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Results

To test the effect of the boundary condition correction operator we used a constant velocity (1000 m/s) model, in which the $\Gamma$ correction is zero. Because the correction only applies to the source wavefield, we did not go through the imaging condition. We generated a shot at the surface, downward propagated it until z=375m, and then picked the maximum amplitudes at each x location at that depth. Figure [*] shows that the operator $\frac{1}{2}\Lambda^{-1}$ brings the amplitudes much closer to the analytically computed curve both for mixed domain and for finite difference. In the case of the mixed-domain method (top and middle panels), the amplitudes are practically as good as those obtained with two-way wave equation modeling. In the case of finite difference (bottom panel), artifacts interfere constructively and destructively with the true events and lead to amplitude variations along the hyperbola. However, even with artifacts, the corrected amplitudes are closer to the analytical ones than the uncorrected ones.

 
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Figure 1
Amplitudes computed in the following ways: Top panel - Curve 0: analytically; Curve 1: mixed domain with $\Lambda^{-1}$ correction; Curve 2: two-way wave equation modeling; Curve 3: mixed domain without $\Lambda^{-1}$ correction. Middle panel - normalized to the analytical curve: Curve 0: analytically; Curve 1: mixed domain with $\Lambda^{-1}$ correction; Curve 2: two-way wave equation modeling. Bottom panel - Curve 0: analytically; Curve 1: finite-difference with $\Lambda^{-1}$correction; Curve 2: finite-difference without $\Lambda^{-1}$correction.
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The $\Gamma$ correction was tested the same way: by comparing the amplitudes from a shot, propagated until z = 1125m in a v = 1000 m/s + 2z medium, for which amplitude curves could be computed analytically (Appendix C). The most meaningful comparison is with the WKBJ amplitude correction Clayton and Stolt (1981):
\begin{displaymath}
\mbox{WKBJ Correction Factor} = \sqrt{\frac{k_z(z=z_{max})}{...
 ...-k_x^2}{\frac{\omega^2}{v_{z=0}^2}-k_x^2}\right)^{\frac{1}{4}}.\end{displaymath} (29)
In v(z) the $\Gamma$ correction has the same effect as the WKBJ correction, but unlike it, it can also be applied in a v(x,z) medium. The differences at large angles between the $\Gamma$ and the WKBJ result can be attributed to the truncations in the Taylor expansions in the $\Gamma$ term.

 
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Figure 2
Amplitudes of a shot propagated by split-step, with the following corrections: Curve 0: $\Lambda^{-1}$ correction only; Curve 1: Both $\Lambda^{-1}$ and $\Gamma$ corrections; Curve 2: $\Lambda^{-1}$ and WKBJ corrections
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We then downward continued a shot until z=2000m through the velocity model shown in Figure [*]. Downward continuation was performed with both split-step (left panels) and finite differences (right panels), without applying any correction (upper panels), with a $\Lambda^{-1}$ boundary condition only (middle panels) and with both $\Lambda^{-1}$ and $\Gamma$ corrections (lower panels). The corrections result in phase changes and in a much more uniform repartition of amplitudes.

 
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Figure 3
V(x,z) velocity model
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next up previous print clean
Next: Conclusions Up: Vlad et al.: Improving Previous: Mixed domain
Stanford Exploration Project
7/8/2003