Methods of multiparameter inversion of seismic data using the acoustic and elastic born approximations

This thesis presents two different forms of the Born approximations for acoustic and elastic wavefields and discusses their application to the inversion of seismic data. The Born approximation is valid for small amplitude heterogeneities superimposed over a slowly varying background. The first method is related to frequency-wavenumber migration methods. It is shown to properly recover two independent acoustic parameters within the bandpass of the source time function of the experiment for contrasts of about 5 percent from data generated using an exact theory for flat interfaces. The independent determination of two parameters is shown to depend on the angle coverage of the medium. For surface data, the impedance profile is well recovered.

The second method explored is mathematically similar to iterative tomographic methods recently introduced in the geophysical literature. Its basis is an integral relation between the scattered wavefield and the medium parameters obtained after applying a far-field approximation to the first-order Born approximation. The Davidon-Fletcher-Powell algorithm is used since it converges faster than the steepest descent method. It consists essentially of successive backprojections of the recorded wavefield, with angular and propagation weighing coefficients for density and bulk modulus. After each backprojection, the forward problem is computed and the residual evaluated. Each backprojection is similar to a before-stack Kirchhoff migration and is therefore readily applicable to seismic data. Several examples of reconstruction for simple point scatterer models are performed. Recovery of the amplitudes of the anomalies are improved with successive iterations. Iterations also improve the sharpness of the images.

The elastic Born approximation, with the addition of a far-field approximation is shown to correspond physically to a sum of WKBJ-asymptotic scattered rays. Four types of scattered rays enter in the sum, corresponding to P-P, P-S, S-P and S-S pairs of incident-scattered rays. Incident rays propagate in the background medium, interacting only once with the scatterers. Scattered rays propagate as if in the background medium, with no interaction with the scatterers. An example of P-wave impedance inversion is performed on a VSP data set consisting of three offsets recorded in two wells.

A measurement of the proton elastic form factors for 1 ≤ Q^2 ≤ 3 (GeV/c)^2

We report measurements of the proton form factors, G^p_E and G^p_M, extracted from elastic electron scattering in the range 1 ≤ Q^2 ≤ 3 (GeV/c)^2 with uncertainties of <15% in G^p_E and <3% in G^p_M. The results for G^p_E are somewhat larger than indicated by most theoretical parameterizations. The ratio of Pauli and Dirac form factors, Q^2(F^p_2/F^p_1), is lower in value and demonstrates less Q^2 dependence than these parameterizations have indicated. Comparisons are made to theoretical models, including those based on perturbative QCD, vector-meson dominance, QCD sum rules, and diquark constituents to the proton. A global extraction of the form factors, including previous elastic scattering measurements, is also presented.

Edge function method applied to thin plates resting on Winkler foundation

The Edge Function method formerly developed by Quinlan⁽²⁵⁾ is applied to solve the problem of thin elastic plates resting on spring supported foundations subjected to lateral loads the method can be applied to plates of any convex polygonal shapes, however, since most plates are rectangular in shape, this specific class is investigated in this thesis. The method discussed can also be applied easily to other kinds of foundation models (e.g. springs connected to each other by a membrane) as long as the resulting differential equation is linear. In chapter VII, solution of a specific problem is compared with a known solution from literature. In chapter VIII, further comparisons are given. The problems of concentrated load on an edge and later on a corner of a plate as long as they are far away from other boundaries are also given in the chapter and generalized to other loading intensities and/or plates springs constants for Poisson's ratio equal to 0.2

Part I - The radiation of elastic waves from a spherical cavity in a half space. Part II - Precision determination of focal depths and epicenters of earthquakes.

Part I: The dynamic response of an elastic half space to an explosion in a buried spherical cavity is investigated by two methods. The first is implicit, and the final expressions for the displacements at the free surface are given as a series of spherical wave functions whose coefficients are solutions of an infinite set of linear equations. The second method is based on Schwarz's technique to solve boundary value problems, and leads to an iterative solution, starting with the known expression for the point source in a half space as first term. The iterative series is transformed into a system of two integral equations, and into an equivalent set of linear equations. In this way, a dual interpretation of the physical phenomena is achieved. The systems are treated numerically and the Rayleigh wave part of the displacements is given in the frequency domain. Several comparisons with simpler cases are analyzed to show the effect of the cavity radius-depth ratio on the spectra of the displacements.

Part II: A high speed, large capacity, hypocenter location program has been written for an IBM 7094 computer. Important modifications to the standard method of least squares have been incorporated in it. Among them are a new way to obtain the depth of shocks from the normal equations, and the computation of variable travel times for the local shocks in order to account automatically for crustal variations. The multiregional travel times, largely based upon the investigations of the United States Geological Survey, are confronted with actual traverses to test their validity.

