The 45 diffraction equation differs from the 15 equation by the inclusion of a -derivative. Luckily this derivative fits on the six-point differencing star
Figure 10 Figure 2 including the 45 term, , for the collapsing spherical wave. What changes must be made to subroutine wavemovie() to get this result? Mark an X at the theoretical focus location.
Figure 11 The accuracy of the x-derivative may be improved by a technique that is analyzed in IEI p 262-265. Briefly, instead of representing by the tridiagonal matrix with (-1,2,-1) on the main diagonal, you use . Modify the extrapolation analysis by multiplying through by the denominator. Make the necessary changes to the 45 collapsing wave program. Left without 1/6 trick; right, with 1/6 trick.
Theory predicts that in two dimensions, waves going through a focus suffer a 90 phase shift. You should be able to notice that a symmetrical waveform is incident on the focus, but an antisymmetrical waveform emerges. This is easily seen in Figure 11.
In migrations, waves go just to a focus, not through it. So the migration impulse response in two dimensions carries a 45 phase shift. Even though real life is three dimensional, the two-dimensional response is appropriate for migrating seismic lines where focusing is presumed to arise from cylindrical, not spherical, reflectors.