Complexity of rupture propagation in large earthquakes in relation to tectonic environment

Complexity in the earthquake rupture process can result from many factors. This study investigates the origin of such complexity by examining several recent, large earthquakes in detail. In each case the local tectonic environment plays an important role in understanding the source of the complexity.

Several large shallow earthquakes (M_s > 7.0) along the Middle American Trench have similarities and differences between them that may lead to a better understanding of fracture and subduction processes. They are predominantly thrust events consistent with the known subduction of the Cocos plate beneath N. America. Two events occurring along this subduction zone close to triple junctions show considerable complexity. This may be attributable to a more heterogeneous stress environment in these regions and as such has implications for other subduction zone boundaries.

An event which looks complex but is actually rather simple is the 1978 Bermuda earthquake (M_s ~ 6). It is located predominantly in the mantle. Its mechanism is one of pure thrust faulting with a strike N 20°W and dip 42°NE. Its apparent complexity is caused by local crustal structure. This is an important event in terms of understanding and estimating seismic hazard on the eastern seaboard of N. America.

A study of several large strike-slip continental earthquakes identifies characteristics which are common to them and may be useful in determining what to expect from the next great earthquake on the San Andreas fault. The events are the 1976 Guatemala earthquake on the Motagua fault and two events on the Anatolian fault in Turkey (the 1967, Mudurnu Valley and 1976, E. Turkey events). An attempt to model the complex P-waveforms of these events results in good synthetic fits for the Guatemala and Mudurnu Valley events. However, the E. Turkey event proves to be too complex as it may have associated thrust or normal faulting. Several individual sources occurring at intervals of between 5 and 20 seconds characterize the Guatemala and Mudurnu Valley events. The maximum size of an individual source appears to be bounded at about 5 x 10²⁶ dyne-cm. A detailed source study including directivity is performed on the Guatemala event. The source time history of the Mudurnu Valley event illustrates its significance in modeling strong ground motion in the near field. The complex source time series of the 1967 event produces amplitudes greater by a factor of 2.5 than a uniform model scaled to the same size for a station 20 km from the fault.

Three large and important earthquakes demonstrate an important type of complexity --- multiple-fault complexity. The first, the 1976 Philippine earthquake, an oblique thrust event, represents the first seismological evidence for a northeast dipping subduction zone beneath the island of Mindanao. A large event, following the mainshock by 12 hours, occurred outside the aftershock area and apparently resulted from motion on a subsidiary fault since the event had a strike-slip mechanism.

An aftershock of the great 1960 Chilean earthquake on June 6, 1960, proved to be an interesting discovery. It appears to be a large strike-slip event at the main rupture's southern boundary. It most likely occurred on the landward extension of the Chile Rise transform fault, in the subducting plate. The results for this event suggest that a small event triggered a series of slow events; the duration of the whole sequence being longer than 1 hour. This is indeed a "slow earthquake".

Perhaps one of the most complex of events is the recent Tangshan, China event. It began as a large strike-slip event. Within several seconds of the mainshock it may have triggered thrust faulting to the south of the epicenter. There is no doubt, however, that it triggered a large oblique normal event to the northeast, 15 hours after the mainshock. This event certainly contributed to the great loss of life-sustained as a result of the Tangshan earthquake sequence.

What has been learned from these studies has been applied to predict what one might expect from the next great earthquake on the San Andreas. The expectation from this study is that such an event would be a large complex event, not unlike, but perhaps larger than, the Guatemala or Mudurnu Valley events. That is to say, it will most likely consist of a series of individual events in sequence. It is also quite possible that the event could trigger associated faulting on neighboring fault systems such as those occurring in the Transverse Ranges. This has important bearing on the earthquake hazard estimation for the region.

On Ultrafast Time-Domain TeraHertz Spectroscopy in the Condensed Phase: Linear Spectroscopic Measurements of Hydrogen-Bond Dynamics of Astrochemical Ice Analogs and Nonlinear TeraHertz Kerr Effect Measurements of Vibrational Quantum Beats

