Abstract The problem of constructing a numerically realizable model of a three-dimensional homogeneous random field in a layer 0 < z < H with given one-dimensional distribution and correlation function of the integral over coordinate z is solved. The gamma distribution with shape parameter ν and scale parameter θ is used in the work. An aggregate of n independent elementary horizontal layers of thickness h = H/n vertically shifted by a random value uniformly distributed in the interval (0, h) is considered as a basic model. For each elementary random field, the normalized correlation function of the corresponding integral over z coincides with the given one, the gamma distribution with parameters depending on the number of horizontal layers is used as a one-dimensional distribution. It is proved that for the constructed model the normalized correlation function of the integral over z coincides with the given normalized ‘horizontal’ correlation function, and the parameters of the one-dimensional distribution asymptotically converge to given values for n → + ∞, but the corresponding mathematical expectation and variance coincide exactly with given values. To extend the class of possible models, an additional randomization of the basic model is considered. In the conclusion the results of computations for a realistic version of the problem are presented.

中文翻译：

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具有给定一维积分分布的随机场模拟

摘要 解决了在给定坐标z 上积分的一维分布和相关函数的情况下，在0 < z < H 层中构建三维均匀随机场的数值可实现模型的问题。工作中使用了具有形状参数 ν 和尺度参数 θ 的伽马分布。将厚度为 h = H/n 的 n 个独立基本水平层的集合垂直移动均匀分布在区间 (0, h) 中的随机值，被视为基本模型。对于每个基本随机场，z 上相应积分的归一化相关函数与给定的相关函数一致，参数取决于水平层数的伽马分布用作一维分布。证明对于构建的模型，z 上积分的归一化相关函数与给定的归一化“水平”相关函数重合，一维分布的参数渐近收敛到给定的 n → + ∞ 值，但相应的数学期望和方差与给定值完全一致。为了扩展可能模型的类别，考虑了基本模型的额外随机化。在结论中，给出了该问题的现实版本的计算结果。但相应的数学期望和方差与给定值完全一致。为了扩展可能模型的类别，考虑了基本模型的额外随机化。在结论中，给出了该问题的现实版本的计算结果。但相应的数学期望和方差与给定值完全一致。为了扩展可能模型的类别，考虑了基本模型的额外随机化。在结论中，给出了该问题的现实版本的计算结果。