We present a statistically and computationally efficient spectral-domain maximum-likelihood procedure to solve for the structure of Gaussian spatial random fields within the Matern covariance hyperclass. For univariate, stationary, and isotropic fields, the three controlling parameters are the process variance, smoothness, and range. The debiased Whittle likelihood maximization explicitly treats discretization and edge effects for finite sampled regions in parameter estimation and uncertainty quantification. As even the best parameter estimate may not be good enough, we provide a test for whether the model specification itself warrants rejection. Our results are practical and relevant for the study of a variety of geophysical fields, and for spatial interpolation, out-of-sample extension, kriging, machine learning, and feature detection of geological data. We present procedural details and high-level results on real-world examples.

翻译：我们提出了一种统计与计算高效的谱域最大似然方法，用于求解Matérn协方差超类内高斯空间随机场的结构。对于单变量、平稳且各向同性的随机场，其三个控制参数为过程方差、平滑度与范围。去偏Whittle似然最大化在参数估计与不确定性量化中，明确处理了有限采样区域内的离散化与边缘效应。鉴于即使最优参数估计亦可能不足够理想，我们提供了一种检验方法，用于判断模型设定本身是否应被拒绝。我们的研究结果对于多种地球物理场的研究，以及空间插值、样本外延拓、克里金法、机器学习与地质数据的特征检测具有实际意义。我们通过实际案例展示了具体实施细节与高层次结果。