The fast Fourier transform-moving average (FFT-MA) is an efficient method for the generation of geostatistical simulations. The method relies on the calculation of a filter operator based on the covariance function of interest and the convolution of the filter with a white noise, to generate multiple realizations of spatially correlated variables. In this work, a revisited mathematical formulation of the FFT-MA method, with the exact expression of the filter, is presented. The proposed derivation of the filter is based on the Wiener–Khinchin theorem and the application of the Fourier transform in a discrete domain. In the specific case of white noise, the proposed formulation leads to the same expression of the traditional algorithm. However, the method can be applied to other types of noise. The proposed technique allows the calculation of a specific filter that imposes an exact covariance function on the noise. Therefore, the experimental covariance function is exactly equal to the theoretical one, which is not the case for many common simulation techniques due to the limited sample size. Applications of the FFT-MA method to synthetic and real data sets, including exact interpolation, hard data conditioning and correlated simulations from cross-correlated noises, are also presented.

中文翻译：

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FFT移动平均的重新制定和应用

快速傅里叶变换移动平均值（FFT-MA）是生成地统计模拟的有效方法。该方法依赖于基于感兴趣的协方差函数和滤波器与白噪声的卷积的滤波器算符的计算，以生成空间相关变量的多种实现。在这项工作中，提出了FFT-MA方法的重新数学公式，并带有滤波器的精确表达式。拟议的滤波器推导基于Wiener-Khinchin定理和傅立叶变换在离散域中的应用。在白噪声的特定情况下，提出的公式导致了传统算法的相同表达。但是，该方法可以应用于其他类型的噪声。所提出的技术允许计算特定滤波器，该滤波器对噪声施加精确的协方差函数。因此，实验协方差函数与理论协方差函数完全相等，由于样本数量有限，许多常见的仿真技术都没有这种情况。还介绍了FFT-MA方法在合成和真实数据集上的应用，包括精确插值，硬数据条件处理和互相关噪声的相关模拟。