Two roads diverged in a wood, and I took the one less traveled by, And that has made all the difference. —Robert Frost
Attenuation is a reduction in the energy of a traveling wave as it propagates through a medium. Attenuation — the falloff of a wave’s energy with distance — has three main causes: (1) transmission loss at interfaces because of reflection, diffraction, mode conversion, and scattering (Bowman, 1955); (2) geometric divergence effects as waves spread out from a source; and (3) absorption, which is the conversion of kinetic energy into heat by friction (note that kinetic energy is the energy of motion). Writers do not always distinguish between the terms attenuation and absorption. Here, we refer to attenuation in the general sense of energy loss to any cause, and we use the term absorption in the special sense of energy loss to heat.
Transmission loss is a wave’s energy loss as the wave travels through an interface. In transmission loss, the energy that is lost is diverted from the traveling wave of interest. There is no loss of total kinetic energy because the lost energy merely travels somewhere else. For example, when a wave meets an interface, some energy is reflected back from the interface, and only part of the wave’s energy is transmitted though the interface.
Mode conversion is the conversion of P-wave energy into S-wave energy or vice versa. Mode conversion occurs when a wave arrives at an interface at an obliquely incident angle to that interface. Converted waves divert energy away from the given wave.
The energy of a wave in a homogeneous material is proportional to the square of its amplitude (which can be expressed in terms of pressure, particle velocity, or particle acceleration). A point source produces a spherical wave that spreads out from the source. The energy is distributed over the area of the sphere, which increases as the square of the sphere’s radius. Thus, the wave’s energy per unit area varies inversely as the square of the distance from the source. The wave’s amplitude is proportional to the square root of the energy per unit area. Thus, the wave’s amplitude is inversely proportional to the distance the wave has traveled. Spherical spreading — that is, the loss of amplitude because a wave spreads out — is one cause of attenuation. Such energy loss is called geometric divergence.
The third cause of attenuation is energy loss from absorption. Absorption is the result of frictional dissipation of elastic energy into heat. The loss from absorption turns out to be approximately exponential with distance. The second and third mechanisms of attenuation] (i.e., spherical spreading and absorption) can be combined in the equation
Here, A is the amplitude at distance x from the source, is the amplitude of distance zero, and is the absorption coefficient. The amplitude of a seismic wave falls off with distance x from the source, in accordance with spherical spreading. In equation 1, the factor indicates spherical spreading, whereas the coefficient indicates the absorption.
The value of the absorption coefficient depends on the material. Absorption of elastic waves in rocks represents an important problem in seismic exploration. Absorption constants have been measured for a variety of earth materials. However, the mechanism for absorption in many types of rocks, particularly in softer sedimentary rocks or in fluid-filled porous rocks, is still a subject of investigation.
Extensive laboratory work with dry rocks has shown that the absorption coefficient tends to be proportional to the first power of frequency f. Such a relation indicates that the mechanism of absorption is solid friction associated with the particle motion in the wave. The logarithmic decrement is the logarithm of the ratio of amplitude of any cycle to that of the following one in a train of damped waves. The quality factor Q is related to by the equation . The absorption coefficient is related to the logarithmic decrement by . Let V denote the P-wave velocity. The coefficient is related to the quality factor Q by
Both and Q are used to describe the attenuation characteristics of rocks. From equation 2, we see that waves with higher frequencies tend to be absorbed more than waves with lower frequencies are absorbed. This property gives rise to a progressive lowering of the apparent frequencies as the wave travels. In addition, we see that waves with higher velocities tend to be absorbed less than are waves with lower velocities.
Norman Ricker did important work on determining the shape that the seismic pulse acquires as it propagates through rock (Ricker, 1940, 1941, 1953). His equations predicted the waveform that would be observed after an impulsive signal has traveled a given distance through an absorbing material. Ricker’s work included first-power, second-power, and fourth-power frequency dependence of the absorption coefficient. He found that the wave’s shape for second-power frequency dependence most closely resembled what could be measured at the time in the field. Such a second-power frequency dependence suggested to him a “viscoelastic” frictional-loss mechanism of a type that usually is associated with viscous liquids. From that attenuation law, Ricker gave equations that generated waveforms for ground displacement and for particle velocity. At large distances from the source, such a waveform becomes symmetric. In his honor, this waveform is now called the Ricker wavelet.
Ricker observed such predicted waveforms in the Pierre Shale of Colorado. However, later, more refined experiments conducted in the Pierre Shale indicated that an absorption mechanism that was proportional to the first power of the frequency fit the experimental data far better. This first-power dependence implies solid friction, and today, the solid-friction hypothesis generally is accepted. Even so, the Ricker wavelet continues to be used as a convenient representation of the basic seismic pulse.
There is a large variation of absorption characteristics not only among different rock types but also among members of the same rock type. Let us compute the absorption coefficients using equation 2 with a frequency of 40 Hz. For example, one shale might have a velocity of 3 km/s and an absorption coefficient of 0.70. Another shale might have a velocity of 2 km/s and an absorption coefficient of 2.20. Two sandstones, each with a velocity of 4 km/s, might have absorption coefficients of 0.75 and 1.75, respectively. Two limestones, each with a velocity of 6 km/s, might have absorption coefficients of 0.04 and 0.30, respectively. Although overlaps exist in the ranges of absorption-coefficient values for sedimentary and igneous rocks, sedimentary rocks generally are more absorptive than igneous rocks are.
Schuster (2005) derived an interferometric form of Fermat’s principle that helps us to perform high-resolution estimation of the velocity distribution among deep interfaces. He used Fermat’s interferometric principle to redatum the surface sources and receivers to interface A kinematically. The velocity model above interface A does not need to be known, so the distorting effects of the overburden and statics are eliminated by his target-oriented approach.
- Bowman, R., 1955, Scattering of seismic waves by small inhomogeneities: Ph.D. thesis, Department of Geology and Geophysics, MIT.
- Ricker, N., 1940, The form and nature of seismic waves and the structure of seismograms: Geophysics, 5, 348–366.
- Ricker, N., 1941, A note on the determination of the viscosity of shale from the measurement of wavelet breadth: Geophysics, 6, 254–258.
- Ricker, N., 1953, The form and laws of propagation of seismic wavelets: Geophysics, 18, 10–40.
- Schuster, G. T., 2005, Fermat’s interferometric principle for target-oriented traveltime tomography: Geophysics, 70, no. 4, U47–U50.
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