The role of analogy is a recurrent theme in science. The same mathematics often can be used to describe physical phenomena of different natures. Analogy is one of the strongest methods available with which to explain old phenomena and to discover new ones. To understand the concepts of science, we have to use analogies. The precise relations between mathematical developments and physical discoveries depend on analogies.
Galileo discovered the parabolic path of projectiles and derived the quantitative laws of motion. When he discovered the moons of Jupiter, Galileo noticed that those moons are analogous to the earth’s moon and to the planets circling the sun. He observed the phases of Venus and realized that they were analogous to the phases of the earth’s moon. These analogies helped to confirm the heliocentric theory of Copernicus. Galileo was at the verge of the greatest scientific discovery of all time, but the time was not ripe.
The day Galileo died, Newton was born. Newton made the analogy between the motion of projectiles and the motion of planets or moons orbiting the sun or a planet, respectively. The result was the law of gravity, which governed the trajectory of a cannonball as well as the passage of the earth around the sun (Robinson and Clark, 2008).
Galileo’s astronomical discoveries inspired the generations that followed. The study of light became paramount. In developing the theory of light, scientists constantly used analogies between light waves and mechanical waves or between light rays and streams of particles. René Descartes (1596–1650) described the functioning of the eye and presented a preliminary version of an enormously important scientific concept — the wave theory of light (Clark and Robinson, 1985). Robert Hooke (1635–1703) formulated stress-strain relationships, which established the elastic behavior of solid bodies (Robinson and Clark, 1988b). Hooke proposed that light is a vibratory displacement of the medium through which it propagates at finite speed. Thomas Young was the first to consider shear as an elastic strain. Augustus Jean Fresnel showed that if light were a transverse wave, then one could develop a theory accommodating the polarization of light. Green (1842) illustrated the power of using mathematical analogies in his treatment of elastic (material) waves and light waves.
Much of digital signal processing today is geared toward image processing. People want to see pictures. Digital pictures can reveal much greater detail than the old analog pictures offered. It is important to recognize that the difference between digital and analog pictures is not just cosmetic. The transition to digital imagery is comparable to the transition provided by the introduction of the magnifying glass or the microscope or the telescope. A person could see the moon before, but when Galileo saw the moon through a telescope, it was a different moon. Geophysicists could create pictures of the underground strata with analog records, but now they can create digital pictures of such quality that a whole new world is opened up.
Geophysics benefits from the tremendous body of work on wave motion. The entire theory of seismic migration is justified by analogy]] to the Huygens-Fresnel principle, as put forth in the mathematical fashion of Kirchhoff. Approximate methods now exist for the numerical solution of the wave equation in media whose material parameters arbitrarily vary. Some computer programs are designed specifically to exhibit diffraction effects. Geophysicists use Kirchhoff migration to produce digital pictures of the subterranean earth, and those images have unequaled scope and beauty.
Geophysicists also are active on the mathematical side of wave theory, as exemplified by Aki and Richards (1980) and others. Ricker (1977) extended the theory of seismic wave propagation to the case of viscoelastic media. Recently, Carcione and Cavallini (1995) showed that the 2D Maxwell equations describing the propagation of the transverse magnetic (TM) mode in anisotropic media are mathematically equivalent to the shear-horizontal (SH) wave equation in an anisotropic-viscoelastic solid where attenuation is described with the Maxwell model. Fresnel’s equations represent a classic example of the analogy between shear waves and electromagnetic waves. Carcione and Robinson (2002) investigated the corresponding mathematical analogy between elastic waves and electromagnetic waves. They obtained a complete parallelism for the reflection problem and the refraction problem, considering the most general situation - a situation with the presence of anisotropy and attenuation.
Let us now summarize. According to Huygens, each point of a wavefront can be envisaged as a source of secondary spherical wavelets. Thus, one can determine the progress through space of a wavefront or of any portion of a wavefront. At any particular time, the shape of the wavefront is the envelope of all secondary wavelets. This technique, however, ignores most portions of each secondary wavelet, thus retaining only the portions that are common to the envelope. Because of this limitation, Huygens’ principle alone cannot account for the diffraction process - the prevalence of which is borne out by everyday experience. Sound waves easily bend around large objects such as telephone poles and trees which, on the other hand, cast fairly distinct shadows when illuminated by light. Yet Huygens’ principle is independent of any wavelength considerations and would predict the same wavefront behavior in both situations.
Fresnel resolved that difficulty with his introduction of the interference principle. The corresponding Huygens-Fresnel principle now states that each unobstructed point of a wavefront at a given instant serves as a source of spherical secondary wavelets (of the same spectral content as the primary wave). The amplitude of the wave at any position beyond that point is then the superposition of all such wavelets (considering their amplitudes and relative phases).
Gustav Kirchhoff developed a precise formulation of the Huygens-Fresnel principle based on the wave equation. Geophysicists today use Kirchhoff’s theory to produce digital images of media whose material parameters vary arbitrarily. To fully understand the principles underlying seismic imaging, we must appreciate the limitations that diffraction imposes on system performance.
Two or more superimposed waves clearly produce a wave that is different from either wave alone. The superposition of waves and the resulting interference phenomena are fundamental to the study of diffraction. Longitudinal sound waves can interfere to produce beats. In a similar way, the interference of transverse light waves produces colors. Constructive and destructive interference are sketched in Figure 23. The interference of water waves is a common sight. In some places, crests overlap crests, and in other places, crests overlap troughs. There is no clear physical distinction between the phenomena of interference and diffraction. It has become customary, however, to speak of interference when referring to the superposition of a few waves and to speak of diffraction when a great many or an infinite number of waves are superposed. >
Seismic waves, with wavelengths in the range of several hundred meters, easily can bend around such obstacles as pinchouts and disconformities. A negative consequence of diffraction is the limits it imposes on the size of seismic exploration targets. Seismic signals originating from sufficiently small objects become less and less well defined as the object’s size approaches the wavelength of the seismic waves illuminating it. If the object is smaller than a single wavelength, then a combination of careful data collection, careful data processing, and careful data interpretation is necessary. However, no amount of theory can completely defeat this fundamental limit of diffraction.
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Also in this chapter
- 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
- Ray theory
- Fermat’s principle
- Fermat’s principle and reflection and refraction
- Appendix A: Exercise
- Robinson, E. A., and R. D. Clark, 2008, Isaac Newton and the birth of geophysics: The Leading Edge, 27, no. 2, 159–161.
- Clark, R. D., and E. A. Robinson, 1985, Descartes as geophysicist: The Leading Edge, 4, no. 8, 32–35.
- Robinson, E. A., and R. D. Clark, 1988b, Elasticity: Hooke’s law: The Leading Edge, 7, no. 8, 58–60.
- Aki, K., and P. G. Richards, 1980, Quantitative seismology: Freeman.
- Ricker, N., 1977, Transient waves in visco-elastic media: Elsevier.
- Carcione, J. M., and F. Cavallini, 1995, On the acoustic-electromagnetic analogy: Wave Motion, 21, no. 2, 321–346.
- Carcione, J. M., and E. A. Robinson, 2002, On the acoustic-electromagnetic analogy for the reflection-refraction problem: Studia Geophysica et Geodaetica, 46, 149–162.
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- Alfredo, Carlos, and Clebsch Kuhn. “The Geological Evolution of the Paraguayan Chaco.” TTU DSpace Home. Texas Tech University, August 1, 1991. https://ttu-ir.tdl.org/handle/2346/9214?show=full.