A Monte Carlo Background Covariance Localization Method for an Ensemble–Variational Assimilation System

Abstract

Spurious long-distance correlations in estimates of the background error covariance can deteriorate the performance of ensemble-based data assimilation methods. In this study, a localization method, called Monte Carlo (MC) localization, is presented to remove these correlations. It is particularly useful for use in high-dimensional ensemble–variational data assimilation systems. In this method, raw ensemble members are truncated by multiplying them with functions having compact support. This creates a larger ensemble, in which points spaced farther apart than the size of the compact support have zero correlation. The localized background error covariance is then estimated as the sample covariance of this larger ensemble. It is hypothesized that this localized background error covariance can be approximated by the MC approximation method using a limited set of the truncated ensemble members. This hypothesis is tested here on a grid with 1001 grid points and assuming a Gaussian true background error covariance. It is found that the mean relative error has an upper bound that scales with the inverse square root of the number of truncated ensemble members. In the case studied the size of the support for which the localized background covariance best approximates the true background covariance increases with increasing number of raw ensemble members and is close to 4 times the standard deviation of the Gaussian when 20 raw ensemble members are used. In the Fourier space the localization manifests itself as a convolution resulting in smoothing of the power spectral density of the ensemble members.