Much of the chemistry that affects life on planet Earth occurs in the condensed phase. The TeraHertz (THz) or far-infrared (far-IR) region of the electromagnetic spectrum (from 0.1 THz to 10 THz, 3 cm^-1 to 300 cm^-1, or 3000 μm to 30 μm) has been shown to provide unique possibilities in the study of condensed-phase processes. The goal of this work is to expand the possibilities available in the THz region and undertake new investigations of fundamental interest to chemistry. Since we are fundamentally interested in condensed-phase processes, this thesis focuses on two areas where THz spectroscopy can provide new understanding: astrochemistry and solvation science. To advance these fields, we had to develop new instrumentation that would enable the experiments necessary to answer new questions in either astrochemistry or solvation science. We first developed a new experimental setup capable of studying astrochemical ice analogs in both the TeraHertz (THz), or far-Infrared (far-IR), region (0.3 - 7.5 THz; 10 - 250 cm^-1) and the mid-IR (400 - 4000 cm^-1). The importance of astrochemical ices lies in their key role in the formation of complex organic molecules, such as amino acids and sugars in space. Thus, the instruments are capable of performing variety of spectroscopic studies that can provide especially relevant laboratory data to support astronomical observations from telescopes such as the Herschel Space Telescope, the Stratospheric Observatory for Infrared Astronomy (SOFIA), and the Atacama Large Millimeter Array (ALMA). The experimental apparatus uses a THz time-domain spectrometer, with a 1750/875 nm plasma source and a GaP detector crystal, to cover the bandwidth mentioned above with ~10 GHz (~0.3 cm^-1) resolution.

Using the above instrumentation, experimental spectra of astrochemical ice analogs of water and carbon dioxide in pure, mixed, and layered ices were collected at different temperatures under high vacuum conditions with the goal of investigating the structure of the ice. We tentatively observe a new feature in both amorphous solid water and crystalline water at 33 cm^-1 (1 THz). In addition, our studies of mixed and layered ices show how it is possible to identify the location of carbon dioxide as it segregates within the ice by observing its effect on the THz spectrum of water ice. The THz spectra of mixed and layered ices are further analyzed by fitting their spectra features to those of pure amorphous solid water and crystalline water ice to quantify the effects of temperature changes on structure. From the results of this work, it appears that THz spectroscopy is potentially well suited to study thermal transformations within the ice.

To advance the study of liquids with THz spectroscopy, we developed a new ultrafast nonlinear THz spectroscopic technique: heterodyne-detected, ultrafast THz Kerr effect (TKE) spectroscopy. We implemented a heterodyne-detection scheme into a TKE spectrometer that uses a stilbazoiumbased THz emitter, 4-N,N-dimethylamino-4-N-methyl-stilbazolium 2,4,6-trimethylbenzenesulfonate (DSTMS), and high numerical aperture optics which generates THz electric field in excess of 300 kV/cm, in the sample. This allows us to report the first measurement of quantum beats at terahertz (THz) frequencies that result from vibrational coherences initiated by the nonlinear, dipolar interaction of a broadband, high-energy, (sub)picosecond THz pulse with the sample. Our instrument improves on both the frequency coverage, and sensitivity previously reported; it also ensures a backgroundless measurement of the THz Kerr effect in pure liquids. For liquid diiodomethane, we observe a quantum beat at 3.66 THz (122 cm^-1), in exact agreement with the fundamental transition frequency of the υ4 vibration of the molecule. This result provides new insight into dipolar vs. Raman selection rules at terahertz frequencies.

To conclude we discuss future directions for the nonlinear THz spectroscopy in the Blake lab. We report the first results from an experiment using a plasma-based THz source for nonlinear spectroscopy that has the potential to enable nonlinear THz spectra with a sub-100 fs temporal resolution, and how the optics involved in the plasma mechanism can enable THz pulse shaping. Finally, we discuss how a single-shot THz detection scheme could improve the acquisition of THz data and how such a scheme could be implemented in the Blake lab. The instruments developed herein will hopefully remain a part of the groups core competencies and serve as building blocks for the next generation of THz instrumentation that pushes the frontiers of both chemistry and the scientific enterprise as a whole.

Response of linear, viscous damped systems to excitations having time-varying frequency

The response of linear, viscous damped systems to excitations having time-varying frequency is the subject of exact and approximate analyses, which are supplemented by an analog computer study of single degree of freedom system response to excitations having frequencies depending linearly and exponentially on time.

The technique of small perturbations and the methods of stationary phase and saddle-point integration, as well as a novel bounding procedure, are utilized to derive approximate expressions characterizing the system response envelope—particularly near resonances—for the general time-varying excitation frequency.

Descriptive measurements of system resonant behavior recorded during the course of the analog study—maximum response, excitation frequency at which maximum response occurs, and the width of the response peak at the half-power level—are investigated to determine dependence upon natural frequency, damping, and the functional form of the excitation frequency.

The laboratory problem of determining the properties of a physical system from records of its response to excitations of this class is considered, and the transient phenomenon known as “ringing” is treated briefly.

