UBC Theses and Dissertations
Linear systems theory applied to a horizontally layered crust Jensen, Oliver George
Elastic wave propagation in a multilayered crust with causally or acausally attenuating layers is formulated directly in terms of linear systems theory. The solution of the linear systems analog determines the wave motions at the free surface, motions of all internal boundaries and the waveforms propagated into the mantle in response to plane P or S waves generated within the layering or entering into the crust. The particular problem solved is the determination of the response of an n-layered crust to teleseismic P or S waves incident with arbitrary angle at the crustal base. Trivial extensions of this problem would allow reflection solutions. Numerical solutions have been accomplished in both the frequency domain and the time domain. The Fourier solution restricted to a non-attenuating crust is equivalent to the standard Haskell matrix solution of elastic wave equations. Direct time domain solutions allow the syntheses of seismograms considering internal crustal absorption. For demonstration of the utility and advantages of the theory, the linear systems formulation has been applied to studies of the elastic wave response of the central Alberta crust using P waves from 6 teleseismic events. Frequency domain comparisons between the theoretical and experimental spectral amplitude V/H ratios have shown that, although the theoretical effect of attenuation within the crust can be considerable, little improvement in correlations between theory and experiment has been achieved by considering plausible crustal absorption models. Although significant similarities between the theoretical and experimental V/H ratios were found below 2 Hz, little correlation was apparent at higher frequencies. Background and scattering noise partly contributed to this effect and it is also probable that insufficient detail and accuracy was available for the model crustal sections. Time domain synthetic seismograms have been determined which well correspond to the early P coda of several of the experimental records. Assumptions on the event source motions and the mantle properties are required to determine incident P waveforms for these solutions. Causal attenuation within the crustal layering was included. Correlations between the synthetic seismograms and the experimental records has been found to decrease rapidly with time following the P onset. It is suggested that this effect is primarily due to background noise and possible scattering of the waves within the crust. Furthermore, it is probable that the waveforms used for these solutions based on the source motion and mantle attenuation assumptions were not sufficient. A major apparent advantage of the new formulation is that causal and acausal attenuation solutions are permitted in both the frequency and time domains. Also, the large body of communications theory mathematics can now be applied directly to the propagation problem and could prove useful in attempts at the solution of the non-normal incidence inverse problem.
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