Seminar in Seismic Tomography
Instructor: Charles J. Ammon
Exercise 03

Another Two-Dimensional Tomography Laboratory - Smoothing

Objective: To extend our earlier two dimensional problem to all for the includesion of smoothing the solution. Smoothing is another common constraint used to stabilize tomography probelms and can be applied in a linear fashion.

Procedure: This is an extension of the last laboratory, and all the tools and skills you developed during that work will help you here.

The problem that you will solve (and the solution) is shown in the following figure (It's the same as Exercise 02). The solution and the paths:


From the previous exrecises you should have an appreciation for what parts of the model are resolved well. In this exercise you will no longer seek the minimum length solution, but the smoothest solution. To estimate the smoothest model that fits the observations we will construct a metric of model roughness and minimize it.

Use MATLAB to complete the following calculations. Submit a write-up of your work that summarizes the calculations and includes key figures - an implicit requirement for each exercise is that you intelligently describe the results and the implications of each experiments. Compare the results with the results of exercise 02.

A) Solve the tomography problem with smoothness constraints using the "generalized inverse approach" or "damped least-squares method" described in Aki, Christofferson, and Husbeye (1977). Construct a trade-off curve showing the variation in misfit and model roughness.
B) Add 5% of random noise to the travel times and part A and choose a preferred model. Show the trade-off curve between truncation fraction, or damping value and model length (insure that your range in fit to the observations makes sense in light of the level of noise you added to the signals).
C) Check the resolution using "spike tests" with a 10% spike located in the third column and third row from the upper left of the image. Perform another spike test for the cell in the sixth row and eighth column from the upper left.


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Prepared by: Charles J. Ammon