The xxz derivative

The 45$^\circ$ diffraction equation differs from the 15$^\circ$ equation by the inclusion of a $\partial^3 / \partial x^2 \partial z$ -derivative. Luckily this derivative fits on the six-point differencing star

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So other than modifying the six coefficients on the star, it adds nothing to the computational cost. Using this extra term allows in programs like subroutine wavemovie() yields wider angles.

 

Mfortyfive90 Figure 10 Figure 2 including the 45$^\circ$ term, $\partial_{xxz}$, 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.

 

Mhi45b90 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 $k_x^2 \,\Delta x^2$ by the tridiagonal matrix ${\bf T}$ with (-1,2,-1) on the main diagonal, you use ${\bf T} / ( {\bf I} - {\bf T} / 6 )$. Modify the extrapolation analysis by multiplying through by the denominator. Make the necessary changes to the 45$^\circ$ 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$^\circ$ 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$^\circ$ 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.

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