Classical geostatistics encodes spatial dependence by prescribing variograms or covariance kernels on Euclidean domains, whereas the SPDE--GMRF paradigm specifies Gaussian fields through an elliptic precision operator whose inverse is the corresponding Green operator. We develop an operator-based formulation of Gaussian spatial random fields on bounded domains and manifolds with internal interfaces, treating boundary and transmission conditions as explicit components of the statistical model. Starting from coercive quadratic energy functionals, variational theory yields a precise precision--covariance correspondence and shows that variograms are derived quadratic functionals of the Green operator, hence depend on boundary conditions and domain geometry. Conditioning and kriging follow from standard Gaussian update identities in both covariance and precision form, with hard constraints represented equivalently by exact interpolation constraints or by distributional source terms. Interfaces are modelled via surface penalty terms; taking variations produces flux-jump transmission conditions and induces controlled attenuation of cross-interface covariance. Finally, boundary-driven prediction and domain reduction are formulated through Dirichlet-to-Neumann operators and Schur complements, providing an operator language for upscaling, change of support, and subdomain-to-boundary mappings. Throughout, we use tools standard in spatial statistics and elliptic PDE theory to keep boundary and interface effects explicit in covariance modeling and prediction.

翻译：经典地统计学通过在欧几里得域上设定变异函数或协方差核来编码空间依赖性，而SPDE-GMRF范式则通过椭圆精度算子（其逆为相应的格林算子）来指定高斯场。我们针对具有内部界面的有界域和流形，发展了一种基于算子的高斯空间随机场表述，将边界条件与传输条件作为统计模型的显式组成部分进行处理。从强制二次能量泛函出发，变分理论给出了精确的精度-协方差对应关系，并表明变异函数是格林算子的导出二次泛函，因而依赖于边界条件与域几何。条件化与克里金插值遵循协方差形式和精度形式下的标准高斯更新恒等式，其中硬约束可通过精确插值约束或分布源项等价表示。界面通过表面惩罚项建模；对其取变分可导出通量跳跃传输条件，并引起跨界面协方差的受控衰减。最后，边界驱动的预测与域约简通过狄利克雷-诺伊曼算子与舒尔补进行形式化，为升尺度、支撑集变换及子域-边界映射提供了算子语言。本文始终运用空间统计学与椭圆偏微分方程理论中的标准工具，在协方差建模与预测中保持边界与界面效应的显式表达。