MA 590  Homework 2 

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MA 590
Homework 2
1 (25 points) Consider the vertical seismic profiling problem, where a downward-propagating
seismic wavefront is generated by a source on the surface and the waves are sensed using
seismometers in a borehole (see Example 1.3 in Aster et al., 2019).
The observed travel time t at depth z can be modeled as
t(z) = ˆ ∞
s(ξ)H(z − ξ)dξ (1)
where s(z) denotes the vertical slowness (reciprocal of velocity) and the kernel H is the
Heaviside step function, which is equal to 1 for nonnegative arguments and 0 for negative
arguments. Assume we have n = 100 equally spaced seismic sensors located at depths of
z = 0.2, 0.4, . . . , 20 m, and we want to estimate n corresponding equal length seismic slowness values for 0.2 m intervals having midpoints at z − 0.1 m.
(a) Calculate the appropriate system matrix G for discretizing the integral equation (1) using
the midpoint rule.
(b) For a seismic velocity model having a linear depth gradient specified by
v = v0 + kz (2)
where the velocity at z = 0 is v0 = 1 km/s and the gradient is k = 40 m/s per m, calculate
the true slowness values, strue, at the midpoints of the n intervals. Additionally, integrate
the corresponding slowness function for (2) using (1) to calculate a noiseless synthetic data
vector, y, of predicted seismic travel times at the sensor depths.
(c) Solve for the slowness, s, as a function of depth using your G matrix from part (a) and
analytically calculated noiseless travel times from part (b) by using the MATLAB backslash
operator (see MATLAB help for \ ). Compare your results graphically with strue.
(d) Generate a noisy travel time vector where independent normally distributed noise with a
standard deviation of 0.05 ms is added to the elements of y. Resolve the system for s and
again compare your results graphically with strue. How has the result changed?
(e) Repeat the problem using n = 4 sensor depths and corresponding equal length slowness
intervals. Is the recovery of the true solution improved? Discuss, considering the condition
number of your G matrices.
Note: For any of the above problems for which you use MATLAB to help you solve, you must
submit your code/.m-files as part of your work. Your code must run in order to receive full credit.
If you include any plots, make sure that each has a title, axis labels, and readable font size, and
include the final version of your plots as well as the code used to generate them.


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