In reflection seismology, the anelastic attenuation factor or the seismic quality factor or Q quantifies the effects of anelastic attenuation on the seismic wavelet caused by fluid movement and grain boundary friction. It is defined as $$Q = 2{\pi}\left ( \frac{E}{{\delta}E} \right )$$; where $$\frac{E}{{\delta}E}$$ is the fraction of energy lost per cycle.[1] Its behaviour said to be dispersive because the rate of attentuation increases with frequency.

## Introduction

As a seismic wave propagates through a medium, the elastic energy associated with the wave is gradually absorbed by the medium, eventually ending up as heat energy. This is known as absorption (or anelastic attenuation) and will eventually cause the total disappearance of the seismic wave.[2]

## Why Measure Q?

Q preferentially attenuates higher frequencies, resulting in the loss of signal resolution as the seismic wave propagates. Quantitative seismic attribute analysis of amplitude versus offset effects is complicated by anelastic attenuation because it is superimposed upon the AVO effects.[3] The rate of anelastic attenuation itself also contains additional information about the lithology and reservoir conditions such as porosity, saturation and pore pressure so it can be used as a useful reservoir characterization tool.[4]

Therefore if Q can be accurately measured then it can be used for both compensation for the loss of information in the data and for seismic attribute analysis.

## Measurement of Q

### Spectral Ratio Method[5]

The geometry of a zero-offset vertical seismic profile (VSP) makes it an ideal survey to use for the calculation of Q using the spectral ratio method. This is because of the coincident raypaths that traverse a given rock layer, ensuring that the only path difference between two reflected waves (one from the top of the interval and one from the bottom) is the interval of interest. Stacked surface seismic reflection traces would offer similar signal-to-noise ratio over a much larger area but cannot be used with this method because every sample represents a different raypath and therefore will have experienced different attenuation effects.[6]

Seismic wavelets captured before and after traversing a medium with seismic quality factor, Q, on coincident raypaths will have amplitudes that are related as follows:

$\frac{A_{final}}{A_{initial}}=R.G.e^{{\frac{-{\pi}ft}{Q}}}$;

where $${A_{final}}$$ and $${A_{initial}}$$ are the amplitudes at frequency $$f$$ after and before traversing the medium; $$R$$ is the reflection coefficient; $$G$$ is the geometrical spreading factor and $$t$$ is the time taken to traverse the medium.

Taking logarithms of both sides and rearranging:

$ln\left ( \frac{A_{final}}{A_{initial}} \right )=\left ( \frac{-{\pi}t}{Q} \right )f +ln(RG)$

$Y = m X + C$

This equation shows that if the logarithm of the spectral ratio of the amplitudes before and after traversing the medium is plotted as a function of frequency, it should yield a linear relationship with an intercept measuring the elastic losses (R and G) and the gradient measuring the inelastic losses, which can be used to find Q.

## References

1. Sheriff, R. E., Geldart, L. P., (1995), 2nd Edition. Exploration Seismology. Cambridge University Press.
2. Toksoz, W.M., & Johnston, D.H. 1981. Seismic Wave Attenuation. SEG.
3. Dasgupta, R., & Clark, R.A. (1998) Estimation of Q from surface seismic reflection data. Geophysics, 63, 2120-2128
4. Enhanced seismic Q compensation, Raji, W.O., Rietbrock, A. 2011. SEG Expanded Abstracts 30, 2737
5. Tonn, R. 1991. The determination of seismic quality factors Q from VSP data: A comparison of different computational methods. Geophys. Prosp., 39, 1-27.
6. Dasgupta, R., & Clark, R.A. (1998) Estimation of Q from surface seismic reflection data. Geophysics, 63, 2120-2128