GEOL0030 Seismology II
Coursework 1: Set Week 3, to be submitted Wednesday Week 7 via Moodle
Please note that marks are awarded not just for getting the correct answer, but also for the quality of the answer, i.e. for the explanation and for the way it is set out. All necessary proofs and auxiliary calculations must be included in your answers.
Use appropriate values for any required parameters. Answer ALL questions. All questions carry equal marks.
1. Boxcar functions are often used to describe the time history of earthquakes.
(a) Calculate analytically the Fourier transform. of a boxcar function:
f(t) = 1 , for –T < t < T,
= 0 , otherwise [4%]
(b) Using Matlab calculate the Fourier transform. of a discrete version of the boxcar function f(t) considered in question 1(a) for a reasonable value of T of your choice. Plot the real and imaginary parts of the solution. [4%]
(c) Using Matlab, build plots comparing the solutions obtained on 1(b) with the analytical solutions obtained in 1(a). [2%]
(d) Using Matlab, compute and plot the convolution between two boxcar functions with T=3 s and T=20s. Plot its spectrum and discuss its characteristics. [5%]
2.
Consider Love waves in a medium that comprises one layer of thickness h of material with shear wave velocity b1 underlain by a half-space of material with a higher velocity b2. Love waves can be treated as the constructive interference between SH waves incident on the interface between the layer and the half-space, with an incidence angle larger than the critical angle.
(a) Write the mathematical expression of the Love wave displacement in the layer as the sum of an upgoing and a downgoing plane wave. [2%]
(b) In the half-space we only need to consider one upgoing plane wave. Write the corresponding mathematical expression. [2%]
(c) Since seismic surface waves travel along the Earth’s surface, their energy is trapped near the surface and hence their displacement must decay as z →∞ . Derive the relationship between apparent velocity and shear velocity imposed by this radiation condition. [3%]
(d) Write the mathematical expressions describing the appropriate boundary conditions at the free surface and at the interface between the layer and the half-space. [3%]
(e) Using the boundary conditions from 2(d) along with the expressions in 1(a)-1(c) derive the relationships between the amplitude coefficients of the displacement in the layer and in the half-space. These relationships should involve the shear modulus in the layers, the wavenumber and the thickness of the layer. [4%]
(f) Using the results obtained in 2(e) to obtain the Love wave dispersion relation. [2%]
(g) Using reasonable values for the various parameters involved in the dispersion relation in 2(f) to model the Earth’s crust, determine the fundamental mode values of ω that satisfy the Love wave dispersion relation at values of phase velocity of 3.8, 4.0, 4.2 and 4.4 km/s. Sketch the c(T) dispersion curve. [4%]
(h) Determine the values of ω that satisfy the Love wave dispersion relation for the first higher mode. Add sketches of the corresponding dispersion curve to the diagram built in (g). [3%]
(i) Discuss the differences between the fundamental and higher mode Love wave dispersion curves plotted in (g) and (h) in the context of their sensitivity to shear wave velocity. [3%]
3.
(a) Assume the following expressions for the phase velocity as a function of wavelength (l):
and
where c0, c1, a, b and λ0 are positive constants.
Sketch the corresponding dispersion curve. [6%]
(b) Give the expressions of the group velocity as a function of wavelength and roughly sketch the group velocity curve. [10%]
Table 1. Measured wave periods of various spheroidal and toroidal modes.
4. Using the relation between modes and traveling waves and the information in Table 1, answer the following questions:
(a) Since the mode 0T2 samples the mantle quite uniformly, assume that the phase velocity appropriate for this mode is the average mantle shear wave velocity in PREM and find the wave period that you would expect. Discuss how your result compares to the actual wave period in Table 1. [5%]
(b) Determine the phase velocity for three modes with similar periods: 4T67, 10T40 and 13T7. Interpret the differences obtained. [5%]
(c) Determine the phase velocities and wavelengths of waves corresponding to the modes 0S3, 0S30 and 0S130. Interpret the trend of the velocities. Which of these modes would you expect to be most affected by lateral heterogeneity in the Earth, and why? [5%]
5. Explain the main principles of the following methods used to calculate theoretical seismograms:
(a) The reflectivity method;
(b) The normal mode summation method;
(c) The finite differences method.
Discuss and compare the strengths and limitations of these methods for applications involving imaging the Earth’s deep interior. [18%]
6. Define mathematically the resolution of an inverse problem and discuss its physical meaning, with examples. [10%]