Variational Deconvolution of Conically Scanned Passive Microwave Observations With Error Quantification

The deconvolution of potentially cloud-affected passive microwave brightness temperatures is an important step for utilization in direct data assimilation in cloud-resolving numerical weather prediction (NWP) models for the purpose of improving model initial conditions. Geophysical retrieval algorithms, such as precipitation rate retrievals, also benefit from consistent resolution across channels. In this paper, we explore how to derive the posterior error estimates that are required for ingestion into data assimilation models or end-to-end error-quantified retrieval algorithms. To this end, we present a minimum variance, best linear-unbiased estimator approach that seeks an optimal estimate of the apparent (i.e., without the effects of antenna pattern convolution) brightness temperatures by iteratively minimizing a cost function measuring the lack of fit between observations and departures from a first guess. Both the observation and first-guess departure terms are weighed by a corresponding covariance term that estimates their relative uncertainty. The first-guess uncertainty, a Bayesian prior “belief” in the spread of the first-guess error, is estimated using geophysical fields from an NWP model in a radiative transfer model plus an antenna pattern forward operator, then iteratively improved using the posterior deconvolved brightness temperatures of actual special sensor microwave imager/sounder observations. The error for the posterior distribution, subject to the initial belief, is derived. The error-quantified results are shown to increase the spatial resolution of microwave observations.