It is shown that system resonant behavior, as portrayed by the above measurements and expressions, is relatively insensitive to the specifics of the excitation frequency-time relation and may be described to good order in terms of parameters combining system properties with the time derivative of excitation frequency evaluated at resonance.

One of these parameters is shown useful for predicting whether or not a given excitation having a time-varying frequency will produce strong or subtle changes in the response envelope of a given system relative to the steady-state response envelope. The parameter is shown, additionally, to be useful for predicting whether or not a particular response record will exhibit the “ringing” phenomenon.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Optimum linear coding for additive noise systems using information feedback

The feedback coding problem for Gaussian systems in which the noise is neither white nor statistically independent between channels is formulated in terms of arbitrary linear codes at the transmitter and at the receiver. This new formulation is used to determine a number of feedback communication systems. In particular, the optimum linear code that satisfies an average power constraint on the transmitted signals is derived for a system with noiseless feedback and forward noise of arbitrary covariance. The noisy feedback problem is considered and signal sets for the forward and feedback channels are obtained with an average power constraint on each. The general formulation and results are valid for non-Gaussian systems in which the second order statistics are known, the results being applicable to the determination of error bounds via the Chebychev inequality.

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 multiple scattering problem

The present work deals with the problem of the interaction of the electromagnetic radiation with a statistical distribution of nonmagnetic dielectric particles immersed in an infinite homogeneous isotropic, non-magnetic medium. The wavelength of the incident radiation can be less, equal or greater than the linear dimension of a particle. The distance between any two particles is several wavelengths. A single particle in the absence of the others is assumed to scatter like a Rayleigh-Gans particle, i.e. interaction between the volume elements (self-interaction) is neglected. The interaction of the particles is taken into account (multiple scattering) and conditions are set up for the case of a lossless medium which guarantee that the multiple scattering contribution is more important than the self-interaction one. These conditions relate the wavelength λ and the linear dimensions of a particle a and of the region occupied by the particles D. It is found that for constant λ/a, D is proportional to λ and that |Δχ|, where Δχ is the difference in the dielectric susceptibilities between particle and medium, has to lie within a certain range.

The total scattering field is obtained as a series the several terms of which represent the corresponding multiple scattering orders. The first term is a single scattering term. The ensemble average of the total scattering intensity is then obtained as a series which does not involve terms due to products between terms of different orders. Thus the waves corresponding to different orders are independent and their Stokes parameters add.

The second and third order intensity terms are explicitly computed. The method used suggests a general approach for computing any order. It is found that in general the first order scattering intensity pattern (or phase function) peaks in the forward direction Θ = 0. The second order tends to smooth out the pattern giving a maximum in the Θ = π/2 direction and minima in the Θ = 0 , Θ = π directions. This ceases to be true if ka (where k = 2π/λ) becomes large (> 20). For large ka the forward direction is further enhanced. Similar features are expected from the higher orders even though the critical value of ka may increase with the order.

The first order polarization of the scattered wave is determined. The ensemble average of the Stokes parameters of the scattered wave is explicitly computed for the second order. A similar method can be applied for any order. It is found that the polarization of the scattered wave depends on the polarization of the incident wave. If the latter is elliptically polarized then the first order scattered wave is elliptically polarized, but in the Θ = π/2 direction is linearly polarized. If the incident wave is circularly polarized the first order scattered wave is elliptically polarized except for the directions Θ = π/2 (linearly polarized) and Θ = 0, π (circularly polarized). The handedness of the Θ = 0 wave is the same as that of the incident whereas the handedness of the Θ = π wave is opposite. If the incident wave is linearly polarized the first order scattered wave is also linearly polarized. The second order makes the total scattered wave to be elliptically polarized for any Θ no matter what the incident wave is. However, the handedness of the total scattered wave is not altered by the second order. Higher orders have similar effects as the second order.

If the medium is lossy the general approach employed for the lossless case is still valid. Only the algebra increases in complexity. It is found that the results of the lossless case are insensitive in the first order of k_imD where k_im = imaginary part of the wave vector k and D a linear characteristic dimension of the region occupied by the particles. Thus moderately extended regions and small losses make (k_imD)² ≪ 1 and the lossy character of the medium does not alter the results of the lossless case. In general the presence of the losses tends to reduce the forward scattering.

