We study how sampling geometry contributes to uncertainty in modeling spatial geophysical observations as sampled random fields characterized by stationary, isotropic, parametric covariance functions. We incorporate the signature of discrete spatial sampling patterns into an asymptotically unbiased spectral maximum-likelihood estimation method along with analytical uncertainty calculation. We illustrate the broad applicability of our modeling through synthetic and real data examples with sampling patterns that include irregularly bounded contiguous region(s) of interest, structured sweeps of instrumental measurements, and missing observations dispersed across the domain of a field, from which contiguous patches are generally favorable. We find through asymptotic studies that allocating samples following a growing-domain strategy rather than a densifying, infill scheme best reduces estimator bias and (co)variance, whether the field has been sampled regularly or not. As our modeling assumptions, too, shape how (well) an observed random field can be characterized, we study the effect of covariance parameters assumed a priori. We demonstrate the desirable behavior of the general Matern class and show how to interrogate goodness-of-fit criteria to detect departures from the null hypothesis of Gaussianity, stationarity, and isotropy.

翻译：我们研究采样几何如何影响将空间地球物理观测建模为以平稳、各向同性、参数化协方差函数为特征的采样随机场时的不确定性。我们将离散空间采样模式的特征融入渐近无偏谱最大似然估计方法，并辅以解析不确定性计算。通过涵盖不规则有界连续感兴趣区域、结构化仪器测量扫描以及分散在域中（通常以连续斑块为佳）的缺失观测等采样模式的合成与真实数据示例，我们展示了所提模型的广泛适用性。渐近研究表明，无论场是否被规则采样，采用增长域策略而非致密填充方案分配样本，最能减少估计器偏差和（协）方差。由于我们的建模假设也塑造了观测随机场被表征的方式（及效果），我们研究了先验假设的协方差参数的影响。我们展示了广义马特恩类的理想特性，并演示了如何通过检验拟合优度准则来检测对高斯性、平稳性和各向同性零假设的偏离。