Seminar in Seismic Tomography
Instructor: Charles J. Ammon
Exercise 01

An Exercise in One-Dimensional Tomography

Objective: To thoroughly study a simple, intuitive travel time tomography problem using MATLAB. Although obviously an oversimplification, this simple example can be used to exercise the mathematics underlying more interesting tomography problems.

You should complete the following calculations and numerical experiments using MATLAB. The exercise is similar to the one we did last semester in the evening session on MATLAB and inverse theory. Review your notes from those lectures before you begin.

The problem that you will solve is illustrated below. The slowness values are (1.2, 0.8, 1.0, 1.2) but you must try to estimate them as if your don't know the answer! Do you think you can?

The travel times for the four ray paths are given in the following table:

Ray Path
Travel Time (s)

A) Set up the matrix equations for the tomography problem posed in the cartoon. That is, express the expressions for the travel times in terms of
Ls = t

where L is a matrix with paths lengths, s is the unknown slowness vector, and t is a column vector that contains the travel times.

Use MATLAB to perform the following calculations.

B) Solve the tomography problem using the "generalized inverse approach" describe in Aki, Christofferson, and Husbeye (1977). You should look at Jackson (1972) to see more explicit details on how to compute the inverse operator using the singular value decomposition of L.
C) Solve the same problem using the stochastic, or damped least-squares method also described in Aki, Christofferson, and Husbeye (1977). Try a range of damping factors and make a tradeoff between damping model length and match to the observations. Recommend a preferred model based on this curve. Compute the resolution matrix for the damped least-squares approach and comment on how resolution is affected by the magnitude of the damping parameter.
D) Add a small fraction (try 5% and 15%) of random noise to the travel times and repeat C).
E) In this exercise you will check the resolution using "spike tests". The idea is to construct "target" models for a series of inversions that directly investigate the resolution of each slowness cell. Here's the algorithm to complete this:

For each cell
Construct the "target" model
Set the slowness of the cell to 1.05 Set the slowness of the other cells to 1.00
Compute the travel times through the target model (t = L s) Invert these travel times for an estimate of s.
end loop over cells

Plot the results of the inversion using a histogram (one column for each cell).

Which cells are best constrained? How does the information from this exercise compare with the resolution matrix from parts C) and D)?

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Prepared by: Charles J. Ammon