Transport-oriented tools for weather forecast verification

该项目的目标是开发一种实用的方法，用于考虑到大气场的传输的预测验证。从本质上讲，我们希望在预测和（重建的）观测场（如累积降雨量）之间建立一个误差度量，以更好地奖励正确预测但发生在错误地点的事件（例如，由于大尺度风的误差）。我们想设计的指标，考虑运输字段与流图之前，采取的差异：该度量则是差的大小和“能量”的量的加权和。像这样的空间方法大约在25年前开始发展，以应对在验证更高分辨率的预报时遇到的困难，这些预报往往没有显示出比它们更高的技能。由于“双重惩罚”，分辨率相对较低。最近的空间方法的相互比较提供了一套全面的测试案例，用于评估新的空间方法的属性，提供了方便和公平地比较方法的手段。该项目将利用这些相互比较项目提供的理想化和真实的案例，对这些属性进行分类，并将其与其他已经研究过的方法进行比较。最佳运输有着悠久的历史，从Monge开始，他考虑了如何最佳地将质量的空间分布从一种配置重新安排到另一种配置的问题。最优输送最初是通过与霍斯金斯、卡伦、舒茨等人揭示的半地转方程的联系而引起气象学家的注意的。该主题在20世纪80年代由Brenier从数学的角度恢复，并且该领域从21世纪初由Benamou及其同事成为数值分析的主题，专注于图像分析和机器学习的应用。我们现在提出这个项目的原因是Benamou更广泛的社区最近对Sinkhorn分歧的工作的出现，这是优化问题的规范化。Sinkhorn分歧促进非常快速的可扩展的解决方案，并可以很容易地适应“不平衡”的运输问题，这是必不可少的领域，不局部保守（如表面量，如表面温度，例如）。这种新的数学和计算技术已经将传输方法从学术追求转变为准备应用于挑战大规模数据集的工具。在这个项目中，我们将设计一种方法，预测指标使用Sinkhorn分歧，移植这项技术从成像科学。

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 theflow 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 additionalskill 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-comparisonprojects, 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 thefield 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 froman 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.