I. Non-linear effects in a self-consistent calculation of SU_3 symmetry breaking in strong interaction coupling constants. II. Direct-channel reggeization of strong interaction scattering amplitudes

The thesis is divided into two parts. Part I generalizes a self-consistent calculation of residue shifts from SU₃ symmetry, originally performed by Dashen, Dothan, Frautschi, and Sharp, to include the effects of non-linear terms. Residue factorizability is used to transform an overdetermined set of equations into a variational problem, which is designed to take advantage of the redundancy of the mathematical system. The solution of this problem automatically satisfies the requirement of factorizability and comes close to satisfying all the original equations.

Part II investigates some consequences of direct channel Regge poles and treats the problem of relating Reggeized partial wave expansions made in different reaction channels. An analytic method is introduced which can be used to determine the crossed-channel discontinuity for a large class of direct-channel Regge representations, and this method is applied to some specific representations.

It is demonstrated that the multi-sheeted analytic structure of the Regge trajectory function can be used to resolve apparent difficulties arising from infinitely rising Regge trajectories. Also discussed are the implications of large collections of "daughter trajectories."

Two things are of particular interest: first, the threshold behavior in direct and crossed channels; second, the potentialities of Reggeized representations for us in self-consistent calculations. A new representation is introduced which surpasses previous formulations in these two areas, automatically satisfying direct-channel threshold constraints while being capable of reproducing a reasonable crossed channel discontinuity. A scalar model is investigated for low energies, and a relation is obtained between the mass of the lowest bound state and the slope of the Regge trajectory.

Non-Linear Scale Interactions in a Forced Turbulent Boundary Layer

This thesis explores the dynamics of scale interactions in a turbulent boundary layer through a forcing-response type experimental study. An emphasis is placed on the analysis of triadic wavenumber interactions since the governing Navier-Stokes equations for the flow necessitate a direct coupling between triadically consist scales. Two sets of experiments were performed in which deterministic disturbances were introduced into the flow using a spatially-impulsive dynamic wall perturbation. Hotwire anemometry was employed to measure the downstream turbulent velocity and study the flow response to the external forcing. In the first set of experiments, which were based on a recent investigation of dynamic forcing effects in a turbulent boundary layer, a 2D (spanwise constant) spatio-temporal normal mode was excited in the flow; the streamwise length and time scales of the synthetic mode roughly correspond to the very-large-scale-motions (VLSM) found naturally in canonical flows. Correlation studies between the large- and small-scale velocity signals reveal an alteration of the natural phase relations between scales by the synthetic mode. In particular, a strong phase-locking or organizing effect is seen on directly coupled small-scales through triadic interactions. Having characterized the bulk influence of a single energetic mode on the flow dynamics, a second set of experiments aimed at isolating specific triadic interactions was performed. Two distinct 2D large-scale normal modes were excited in the flow, and the response at the corresponding sum and difference wavenumbers was isolated from the turbulent signals. Results from this experiment serve as an unique demonstration of direct non-linear interactions in a fully turbulent wall-bounded flow, and allow for examination of phase relationships involving specific interacting scales. A direct connection is also made to the Navier-Stokes resolvent operator framework developed in recent literature. Results and analysis from the present work offer insights into the dynamical structure of wall turbulence, and have interesting implications for design of practical turbulence manipulation or control strategies.

On the uniqueness of singular solutions to boundary-initial value problems in linear elastodynamics

This investigation deals with certain generalizations of the classical uniqueness theorem for the second boundary-initial value problem in the linearized dynamical theory of not necessarily homogeneous nor isotropic elastic solids. First, the regularity assumptions underlying the foregoing theorem are relaxed by admitting stress fields with suitably restricted finite jump discontinuities. Such singularities are familiar from known solutions to dynamical elasticity problems involving discontinuous surface tractions or non-matching boundary and initial conditions. The proof of the appropriate uniqueness theorem given here rests on a generalization of the usual energy identity to the class of singular elastodynamic fields under consideration.

Following this extension of the conventional uniqueness theorem, we turn to a further relaxation of the customary smoothness hypotheses and allow the displacement field to be differentiable merely in a generalized sense, thereby admitting stress fields with square-integrable unbounded local singularities, such as those encountered in the presence of focusing of elastic waves. A statement of the traction problem applicable in these pathological circumstances necessitates the introduction of "weak solutions'' to the field equations that are accompanied by correspondingly weakened boundary and initial conditions. A uniqueness theorem pertaining to this weak formulation is then proved through an adaptation of an argument used by O. Ladyzhenskaya in connection with the first boundary-initial value problem for a second-order hyperbolic equation in a single dependent variable. Moreover, the second uniqueness theorem thus obtained contains, as a special case, a slight modification of the previously established uniqueness theorem covering solutions that exhibit only finite stress-discontinuities.