INTRODUCTION

In all our work with seismic refraction and with reflection up till today we have worked primarily with travel-times. Today we’ll take at look at some of the factors that determine amplitudes of the waves.

RAYLEIGH REFLECTION COEFFICIENTS

Let’s consider a plane wave approaching a sharp, planar interface between two media. What determines the amplitude of the reflected wave and the transmitted (refracted wave)? We’ll express the results in terms of acoustic impedance; this is the product of density times velocity.

The amplitude of the reflected wave relative to the incident wave is given by the difference in the acoustic impedances divided by the sum of the acoustic impedances of the two media. This ratio is called the "Rayleigh reflection coefficient" after the work of Lord Rayleigh (John Strutt) in the nineteenth century.(For non-normal incidence we use the more complex Zoeppritz equations).

The relative amplitude of the transmitted wave continuing into the second medium is given by the ratio of twice the impedance in the incident layer divided by the sum of the two acoustic impedances. This ratio is called the "Rayleigh transmission coefficient".

The Rayleigh reflection coefficients hold for both compressional waves and shear waves. Just use the appropriate velocity in the acoustic impedance.

EXAMPLES OF RAYLEIGH REFLECTION COEFFICIENTS

Let’s suppose the acoustic impedance of the second layer is much, much higher than that of the first layer (layer with incident wave). A common example occurs in single-channel reflection profiling on the continental shelf of New Hampshire where much crystalline bedrock outcrops. For water the density is about 1000 kg/m^3 and the sound wave velocity is about 1.5 km/s. For granite the values might be about 2600 kg/m^3 and 6 km/s. Thus the relection coefficient is 0.83 and the transmission coefficient is 0.17. This means that we get a very strong reflection and that the wave continuing down into the rock is weak. This could pose a problem if we wish to "see" into the granite. As we’ll soon see this also means that there will be very strong water layer multiple echoes. In the limit of increasing impedance ratio the reflection coefficient approaches one and the transmission coefficient, zero.

In a second important case the impedances of the two media are almost the same. This is usual in a stack of sedimentary rock or sediment layers. In this case the reflected wave amplitude is almost zero. The sign depends on which impedance is larger. The transmitted wave, in contrast, retains most of its amplitude (close to one). In the limit as the impedances become equal the reflection amplitude is zero and the transmitted amplitude is one.

In the third important case the impedance of the incident layer is much greater than that of the other layer. Note that it doesn’t matter which layer is above and which is below. What matters is layer in which the incident wave is located. A common geological example deals with an incident wave in water reflecting from air at the sea surface. Here the water echo is strong with a reflection coefficient of almost minus one. The negative sign means that the phase of the echo is reversed 180 degrees with respect to the incident wave. The transmitted wave amplitude is almost two. It turns out that this wave transmitted into the air has very little energy despite it’s large amplitude (see below).

These amplitude relationships are similar to those for a wave propagated along a rope (the phase relations are a little different). Our first seismic case resembles what happens when we shake a light, flexible rope joined to a heavy, stiff one: the waves in the first rope are large whereas the heavy rope barely moves.

The second case corresponds to an almost uniform rope. Here the waves just travel along the rope with change or reflection. We have an almost perfect impedance match.

The third case shows behaviour similar to that of a heavy, stiff rope attached to a light, flexible one. Here the waves in the light rope are much larger than the incident waves. This case is somewhat analogous to a fly rod and fishing line.

ENERGY IN A WAVE

For a plane wave in one dimension we see that the energy per unit volume alternates between purely kinetic and purely potential. This action resembles simple harmonic motion of a weight on a spring. The energy per unit volume equals one half of the density times amplitude squared times angular frequency squared. This density factor explains why a wave in the air can have a very large amplitude and a very low energy (density of air equals about one kg.m^3).

ENERGY FLUX

The energy traveling with the wave or carried by the wave is called the energy flux. It equals the energy passing through some unit area per unit time. It is analogous to discharge of a stream. The energy flux equals the energy per unit volume times the seismic wave velocity.

Using the equation for energy flux we can easily show that the Rayleigh reflection coefficients are consistent with conservation of energy. In other words the energy flux of the incident wave ("in") equals the sum of the energy fluxes of the reflected and transmitted waves ("out").

OTHER FACTORS INFLUENCING AMPLITUDES

The amplitude of a wave spreading in a spherical pattern decreases inversely with distance. In contrast, a wave spreading in a circular pattern , as in the SOFAR channel of the ocean, has amplitude inversely proportional to the square root of the distance. Thus the sound channel waves travel with less loss of amplitude. What amplitude decay occurs in an old-fashioned speaking tube as on the "Titanic"?

As real waves travel some of their energy is converted to heat and so the waves get weaker even if they don’t spread at all. This attenuation or absorption is exponential with distance. The attenuation coefficient increases linearly with frequency so that high frequency waves decay more rapidly than low frequency ones. As we’ll see in the next class this differential decay leads to loss of resolution with depth.

Amplitudes also are affected by reflector shape. A dome or anticline tends to defocus reflected waves and thus to weaken echoes. A basin or syncline, on the other hand) tends to focus waves and leads to strong echoes. This phenomenon is similar to that of a mirror or satellite television antenna.

RECOMMENDED READING

Strutt, J. W., 1877 (1937 reprint), "The Theory of Sound", Macmillan and Company, London, in two volumes. This classic text by Lord Rayleigh includes many derivations of great value even today. It also describes ingenious experiments carried out before the age of electronic instrumentation. He also goes into great depth on musical theory.