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To test the effect of the boundary condition
correction operator we used a constant velocity (1000 m/s) model, in
which the 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 brings the amplitudes much
closer to the analytically computed curve both for mixed domain and for
finite difference. In the case of the mixeddomain method (top and
middle panels), the
amplitudes are practically as good as those obtained with twoway 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.
laminus
Figure 1 Amplitudes computed in the following
ways: Top panel  Curve 0: analytically; Curve 1:
mixed domain with correction; Curve 2: twoway wave
equation modeling; Curve 3: mixed domain without
correction. Middle panel  normalized to the analytical
curve: Curve 0: analytically; Curve 1:
mixed domain with correction; Curve 2: twoway wave
equation modeling. Bottom panel  Curve 0:
analytically; Curve 1: finitedifference with correction; Curve 2: finitedifference without correction.
The 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):
 
(29) 
In v(z) the 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 and the
WKBJ result can be attributed to the truncations in the Taylor
expansions in the term.
crscmp
Figure 2 Amplitudes of a shot propagated
by splitstep, with the following corrections: Curve 0:
correction only; Curve 1: Both and
corrections; Curve 2: and WKBJ
corrections

 
We then downward continued a shot until z=2000m through the velocity
model shown in Figure
. Downward continuation was performed with both splitstep
(left panels) and finite differences (right panels), without applying
any correction (upper panels), with a boundary
condition only (middle panels) and with both and
corrections (lower panels). The corrections result in
phase changes and in a much more uniform repartition of
amplitudes.
symes_v
Figure 3 V(x,z) velocity model

 
Next: Conclusions
Up: Vlad et al.: Improving
Previous: Mixed domain
Stanford Exploration Project
7/8/2003