Book Chapter

# Weak-contrast edge and vertex diffractions in anisotropic elastic media

By
Martin Tygel
Martin Tygel
a
b
The Norwegian University of Science and Technology, Department of Petroleum Engineering and Applied Geophysics, S.P. Andersensv. 15A, N-7034 Trondheim, Norway
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Bjørn Ursin
Bjørn Ursin
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Published:
January 01, 2016

## Abstract

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.

### Contents

Society of Exploration Geophysicists Geophysics Reprint Series

# Seismic Diffraction

Kamil Klem-Musatov
Kamil Klem-Musatov
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Henning Hoeber
Henning Hoeber
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Michael Pelissier
Michael Pelissier
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Tijmen Jan Moser
Tijmen Jan Moser
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Society of Exploration Geophysicists
Volume
30
ISBN electronic:
9781560803188
Publication date:
January 01, 2016

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