  For modelling geophysical systems, large-scale processes are described
through a set of coarse-grained dynamical equations while small-scale processes
are represented via parameterizations. This work proposes a method for
identifying the best possible stochastic parameterization from noisy data.
State-the-art sequential estimation methods such as Kalman and particle filters
do not achieve this goal succesfully because both suffer from the collapse of
the parameter posterior distribution. To overcome this intrinsic limitation, we
propose two statistical learning methods. They are based on the combination of
two methodologies: the maximization of the likelihood via
Expectation-Maximization (EM) and Newton-Raphson (NR) algorithms which are
mainly applied in the statistic and machine learning communities, and the
ensemble Kalman filter (EnKF). The methods are derived using a Bayesian
approach for a hidden Markov model. They are applied to infer deterministic and
stochastic physical parameters from noisy observations in coarse-grained
dynamical models. Numerical experiments are conducted using the Lorenz-96
dynamical system with one and two scales as a proof-of-concept. The imperfect
coarse-grained model is modelled through a one-scale Lorenz-96 system in which
a stochastic parameterization is incorpored to represent the small-scale
dynamics. The algorithms are able to identify an optimal stochastic
parameterization with a good accuracy under moderate observational noise. The
proposed EnKF-EM and EnKF-NR are promising statistical learning methods for
developing stochastic parameterizations in high-dimensional geophysical models.
