GEOL 706
LAB 8 - Paraxial Extrapolation

Reading: Claerbout IEI, Chapter 2. Start looking at Chapter 3.

Page numbers are given in the order [page in printed book, number appearing on the page in the PDF, consecutive page number in the PDF].

Most of this lab involves making changes to ``'' (as well as the ctris() complex tridiagonal matrix solver method included). To turn in the first frame of a movie, put the Plane Index slider on 0, then activate your machine's screen grab and print method. Remember that you have to Control-C or Command-Q out of the extrap application, not just close the plot movie window.

The following exercises are from Claerbout, p. [107-109, 96-104, 100.46-150.4]. For each, turn in your changes to ``'' and the first frame of the resulting movie. Do not accumulate changes as you progress through the exercises; just make the changes specified to the original program, after copying it to a file with a new name, and renaming the class at its declaration line.

1. [ex. 1, fig. 4.4], p. [107, 97, 47]. Specify program changes that give an initial plane wave propagating downward at an angle of 15 degrees to the right of vertical. To the right means in the positive-x direction.

2. [ex. 2, fig. 4.6], p. [108, 100, 50]. Given that the domain of computation is 0 < x <= xmax and 0 < z <= zmax , how would you modify the initial conditions at z=0 to simulate a point source at (x, z) = (xmax/3, -zmax/2)? Try it.

3. ex. [ex. 3, fig. 4.7], p. [108, 101, 1]. Modify the program so that zero-slope side boundaries are replaced by zero-value side boundaries. What changes? (The changes are easiest to see using one of the altered sources from exercise 1 or 2 above.)

4. [ex. 4, fig. 4.9], p. [108, 102, 2]. Incorporate the 45 degree term partial xxz for the collapsing spherical wave. Use zero-slope sides. Compare your result with the 15 degree result obtained via the original program. Mark an X at the theoretical focus location. Remember that the 45 degree extrapolator fits on the same differencing star as the 15 degree. What differences in wave propagation can you see? What are the approximate coordinates of the focus in each case? Should it appear to change between the 15 degree and 45 degree solutions? Why?

5. [ex. 5, fig. 4.8], fi, p. [108, 102, 2]. Make changes to the program to include a thin-lens term with a lateral velocity change of 40% across the frame produced by a constant slowness gradient. The easiest way is to split the calculation and do the thin-lens extrapolation analytically at each z-step, right after the diffraction part. Identify other parts of the program which are affected by lateral velocity variation. You need not make these other changes. Why are they expected to be small?

6. [ex. 6, fig. 4.5], p. [108, 100, 50]. Observe and describe various computational artifacts by testing the program using a point source at (x, z) = (xmax/2, 0). Such a source is rich in the high spatial frequencies for which difference equations do not mimic their differential counterparts.

7. [ex. 7, 4.11], p. [109, 104, 4]. Section [4.4, 9.1] (page [267, 259, 9]) explains how to absorb energy at the side boundaries. Make the necessary changes to the program. We went over this in class too. Use one of the expanding-source configurations to show that your absorbing boundaries work well.

8. The program ``'' performs omega-domain migration by the method in Claerbout's IEI on page [111, 103-106, 3-6]. Where are the original impulses? Move them in x to x = 1/2 xmax. Turn in the migrated section. What happened? Why were the impulses placed where they were originally?

9. Change ``'' to incorporate simple absorbing boundary conditions. Turn in the migration for data impulses at x = 1/2 xmax.

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