One persistent problem with ensemble weather forecasts is that the spread of the forecasts is generally much smaller than the root mean square prediction error. Some forecast centers have implemented stochastic parameterizations in order to increase the forecast spread, but without any real guidance from data. This work demonstrates that data assimilation can be used to provide parameter distributions for use by stochastic parameterization schemes. It is shown that when the system of interest is stochastic the expected variability of a stochastic parameter is somewhat overestimated when a deterministic model is employed for parameter estimation. However, this overestimation is ameliorated through application of the dynamical Central Limit Theorem, and good estimates of both the first and second moments of the stochastic parameter can be obtained. It is also shown that the overestimated variability information can be utilized to construct a hybrid stochastic/deterministic integration scheme that is able to accurately approximate the evolution of the true stochastic system. That is, data assimilation can properly exploit observed data properties to eliminate the need to rewrite a numerical forecast model with computationally expensive stochastic integration schemes.

The techniques described in this work may be particularly valuable in accounting for the effects of unresolved time and space scales in numerical climate models, which generally must be run with lower spatial and temporal resolution than numerical weather prediction models.