A solid consists of many small particles. Elastic waves transfer energy from one point to another by the movements of those small particles. The particles themselves are not transported; they only vibrate with small displacements about their respective equilibrium (or rest) positions. Their vibrational movements can propagate energy in the form of a wave. A wave is an energy-transport phenomenon that transports energy through a medium without transporting matter. Any particular small particle simply vibrates about its own rest position while the wave is passing, and it returns to its rest position and remains there after the wave has passed. The initial energy that created the wave spreads out with the wave. An electromagnetic wave is nature’s way of transporting energy from one place to another at the speed of light.
A homogeneous, isotropic rock has two critical physical parameters: its density and its wave velocity v. The product of these two parameters gives the acoustic impedance or characteristic impedance Z; that is, the acoustic impedance is . Next, we shall learn how the reflection coefficient and the transmission coefficient at an interface depend on the respective impedances on each side of the interface.
Most seismic interpretation is done in terms of models involving the propagation of P-waves (also called compressional waves) through a sequence of rock formations. In many cases, but by no means in all cases, it is appropriate as a first approximation to assume that each formation is homogeneous and isotropic.
A seismic wave represents an interchange between kinetic and potential energy. The kinetic energy comes from the physical motion of the particles, and the potential energy comes from their relative positions with respect to the interparticle elastic (or restoring) forces. The rest position of each particle can be taken as the origin of a 3D coordinate system. The particle vibrates around this rest position. At any instant, we can measure in principle the particle’s velocity and, if we wish, the particle’s acceleration. We also can measure the degree of compression of the particles in the form of a pressure or stress.
In the case of a homogeneous isotropic solid, physical theory tells us that we need only two quantities for a complete specification of the wave motion resulting from the particle motion. In seismic work, it is convenient to take the particle velocity and the pressure as these two quantities. Particle velocity is a vector in a 3D coordinate system, whereas pressure is a scalar. To simplify, we shall restrict ourselves to P-waves traveling in the vertical direction, so particle velocity becomes a scalar too.
A seismic trace is a graph of amplitude versus time. Each trace is equal to the sum of the downgoing wave motion plus the upcoming wave motion at the sensor. In marine work, the hydrophone measures pressure, so the amplitude of a marine seismic trace indicates pressure. In land work, the geophone measures particle velocity, so the amplitude of a land seismic trace represents particle velocity. In a given homogeneous isotropic medium, let V (which we simply call the particle-velocity disturbance) denote the solution of the wave equation for particle velocity. Let p (which we simply call the pressure disturbance) denote the solution of the wave equation for pressure. Let D denote the downgoing component of the particle-velocity disturbance, and let U denote the upgoing component of the particle-velocity disturbance. Similarly, let d denote the downgoing component of the pressure disturbance, and let u denote the upgoing component of the pressure disturbance. Thus, we have the two equations V = D + U and p = d + u for particle velocity and for pressure, respectively.
Various conventions are used for pressure waves and particle-velocity waves. Let us use the Berkhout convention (Berkhout, 1987). Berkhout’s first equation, d = ZD, says that the downgoing pressure wave has the same polarity as does the downgoing particle-velocity wave, and it says that the two are related by a scale factor given by the acoustic impedance Z. His second equation, u = –ZU, says that the upgoing pressure wave has the opposite polarity to that of the upgoing particle-velocity wave, and it says that both also are related by the same scale factor.
- Berkhout, A. J., 1987, Applied seismic wave theory: Elsevier.
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Also in this chapter
- Reflection coefficients and transmission coefficients
- Ghost reflection
- Layer-cake model
- Synthetic seismogram without multiples
- Water reverberations
- Synthetic seismogram with multiples
- Small and white reflection coefficients
- Appendix H: Exercises