High frequency wave propagation in the earth: theory and observation

The wave-theoretical analysis of acoustic and elastic waves refracted by a spherical boundary across which both velocity and density increase abruptly and thence either increase or decrease continuously with depth is formulated in terms of the general problem of waves generated at a steady point source and scattered by a radially heterogeneous spherical body. A displacement potential representation is used for the elastic problem that results in high frequency decoupling of P-SV motion in a spherically symmetric, radially heterogeneous medium. Through the application of an earth-flattening transformation on the radial solution and the Watson transform on the sum over eigenfunctions, the solution to the spherical problem for high frequencies is expressed as a Weyl integral for the corresponding half-space problem in which the effect of boundary curvature maps into an effective positive velocity gradient. The results of both analytical and numerical evaluation of this integral can be summarized as follows for body waves in the crust and upper mantle:

1) In the special case of a critical velocity gradient (a gradient equal and opposite to the effective curvature gradient), the critically refracted wave reduces to the classical head wave for flat, homogeneous layers.

2) For gradients more negative than critical, the amplitude of the critically refracted wave decays more rapidly with distance than the classical head wave.

3) For positive, null, and gradients less negative than critical, the amplitude of the critically refracted wave decays less rapidly with distance than the classical head wave, and at sufficiently large distances, the refracted wave can be adequately described in terms of ray-theoretical diving waves. At intermediate distances from the critical point, the spectral amplitude of the refracted wave is scalloped due to multiple diving wave interference.

These theoretical results applied to published amplitude data for P-waves refracted by the major crustal and upper mantle horizons (the Pg, P*, and Pn travel-time branches) suggest that the 'granitic' upper crust, the 'basaltic' lower crust, and the mantle lid all have negative or near-critical velocity gradients in the tectonically active western United States. On the other hand, the corresponding horizons in the stable eastern United States appear to have null or slightly positive velocity gradients. The distribution of negative and positive velocity gradients correlates closely with high heat flow in tectonic regions and normal heat flow in stable regions. The velocity gradients inferred from the amplitude data are generally consistent with those inferred from ultrasonic measurements of the effects of temperature and pressure on crustal and mantle rocks and probable geothermal gradients. A notable exception is the strong positive velocity gradient in the mantle lid beneath the eastern United States (2 x 10^-3 sec^-1), which appears to require a compositional gradient to counter the effect of even a small geothermal gradient.

New seismic-refraction data were recorded along a 800 km profile extending due south from the Canadian border across the Columbia Plateau into eastern Oregon. The source for the seismic waves was a series of 20 high-energy chemical explosions detonated by the Canadian government in Greenbush Lake, British Columbia. The first arrivals recorded along this profile are on the Pn travel-time branch. In northern Washington and central Oregon their travel time is described by T = Δ/8.0 + 7.7 sec, but in the Columbia Plateau the Pn arrivals are as much as 0.9 sec early with respect to this line. An interpretation of these Pn arrivals together with later crustal arrivals suggest that the crust under the Columbia Plateau is thinner by about 10 km and has a higher average P-wave velocity than the 35-km-thick, 62-km/sec crust under the granitic-metamorphic terrain of northern Washington. A tentative interpretation of later arrivals recorded beyond 500 km from the shots suggests that a thin 8.4-km/sec horizon may be present in the upper mantle beneath the Columbia Plateau and that this horizon may form the lid to a pronounced low-velocity zone extending to a depth of about 140 km.

Real-time load-side control of electric power systems

Two trends are emerging from modern electric power systems: the growth of renewable (e.g., solar and wind) generation, and the integration of information technologies and advanced power electronics. The former introduces large, rapid, and random fluctuations in power supply, demand, frequency, and voltage, which become a major challenge for real-time operation of power systems. The latter creates a tremendous number of controllable intelligent endpoints such as smart buildings and appliances, electric vehicles, energy storage devices, and power electronic devices that can sense, compute, communicate, and actuate. Most of these endpoints are distributed on the load side of power systems, in contrast to traditional control resources such as centralized bulk generators. This thesis focuses on controlling power systems in real time, using these load side resources. Specifically, it studies two problems.

