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

arXiv: https://arxiv.org/abs/2107.07475 · HTML: https://mlsanity.com/paper/2107.07475/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Jul 2021.

## Built on by (most cited first, 4 of 4)

- [Ensemble Kalman filter in latent space using a variational autoencoder pair](https://mlsanity.com/paper/2502.12987/) (2025)
- [Exact nonlinear state estimation](https://mlsanity.com/paper/2310.10976/) (2023)
- [Non-Sequential Ensemble Kalman Filtering using Distributed Arrays](https://mlsanity.com/paper/2311.12909/) (2023)
- [A competitive baseline for deep learning enhanced data assimilation using conditional Gaussian ensemble Kalman filtering](https://mlsanity.com/paper/2409.14300/) (2024)

## Abstract (arXiv)

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.