Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references (leaves 166-173). Also available in an electronic version.

The propagation of seismic waves may be described in the space-frequency domain by the Rayleigh-Sommerfeld convolution integral. The kernel of this integral is called a spatial wavelet and it embodies the physics and geometry of the propagation problem. The concepts of spatial convolution and spatial wavelet are simple and are similar to other topics studied by geophysicists. With a view to understanding these concepts, some aspects of spatial wavelets and their application to two-dimensional, zero-offset, acoustic seismic modelling were investigated. In studying the spatial wavelet, two topics in particular were examined: spatial aliasing and wavelet truncation. Spatial aliasing arises from the need to compute a discrete wavelet for implementation on a computer. This problem was solved by using an analytic expression for the spatial wavelet in the Fourier (wavenumber) domain. In the wavenumber domain the wavelet was windowed by a fourth order Butterworth operator, which removed aliasing. This technique is simple and flexible in its use. The second problem of wavelet truncation is due to the necessity of having a wavelet of finite length. A length limiting scheme based upon on the energy content of a wavelet was developed. It was argued that if that if a large portion of the wavelet energy was contained in a finite number of samples, then truncation at that sample would incur a minimal loss of information. Numerical experiments showed this to be true. The smallest length wavelet was found to depend on temporal frequency, medium velocity and extrapolation increment. The combined effects of these two solutions to the practical problem of computing a spatial wavelet resulted in two drawbacks. First, the wavelets provide modelling capabilities up to structural dips of 30 degrees. Second, there is a potential for instability due to recursive application of the wavelet. However, neither of these difficulties hampered the modelling of fairly complex structures. The spatial wavelet concept was applied to seismic modelling for media of varying complexity. Homogeneous velocity models were used to demonstrate diffraction evolution, dip limitations and imaging of curved structures. The quality of modelling was evaluated by migrating the modelled data to recover the time-image model of the reflection structure. Migrations of dipping and synform structures indicated that the modelled results were of a high calibre. Horizontally stratified velocity models were also examined for dipping and synform structures. Modelling these reflection structures showed that the introduction of a depth variable velocity profile has a tremendous influence on the synthetic seismic section. Again, migration proved that the quality of the data was excellent. Finally, the spatial wavelet algorithm was extended to the case of laterally varying velocity structures. The effects of space variant spatial convolution in the presence of a smoothed velocity field were examined. Smoothed velocity fields were computed by a simple weighted averaging procedure. The weighting function used was a decaying exponential whose decay rate determined the amount of smoothing. Seis-mograms computed for this case showed that the algorithm gave smoother and more continuous reflection signatures when the velocity field has been smoothed so that the largest lateral velocity gradient corresponded to the lower end of the temporal frequency band of the spatial wavelets. In this respect, the results are similar to those of geometric ray theory. Also, the travel times of these models compared favourably with those of ray tracings. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

The history of diffraction theory, exact frequency domain solutions and selected past time domain solutions are briefly reviewed. Exact time domain solutions for scattering of plane, cylindrical and spherical acoustic waves by a half plane are derived by inverse Fourier transforming the frequency domain integral solutions. The solutions consist of two diffraction terms, a reflected term and a direct term. The diffracting edge induces step function discontinuities in the direct and reflected terms at two shadow boundaries. At each boundary, the associated diffraction term reaches a maximum amplitude of half the geometrical optics term and has a signum function discontinuity, so that the total field remains continuous. A physical interpretation is developed in terms of Huygen's principle near the diffracting edge. Solutions for practical point source configurations are evaluated by numerically convolving the impulse diffraction responses with a wavelet. The numerical problems presented by convolution with a singular, truncated operator are solved by analytically derived correction techniques, which are favourably compared to those used by earlier authors. The diffraction solution collapses into a compact discretized formulation. The half plane is shown to be a limiting form of wedge solutions, which can thus be computed using similar algorithms. Two zero offset sections are produced and compared to approximate Kirchhoff integral solutions. The exact diffraction hyperbola is noticeably non-symmetric, with higher amplitudes on the reflector side of the edge. Near the apex of the hyperbola the Kirchhoff solution is nearly equivalent to the exact diffraction term symmetric in amplitude about the reflection shadow boundary but fails to describe the other, low amplitude, term equivalent to half the response of a line scatterer. The differences are more noticeable on the flanks of the hyperbola, where the two terms are comparable in amplitude, and at shallow depths, due to an aperture effect. Increasing either the depth of the edge or the angle of the seismic line to the normal to the edge results in a flatter diffraction hyperbola showing little amplitude variation with moveout. As the seismic line becomes parallel to the edge the diffraction curve becomes flat and is indistinguishable from a reflection event. At great depth diffraction events may be mistaken for reflection events as well. Examples of CDP and CSP gathers, when compared to the Common Offset (CO) gathers, demonstrate that CO gathers are optimal for diffraction processing. Also, since the diffraction moveout and reflection moveout curves differ widely except for depth points near the edge, normal moveout stacking will distort the diffractions and diffraction stacking is essential to retain diffraction information. Strips of varying width are modelled by superposition of half planes to demonstrate resolution effects and show that the limit of a strip is a line scatterer. A dipping strip and an offset half plane model are produced and added for later comparison with wedge solutions. / Science, Faculty of / Earth, Ocean and Atmospheric Sciences, Department of / Graduate

No description available.

published_or_final_version / Applied Geosciences / Master / Master of Science

No description available.

Graduation date: 1997

Graduation date: 1987 / Best scan available for figures.

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Earth and Planetary Science, 1982. / Supervised by Sean C. Solomon. Vita. Includes bibliographical references (leaves 312-321).

Thesis (Ph. D.)--Massachusetts Institute of Technology and Woods Hole Oceanographic Institution, 1982. / Includes bibliographical references (p. 312-321).