In the task of predicting spatio-temporal fields in environmental science using statistical methods, introducing statistical models inspired by the physics of the underlying phenomena that are numerically efficient is of growing interest. Large space-time datasets call for new numerical methods to efficiently process them. The Stochastic Partial Differential Equation (SPDE) approach has proven to be effective for the estimation and the prediction in a spatial context. We present here the advection-diffusion SPDE with first order derivative in time which defines a large class of nonseparable spatio-temporal models. A Gaussian Markov random field approximation of the solution to the SPDE is built by discretizing the temporal derivative with a finite difference method (implicit Euler) and by solving the spatial SPDE with a finite element method (continuous Galerkin) at each time step. The ''Streamline Diffusion'' stabilization technique is introduced when the advection term dominates the diffusion. Computationally efficient methods are proposed to estimate the parameters of the SPDE and to predict the spatio-temporal field by kriging, as well as to perform conditional simulations. The approach is applied to a solar radiation dataset. Its advantages and limitations are discussed.

翻译：在环境科学中利用统计方法预测时空场时，采用受底层物理现象启发且数值高效的统计模型日益受到关注。大尺度时空数据集需要新型数值方法进行高效处理。随机偏微分方程方法已在空间背景下被证明对估计和预测有效。本文提出带有时间一阶导数的对流扩散SPDE模型，该模型定义了一类广泛的不可分离时空模型。通过有限差分法（隐式欧拉格式）离散时间导数，并在每个时间步采用有限元法（连续伽辽金法）求解空间SPDE，构建了SPDE解的高斯马尔可夫随机场近似。当对流项占主导地位时，引入"流线扩散"稳定化技术。提出了计算高效的方法用于估计SPDE参数、通过克里金法预测时空场以及执行条件模拟。该方法被应用于太阳辐射数据集，并讨论了其优势与局限性。