The wave equation

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Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
DigitalImaging.png
Series Geophysical References Series
Title Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author Enders A. Robinson and Sven Treitel
Chapter 3
DOI http://dx.doi.org/10.1190/1.9781560801610
ISBN 9781560801481
Store SEG Online Store

The wave equation is based on two fundamental laws. Hooke’s law says that stress is proportional to strain, and Newton’s law says that force equals mass times acceleration. From the wave equation, we can predict the existence of compressional waves and shear waves and their properties. These properties include Snell’s law of reflection and refraction, the partition of energy at an interface into compressional and shear components, the generation of surface waves and their characteristics, the diffraction of waves, the attenuation of waves as they travel in the earth, and many other facts of wave propagation. All migration procedures invoke the wave equation (Loewenthal et al., 1976[1]). A simplified model based on raypaths obeying Snell’s law at interfaces and following least-time paths has become a standard model used in geophysics.

What is a plane wave? A plane wave is perhaps the simplest example of a 3D wave. It exists at a given instant and can be visualized as a propagating plane surface of constant phase, so that the plane surface remains perpendicular to a given direction of propagation. A plane wave is essentially one-dimensional because spatial variation occurs only along the direction of propagation. We have quite practical reasons for studying this sort of disturbance, one of which is that an actual observed spherical wave can be decomposed into its constituent plane waves. This procedure is called plane-wave decomposition, and it is useful because plane waves have much simpler properties than spherical waves do. The special significance of this approach is that any 3D wave can be expressed as a combination of plane waves, each having a distinct amplitude and direction of propagation. Of all the 3D waves, only the plane wave (whether it is sinusoidal or not) moves through space with an unchanging profile.

When the 3D wave equation is written in terms of Cartesian coordinates (x,y,z), the position variables x, y, and z appear symmetrically — a fact to be kept in mind. Cartesian coordinates are suited particularly for describing plane waves. However, as various physical situations arise, we often can take better advantage of existing symmetries by using other coordinate representations.

What is a ray? There are several ways to think about waves. One way is to think of wave motion in terms of rays. The concept of a ray is very useful. Picture the wave as traveling in very narrow pencils, or beams, called rays. This ray model gives an understanding of various wave phenomena, especially reflection and refraction. A ray is a line drawn in space corresponding to the direction of flow of radiant energy. As such, it is a mathematical device rather than a physical entity. In practice, we sometimes can produce very narrow beams, or pencils, and we might imagine a ray to be the unattainable limit on the narrowness of such a beam. Within homogeneous isotropic materials, a ray is a straight line because, by symmetry, there is no preferred direction that would cause the ray to form a curved path. Rays are bent at the interface between two media with different velocities, and rays can be curved in a medium in which velocity is a function of position.

What is a wavefront? The dual of the ray is the wavefront (Figure 3). Consider the wavefront representing the front surface of the wave traveling away from an explosive source. We can visualize this wavefront physically and we can measure it within narrow limits. For example, when we look at the waves emanating from a point at which we have dropped a stone into a still pond, we see the wave motion traveling outward as wavefronts, and the raypaths are only mental constructions. Likewise, in seismic interpretation, the rays are visualized only as mathematical abstractions, whereas wavefronts can be seen on the seismic sections as reflection events. For that reason, we turn our attention to the geometric study of wavefronts.

Figure 3.  A ray and a wavefront.

Points at which a single ray intersects a set of wavefronts are called corresponding points. Evidently, the separation in time between any two corresponding points on any two sequential wavefronts is identical. In other words, if wavefront S transforms into after a time , the distance between corresponding points on any and all rays will be traversed in that same time . This obviously will be true even if the wavefronts pass from one homogeneous isotropic medium into another. This simply means that each point on S can be imagined as following the path of a ray to arrive at in the same time . Within a homogeneous isotropic medium, the velocity of propagation is identical in all directions. In such a medium, the spatial separation between two wavefronts, measured along a ray, must be the same everywhere.

A wavefront can be defined as the surface (in three dimensions) or curve (in two dimensions) over which the phase of a traveling-wave disturbance is the same. In an isotropic medium — one whose properties are the same in all directions — rays form trajectories that are orthogonal to the wavefronts. That is to say, rays are lines normal to the wavefronts at every point of intersection. Evidently, in such a medium, a ray follows the direction of propagation. However, that is not true in anisotropic materials.

A wavefront also can refer to the leading edge of a waveform. A wavefront chart (Figure 4) is a plot of horizontal distance x versus depth z on which wavefronts have been drawn emanating from a source. The wavefronts show the position of a traveling seismic disturbance at successive times. The shapes of the wavefronts depend on the velocity distribution of the rocks. On such charts, raypaths corresponding to different move-outs also can be drawn. These raypaths are perpendicular to the wavefronts for isotropic media. Sometimes, wavefront charts are drawn that are specific for a particular location where velocity varies laterally; in such cases, the wavefront charts are asymmetric. For the case in which velocity varies only with depth, the wavefront chart is symmetric with respect to the vertical raypath.

