A comparison of nonlinear extensions to the ensemble Kalman filter: Gaussian Anamorphosis and Two-Step Ensemble Filters

TL;DR

This paper compares nonlinear extensions of the ensemble Kalman filter, specifically Gaussian anamorphosis and two-step filters, demonstrating their advantages in non-Gaussian settings through experiments with the Lorenz-96 model.

Contribution

It introduces the iRHF method, a new two-step filter, and analyzes its performance relative to existing methods like RHF, GA-EnKF, and EnKF in nonlinear, non-Gaussian scenarios.

Findings

RHF and iRHF outperform EnKF and GA-EnKF in nonlinear, non-Gaussian cases.

iRHF is more accurate than RHF at small ensemble sizes.

Experiments with Lorenz-96 model validate the effectiveness of the proposed methods.

Abstract

Ensemble Kalman filters are based on a Gaussian assumption, which can limit their performance in some non-Gaussian settings. This paper reviews two nonlinear, non-Gaussian extensions of the Ensemble Kalman Filter: Gaussian anamorphosis (GA) methods and two-step updates, of which the rank histogram filter (RHF) is a prototypical example. GA-EnKF methods apply univariate transforms to the state and observation variables to make their distribution more Gaussian before applying an EnKF. The two-step methods use a scalar Bayesian update for the first step, followed by linear regression for the second step. The connection of the two-step framework to the full Bayesian problem is made, which opens the door to more advanced two-step methods in the full Bayesian setting. A new method for the first part of the two-step framework is proposed, with a similar form to the RHF but a different motivation, called the `improved RHF' (iRHF). A suite of experiments with the Lorenz-`96 model demonstrate situations where the GA-EnKF methods are similar to EnKF, and where they outperform EnKF. The experiments also strongly support the accuracy of the RHF and iRHF filters for nonlinear and non-Gaussian observations; these methods uniformly beat the EnKF and GA-EnKF methods in the experiments reported here. The new iRHF method is only more accurate than RHF at small ensemble sizes in the experiments reported here.