Figure 2 shows a vertically and horizontally variable but smooth background velocity model whose velocity ranges from 4000 m/s to 5000 m/s. This model represents the results of transmission traveltime tomography. The crosswell survey with this velocity model is taken in a subsurface segment of the wells. The survey depth interval is 200 m. The two wells are separated by 100 m. The wavelet used in the finite-difference modeling is a Ricker wavelet whose frequency ranges from 200 Hz to 2000 Hz (Lazaratos et al., 1991), with the fundamental frequency being 1000 Hz. The dominant wavelength is 4 m; the smallest wavelength 2 m. The horizontal and vertical grid spacing is 0.5 m. Thus the spatial sampling interval is 4 points per wavelength for the smallest wavelength, which is a very coarse sampling interval. If second-order finite-differences were used, this spatial sampling would create severe grid dispersion. For the fourth-order-in-time, tenth-order-in-space finite differencing scheme employed in this paper, the numerical accuracy is very good.
Seismic modeling has been dominated by velocity models of velocity blocks with ``knife-cut'' sharp velocity contrast interfaces. These models do not correspond with the well logs and the field geology one sees. One never sees a well log with sharp contrast interfaces and never sees sharp contrast interfaces between different sedimentary rocks in field geology. Moreover, these models violate the theory of seismic data processing, which asserts that the low-wavenumber velocity, which is the smooth velocity background, governs the wave propagation, and the high-wavenumber velocity, which is the reflectivity, generates the diffractions. Inversion for velocity consists of two parts. The low-wavenumber velocity is inverted by transmission traveltime tomography, the high-wavenumber velocity by migration. What I did in this paper, I made the velocity model by superimposing high-wavenumber velocity contrast heterogeneity in a smooth background velocity model.