The efficient simulation of Gaussian random fields with flexible correlation structures is fundamental in spatial statistics, machine learning, and uncertainty quantification. In this work, we revisit the \emph{spectral turning-bands} (STB) method as a versatile and scalable framework for simulating isotropic Gaussian random fields with a broad range of covariance models. Beyond the classical Matérn family, we show that the STB approach can be extended to two recent and flexible correlation classes that generalize the Matérn model: the Bummer-Tricomi model, which allows for polynomially decaying correlations and long-range dependence, and the Gauss-Hypergeometric model, which admits compactly supported correlations, including the Generalized Wendland family as a special case. We derive exact stochastic representations for both families: a Beta-prime mixture formulation for the Kummer-Tricomi model and complementary Beta- and Gasper-mixture representations for the Gauss-Hypergeometric model. These formulations enable exact, numerically stable, and computationally efficient simulation with linear complexity in the number of spectral components. Numerical experiments confirm the accuracy and computational stability of the proposed algorithms across a wide range of parameter configurations, demonstrating their practical viability for large-scale spatial modeling. As an application, we use the proposed STB simulators to perform parametric bootstrap for standard error estimation and model selection under weighted pairwise composite likelihood in the analysis of a large climate dataset.

翻译：在空间统计、机器学习及不确定性量化领域，具有灵活相关结构的高斯随机场的高效模拟是基础性课题。本文重新审视了谱转向带方法，将其作为可模拟宽泛协方差模型各向同性高斯随机场的通用可扩展框架。除经典Matérn族外，我们证明STB方法可扩展至两类近期提出的、推广Matérn模型的灵活相关类：允许多项式衰减相关和长程依赖的Bummer-Tricomi模型，以及可容纳紧支撑相关性（包含作为特例的广义Wendland族）的Gauss-Hypergeometric模型。我们为这两类模型推导出精确随机表示：针对Kummer-Tricomi模型提出Beta-prime混合公式，针对Gauss-Hypergeometric模型提出互补Beta-与Gasper-混合表示。这些公式使得在谱分量数量上具有线性复杂度的精确、数值稳定且计算高效的模拟成为可能。数值实验验证了所提算法在广泛参数配置下的精度与计算稳定性，证明了其在大规模空间建模中的实用可行性。作为应用，我们利用所提出的STB模拟器对大型气候数据集进行加权成对复合似然分析，通过参数自助法估计标准误差并执行模型选择。