Accounting for Non-Gaussianity in Background Errors Associated with Cloud-Related Variables

All of the hydrometeors associated with cloud prediction cannot go negative, and as such this constraint must be adhered to in the data assimilation system.  Unfortunately the Gaussian fits all representation cannot prevent the increments from being too negative to make the analysis state becoming unphysical.

Theory developed at the Cooperative Institute for Research in the Atmosphere has been designed to prevent the unphysical increments from occurring due to the Bayesian problem begin recast with a lognormal distribution for the positive definite variables.  This lognormal distribution has been combined with a Gaussian distribution to form a mixed distribution that models the covariances between Gaussian and lognormal random variables.

Given the associated cost function in a variational formulation, we present an outline of the initial implementation of the median based mixed distribution approach in the Gridpoint Statistical Interpolation system for the Rapid Refresh associated with assimilating microwave radiances and hydrometeors.