Weak-contrast edge and vertex diffractions in anisotropic elastic media
Martin Tygel, Bjørn Ursin, 2016. "Weak-contrast edge and vertex diffractions in anisotropic elastic media", Seismic Diffraction, Kamil Klem-Musatov, Henning Hoeber, Michael Pelissier, Tijmen Jan Moser
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Diffraction phenomena caused by edges and vertices are important in a number of studies and applications of wave propagation methods. Diffractions play a fundamental role, for example, in the interpretation of seismic data from faults and other complex geological structures in which petroleum reservoirs are located.
The geometric theory of diffraction provides a good description of the diffraction of scalar waves from an edge in a homogeneous medium assuming free or rigid boundary conditions. The solution becomes infinite at the shadow boundary between the illuminated zone and the shadow zone, so that, within this region, a boundary layer solution must be used. This can be achieved by analytic continuation of the geometrical ray solution or, equivalently, using a one-dimensional or two-dimensional diffusion equation for modelling the amplitude of the diffracted wave.
An alternative, older method for obtaining asymptotic expressions for different parts of the wavefield is by asymptotic expansion of the Kirchhoff integral. The authors have developed a new reciprocal surface-scattering integral by applying the divergence theorem to the Born volume integral. This new integral is called the Born-Kirchhoff (BK) integral. The scattering surface is a finite sum of smooth surfaces separated by smooth curves with a finite number of corners. The BK integral is now a sum of integrals over each smooth surface. These integrals are evaluated by the method of stationary phase, resulting in specularly reflected waves and boundary diffracted waves represented by a line integral along the boundary of each surface. Further asymptotic evaluation of these line integrals results in expressions for edge-and corner-diffracted waves. Smooth approximations of the wavefield are given for the case that a reflection point and an edge-diffraction point are close to each other, and when an edge-diffraction point and a corner-diffraction point are close to each other.
These formulas can be cascaded to provide asymptotic expressions for multiple converted, reflected, transmitted and diffracted waves in anisotropic, inhomogeneous, elastic media. In the case that the rays are well separated from all shadow zones, the new expressions satisfy reciprocity.
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The use of diffraction imaging to complement the seismic reflection method is rapidly gaining momentum in the oil and gas industry. As the industry moves toward exploiting smaller and more complex conventional reservoirs and extensive new unconventional resource plays, the application of the seismic diffraction method to image sub-wavelength features such as small-scale faults, fractures and stratigraphic pinchouts is expected to increase dramatically over the next few years. “Seismic Diffraction” covers seismic diffraction theory, modeling, observation, and imaging. Papers and discussion include an overview of seismic diffractions, including classic papers which introduced the potential of diffraction phenomena in seismic processing; papers on the forward modeling of seismic diffractions, with an emphasis on the theoretical principles; papers which describe techniques for diffraction mathematical modeling as well as laboratory experiments for the physical modeling of diffractions; key papers dealing with the observation of seismic diffractions, in near-surface-, reservoir-, as well as crustal studies; and key papers on diffraction imaging.