Ensemble Square Root Filters*

TLDR

Ensemble data‑assimilation methods estimate states by combining observations with low‑rank representations of forecast and analysis error covariances, and deterministic updates are implementations of Kalman square‑root filters. The study compares three recently proposed ensemble data‑assimilation methods by exploiting the non‑uniqueness of deterministic transformations in square‑root Kalman filters. The authors transform the forecast ensemble into an analysis ensemble either stochastically, treating observations as random variables, or deterministically, enforcing the Kalman filter analysis error‑covariance equation.

Abstract

Ensemble data assimilation methods assimilate observations using state-space estimation methods and low-rank representations of forecast and analysis error covariances. A key element of such methods is the transformation of the forecast ensemble into an analysis ensemble with appropriate statistics. This transformation may be performed stochastically by treating observations as random variables, or deterministically by requiring that the updated analysis perturbations satisfy the Kalman filter analysis error covariance equation. Deterministic analysis ensemble updates are implementations of Kalman square root filters. The nonuniqueness of the deterministic transformation used in square root Kalman filters provides a framework to compare three recently proposed ensemble data assimilation methods.