Transport-oriented tools for weather forecast verification

该项目的目标是发展一种考虑到大气场传输的实际预报验证方法。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）之间开发一个误差度量，以便更好地奖励正确预测事件但发生在错误地点的情况（例如，由于大规模风的误差）。我们想要设计度量，考虑在取差值之前用流图传输场：度量是差值大小和生成流图所需的“能量”的加权和。像这样的空间方法大约在25年前开始发展，以应对在验证高分辨率预测时遇到的困难，由于“双重惩罚”，高分辨率预测往往没有比粗分辨率预测显示出额外的技能。最近空间方法的相互比较为评估新空间方法的特性提供了一套全面的测试用例，为方便、公平地比较方法提供了手段。该项目将利用这些相互比较项目提供的理想化和真实案例，对这些属性进行分类，并将其与其他研究方法进行比较。最优输运有很长的历史，从Monge开始，他考虑了如何从一种配置到另一种配置的最佳重新安排质量的空间分布的问题。气象学家通过与霍斯金斯、卡伦、舒茨等人所揭示的半营养方程式的联系，首次注意到最优运输。20世纪80年代，Brenier从数学的角度复兴了这个主题，从21世纪初开始，Benamou和他的同事们把这个领域变成了数值分析的主题，重点是在图像分析和机器学习中的应用。我们现在提出这个项目的原因是Benamou更广泛的社区最近在Sinkhorn分歧方面的工作的出现，这是优化问题的规范化。Sinkhorn散度促进了非常快速的可扩展求解器，并且可以很容易地适应于“不平衡”输运问题，这对于局部不守恒的场（例如表面温度等表面量）是必不可少的。这种新的数学和计算技术已经将传输方法从一种学术追求转变为一种工具，可以应用于挑战大规模数据集。在这个项目中，我们将设计一种利用下沉角散度预测指标的方法，将这项技术从成像科学中移植过来。

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.