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The subtle effect of attenuation

When you wake up in the morning, you look at yourself in the mirror — at least I hope you do! You are able to see yourself because light rays are reflected when they hit the opaque but shiny back surface of the mirror. Later, when you are on a bus or train, you see yourself reflected in the window, but not as sharply as in the mirror. That is because only a small percentage of light rays reflect back; most of them transmit through the glass. That is why others outside the bus can see you.

The same principles apply to the seismic waves in the earth. The following famous equation for reflection coefficient R shows how much of the wave amplitude is reflected from a normally incident seismic ray when it hits a boundary:

$$R=\frac{\rho_{2}V_{2}-\rho_{1}V_{1}}{\rho_{2}V_{2}+\rho_{1}V_{1}}$$

where ρ and V are density and P-wave velocity, respectively. The subscripts represent the layers across the boundary, with 1 being the upper layer. The product of the velocity and density is called the acoustic impedance. The equation shows that when there is an acoustic impedance contrast between the two layers, there will be reflection. But this equation is actually an approximation Research shows that there is another contrast that contributes to the reflection of the waves and rays in exploration seismology, and that is attenuation contrast.

Attenuation is the loss of energy in the form of heat. Particles oscillate in their place without flowing from one location to another, to allow the seismic waves to go through. These particles turn some of the seismic wave energy into heat due to friction forces. You can think of this energy loss as a sort of currency exchange fee. Every time you exchange one currency for another, the bank charges a transaction fee, which is deducted from your money. If you do the exchange several times, you lose more and more of your money. As an experiment, start with some of your local currency and exchange it to a few foreign currencies in turn and then convert it back to your local money. You will see that a significant portion of the money is gone! This loss, in seismic terms, is called attenuation and is measured through a parameter called quality factor Q. In fact Q is inversely proportional to the attenuation.

When we account for attenuation as well as acoustic impedance, Lines et al. (2008) showed that we can write a more accurate version of the reflection co­efficient equation:

$$R=\frac{\rho_{2}V_{2}(1+\frac{i}{2Q_{2}})-\rho_{1}V_{1}(1+\frac{i}{2Q_{1}})}{\rho_{2}V_{2}(1+\frac{i}{2Q_{2}})+\rho_{1}V_{1}(1+\frac{i}{2Q_{1}})}$$

It results in a complex reflection coefficient. This equation reduces to the simpler form when there is no contrast in the quality factors, i.e. when Q1 = Q2 .

Imagine a two layer model with the following properties: ρ1 = ρ2, V1 = 2000 m/s, V2 = 3500 m/s, Q1 = 40, and Q2 = 6.28.The reflection coefficient R calculated using the simple equation opposite is 0.273, which means the amplitude of the reflected wave is 0.273 times the amplitude of the incident wave. The result including the attenuation effect is the complex number (0.274 + 0.031i), which means the reflected amplitude is 0.276 times that of the incident wave (0.276 is the modulus or magnitude of the complex reflection coefficient), and the phase rotation between the incident and reflected waves is 6.4° (the argument or phase angle of the complex number).

Since the difference between the predicted amplitudes and phases of the re­flected waves calculated from two equations is small (in real cases, seismic noise makes the relative effect even smaller), it is within reason to say that the familiar equation is a good approximation of the reflection coefficient for most scenarios. But Q is in there somewhere!

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