Further Extensions
The EnKF version described here involves randomization of data. For filters without randomization of data, see.
Since the ensemble covariance is rank deficient (there are many more state variables, typically millions, than the ensemble members, typically less than a hundred), it has large terms for pairs of points that are spatially distant. Since in reality the values of physical fields at distant locations are not that much correlated, the covariance matrix is tapered off artificially based on the distance, which gives rise to localized EnKF algorithms. These methods modify the covariance matrix used in the computations and, consequently, the posterior ensemble is no longer made only of linear combinations of the prior ensemble.
For nonlinear problems, EnKF can create posterior ensemble with non-physical states. This can be alleviated by regularization, such as penalization of states with large spatial gradients.
For problems with coherent features, such as hurricanes, thunderstorms, firelines, squall lines, and rain fronts, there is a need to adjust the numerical model state by deforming the state in space (its grid) as well as by correcting the state amplitudes additively. In Data Assimilation by Field Alignment, Ravela et al. introduce the joint position-amplitude adjustment model using ensembles, and systematically derive a sequential approximation which can be applied to both EnKF and other formulations. Their method does not make the assumption that amplitudes and position errors are independent or jointly Gaussian, as others do. The morphing EnKF employs intermediate states, obtained by techniques borrowed from image registration and morphing, instead of linear combinations of states.
EnKFs rely on the Gaussian assumption, though they are of course used in practice for nonlinear problems, where the Gaussian assumption may not be satisfied. Related filters attempting to relax the Gaussian assumption in EnKF while preserving its advantages include filters that fit the state pdf with multiple Gaussian kernels, filters that approximate the state pdf by Gaussian mixtures, a variant of the particle filter with computation of particle weights by density estimation, and a variant of the particle filter with thick tailed data pdf to alleviate particle filter degeneracy.
Read more about this topic: Ensemble Kalman Filter
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