1. Introduction
Over the past 15 years, scattering of seismic waves in random media in two dimensions has
Ž been studied numerically Frankel and Clayton,
1984, 1986; Gibson and Levander, 1988; Ikelle .
et al., 1993; Roth and Korn, 1993 . The random media generally consist of a deterministic back-
ground velocity plus a random component de- fined by its wave number spectra. Gaussian,
exponential and self-similar random media are often used in these scattering studies since these
are easily defined by their respective wave num- ber spectra. Self-similar media, however, appear
to best describe the spectral content of velocity
Ž logs from boreholes Holliger, 1997; Dolan and
. Bean, 1997; Leary, 1997 . Isotropic self-similar
media are completely described by the standard deviation of the velocity fluctuations, correla-
tion distance and Hurst number.
Analytical expressions for the amplitude de- cay of a forward propagating wave due to scat-
tering may be estimated using single scattering Ž
. theory Chernov, 1960 . An important parame-
ter in the analytical expressions for scattering Q is u
, the minimum angle that energy must be
min
scattered so as not to be considered as contribut- ing to the forward propagating wave. At high
wave numbers, 1rQ can vary by several orders of magnitude depending upon the choice u
.
min
In order to estimate scattering Q from borehole velocity logs it is, therefore, important to know
what u to use in the analytical expressions.
min
Comparison of numerical simulations with the analytical functions is one method for determin-
ing u . For 2-D media, Frankel and Clayton
min
Ž .
Ž .
1986 and Roth and Korn 1993 estimated u
min
to be about 308 for Gaussian and self-similar Ž
. media. Sato 1982 derived a value of 298 from
a theoretical viewpoint. In this paper, we perform a similar compari-
son in 2-D as in the previous studies, but with significantly larger models and more synthetic
data, which give better determined estimates. We also perform a 3-D comparison, which has
not been done before, to investigate if a differ- ence between 2-D and 3-D scattering simula-
tions can be observed, or if 2-D is enough when studying scattering.
In the first part of this paper, we discuss the properties of random media and how they are
generated. We will in detail investigate the dif- ferences between continuous and discrete media
and point at the importance of this difference. In the second part, we generate synthetic seismo-
Ž .
grams for vertical seismic profiling VSP ac- quisition geometries in Gaussian and self-simi-
lar 2-D and 3-D acoustic random media. The Hurst number and standard deviation of the
velocity fluctuations of the random media are systematically varied. From the seismograms,
we calculate the scattering attenuation observed in the finite difference seismograms and com-
pare these results with analytical formulas ŽChernov, 1960; Wu, 1982; Frankel and Clay-
. ton, 1986 .
2. Random media
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