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

Most of this lab involves making changes to ``extrap.java'' (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. For each, turn in your changes to ``extrap.java'' 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, p. 107. 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, p. 108. 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. 3, p. 108. 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, p. 108. Incorporate the 45 degree term for the collapsing spherical wave. Use zero-slope sides. Compare your result with the 15 degree result obtained via the original extrap.java 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, p. 108. 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, p. 108. 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, p. 109. Section 4.4 (page 267) 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 ``wmig.java'' performs -domain migration by the method in Claerbout on page 111. 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 ``wmig.java'' to incorporate simple absorbing boundary conditions. Turn in the migration for data impulses at x = 1/2 xmax.

Possible Projects

• Construct a migration routine that includes the thin lens term. Use it to migrate real and synthetic datasets, and explain artifacts and areas of applicability.

• Construct a routine to model zero-offset data for complex structure and using paraxial extrapolators. Compare the results to full-wave calculations.

• Alter the extrapolation and frequency-domain migration programs to employ the stable, explicit Dufort-Frankel difference scheme. Compare the accuracy and computation speed of this method to those of the programs using the implicit scheme.

• Beginning with wmig.java, investigate the accuracy of wave-field normal moveout correction and stacking on a common-midpoint, multi-offset record. Produce stacked sections from both real and synthetic datasets using both ray- and wave-based methods. Compare the robustness of the two techniques in the presence of noise and velocity variations.