(t,x,z)-Space, 45 degree equation

The 45$^\circ$ migration is a little harder than the 15$^\circ$ migration because the operator in the time domain is higher order, but the methods are similar to those of the 15$^\circ$ equation and the recursive dip filter. The straightforward approach is just to write down the differencing stars. When I did this kind of work I found it easiest to use the Z-transform approach where $1 /( -\, i \omega \Delta t)$ is represented by the bilinear transform ${1\over 2} (1+Z)/(1-Z)$.There are various ways to keep the algebra bearable. One way is to bring all powers of Z to the numerator and then collect powers of Z. Another way, called the integrated approach, is to keep 1/(1-Z) with some of the terms. Terms including 1/(1-Z) are represented in the computer by buffers that contain the sum from infinite time to time t. The Z-transform approach systematizes the stability analysis.

EXERCISES:

1. Alter the program time15 so that it does migration. The delta-function inputs should turn into approximate semicircles.
2. Perform major surgery on the program time15 so that it becomes a low-pass dip filter.
3. Consider a 45$^\circ$ migration program in the space of $(z,\,t, k_x )$.Find the coefficients in a 6-point differencing star, three points in time and two points in depth. For simplicity, take v=1, $\Delta t=1$, and $\Delta z=1$.Suppose this analysis were transformed into the x-domain ($\Delta x = 1$)by replacing k_x² with ${\bf T}$.What set of tridiagonal equations would have to be solved?