# Performance analysis of local ensemble Kalman filter

## TL;DR

This paper provides a rigorous analysis of the local ensemble Kalman filter (LEnKF) for linear systems, establishing conditions for error control and revealing an intrinsic inconsistency due to localization, supported by numerical validation.

## Contribution

It offers the first rigorous theoretical analysis of LEnKF error behavior for linear systems with localized structures and sparse observations.

## Key findings

- Filter error dominated by ensemble covariance under certain conditions
- Stable localized structure is necessary for controlling localization inconsistency
- Numerical validation confirms theoretical predictions

## Abstract

Ensemble Kalman filter (EnKF) is an important data assimilation method for high dimensional geophysical systems. Efficient implementation of EnKF in practice often involves the localization technique, which updates each component using only information within a local radius. This paper rigorously analyzes the local EnKF (LEnKF) for linear systems, and shows that the filter error can be dominated by the ensemble covariance, as long as 1) the sample size exceeds the logarithmic of state dimension and a constant that depends only on the local radius; 2) the forecast covariance matrix admits a stable localized structure. In particular, this indicates that with small system and observation noises, the filter error will be accurate in long time even if the initialization is not. The analysis also reveals an intrinsic inconsistency caused by the localization technique, and a stable localized structure is necessary to control this inconsistency. While this structure is usually taken for granted for the operation of LEnKF, it can also be rigorously proved for linear systems with sparse local observations and weak local interactions. These theoretical results are also validated by numerical implementation of LEnKF on a simple stochastic turbulence in two dynamical regimes.