Statistical simulation of seismic noise in a multidimensional area in determining frequency characteristics of geological media

The paper deals with the theory and methods of statistical simulation of random processes and fields based on their spectral decomposition and Kotelnikov-Shennon modified interpolation sums, as well as applying these methods for environmental geophysical monitoring. Statistical simulation of multivariate random fields (those homogeneous in time and homogeneous isotropic in n other variables) are considered to be essential for seismological research into frequency characteristics of geological media. A statistical model and a numerical algorithm of simulating random fields are built on the basis of Kotelnikov-Shennon modified interpolation decomposition to generate adequate realizations of seismic noise. The paper examines real-valued random fields ξ(t,x),tϵÎR,xÎRn, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. It also considers approximation of random fields by the random fields with a bounded spectrum.There is made an analogue of the Kotelnikov–Shannon theorem for random fields with a bounded spectrum. Besides, there are obtained estimates of the mean-square approximation of random fields in the space R´Rn by a model constructed with the help of spectral decomposition and Kotelnikov–Shannon interpolation formula. The paper provides a mechanism for statistical simulation of Gaussian random fields with a bounded spectrum; namely, those homogeneous in time and homogeneous isotropic ones relative to spatial variables in the multidimensional space. Proved have been the theorems of the mean-square approximation of random fields (those homogeneous in time and homogeneous isotropic ones relative to n- other variables) by special partial sums. A simulation method was used to formulate an algorithm of numerical simulation by means of these theorems. There are also considered ways to carry out spectral analysis of generated seismic noise realizations. Finally, there have been developed universal methods of statistical simulation (Monte Carlo methods) of multi-parameter seismology data for generating seismic noise on 2D and 3D grids of the required detail and regularity.