Spatial process models popular in geostatistics often represent the observed data as the sum of a smooth underlying process and white noise. The variation in the white noise is attributed to measurement error, or micro-scale variability, and is called the "nugget". We formally establish results on the identifiability and consistency of the nugget in spatial models based upon the Gaussian process within the framework of in-fill asymptotics, i.e. the sample size increases within a sampling domain that is bounded. Our work extends results in fixed domain asymptotics for spatial models without the nugget. More specifically, we establish the identifiability of parameters in the Mat\'ern covariance function and the consistency of their maximum likelihood estimators in the presence of discontinuities due to the nugget. We also present simulation studies to demonstrate the role of the identifiable quantities in spatial interpolation.

翻译：地理统计学中普遍使用的空间过程模型通常将观测数据表示为平滑潜在过程与白噪声之和。白噪声的变异性归因于测量误差或微观尺度变异性，称为"金块项"。我们在渐进填充抽样框架下（即采样域有界时样本量增加）正式建立了基于高斯过程的空间模型中金块项的可识别性与相合性结果。本研究将无金块项空间模型的固定域渐近理论进行了拓展。具体而言，我们证明了包含金块项导致间断性的马特恩协方差函数参数的可识别性，及其最大似然估计的相合性。同时通过数值模拟实验，阐释了可识别量在空间插值中的作用。