[Solved] MA590 Homework 2

Problems

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

(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.

• Calculate the appropriate system matrix G for discretizing the integral equation (1) using the midpoint rule.
• 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, s_true, 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.

• 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 s_true.
• Generate a noisy travel time vector where independent normally distributed noise with astandard deviation of 0.05 ms is added to the elements of y. Resolve the system for s and again compare your results graphically with s_true. How has the result changed?
• 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

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