Observation and interpretation of tectonic strain release mechanisms

In four chapters various aspects of earthquake source are studied.

Chapter I

Surface displacements that followed the Parkfield, 1966, earthquakes were measured for two years with six small-scale geodetic networks straddling the fault trace. The logarithmic rate and the periodic nature of the creep displacement recorded on a strain meter made it possible to predict creep episodes on the San Andreas fault. Some individual earthquakes were related directly to surface displacement, while in general, slow creep and aftershock activity were found to occur independently. The Parkfield earthquake is interpreted as a buried dislocation.

Chapter II

The source parameters of earthquakes between magnitude 1 and 6 were studied using field observations, fault plane solutions, and surface wave and S-wave spectral analysis. The seismic moment, M_O, was found to be related to local magnitude, M_L, by log M_O = 1.7 M_L + 15.1. The source length vs magnitude relation for the San Andreas system found to be: M_L = 1.9 log L - 6.7. The surface wave envelope parameter AR gives the moment according to log M_O = log AR₃₀₀ + 30.1, and the stress drop, τ, was found to be related to the magnitude by τ = 0.54 M - 2.58. The relation between surface wave magnitude M_S and M_L is proposed to be M_S = 1.7 M_L - 4.1. It is proposed to estimate the relative stress level (and possibly the strength) of a source-region by the amplitude ratio of high-frequency to low-frequency waves. An apparent stress map for Southern California is presented.

Chapter III

Seismic triggering and seismic shaking are proposed as two closely related mechanisms of strain release which explain observations of the character of the P wave generated by the Alaskan earthquake of 1964, and distant fault slippage observed after the Borrego Mountain, California earthquake of 1968. The Alaska, 1964, earthquake is shown to be adequately described as a series of individual rupture events. The first of these events had a body wave magnitude of 6.6 and is considered to have initiated or triggered the whole sequence. The propagation velocity of the disturbance is estimated to be 3.5 km/sec. On the basis of circumstantial evidence it is proposed that the Borrego Mountain, 1968, earthquake caused release of tectonic strain along three active faults at distances of 45 to 75 km from the epicenter. It is suggested that this mechanism of strain release is best described as "seismic shaking."

Chapter IV

The changes of apparent stress with depth are studied in the South American deep seismic zone. For shallow earthquakes the apparent stress is 20 bars on the average, the same as for earthquakes in the Aleutians and on Oceanic Ridges. At depths between 50 and 150 km the apparent stresses are relatively high, approximately 380 bars, and around 600 km depth they are again near 20 bars. The seismic efficiency is estimated to be 0.1. This suggests that the true stress is obtained by multiplying the apparent stress by ten. The variation of apparent stress with depth is explained in terms of the hypothesis of ocean floor consumption.

Spectral density of first order Piecewise linear systems excited by white noise

The Fokker-Planck (FP) equation is used to develop a general method for finding the spectral density for a class of randomly excited first order systems. This class consists of systems satisfying stochastic differential equations of form ẋ + f(x) = m/Ʃ/j = 1 h_j(x)n_j(t) where f and the h_j are piecewise linear functions (not necessarily continuous), and the n_j are stationary Gaussian white noise. For such systems, it is shown how the Laplace-transformed FP equation can be solved for the transformed transition probability density. By manipulation of the FP equation and its adjoint, a formula is derived for the transformed autocorrelation function in terms of the transformed transition density. From this, the spectral density is readily obtained. The method generalizes that of Caughey and Dienes, J. Appl. Phys., 32.11.

This method is applied to 4 subclasses: (1) m = 1, h₁ = const. (forcing function excitation); (2) m = 1, h₁ = f (parametric excitation); (3) m = 2, h₁ = const., h₂ = f, n₁ and n₂ correlated; (4) the same, uncorrelated. Many special cases, especially in subclass (1), are worked through to obtain explicit formulas for the spectral density, most of which have not been obtained before. Some results are graphed.

Dealing with parametrically excited first order systems leads to two complications. There is some controversy concerning the form of the FP equation involved (see Gray and Caughey, J. Math. Phys., 44.3); and the conditions which apply at irregular points, where the second order coefficient of the FP equation vanishes, are not obvious but require use of the mathematical theory of diffusion processes developed by Feller and others. These points are discussed in the first chapter, relevant results from various sources being summarized and applied. Also discussed is the steady-state density (the limit of the transition density as t → ∞).

A mathematical and experimental investigation of microcycle spectral estimates of semiconductor flicker noise

The experimental portion of this thesis tries to estimate the density of the power spectrum of very low frequency semiconductor noise, from 10^-6.3 cps to 1. cps with a greater accuracy than that achieved in previous similar attempts: it is concluded that the spectrum is 1/f^α with α approximately 1.3 over most of the frequency range, but appearing to have a value of about 1 in the lowest decade. The noise sources are, among others, the first stage circuits of a grounded input silicon epitaxial operational amplifier. This thesis also investigates a peculiar form of stationarity which seems to distinguish flicker noise from other semiconductor noise.

In order to decrease by an order of magnitude the pernicious effects of temperature drifts, semiconductor "aging", and possible mechanical failures associated with prolonged periods of data taking, 10 independent noise sources were time-multiplexed and their spectral estimates were subsequently averaged. If the sources have similar spectra, it is demonstrated that this reduces the necessary data-taking time by a factor of 10 for a given accuracy.

In view of the measured high temperature sensitivity of the noise sources, it was necessary to combine the passive attenuation of a special-material container with active control. The noise sources were placed in a copper-epoxy container of high heat capacity and medium heat conductivity, and that container was immersed in a temperature controlled circulating ethylene-glycol bath.

Other spectra of interest, estimated from data taken concurrently with the semiconductor noise data were the spectra of the bath's controlled temperature, the semiconductor surface temperature, and the power supply voltage amplitude fluctuations. A brief description of the equipment constructed to obtain the aforementioned data is included.

The analytical portion of this work is concerned with the following questions: what is the best final spectral density estimate given 10 statistically independent ones of varying quality and magnitude? How can the Blackman and Tukey algorithm which is used for spectral estimation in this work be improved upon? How can non-equidistant sampling reduce data processing cost? Should one try to remove common trands shared by supposedly statistically independent noise sources and, if so, what are the mathematical difficulties involved? What is a physically plausible mathematical model that can account for flicker noise and what are the mathematical implications on its statistical properties? Finally, the variance of the spectral estimate obtained through the Blackman/Tukey algorithm is analyzed in greater detail; the variance is shown to diverge for α ≥ 1 in an assumed power spectrum of k/|f|^α, unless the assumed spectrum is "truncated".