Examining Entropic Unbalanced Optimal Transport and Sinkhorn Divergences for Spatial Forecast Verification

An optimal transport (OT) problem seeks to find the cheapest mapping between two distributions with equal total density, given the cost of transporting density from one place to another. Unbalanced OT allows for different total density in each distribution. This is the typical setting for precipitation forecast and observation data, when considering the densities as accumulated rainfall, or intensity. True OT problems are computationally expensive, however through entropic regularisation it is possible to obtain an approximation maintaining many of the underlying attributes of the true problem. In this work, entropic unbalanced OT and its associated Sinkhorn divergence are examined as a spatial forecast verification method for precipitation data. The latter being a novel introduction to the forecast verification literature. It offers many attractive features, such as morphing one field into another, defining a distance between fields and providing feature based optimal assignment. This method joins the growing research by the Spatial Forecast Verification Methods Inter-Comparison Project (ICP) which aims to unite spatial verification approaches. After testing this methodology's behaviour on numerous ICP test sets, it is found that the Sinkhorn divergence is robust against the common double penalty problem (a form of phase error), on average aligns with expert assessments of model performance, and allows for a variety of novel pictorial illustrations of error. It provides informative summary scores, and has few limitations to its application. Combined, these findings place unbalanced entropy regularised optimal transport and the Sinkhorn divergence as an informative method which follows geometric intuition.