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Book Chapter

Ray theory

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Published:
January 01, 1980

Abstract

Suppose that the layers of the model in Figure 2-1 are inhomogeneous and isotropic; then both P- and S-wave velocities are functions of space coordinates but are independent of the direction of wave propagation. {Anisotropic media, in which velocities depend on the direction of wave propagation at each point, are considered only briefly in Chapter 9.) Isotropic media in solid contact with one another can host two independent types of waves: One propagates with the local velocity of compressional waves, the other with the local velocity of shear waves.

The wavefront of an elementary wave can be described by the moving surface t = τ (x0, y0, z0) defined with respect to some arbitrary right-hand [x0, y0, z0] coordinate system (Figure 2-1). The function τ(x0, y0, z0) is known as the phase function. It satisfies the eikonal equation, a first-order partial differential equation of fundamental importance, for it leads directly to the concept of rays (Officer, 1974). Trajectories orthogonal to moving wavefronts are designated as rays. These rays can as well be obtained as the extremals of Fermat's functional, which represents the traveltime between any two designated points. Strictly speaking, the concept of a ray is appropriate only if energy propagates along a particular path, as in the case of a geometrical optics solution to the wave equation (Kline and Kay, 1965). Such solutions are obtained from the wave equation when the propagating pulse has wavelengths that are small in comparison with the structural dimensions of a model (e.g., radii

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Interval Velocities from Seismic Reflection Time Measurements

Peter Hubral
Peter Hubral
Bundesanstalt fiir Geowissenschaften und Rohstoffe Rohstoffe Hannover, West Germany
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Theodor Krey
Theodor Krey
Consultant Prakla-Seismos Hannover, West Germany
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L.Larner Kenneth
L.Larner Kenneth
Western Geophysical Company Houston, Texas
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Society of Exploration Geophysicists
ISBN electronic:
9781560802501
Publication date:
January 01, 1980

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