Figure 4.  Wavefronts and raypaths.

An idealized point source is one for which the radiation emanating from it streams out radially and uniformly in all directions. The source is said to be isotropic, and the resulting wavefronts are again concentric spheres that increase in diameter as they expand into the surrounding space. One solution of the wave equation gives a spherical wave progressing radially outward from the origin at a constant velocity and having an arbitrary functional form. Another solution is given by the case in which the wave converges toward the origin.

Notice that the amplitude of any spherical wavefront is a function of its distance from the center because the radial term serves as an extrinsic attenuation factor. This extrinsic attenuation factor is a direct consequence of energy conservation, a phenomenon known as geometric spreading. That is, unlike a plane wave, a spherical wave decreases in amplitude as it expands and travels away from its source, thereby changing its profile. As a spherical wavefront propagates outward, its radius increases. Far enough away from the source, a small area of the wavefront therefore will resemble closely a portion of a plane wave (Figure 5).

Figure 5.  Spherical spreading. Energy flowing across the upper area later flows across the lower area. Explain why the two figures are different. It is necessary to correct the amplitude for spherical spreading. The correction is obtained by multiplying the amplitude by the distance traveled. The correction can be obtained approximately by multiplying the amplitude by the traveltime.

Much of current seismic processing methodology is based on the plane-wave approximation to a spherical wave in particular as well as to other types of curved wavefronts in general. The temporal frequency, or the number of cycles per unit time, is the Fourier dual for the time variable. The spatial frequency, or wavenumber, is the Fourier dual for the space variable. The wavenumber gives the number of cycles per unit distance.

When irregular discontinuities exist in a nonuniform medium, they give rise to complex reflection, refraction, and diffraction phenomena. Huygens’ principle (or construction) is particularly useful for treating problems of this type. Huygens’ principle states that every point on a primary wavefront serves as the source of spherical secondary wavelets, and the primary wavefront at some later time is the envelope of these wavelets. Moreover, the wavelets advance with a speed and frequency equal to those of the primary wave at each point in space. If the medium is homogeneous, the wavelets can be constructed with finite radii, whereas if the medium is inhomogeneous, the wavelets must have infinitesimal radii.

Huygens’ construction allows a backward-traveling wave moving toward the source — something that is not observed. This difficulty was taken care of theoretically by Fresnel and Kirchhoff, who showed that only the forward-moving wave can exist in the real world. Thus, we merely can use the forward waves when applying Huygens’ construction.

Seismic analysis almost always is carried out through a process of abstraction in which the data are reduced to the essentials that admit of mathematical treatment. The accuracy required in the results influences the extent to which such reductions are made, and that accuracy is limited by the reliability of the data. Today, analytic methods are implemented with the digital computer, and their relative values are judged by their speed, reliability, and clarity.

Two important processes in seismic data processing are stacking and migration (Berkhout and de Jong, 1981[2]). Much can be learned about them by geometric lines of thought. In dealing with stacking and migration from a geometric point of view, one can take either the raypath concept or the wavefront concept as the point of departure. In geometric seismology (i.e., the theory of seismic waves with wavelengths that are small compared with the undulations of the subsurface interfaces), a ray can be visualized as a narrow beam along which energy is transported. This concept is especially valid in treating a long series of waves with wavelengths on the order of a fraction of the dimension of any object encountered.

However, when the dimension of some obstacle is of the same order of magnitude as the wavelength, then the phenomenon of diffraction occurs. In the case of an impulsive source (e.g., a dynamite explosion), the seismic energy is transported in the form of a short compressional wavelet whose breadth is of the same order of magnitude as the geologic bodies whose shape we wish to determine. Thus, within the usual distance ranges, velocities, and frequencies encountered in seismic work, we do not have pure raypaths in the form of infinitely narrow beams, so we must take into account diffraction effects.


References

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  1. Loewenthal, D., L. Lu, R. Roberson, and J. Sherwood, 1976, The wave equation applied to migration: Geophysical Prospecting, 24, 380–399.
  2. Berkhout, A. J., and B. A. de Jong, 1981, Recursive migration in three dimensions: Geophysical Prospecting, 29, 758–781.
  3. Whaley, J., 2017, Oil in the Heart of South America, https://www.geoexpro.com/articles/2017/10/oil-in-the-heart-of-south-america], accessed November 15, 2021.
  4. Wiens, F., 1995, Phanerozoic Tectonics and Sedimentation of The Chaco Basin, Paraguay. Its Hydrocarbon Potential: Geoconsultores, 2-27, accessed November 15, 2021; https://www.researchgate.net/publication/281348744_Phanerozoic_tectonics_and_sedimentation_in_the_Chaco_Basin_of_Paraguay_with_comments_on_hydrocarbon_potential
  5. 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.