A stochastic wavefield solution of the acoustic wave equation based on the random Fourier–Stieltjes increments

Abstract

In this paper we investigate the problem of compressional wave seismic propagation in random media. This problem is important because almost all geologic media is spatially heterogeneous by nature, consisting of a random agglomerate of many-sized rocks, soil and strata. In our formulation, a plane-harmonic seismic wave propagates in a medium having random material properties in the vertical direction. The random field representation is introduced through the intrinsic rock physical properties of the elastic medium. Each of these intrinsic properties is assumed to have a log-normal probability density function, and the random field representation is expressed in terms of these log-normal probability density functions. The constitutive law, the mass balance, and the moment balance equations are written in the Fourier–Stieltjes representation using random Lamé coefficients and random mass density. The stochastic wave equation is developed by introducing a perturbation approach based on an infinite series expansion of both random coefficients and the displacement solution in terms of σ-parameters (standard deviations of the random material properties). The method yields an integral representation of the displacement wavefield based on the Green's function. This representation is expressed in terms of the random rock physical properties of the medium. The key feature of this paper is that we have expressed the solution as a function of statistical parameters of 1D random medium, including the second order moments. Contrary to most previous derivations, the solutions can also simulate the coda and can be easily extended to simulate waves propagating in 2D and 3D random media. To test the displacement wave solution, synthetic seismograms and dispersion due to scattering effects were calculated for stiffness and density fluctuations of the random medium. This paper is the underlying foundation for the development of the effective propagation vector of acoustic waves in randomly heterogeneous media. This development is presented in a companion paper. In this case, an analytical expression is obtained using a second order perturbation solution. The solution is obtained in terms of the standard deviations of the density and the Young's modulus, respectively, as well as the cross-correlation coefficient and an integral that includes the spectral density and a kernel. In addition, this paper introduces practical expressions for the calculation of the effective attenuation and phase velocity of waves in randomly heterogeneous media. In this companion paper the solution is applied to interpret phase velocity curves that were obtained from interwell acoustic data recorded at Buckhorn test site, Illinois. The objective in this case is to be able to simulate the effect of scattering and intrinsic attenuation associated with acoustic waves in randomly heterogeneous media.