(1) Distributed load-side frequency control: We establish a mathematical framework to design distributed frequency control algorithms for flexible electric loads. In this framework, we formulate a category of optimization problems, called optimal load control (OLC), to incorporate the goals of frequency control, such as balancing power supply and demand, restoring frequency to its nominal value, restoring inter-area power flows, etc., in a way that minimizes total disutility for the loads to participate in frequency control by deviating from their nominal power usage. By exploiting distributed algorithms to solve OLC and analyzing convergence of these algorithms, we design distributed load-side controllers and prove stability of closed-loop power systems governed by these controllers. This general framework is adapted and applied to different types of power systems described by different models, or to achieve different levels of control goals under different operation scenarios. We first consider a dynamically coherent power system which can be equivalently modeled with a single synchronous machine. We then extend our framework to a multi-machine power network, where we consider primary and secondary frequency controls, linear and nonlinear power flow models, and the interactions between generator dynamics and load control.

(2) Two-timescale voltage control: The voltage of a power distribution system must be maintained closely around its nominal value in real time, even in the presence of highly volatile power supply or demand. For this purpose, we jointly control two types of reactive power sources: a capacitor operating at a slow timescale, and a power electronic device, such as a smart inverter or a D-STATCOM, operating at a fast timescale. Their control actions are solved from optimal power flow problems at two timescales. Specifically, the slow-timescale problem is a chance-constrained optimization, which minimizes power loss and regulates the voltage at the current time instant while limiting the probability of future voltage violations due to stochastic changes in power supply or demand. This control framework forms the basis of an optimal sizing problem, which determines the installation capacities of the control devices by minimizing the sum of power loss and capital cost. We develop computationally efficient heuristics to solve the optimal sizing problem and implement real-time control. Numerical experiments show that the proposed sizing and control schemes significantly improve the reliability of voltage control with a moderate increase in cost.

Design and Applications of a Decade-Spanning Terahertz Frequency Comb Spectrometer: Doppler-limited Rotational Spectroscopy of Methanol and Methanol-OD

This thesis details the design and applications of a terahertz (THz) frequency comb spectrometer. The spectrometer employs two offset locked Ti:Sapphire femtosecond oscillators with repetition rates of approximately 80 MHz, offset locked at 100 Hz to continuously sample a time delay of 12.5 ns at a maximum time delay resolution of 15.6 fs. These oscillators emit continuous pulse trains, allowing the generation of a THz pulse train by the master, or pump, oscillator and the sampling of this THz pulse train by the slave, or probe, oscillator via the electro-optic effect. Collecting a train of 16 consecutive THz pulses and taking the Fourier transform of this pulse train produces a decade-spanning frequency comb, from 0.25 to 2.5 THz, with a comb tooth width of 5 MHz and a comb tooth spacing of ~80 MHz. This frequency comb is suitable for Doppler-limited rotational spectroscopy of small molecules. Here, the data from 68 individual scans at slightly different pump oscillator repetition rates were combined, producing an interleaved THz frequency comb spectrum, with a maximum interval between comb teeth of 1.4 MHz, enabling THz frequency comb spectroscopy.

The accuracy of the THz frequency comb spectrometer was tested, achieving a root mean square error of 92 kHz measuring selected absorption center frequencies of water vapor at 10 mTorr, and a root mean square error of 150 kHz in measurements of a K-stack of acetonitrile. This accuracy is sufficient for fitting of measured transitions to a model Hamiltonian to generate a predicted spectrum for molecules of interest in the fields of astronomy and physical chemistry. As such, the rotational spectra of methanol and methanol-OD were acquired by the spectrometer. Absorptions from 1.3 THz to 2.0 THz were compared to JPL catalog data for methanol and the spectrometer achieved an RMS error of 402 kHz, improving to 303 kHz when excluding low signal-to-noise absorptions. This level of accuracy compares favorably with the ~100 kHz accuracy achieved by JPL frequency multiplier submillimeter spectrometers. Additionally, the relative intensity performance of the THz frequency comb spectrometer is linear across the entire decade-spanning bandwidth, making it the preferred instrument for recovering lineshapes and taking absolute intensity measurements in the THz region. The data acquired by the Terahertz Frequency Comb Spectrometer for methanol-OD is of comparable accuracy to the methanol data and may be used to refine the fit parameters for the predicted spectrum of methanol-OD.