Transport-oriented tools for weather forecast verification

The goal of this project is to develop a practical methodology for forecast verification that takes into account the transportof atmospheric fields. In essence, we would like to develop an error metric between forecast and (reconstructed) observed fields (such as accumulated rainfall) that better rewards the situation where an event was correctly predicted but occurred in the wrong place (because of errors in the large scale winds, for example). We would like to design metrics that consider transporting the field with a flow map before taking the difference: the metric is then a weighted sum of the size of the difference and the amount of "energy" required to generate the
flow map.

Spatial methods like this began to be developed around 25 years ago in response to the difficulties encountered in verifying higher resolution forecasts which tended to show no additional
skill over their coarser resolution counterparts, due to the "double penalty". A recent intercomparison of spatial methods provides a comprehensive set of test cases for evaluating the properties of new spatial methods, providing the means for comparing methods easily and equitably. The project will utilise the idealised and real cases provided by these inter-comparison
projects, to catalogue the properties and compare them to other methods that have been studied.

Optimal transport has a long history, starting with Monge, who considered the problem of how to optimally rearrange a spatial distribution of mass from one configuration to another. Optimal transport first came to the attention of meteorologists through the connections with the semigeostrophic equations exposed by Hoskins, Cullen, Schutts and others. The subject was revived from the mathematical point of view by Brenier in the 1980s, and the
field became a topic of numerical analysis from the early 2000s by Benamou and coworkers, with a focus on applications in image analysis and machine learning. The reason that we are proposing this project now is the emergence of recent work from Benamou's wider community on Sinkhorn divergences, which are are regularisations of the optimisation problem. Sinkhorn divergences facilitate very fast scalable solvers, and can be easily adapted to ``unbalanced'' transport problems, which are essential for fields that are not locally conserved (such as surface quantities like surface temperature, for example). This new mathematical and computational technology has transformed transport methods from
an academic pursuit into tools that are ready for application to challenging large scale datasets. In this project, we will design a methodology for forecast metrics using Sinkhorn divergences, porting this technology from imaging science.