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 an 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 mean and covariance 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, as they all share the EnKF.

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Kalman filter
For statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a series of measurements observed over time, including statistical noise and other inaccuracies, and produces estimates of unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, who was one of the primary developers of its theory.
Monte Carlo method
Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. Monte Carlo methods are mainly used in three problem classes: optimization, numerical integration, and generating draws from a probability distribution.

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