  Model uncertainty quantification is an essential component of effective data
assimilation. Model errors associated with sub-grid scale processes are often
represented through stochastic parameterizations of the unresolved process.
Many existing Stochastic Parameterization schemes are only applicable when
knowledge of the true sub-grid scale process or full observations of the coarse
scale process are available, which is typically not the case in real
applications. We present a methodology for estimating the statistics of
sub-grid scale processes for the more realistic case that only partial
observations of the coarse scale process are available. Model error
realizations are estimated over a training period by minimizing their
conditional sum of squared deviations given some informative covariates (e.g.
state of the system), constrained by available observations and assuming that
the observation errors are smaller than the model errors. From these
realizations a conditional probability distribution of additive model errors
given these covariates is obtained, allowing for complex non-Gaussian error
structures. Random draws from this density are then used in actual ensemble
data assimilation experiments. We demonstrate the efficacy of the approach
through numerical experiments with the multi-scale Lorenz 96 system using both
small and large time scale separations between slow (coarse scale) and fast
(fine scale) variables. The resulting error estimates and forecasts obtained
with this new method are superior to those from two existing methods.
