A stochastic wavefield solution of the acoustic wave equation based on therandom Fourier-Stieltjes increments

In this paper we investigate the problem of compressional wave seismic prop agation in random media. This problem is important because almost all geolo gic media is spatially heterogeneous by nature, consisting of a random aggl omerate of many-sized rocks, soil and strata. In our formulation, a plane-h armonic seismic wave propagates in a medium having random material properti es in the vertical direction. The random field representation is introduced through the intrinsic rack 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, t he mass balance, and the moment balance equations are written in the Fourie r-Stieltjes representation using random Lame coefficients and random mass d ensity. The stochastic wave equation is developed by introducing a perturba tion approach based on an infinite series expansion of both random coeffici ents and the displacement solution in terms of sigma-parameters (standard d eviations 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 pro perties of the medium. The key feature of this paper is that we have expres sed the solution as a function of statistical parameters of 1D random mediu m, including the second order moments. Contrary to most previous derivation s, the solutions can also simulate the coda and can be easily extended to s imulate waves propagating in 2D and 3D random media. To test the displaceme nt wave solution, synthetic seismograms and dispersion due to scattering ef fects 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 me dia. This development is presented in a companion paper. In this case, an a nalytical expression is obtained using a second order perturbation solution . The solution is obtained in terms of the standard deviations of the densi ty 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 calculat ion of the effective attenuation and phase velocity of waves in randomly he terogeneous media. In this companion paper the solution is applied to inter pret 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 asso ciated with acoustic waves in randomly heterogeneous media. (C) 1999 Elsevi er Science B.V. All rights reserved.