Diffraction

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

In dealing with reflection and refraction, we have considered only smooth interfaces.

However, if an interface contains a sharp bend, special consideration is required. Suppose for instance that a body of material of velocity $v_{1}$ with a sharp edge is embedded in a uniform medium of velocity $v_{2}$, as shown in Figure 22. A ray BA incident on the sharp edge B, according to Huygens’ principle, will disturb point B so that it will act as a point source and emit energy in the form of an expanding circular wavefront. This wavefront is called a diffracted or scattered wave because the raypaths, as shown in the figure, are scattered in all directions with respect to the incident ray BA. Evidence exists on many seismic records that this process, which is expected on the basis of Huygens’ principle, is not merely hypothetical but is real and that the diffracted waves often carry very appreciable and measurable amounts of energy. The situation depicted in Figure 22 is in fact representative of a fault in a subsurface geologic stratum whose velocity is different from that in the surrounding strata. Prominent waves diffracted from the edges of faults often are observed on seismic records. We see from Figure 22 that diffracted rays do not have to obey either the law of reflection or Snell’s law of refraction.

An opaque body placed midway between a screen and a point source casts an intricate shadow made up of bright and dark regions quite unlike anything one might expect from the rules of geometric optics. The effect is a general characteristic of wave phenomena occurring whenever a portion of a wavefront from either a mechanical wave or from an electromagnetic wave is obstructed in some way. If in the course of encountering an obstacle, either transparent or opaque, a region of the wavefront is altered in amplitude or phase, diffraction will occur. The various segments of the wavefront that propagate beyond the obstacle interfere to cause the particular energy-density distribution referred to as the diffraction pattern.

Francesco Grimaldi (1665) observed bands of light within the shadow of a rod illuminated by a small source. In other words, he observed that some light had bent around the rod into the shadow region. Grimaldi gave the name diffraction to this phenomenon. Diffraction refers to the deviation from rectilinear propagation that occurs when waves advance beyond an obstruction, Diffracted sound waves allow sound to be heard around corners. If the length of the obstruction is equal to about one wavelength, then the amount of diffraction becomes so large that the notion of a shadow becomes almost meaningless.

Robert Hooke (1665) was the first to study the colored patterns generated by thin films (such as an oil slick on water). Hooke gave the name interference to this phenomenon. He concluded correctly that the various colors arise from the superposition of (1) light waves that reach the eye by reflection from the top surface and (2) light waves that reach the eye by refraction into the film followed by reflection from the lower surface. Hooke’s thin-film analysis can be used in seismic exploration to account for the multiple reflections produced in a stratified system consisting of horizontal layers (Treitel and Robinson, 1966).

Figure 22.  Diffraction at a sharp bend.

In the general sense, both interference and diffraction refer to effects resulting from the superposition of two or more waves at a given point in space. There is no essential mathematical difference between interference and diffraction. However, in more restricted usage, interference is used to describe effects that result from the superposition of two or more wave trains (as in the case of water-layer reverberations of seismic waves), and diffraction is used to describe interference effects caused by the presence of an aperture or an obstacle in the path of a wave (as in the bending of seismic waves around obstacles). In modern usage, the terms interference and diffraction still refer more or less to the types of problems studied by Hooke and Grimaldi, respectively.

Isaac Newton (1642–1727) intended to build on direct observation and to avoid speculative hypotheses. He thus remained ambivalent for a long while about the actual nature of light. Was light corpuscular, in the form of a stream of particles, as some people maintained? Or was light a wave propagating in an all-pervading medium, the ether? At age 23, Newton began his now famous experiments. In his words, “I procured me a triangular glass prism to try therewith the celebrated phenomena of colors” (Newton, 1672). In time, Newton rejected the wave theory of light. However, the corpuscular theory could not explain diffraction.

The general belief is that the great weight of Newton’s opinion stifled development of the wave theory of light during the entire eighteenth century. However, we believe instead that it took the entire eighteenth century to develop mathematics to the point at which it could handle wave theory. The prominent mathematician Leonhard Euler (1707–1783) was a devotee of the wave theory, and he led the way in the development of mathematics in his time.

