Stochastic wave field solution of the 2D elastic wave equation based on the random Fourier-Stieltjes increments

An analytical solution of the stochastic wave equation is presented to mode l 2D heterogeneous geological environments. In the formulation, a plane-har monic seismic wave propagates in a medium having random elastic properties in the horizontal and vertical directions. The 2D random field representati on is introduced in the stiffness properties of the medium by assuming it h as log-normal probability density functions. The constitutive stress and di splacement laws with the momentum balance equation for total stress yield a partial differential equation, which is developed using a perturbation app roach by assuming a 2D) random geological medium having material heterogene ity randomly distributed in the horizontal x and vertical (z) directions. T he method yields a double integral representation of the displacement wave vector based on the Green's tensor and the Fourier-Stieltjes increments. Th e double integral is reduced to one integral representation by removing the singularities. The final form of the integral is used to construct the sto chastic wave field displacement components expressed in terms of a single i ntegral 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 horizonta l component seismograms based on this random medium indicate a decrease in wave amplitude and wave broadening effects at different depths of the rando m velocity images. The results suggest that the attenuation and dispersion of waves traveling between two wells are caused by the presence of scattere rs observed in the 2D random velocity distribution. Large scatterers produc e strong reflections that are observed in the random horizontal wave field seismograms. In general, the random wave field seismograms show characteris tic seismic signatures that are associated with the structural distribution of the heterogeneities. In particular, random wave field signatures associ ated 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. (C) 2001 Elsevier Science B.V. All rights res erved.