This work is focused on finding G-optimal designs theoretically for kriging models with two-dimensional inputs and separable exponential covariance structures. For design comparison, the notion of evenness of two-dimensional grid designs is developed. The mathematical relationship between the design and the supremum of the mean squared prediction error (SMSPE) function is studied and then optimal designs are explored for both prospective and retrospective design scenarios. In the case of prospective designs, the new design is developed before the experiment is conducted and the regularly spaced grid is shown to be the G-optimal design. Retrospective designs are constructed by adding or deleting points from an already existing design. Deterministic algorithms are developed to find the best possible retrospective designs (which minimizes the SMSPE). It is found that a more evenly spread design under the G-optimality criterion leads to the best possible retrospective design. For all the cases of finding the optimal prospective designs and the best possible retrospective designs, both frequentist and Bayesian frameworks have been considered. The proposed methodology for finding retrospective designs is illustrated with a spatiotemporal river water quality monitoring experiment.

中文翻译：

---

克里金模型的 G 最优网格设计

这项工作的重点是从理论上寻找具有二维输入和可分离指数协方差结构的克里金模型的 G 最优设计。为了进行设计比较，提出了二维网格设计的均匀度概念。研究了设计与均方预测误差 (SMSPE) 函数上限值之间的数学关系，然后针对前瞻性和回顾性设计方案探索最优设计。在前瞻性设计的情况下，新设计是在进行实验之前开发的，并且规则间隔的网格被证明是 G 最优设计。回顾性设计是通过在现有设计中添加或删除点来构建的。开发确定性算法是为了找到最佳的回顾性设计（最大限度地减少 SMSPE）。研究发现，G 最优准则下更均匀分布的设计会带来最佳的回顾性设计。对于寻找最佳前瞻性设计和最佳回顾性设计的所有情况，都考虑了频率论框架和贝叶斯框架。通过时空河流水质监测实验说明了所提出的寻找回顾性设计的方法。