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：在利用统计方法预测环境科学中时空场的任务中，引入受基础现象物理学启发且数值高效的统计模型正日益受到关注。大型时空数据集需要新的数值方法以进行高效处理。随机偏微分方程（SPDE）方法已被证明在空间背景下的估计与预测中是有效的。本文提出了具有时间一阶导数的平流-扩散SPDE，它定义了一大类不可分离的时空模型。通过对时间导数采用有限差分法（隐式欧拉法）进行离散化，并在每个时间步长上使用有限元法（连续伽辽金法）求解空间SPDE，构建了该SPDE解的高斯马尔可夫随机场近似。当平流项主导扩散时，引入了"流线扩散"稳定化技术。本文提出了计算高效的方法来估计SPDE参数、通过克里金法预测时空场以及执行条件模拟。该方法应用于一个太阳辐射数据集，并讨论了其优势与局限性。