Ensemble Kalman filter  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. 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, 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, introduced in 1960, 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. Let us review first the Kalman filter. Let formula_1 denote the formula_2-dimensional state vector of a model, and assume that it has Gaussian probability distribution with mean formula_3 and covariance formula_4, i.e., its pdf is  Here and below, formula_6 means proportional; a pdf is always scaled so that its integral over the whole space is one.