  Model error covariances play a central role in the performance of data
assimilation methods applied to nonlinear state-space models. However, these
covariances are largely unknown in most of the applications. A misspecification
of the model error covariance has a strong impact on the computation of the
posterior probability density function, leading to unreliable estimations and
even to a total failure of the assimilation procedure. In this work, we propose
the combination of the Expectation-Maximization algorithm (EM) with an
efficient particle filter to estimate the model error covariance, using a batch
of observations. Based on the EM algorithm principles, the proposed method
encompasses two stages: the expectation stage, in which a particle filter is
used with the present estimate of the model error covariance as given to find
the probability density function that maximizes the likelihood, followed by a
maximization stage in which the expectation under the probability density
function found in the expectation step is maximized as a function of the
elements of the model error covariance. This novel algorithm here presented
combines the EM with a fixed point algorithm and does not require a particle
smoother to approximate the posterior densities. We demonstrate that the new
method accurately and efficiently solves the linear model problem. Furthermore,
for the chaotic nonlinear Lorenz-96 model the method is stable even for
observation error covariance 10 times larger than the estimated model error
covariance matrix, and also that it is successful in high-dimensional
situations where the dimension of the estimated matrix is 1600.