At last, the wave theory of light was reborn in the hands of Thomas Young (1773–1829). He read papers before the Royal Society extolling wave theory and adding to it a new fundamental concept, the so-called principle of interference (Young, 1804). In effect, Young put the earlier work of Hooke and Grimaldi on a firm mathematical basis. In particular, Young devised his famous two-slit interference experiment.

Augustin Jean Fresnel (1788–1827) was born in Broglie, Normandy. In about 1814, Fresnel became interested in the problem of light and began a brilliant revival of wave theory, although he was unaware of Young’s efforts 13 years earlier. Fresnel ultimately combined the concepts of Huygens’ wave description with the interference principle (Fresnel, 1818 further analyzed Huygens’ principle, and Kirchhoff (1883)).

As we have seen, Huygens’ principle can be described as follows: (1) Every point on a wavefront can be considered to be a center of a secondary disturbance, which gives rise to spherical wavelets. (2) The wavefront at any later instant can be regarded as the envelope of these spherical wavelets.

However, Huygens’ principle does not account for diffraction. Fresnel accounted for diffraction by supplementing Huygens’ principle with the postulate that the secondary wavelets mutually interfere. The result is the Huygens-Fresnel principle, which is a combination of Huygens’ principle with the principle of interference. In other words, Huygens viewed the propagation of a primary wave as a succession of stimulated overlapping and interfering spherical secondary wavelets. These wavelets merge into the advancing primary wave. Fresnel’s theory took on a more mathematical emphasis than Young’s had. As a result, Fresnel calculated the diffraction patterns that arise from various obstacles and apertures, and he satisfactorily accounted for rectilinear propagation in homogeneous isotropic media, thus dispelling Newton’s main objection to the wave theory.

The term diffraction can be defined as any deviation of rays from rectilinear paths that cannot be interpreted as either a reflection or a refraction. The classic description of the diffraction of a light ray is as follows: An aperture in an opaque screen is illuminated by a light source, and the light intensity is observed across a plane some distance behind the screen. The rectilinear theory of light propagation predicts that the shadow behind the screen should be well defined, with sharp borders. However, observation shows that the transition from light to shadow is gradual rather than abrupt. Given an appropriate light source, we can observe some striking results, such as the presence of light and dark fringes extending far into the geometric shadow of the screen. Such effects cannot be explained with a strict ray-based theory of light, which requires rectilinear propagation of light rays without reflection or refraction.

When a wave passes a point in a medium, the resulting disturbance is a source of new wave motion of the same frequency (according to the Huygens-Fresnel principle). Consider waves of water that are incident from the left on a barrier with a narrow opening. First, suppose the opening is small compared with the wavelength of the waves. The water sloshing up and down in the opening acts as a point source of new waves. As a result, concentric waves are produced on the far side of the barrier. As the opening is widened, however, waves produced on the far side no longer emanate from a pointlike source, and the resulting waves are now less circular in shape. When the opening is very wide compared with the wavelength of the waves, the waves incident from the left simply pass through the opening unobstructed and undergo only slight diffraction at the edges. For light waves, this slight diffractive effect blurs the edges of a shadow. If the light source is very small (preferably a point), dark and bright fringes will be seen.

The degree of diffraction depends on the wavelength of the incident wave. For example, AM radio waves are long, ranging from 200 to 6000 m. Thus, such waves readily bend around objects that might otherwise obstruct them. On the other hand, FM radio waves range from only 3 to 4 m in length and do not bend as readily as AM waves do. This is one of the reasons why FM reception can be poor at localities where AM reception is good.

The Huygens-Fresnel principle had some mathematical shortcomings. Gustav Kirchhoff (1883) developed a rigorous theory based directly on the solution of the wave equation. His analysis gave a precise formulation of Huygens’ principle. In its analytic form, Kirchhoff’s solution turns out to be useful only for some of the simplest applications of the wave equation. The difficulty arises because we must require a solution of a partial differential equation that satisfies the boundary conditions imposed by a particular obstacle. Such rigorous solutions are obtainable only in a very few special cases. Exact solutions are rare in the case of boundaries even slightly more complicated than the simplest ones. Before the advent of the digital computer, Kirchhoff’s mathematical work was little used in practice.

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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
• Ray theory
• Fermat’s principle
• Fermat’s principle and reflection and refraction
• Analogy
• Appendix A: Exercise

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