Frequency domain seismic forward modelling: A tool for waveform ...

(Boore, 1972; Kelly et al., 1975). With these modelling techniques, the simulation
of complex media became possible, although due to the simple, low accuracy nite
dierence formulations and the still limited computational resources, realistically
sized models were still out of reach.
As methods became more capable, a number of other development directions
were explored. These included a switch from the earliest, 1 - D earth models
(Abramovici and Alterman, 1965), to 2-D models (Alterman and Karal Jr, 1968),
and nally to 3-D models (Johnson, 1984; Reshef et al., 1988a; Reshef et al., 1988b;
Mora, 1989a). A fundamental limitation of Cartesian 2-D methods is that they do
not accurately simulate the phase and amplitude of eld seismic data, even within
the assumption of a 2-D earth model. Bleistein (1986) suggested a \2.5-D" method
for correcting for phase and amplitude data from point sources using ray trace parameters
later reformulated by Randall (1991) as a nite dierence formulation;
Song and Williamson (1995) suggested a wavenumber transform approach for accounting
for these corrections with nite dierence modelling methods in frequency
domain.
In addition to the progress made in the last decades in extending the dimensionality
of numerical wave equation modelling methods, researchers have also
attempted to model increasingly general physical phenomena.
Methods for the
acoustic wave equation (Michell, 1969; Gazdag, 1981; Virieux, 1986b; Reshef et
al., 1988a; Song and Williamson, 1995), the elastic wave equation (Alterman and
Karal Jr, 1968; Virieux, 1986a; Pratt, 1990a), the visco-elastic wave equation (Kjartansson,
1979; Emmerich and Korn, 1987; Robertsson et al., 1994), the anistropic
wave equation (Mora, 1989a; Carcione et al., 1992; Carcione, 1995) and the poroelastic
wave equation (Zhu and McMechan, 1991; Dai et al., 1995) have all been
developed. While one knows that the earth is 3-D, porous and anisotropic, in production
modelling and inversion choices and compromises must be made. Even if
it were possible to incorporate all these physical eects, dening appropriate model
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