Stochastic process models for spatiotemporal data underlying random fields find substantial utility in a range of scientific disciplines. Subsequent to predictive inference on the values of the random field (or spatial surface indexed continuously over time) at arbitrary space-time coordinates, scientific interest often turns to gleaning information regarding zones of rapid spatial-temporal change. We develop Bayesian modeling and inference for directional rates of change along a given surface. These surfaces, which demarcate regions of rapid change, are referred to as ``wombling'' surface boundaries. Existing methods for studying such changes have often been associated with curves and are not easily extendable to surfaces resulting from curves evolving over time. Our current contribution devises a fully model-based inferential framework for analyzing differential behavior in spatiotemporal responses by formalizing the notion of a ``wombling'' surface boundary using conventional multi-linear vector analytic frameworks and geometry followed by posterior predictive computations using triangulated surface approximations. We illustrate our methodology with comprehensive simulation experiments followed by multiple applications in environmental and climate science; pollutant analysis in environmental health; and brain imaging.

翻译：随机过程模型作为随机场基础的空间-时间数据模型在众多科学领域中具有广泛应用价值。在完成对任意时空坐标点上随机场（或随时间连续索引的空间曲面）的预测推断后，科学研究的关注点往往转向获取关于时空快速变化区域的信息。本文针对给定曲面沿特定方向的变率建立了贝叶斯建模与推断框架。这些界定快速变化区域的曲面被称为"沃姆林"曲面边界。现有研究此类变化的方法多局限于曲线分析，难以推广至随时间演化的曲线所生成的曲面。本研究通过运用经典多重线性向量分析框架与几何学形式化定义"沃姆林"曲面边界概念，并采用三角曲面近似进行后验预测计算，构建了完整的模型驱动推断框架以分析时空响应中的微分行为特征。我们通过综合模拟实验及多领域应用案例验证方法论：环境与气候科学中的污染物健康效应分析；脑成像研究等场景均展示了该框架的有效性。