EnsembleKalmanFilter

Ensemble Kalman Filter (EnKF) is a Monte Carlo approximation of the Kalman filter designed for nonlinear state-space models. It represents the state distribution with a finite ensemble of state vectors, whose sample mean and covariance approximate the posterior mean and covariance. It was introduced by Geir Evensen in 1994 in the context of data assimilation for geophysical systems and has since become a standard tool in meteorology, oceanography, hydrology, and environmental monitoring.

Algorithmically, an ensemble of N state vectors is initialized to represent the prior. At each assimilation cycle, each member is propagated forward by the system dynamics to form a forecast ensemble. The forecast mean and covariance are computed from the ensemble. The Kalman gain is derived from the forecast covariance and the observation error covariance. The analysis update then adjusts each ensemble member toward the observations. In the stochastic EnKF, observation perturbations are added to reflect observation uncertainty; in deterministic variants, a transform is used to update members without perturbing observations. Common variants include the ensemble transform Kalman filter (ETKF), the ensemble adjustment Kalman filter (EAKF), and the local ensemble Kalman filter with localization (LETKF).

Strengths and limitations: EnKF scales better to high-dimensional problems than the classical Kalman filter and naturally handles model noise, but it relies on Gaussian error assumptions and can suffer from sampling errors due to a finite ensemble size. Techniques such as covariance inflation and localization are used to mitigate these issues.

Applications include weather and climate data assimilation, ocean state estimation, groundwater and hydrology, and other nonlinear dynamical systems where state estimation is required.