Optimal Transport Tools for Spatial Forecast Verification

We formulate a novel spatial forecast verification methodology grounded in the geometric principles underlying optimal transport (OT). In its original form OT seeks to minimise the transport of mass between two distributions in space, providing a cost of transportation. Canonical examples come from logistics, such as finding the optimal route to distribute bread from bakeries to cafes. Since its initial formulation, OT theory has found many varying applications from signal processing and economics to meteorology and machine learning, notably through the famed Wasserstein distance. Its success is first due to Kantorovich’s dual formulation and more recently due to novel algorithms and GPU compute. Which combined allow regularised OT problems to be solved efficiently.  

In this work we consider a precipitation field as a measure in 2D space and compute the unbalanced OT distance between a 2D observation and forecast field. By leveraging this geometric formulation, we find a summary metric (the objective), which itself has constituent parts indicating performance. Additionally, the methodology allows us to form transport maps. These maps highlight regions in the field which require the most transport to align with the observation. Alongside providing a visual representation of the error, this provides physical insight into transport error as the map will traverse along geodesics of the underlying space.  This has direct implications for operational forecasters, giving clear, easy to understand illustrations of error, whilst simultaneously providing important information to researchers at forecasting services. This research is supported by UKRI NERC, Imperial’s Grantham Institute and the MET Office UK.