It is shown that several crustal phases provide control enough to obtain good solutions in depth for nuclear explosions, though not all the recording stations are in the region where crustal corrections are considered. The use of the European travel times, to locate the French nuclear explosion of May 1962 in the Sahara, proved to be more adequate than previous work.

A simpler program, with manual crustal corrections, is used to process the Kern County series of aftershocks, and a clearer picture of tectonic mechanism of the White Wolf fault is obtained.

Shocks in the California region are processed automatically and statistical frequency-depth and energy depth curves are discussed in relation to the tectonics of the area.

PP backward elastic scattering between 0.70 and 2.37 GeV/c

̄pp backward elastic scattering has been measured for the cos θ_cm region between – 1.00 and – 0.88 and for the incident ̄p laboratory momentum region between 0.70 and 2.37 GeV/c. These measurements, done in intervals of approximately 0.1 GeV/c, have been performed at the Alternating Gradient Synchrotron at Brookhaven National Laboratory during the winter of 1968. The measured differential cross sections, binned in cos θ_cm intervals of 0.02, have statistical errors of about 10%. Backward dipping exists below 0.95 GeV/c and backward peaking above 0.95 GeV/c. The 180˚ differential cross section extrapolated from our data shows a sharp dip centered at 0.95 GeV/c and a broad hump centered near 1.4 GeV/c. Our data have been interpreted in terms of resonance effects and in terms of diffraction dominance effects.

The influence of the NN* channel on elastic NN scattering

The problem of two channels NN and NN*, coupled through unitarity, is studied to see whether sizable peaks can be produced in elastic nucleon-nucleon scattering due to the opening of a strongly coupled inelastic channel. One-pion-exchange (OPE) interactions are calculated to estimate the NN*→NN* and NN→NN* amplitudes. The OPE production amplitudes are used as the sole dynamical input to drive the multichannel ND^-1 equations in the determinental approximation, and the effect on the J = 2+ (¹D₂) elastic NN scattering amplitude is studied as the width of the unstable N* and strength of coupling to the inelastic channel are varied. A cusp-type enhancement appears in the NN channel near the NN* threshold but for the known value of the N* width the cusp is so “wooly” that any resulting elastic peak is likely to be too broad and diminished in height to be experimentally prominent. A brief survey of current experimental knowledge of the real part of the ¹D₂ NN phase shift near the NN* threshold is given, and the values are found to be much smaller than the nearly “resonant” phase shifts predicted by the coupled channel model.

On the transport properties of fluid-particle flow

The hydrodynamic forces acting on a solid particle in a viscous, incompressible fluid medium at low Reynolds number flow is investigated mathematically as a prerequisite to the understanding of transport processes in two-phase flow involving solid particles and fluid. Viscous interaction between a small number of spherical particles and continuous solid boundaries as well as fluid interface are analyzed under a “point-force” approximation. Non-spherical and elastic spherical particles in a simple shear flow area are then considered. Non-steady motion of a spherical particle is briefly touched upon to illustrate the transient effect of particle motion.

A macroscopic continuum description of particle-fluid flow is formulated in terms of spatial averages yielding a set of particle continuum and bulk fluid equations. Phenomenological formulas describing the transport processes in a fluid medium are extended to cases where the volume concentration of solid particles is sufficiently high to exert an important influence. Hydrodynamic forces acting on a spherical solid particle in such a system, e.g. drag, torque, rotational coupling force, and viscous collision force between streams of different sized particles moving relative to each other are obtained. Phenomenological constants, such as the shear viscosity coefficient, and the diffusion coefficient of the bulk fluid, are found as a function of the material properties of the constituents of the two-phase system and the volume concentration of solid. For transient heat conduction phenomena, it is found that the introduction of a complex conductivity for the bulk fluid permits a simple mathematical description of this otherwise complicated process. The rate of heat transfer between particle continuum and bulk fluid is also investigated by means of an Oseen-type approximation to the energy equation.

Energy inequalities and error estimates for axisymmetric torsion of thin elastic shells of revolution

The problem motivating this investigation is that of pure axisymmetric torsion of an elastic shell of revolution. The analysis is carried out within the framework of the three-dimensional linear theory of elastic equilibrium for homogeneous, isotropic solids. The objective is the rigorous estimation of errors involved in the use of approximations based on thin shell theory.

The underlying boundary value problem is one of Neumann type for a second order elliptic operator. A systematic procedure for constructing pointwise estimates for the solution and its first derivatives is given for a general class of second-order elliptic boundary-value problems which includes the torsion problem as a special case.

