UBC Theses and Dissertations
Derivation and practical application of exact time domain solutions for diffraction of acoustic waves by a half plane Dalton, David Raymond
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.
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