  Data assimilation plays a key role in large-scale atmospheric weather
forecasting, where the state of the physical system is estimated from model
outputs and observations, and is then used as initial condition to produce
accurate future forecasts. The Ensemble Kalman Filter (EnKF) provides a
practical implementation of the statistical solution of the data assimilation
problem and has gained wide popularity as. This success can be attributed to
its simple formulation and ease of implementation. EnKF is a Monte-Carlo
algorithm that solves the data assimilation problem by sampling the probability
distributions involved in Bayes theorem. Because of this, all flavors of EnKF
are fundamentally prone to sampling errors when the ensemble size is small. In
typical weather forecasting applications, the model state space has dimension
$10^{9}-10^{12}$, while the ensemble size typically ranges between $30-100$
members. Sampling errors manifest themselves as long-range spurious
correlations and have been shown to cause filter divergence. To alleviate this
effect covariance localization dampens spurious correlations between state
variables located at a large distance in the physical space, via an empirical
distance-dependent function. The quality of the resulting analysis and forecast
is greatly influenced by the choice of the localization function parameters,
e.g., the radius of influence. The localization radius is generally tuned
empirically to yield desirable results.This work, proposes two adaptive
algorithms for covariance localization in the EnKF framework, both based on a
machine learning approach. The first algorithm adapts the localization radius
in time, while the second algorithm tunes the localization radius in both time
and space. Numerical experiments carried out with the Lorenz-96 model, and a
quasi-geostrophic model, reveal the potential of the proposed machine learning
approaches.
