Description
A (one might say the) fundamental problem in science is finding a mathematical representation, hopefully with predictive or insight benefits, that describes a physical phenomenon of interest. In seismology, one such classic problem is determining the seismic velocity structure of the Earth from travel time or other measurements made on seismograms (seismic tomography). The document below provides notes and an associated exercise to be completed in Matlab.
Audience
Undergraduate
Time
60 minutes lecture followed by a 15 minute Matlab exercise
Supporting Resources
Full document: Notes and Exercise (.doc)
Exercise (excerpted from the full document)
Figure 2. A simple tomography example.
Consider a 4-cube model of a square region, 200 m on a side, where we make travel time measurements, t1, …, t5 in five directions as shown in Figure 2. The slowness (the reciprocal of the seismic velocity) in each region is parameterized as S11, S12, S21, S22, as also depicted in the figure (we parameterize the model in terms of slowness instead of velocity because it results in a linear system of equations, as we’ll see).
Each travel time measurement has a forward model associated with it. For example
t1 = S11 • 100 + S12 • 100 (11)
where the slownesses are specified in seconds/meter and the time is in seconds. The complete (overdetermined) system of (m=5) constraint equations (in n=4 unknowns) is thus:
(12)
where the elements Gij are all specified by the raypath geometry of the experiment. Your assignment is:
1) Find the elements of G.
2) Solve for a least-squares solution using the normal equations (10) and MATLAB if:
t1 = 0.1783 s
t2 = 0.1896 s
t3 = 0.2008 s
t4 = 0.1535 s
t5 = 0.2523 s
Note that there is some random noise in these times, so the system of equations (12) is inconsistent (it has no exact solution).
3) Convert your slownesses to velocities; where is the region seismically faster or slower?
4) Calculate the residual vector, r (6). How well does your model actually fit the data on average?
Author: Rick Aster, Professor of Geophysics, NM Tech (aster "at" ees.nmt.edu)
