EXAMPLES

In the first example we calculate traveltimes for a medium with a velocity gradient linear with respect to depth (the constant-velocity case is trivial in polar coordinates). The velocity at the top of the model is 2 km/s, the one at the bottom 2.5 km/s. The grid is evenly sampled in depth and laterally, with a sample interval of 10 m. Figure 2 shows the traveltime function for a source on the surface at the middle of the model, and Figure 3 displays the difference between that function and the analytical solution. Finite-difference traveltimes are calculated in polar coordinates, and errors accumulate as the radius increases. A simple bilinear interpolation is used in the mapping from and to Cartesian coordinates. Although interpolation errors in that mapping are not large, they cause the distinct pattern visible in Figure 3. The maximum error is about .3 ms (see Figure 5), which is one order of magnitude smaller than the standard time-sampling interval of 4 ms.

 

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Figure 2 Finite-difference traveltime field for a model with a velocity function that is linearly increasing as a function of depth. The source is located at the surface in the middle of the model. The figure displays both an intensity and contour plot of the traveltime field. Low intensities denote small traveltimes; contour lines are drawn at .1 s intervals.

 

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Figure 3 Difference between the traveltime map of Figure 2 and the analytical solution. Higher intensities in the plot represent larger errors.

 

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Figure 4 Difference between a traveltime function calculated with Vidale's plane-wave extrapolation method and the analytical solution.

Figure 4 shows the difference between Vidale's scheme and the analytical solution. We have implemented only the plane-wave extrapolation method, and errors can probably be reduced if a combination of plane and circular wave extrapolation is used. The errors are largest away from the vertical, diagonal, and horizontal directions, where the plane-wave approximation breaks down. The errors at the bottom of the model are of the same magnitude as the errors in the method described here (again see Figure 5).

 

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Figure 5 Errors in the finite-difference traveltime calculations at the bottom of model. The solid line represents the error curve for the method described here, the dashed line denotes the errors in Vidale's scheme.

 

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Figure 6 Wedge model. The grid spacing is $10\times 10$ m. Low intensities denote low velocities. The interfaces between the different layers are represented by solid lines in the figure. The dashed line denotes an imaginary reflector in the bottom layer. Figure 10 shows reflection events that correspond to these reflectors.

 

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Figure 7 Rays traced through a smoothed version of the model in Figure 6.

The next example illustrates the calculations for a more complicated model. The model is shown in Figure 6; it consists of 3 layers and a wedge intrusion. The velocity in the top layer is 2 km/s, the middle layer has a velocity of 1.75 km/s, and the bottom layer's velocity is 2.5 km/s. The velocity in the wedge that intrudes the middle layer from the right is 2.75 km/s. Figure 7 shows the result of tracing rays through a smoothed version of the model. The smoothing causes the rays to bend or turn in regions with a large velocity gradient.

 

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Figure 8 Finite-difference traveltimes calculated for the model of Figure 6. Overlain on the figure are contour lines of the traveltime field. The contour interval is .1 s.

 

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Figure 9 Comparison of the wave field computed by wave-equation modeling with the traveltime field calculated by upwind finite-differences. The intensity plot shows a snapshot of the wave field at .7s; the overlain dashed curve is the .7s-contour of the traveltime function (see Figure 8).

As is obvious from Figure 7, interpolating traveltimes from the rays onto the grid is not easy for this model; some parts of the model are not illuminated by rays, and in some other parts rays cross. However, the finite-difference calculation correctly fills in the problem areas as can be seen in Figure 8: the contour lines in the plot reveal the correct curvature of the wave fronts in the high- and low-velocity regions.

This result is verified in Figure 9, which shows the result of finite-difference wave-equation modeling. The figure displays a snapshot of the wave field at .7 s. Also shown in the figure is the .7s-contour line of the traveltime function (Figure 8). Barring some discrepancies due to the limited bandwidth and dispersion of the source wavelet, the contour exactly follows the first-arrival wave front. In particular note the match between wave field and traveltime function in the upper-right corner of the model, where the refracted wave travels in front of the direct wave.