycle. This result is of significant practical interest since the vertical diffusion operator employs an inexpensive direct solver and thus can be applied on each cycle with a large random sample to obtain a good approximation of the normalization matrix.