The ensemble Kalman filter (EnKF) is a recursive filter suitable for problems with a large number of variables, such as discretizations of partial differential equations in geophysical models. The EnKF originated as a version of the Kalman filter for large problems (essentially, the covariance matrix is replaced by the sample covariance), and it is now an important data assimilation component of ensemble forecasting. EnKF is related to the particle filter (in this context, a particle is the same thing as ensemble member) but the EnKF makes the assumption that all probability distributions involved are Gaussian; when it is applicable, it is much more efficient than the particle filter.
Contents
Introduction
The Ensemble Kalman Filter (EnKF) is a Monte Carlo implementation of the Bayesian update problem: given a probability density function (pdf) of the state of the modeled system (the prior, called often the forecast in geosciences) and the data likelihood, the Bayes theorem is used to obtain the pdf after the data likelihood has been taken into account (the posterior, often called the analysis). This is called a Bayesian update. The Bayesian update is combined with advancing the model in time, incorporating new data from time to time. The original Kalman Filter assumes that all pdfs are Gaussian (the Gaussian assumption) and provides algebraic formulas for the change of the mean and the covariance matrix by the Bayesian update, as well as a formula for advancing the covariance matrix in time provided the system is linear. However, maintaining the covariance matrix is not feasible computationally for high-dimensional systems. For this reason, EnKFs were developed. EnKFs represent the distribution of the system state using a collection of state vectors, called an ensemble, and replace the covariance matrix by the sample covariance computed from the ensemble. The ensemble is operated with as if it were a random sample, but the ensemble members are really not independent – the EnKF ties them together. One advantage of EnKFs is that advancing the pdf in time is achieved by simply advancing each member of the ensemble. For a survey of EnKF and related data assimilation techniques, see G. Evensen.
Kalman filter
Let us review first the Kalman filter. Let
Here and below,
The pdf of the state and the data likelihood are combined to give the new probability density of the system state
The data
with the posterior mean
where
is the so-called Kalman gain matrix.
Ensemble Kalman Filter
The EnKF is a Monte Carlo approximation of the Kalman filter, which avoids evolving the covariance matrix of the pdf of the state vector
Replicate the data
so that each column
form a sample from the posterior probability distribution. [To see this in the scalar case with
The EnKF is now obtained simply by replacing the state covariance
Basic formulation
Here we follow. Suppose the ensemble matrix
where
and
The posterior ensemble
where the perturbed data matrix
Note that since
Since these formulas are matrix operations with dominant Level 3 operations, they are suitable for efficient implementation using software packages such as LAPACK (on serial and shared memory computers) and ScaLAPACK (on distributed memory computers). Instead of computing the inverse of a matrix and multiplying by it, it is much better (several times cheaper and also more accurate) to compute the Cholesky decomposition of the matrix and treat the multiplication by the inverse as solution of a linear system with many simultaneous right-hand sides.
Observation matrix-free implementation
Since we have replaced the covariance matrix with ensemble covariance, this leads to a simpler formula where ensemble observations are directly used without explicitly specifying the matrix
The function
where
and
with
Consequently, the ensemble update can be computed by evaluating the observation function
Implementation for a large number of data points
For a large number
with
gives
which requires only the solution of systems with the matrix
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, although they in practice are used 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.