STOCHASTIC WAVE FIELD SOLUTION OF THE 2D ELASTIC WAVE EQUATION BASED ON THE RANDOM FOURIER-STIELTJES INCREMENTS

Abstract

An analytical solution of the stochastic wave equation is presented to model 2D heterogeneous geological environments. In the formulation, a plane-harmonic seismic wave propagates in a medium having random elastic properties in the horizontal and vertical directions. The 2D random field representation is introduced in the stiffness properties of the medium by assuming it has log-normal probability density functions. The constitutive stress and displacement laws with the momentum balance equation for total stress yield a partial differential equation, which is developed using a perturbation approach by assuming a 2D random geological medium having material heterogeneity randomly distributed in the horizontal (x) and vertical (z) directions. The method yields a double integral representation of the displacement wave vector based on the Green's tensor and the Fourier-Stieltjes increments. The double integral is reduced to one integral representation by removing the singularities. The final form of the integral is used to construct the stochastic wave field displacement components expressed in terms of a single integral that is appropriate for calculations.This paper also describes a numerical approach that predicts the stochastic wave field used to test the applicability of the theory by simulating a 2D randomly heterogeneous geological medium. Synthetic vertical and horizontal component seismograms based on this random medium indicate a decrease in wave amplitude and wave broadening effects at different depths of the random velocity images. The results suggest that the attenuation and dispersion of waves traveling between two wells are caused by the presence of scatterers observed in the 2D random velocity distribution. Large scatterers produce strong reflections that are observed in the random horizontal wave field seismograms. In general, the random wave field seismograms show characteristic seismic signatures that are associated with the structural distribution of the heterogeneities. In particular, random wave field signatures associated with heterogeneous low velocity zones are observed in the simulations. This kind of signature has been observed in crosswell data recorded at the Gypsy test site in Oklahoma.