Ray theory

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	1
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

The concept of a ray is extremely useful. Rays are curves drawn in space, and they correspond to the directions of flow of propagated energy. In other word, rays are flow lines. Being flow lines, rays cannot cross each other. As such, the ray is a mathematical device rather than a physical entity. In practice, we can produce very narrow beams or pencils (for example, a laser beam), and we might imagine a ray to be the unattainable narrowness limit of such a beam. We recall that an isotropic medium is a substance for which each physical property at any point has the same value when measured in different directions. An isotropic medium can be either homogeneous (i.e., consisting of points all of the same kind) or inhomogeneous (not homogeneous). In an isotropic medium, rays constitute orthogonal trajectories of the wavefronts, that is to say, such rays are lines normal to the wavefronts at all points of intersection. In such a medium, a ray is evidently parallel to the propagation vector. However, this ceases to be the case in anisotropic materials (whose properties vary as a function of direction).

Figure 15.  Corresponding points.

Within homogeneous isotropic materials, rays are straight lines. By symmetry, they cannot bend in any preferred direction because no such preferred direction exists. Moreover, because the propagation speed is identical in all directions, the spatial separation between two wavefronts, measured along rays, must be the same everywhere. Points at which a single ray intersects a set of wavefronts are called corresponding points, as for example points A, B, and C in Figure 15. Evidently the separation in time between any two corresponding points on any two sequential wavefronts is identical. In other words, if wavefront Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S_{{\rm l}}} transforms into wavefront $S_{\rm {2}}$ after a time $\Delta t$, the distance between corresponding points on any ray will be traversed in the same time $\Delta t$. This is true even if the wavefronts travel from one homogeneous isotropic medium into another, and it simply means that every point on $S_{\rm {1}}$ can be imagined to follow the path of a ray that arrives at $S_{\rm {2}}$ in the time $\Delta t$.

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Table of Contents (book)

Also in this chapter

• Introduction
• Wavefronts and raypaths
• d’Alembert’s solution
• One-dimensional waves
• Sinusoidal waves
• Phase velocity
• Wave pulses
• Geometric seismology
• The speed of light
• Huygens’ principle
• Reflection and refraction
• Fermat’s principle
• Fermat’s principle and reflection and refraction
• Diffraction
• Analogy
• Appendix A: Exercise

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