Abstrakt
In meteorological and geophysical sciences where the system state is a vector with a big dimension, one widely popular group of methods for data assimilation is formed by filtersbased on Monte-Carlo approximation of the traditional Kalman filter. In these methods, the posterior distribution (of the state given the observed data) is approximated by a set of system states (usually called an ensemble).
Examples of these filters are ensemble Kalman filter (EnKF), ensemble Square-root filter and others. Due to the computational cost, the ensemble is always very small and provides a poor estimate of the covariance matrix - singular and with spurious correlations.
In practice this problem is usually solved by heuristic methods like tapering or shrinkage. We present an ensemble filter that provides a rigorous covariance regularization when the underlying random field is Gaussian Markov.
We use a linear model for the precision matrix (inverse of covariance) and estimate its parameters together with the posterior mean by the score-matching method. This procedure provides an explicit expression for parameter estimates and is significantly faster than e.g. numeric maximization of likelihood.
We show that this method leads to plugging-in the estimated covariance matrix into the classical formula for Kalman filter.