Obtaining precise erosion measurements requires costly fieldwork, making it infeasible to directly survey large domains such as a province or river basin. To extend fieldwork results across such extensive domains, we propose a novel spatial prediction method that treats local erosion distributions as objects in the Wasserstein space. These distributions are mapped into square-integrable trajectories and represented via basis expansion, forming a multivariate random field that captures spatial dependence. By applying local regression and Kriging in this representation, our approach flexibly models and predicts erosion distributions at arbitrary locations. This framework improves prediction for functionals of the distribution, such as the mean and exceedance probabilities. Simulation studies demonstrate that the proposed method outperforms a misspecified parametric alternative and existing Fréchet regression approaches. We illustrate the approach with a detailed erosion analysis in Shaanxi province, China, where local measurements from surveyed watersheds are extended to predict erosion distributions across the entire province using covariates such as land use and elevation.

翻译：精准的侵蚀测量需要昂贵的实地工作，因此直接勘测如省份或流域等广阔区域并不可行。为了将实地工作结果推广到如此广阔的领域，我们提出了一种新颖的空间预测方法，将局部侵蚀分布视为瓦瑟斯坦空间中的对象。这些分布被映射为平方可积轨迹，并通过基展开进行表示，从而形成一个捕捉空间依赖性的多变量随机场。通过在表示中应用局部回归和克里金法，我们的方法能够灵活地建模并预测任意位置的侵蚀分布。该框架改进了对分布函数（例如均值和超越概率）的预测。模拟研究表明，所提出的方法优于错误设定的参数化替代方法和现有的弗雷歇回归方法。我们以中国陕西省的详细侵蚀分析为例进行了说明，在该分析中，通过土地利用和海拔等协变量，将来自调查流域的局部测量结果推广，以预测整个省份的侵蚀分布。