Phase shift

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 10.

 

trick
Figure 10 The accuracy of the x-derivative may be improved by a technique that is analyzed later in chapter . 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.

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.

 

Mabsorbside Figure 11 Chapter explains how to absorb energy at the side boundaries. Make the necessary changes to the program to absorb waves incident on the left-side boundary.