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

The 45 migration is a little harder than the 15 migration because the operator in the time domain is higher order, but the methods are similar to those of the 15 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 is represented by the bilinear transform .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 migration program in the space of .Find the coefficients in a 6-point differencing star, three points in time and two points in depth. For simplicity, take v=1, , and .Suppose this analysis were transformed into the x-domain ()by replacing kx2 with .What set of tridiagonal equations would have to be solved?

Next: INTRODUCTION TO STABILITY Up: FINITE DIFFERENCING IN (t, Previous: You Can't Time Shift
Stanford Exploration Project
10/31/1997