Simulation of a random field with given distribution of one-dimensional integral

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.