# Geometric seismology

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 |

For a geophysicist to carry out sound seismic interpretation, he or she must have a good understanding of the mechanism by which elastic waves are propagated through various materials. He must know how propagation is affected by types of rock formations and structures through which the waves pass. The clearer is his understanding of these physical processes and the better is his knowledge of the local geology, the sounder his resulting interpretation will be. If the recorded events consist of distinct waveforms, reflected or refracted from essentially plane-parallel horizontal subsurface strata, processing and interpretation usually present no great difficulties. However, if the travel paths are influenced by irregular subsurface features, the seismic recordings rapidly become complicated. Under those circumstances, the deleterious effects of scattering, refraction, focusing, and other disturbing influences can be so severe that they render the records extremely difficult to process and interpret.

To gain the most comprehensive knowledge of the details of wave propagation in the earth, it is necessary to study the theory of elasticity (Robinson and Clark, 1988a^{[1]}). This vast subject deals with the elastic properties of physical media and attempts to explain the manner in which solids are deformed under applied stresses. It further seeks to explain the possible types and modes of reactions, both static and dynamic, that result when a solid is acted on by external agencies. It describes in detail how motion, momentum, and energy, initiated by a local disturbance, are propagated to other parts of an extended solid. In particular, this theory provides us with a detailed exposition of the complex processes of elastic wave propagation in solids. The general theory has been a subject for investigation by mathematical physicists for many generations and, as a consequence, it has attained a high degree of completeness. However, the theory rapidly becomes mathematically difficult. Practical seismic applications almost invariably require numerical work on a computer. In exploration geophysics, the seismic waves propagate through subterranean rocks as well as through any overlying water body (Figure 6).

Fortunately, simplifying approaches can explain many of the phenomena associated with seismic wave propagation. In many respects, seismic waves are similar to light waves. During the past several centuries, physicists and mathematicians have built up a very extensive body of principles, based almost entirely on geometric reasoning, to explain what is known about light. This doctrine deals exhaustively with the ideas of raypaths and wavefronts and is known as *geometric optics*. It is readily possible to adapt many of these principles to explain seismic waves as well as acoustic waves. These fields of study are known as *geometric seismology* and *geometric acoustics*. Although this approach does not suffice to explain everything in which we are interested as geophysicists, it quite adequately accounts for many important phenomena. Insofar as it is capable of dealing with problems in seismic wave propagation, it agrees with the dynamic theory of elasticity.

In geologically complex areas where hydrocarbon exploration is to occur, the effectiveness of the seismic method will depend to a large extent on the ability of geophysicists to interpret the more difficult data. This in turn depends largely on a thorough understanding of the mechanisms of seismic wave propagation. The principles of geometric seismology provide good insights into many of the essential physical processes that give rise to the original data.

The geometric theory of wave propagation deals with wavefronts and raypaths. In a homogeneous and isotropic medium, the raypaths are straight lines at right angles to the wavefronts. The movement of a water wave gives a simple example of wave propagation. From a point disturbance on the surface of a pond, for instance, circular waves proceed outward in all directions. The concentric circles are the wavefronts and the radii of the circles are the raypaths. The geometric theory of wave propagation does not deal in any way with such specific rock properties as density, compressibility, and rigidity. In fact, it is incapable of dealing with them. It cannot deal quantitatively with such important matters as motion and momentum of the portions of a body traversed by a wave either. Likewise, the energy content and the shape of the wave, or the distinction between longitudinal and transverse waves, cannot be treated. Of all the physical properties of a rock, a single one - the *velocity* with which the rock transmits elastic waves - is used in the geometric theory.

In many rocks, especially the crystalline types, the velocity at any point is dependent on the direction of propagation of the wave at that point. For instance, the vertical velocity of propagation can differ from the horizontal velocity of propagation at a given locality. When this is the case, the rock is called *anisotropic*. Unless otherwise stated, we will deal only with rocks that are *isotropic* - that is, those for which the propagation velocity of seismic waves is independent of direction. A good example of an anisotropic rock is Iceland spar, which is basically a clear cleaved fragment of a completely colorless (icelike) form of calcite. Iceland spar displays the classic cleavage form of calcite, the rhombohedron, and it best demonstrates the property of calcite known as *double refraction*.

Another type of inhomogeneity exists with which we often will be concerned. The rocks might be isotropic everywhere, yet the absolute velocity can vary from place to place. Normally, the velocity in sedimentary rocks increases with depth. This is called vertical velocity variation. The velocity also can change horizontally, in which case we have horizontal, or lateral, velocity variation. When such variations are encountered, methods must be devised to take them into account properly.

## References

- ↑ Robinson, E. A., and R. D. Clark, 1988a, Elasticity: Stress and strain: The Leading Edge,
**7**, no. 2, 16–19, 29.

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## Also in this chapter

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