Attenuation and Q
Learning objectives
- Define the quality factor Q as the inverse of the fractional energy lost per wave cycle, so low Q means strong attenuation
- Pair attenuation with dispersion: the 1/Q peak sits exactly at the dispersion transition frequency
- Read the standard-linear-solid curve: 1/Q peaks at dV/V (0.05, so Q about 20) at f0, falling off symmetrically in log frequency
- Carry the causality rule: no dispersion, no attenuation, so a Q measured in one band predicts dispersion between bands
The Same Physics, Read as Energy Loss
A rock that disperses must also absorb. That is not a coincidence of rock physics; it is a requirement of causality, and it ties the last section to this one. Where a velocity climbs with frequency, the rock is also converting some of the wave's energy to heat, and the loss is strongest exactly where the climb is steepest. We measure that loss with the quality factor , defined by : the fraction of the wave's stored energy dissipated per cycle, scaled by . A high means a nearly lossless rock; a low means a strongly absorbing one. The two descriptions, velocity that rises with frequency and energy that is lost per cycle, are not two effects a rock happens to have. They are one relaxation mechanism seen two ways, and a standard-linear-solid model produces both from the same parameters.
The Peak Sits on the Step
Take the same relaxation that produced the dispersion step of Part 9.4, with its 5 percent velocity rise and a transition centred at . The attenuation it implies has a clean shape: rises to a peak exactly at , then falls away symmetrically on either side in the logarithm of frequency. And the peak height is fixed by the size of the velocity step. For a fractional velocity rise of 0.05, the peak is 0.05, a minimum of about 20, reached right at the transition. Move away from and the loss drops fast: with the transition at 30 kHz, the seismic band near 30 Hz sees of only about 0.0001, a near 10000, almost perfectly elastic; at 1 kHz is about 0.0033, a near 300; and only as the frequency approaches 30 kHz does swing up to its peak of 0.05. Cross to the far side, out at 1 MHz, and the loss has fallen back to about 0.003, a near 330 again, the symmetric mirror of the low-frequency side. The rock is transparent far from its relaxation and opaque right at it.
One Measurement, Two Predictions
The reason this pairing earns its own section is that it turns one measurement into two. Because attenuation and dispersion come from the same relaxation, a measured at one frequency band constrains how much the velocity must change between bands. If a rock shows strong absorption in a given band, its velocity is stepping there, and the total dispersion between the seismic and sonic measurements can be estimated from the observed rather than guessed. Run the logic the other way and it is just as strong: if a rock is genuinely non-dispersive, with the same velocity in every band, then it cannot attenuate by this mechanism either, because there is no relaxation to lose energy to. No dispersion, no attenuation, and the converse. That reciprocity is the practical payoff. It also sets up the closing problem of the part. We now have every reason a single rock reports different velocities to the lab, the log, and the seismic survey, from layering to fabric to dispersion. The final section gathers those three bands into one reconciliation and asks the only question that matters at the wellsite: which velocity do you carry into your model.
References
- Aki, K., & Richards, P. G. (2002). Quantitative Seismology (2nd ed.). University Science Books.
- Mavko, G., Mukerji, T., & Dvorkin, J. (2009). The Rock Physics Handbook (2nd ed.). Cambridge University Press.