The method used here rests on the construction of “energy inequalities” and on the subsequent deduction of pointwise estimates from the energy inequalities. This method removes certain drawbacks characteristic of pointwise estimates derived in some investigations of related areas.

Special interest is directed towards thin shells of constant thickness. The method enables us to estimate the error involved in a stress analysis in which the exact solution is replaced by an approximate one, and thus provides us with a means of assessing the quality of approximate solutions for axisymmetric torsion of thin shells.

Finally, the results of the present study are applied to the stress analysis of a circular cylindrical shell, and the quality of stress estimates derived here and those from a previous related publication are discussed.

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.

Large plane deformations of thin elastic sheets of neo-hookean material

Large plane deformations of thin elastic sheets of neo-Hookean material are considered and a method of successive substitutions is developed to solve problems within the two-dimensional theory of finite plane stress. The first approximation is determined by linear boundary value problems on two harmonic functions, and it is approached asymptotically at very large extensions in the plane of the sheet. The second and higher approximations are obtained by solving Poisson equations. The method requires modification when the membrane has a traction-free edge.

Several problems are treated involving infinite sheets under uniform biaxial stretching at infinity. First approximations are obtained when a circular or elliptic inclusion is present and when the sheet has a circular or elliptic hole, including the limiting cases of a line inclusion and a straight crack or slit. Good agreement with exact solutions is found for circularly symmetric deformations. Other examples discuss the stretching of a short wide strip, the deformation near a boundary corner which is traction-free, and the application of a concentrated load to a boundary point.

Using Earth Deformation Caused by Surface Mass Loading to Constrain the Elastic Structure of the Crust and Mantle

Surface mass loads come in many different varieties, including the oceans, atmosphere, rivers, lakes, glaciers, ice caps, and snow fields. The loads migrate over Earth's surface on time scales that range from less than a day to many thousand years. The weights of the shifting loads exert normal forces on Earth's surface. Since the Earth is not perfectly rigid, the applied pressure deforms the shape of the solid Earth in a manner controlled by the material properties of Earth's interior. One of the most prominent types of surface mass loading, ocean tidal loading (OTL), comes from the periodic rise and fall in sea-surface height due to the gravitational influence of celestial objects, such as the moon and sun. Depending on geographic location, the surface displacements induced by OTL typically range from millimeters to several centimeters in amplitude, which may be inferred from Global Navigation and Satellite System (GNSS) measurements with sub-millimeter precision. Spatiotemporal characteristics of observed OTL-induced surface displacements may therefore be exploited to probe Earth structure. In this thesis, I present descriptions of contemporary observational and modeling techniques used to explore Earth's deformation response to OTL and other varieties of surface mass loading. With the aim to extract information about Earth's density and elastic structure from observations of the response to OTL, I investigate the sensitivity of OTL-induced surface displacements to perturbations in the material structure. As a case study, I compute and compare the observed and predicted OTL-induced surface displacements for a network of GNSS receivers across South America. The residuals in three distinct and dominant tidal bands are sub-millimeter in amplitude, indicating that modern ocean-tide and elastic-Earth models well predict the observed displacement response in that region. Nevertheless, the sub-millimeter residuals exhibit regional spatial coherency that cannot be explained entirely by random observational uncertainties and that suggests deficiencies in the forward-model assumptions. In particular, the discrepancies may reveal sensitivities to deviations from spherically symmetric, non-rotating, elastic, and isotropic (SNREI) Earth structure due to the presence of the South American craton.

Determination and analysis of the optical constants of thin films of nickel(II) and copper(II) hydrazone complexes by spectroscopic ellipsometry

Thin films of four nickel(II) and copper(II) hydrazone complexes, which will hopefully be used as recording layers for the next-generation of high-density recordable disks, were prepared by using the spin-coating method. Absorption spectra of the thin films on K9 optical glass substrates in the 300-700 nm wavelength region were measured. Optical constants (complex refractive indices N) and thickness d of the thin films prepared on single-crystal silicon substrates in the 275-675 nm wavelength region were investigated on a rotating analyzer-polarizer scanning ellipsometer by fitting the measured ellipsometric angles (Psi(lambda) and Delta(lambda)) with a 3-layer model (Si/dye film/air). The dielectric functions epsilon and absorption coefficients alpha as a function of the wavelength were then calculated. Additionally, a design to achieve high reflectivity and optimum dye film thickness with an appropriate reflective layer was performed with the Film Wizard software using a multilayered model (PC substrate/reflective layer/dye film/air) at 405 nm wavelength.