Bivariate DeepKriging for Large-scale Spatial Interpolation of Wind Fields

High spatial resolution wind data play a crucial role in various fields such as climate, oceanography, and meteorology. However, spatial interpolation or downscaling of bivariate wind fields, characterized by velocity in two dimensions, poses a challenge due to their non-Gaussian nature, high spatial variability, and heterogeneity. While cokriging is commonly employed in spatial statistics for predicting bivariate spatial fields, it is suboptimal for non-Gaussian processes and computationally prohibitive for large datasets. In this paper, we introduce bivariate DeepKriging, a novel method utilizing a spatially dependent deep neural network (DNN) with an embedding layer constructed by spatial radial basis functions for predicting bivariate spatial data. Additionally, we devise a distribution-free uncertainty quantification technique based on bootstrap and ensemble DNN. We establish the theoretical basis for bivariate DeepKriging by linking it with the Linear Model of Coregionalization (LMC). Our proposed approach surpasses traditional cokriging predictors, including those utilizing commonly used covariance functions like the linear model of co-regionalization and parsimonious bivariate Matérn covariance. We demonstrate the computational efficiency and scalability of the proposed DNN model, achieving computation speeds approximately 20 times faster than conventional techniques. Furthermore, we apply the bivariate DeepKriging method to wind data across the Middle East region at 506,771 locations, showcasing superior prediction performance over cokriging predictors while significantly reducing computation time.高空间分辨率风场数据在气候学、海洋学与气象学等诸多领域均发挥着至关重要的作用。然而，以二维风速为表征的双变量风场的空间插值或降尺度任务却面临诸多挑战，这源于其非高斯特性、高空间变异性与异质性。尽管协同克里金（cokriging）在空间统计学中常用于双变量空间场的预测，但对于非高斯过程而言其表现并非最优，且在处理大规模数据集时计算成本高昂，难以落地。本文提出了双变量DeepKriging方法：这是一种全新的预测模型，采用由空间径向基函数构建嵌入层的空间依赖型深度神经网络（DNN），用于双变量空间数据的预测。此外，本文还设计了一种基于Bootstrap（自助法）与集成深度神经网络的无分布依赖不确定性量化技术。本文通过将双变量DeepKriging与共区域化线性模型（LMC）相联结，为其建立了理论基础。所提出的方法性能优于传统协同克里金预测器，包括那些采用常用协方差函数（如共区域化线性模型与简约双变量Matérn协方差）的预测器。本文验证了所提出的深度神经网络模型的计算效率与可扩展性，其计算速度相较传统方法提升约20倍。此外，本文将双变量DeepKriging方法应用于中东地区506771个测点的风场数据，结果显示其预测性能优于协同克里金预测器，同时大幅缩短了计算耗时。