Inverse Q filtering
Frequency attenuation caused by the intrinsic attenuation in rocks was discussed in entries dedicated to gain applications and the convolutional model in the time domain. Attenuation causes loss of high frequencies in the propgating waveform with increasing traveltime. This gives rise to a nonstationary behavior in the shape of the wavelets associated with reflection events at different times.
Wave attenuation usually is described by a dimensionless factor Q, which is defined by the ratio of the mean stored energy to the energy loss over a period of time that is equivalent to one cycle of a frequency component of the waveform . Time-variant deconvolution and time-variant spectral whitening discussed in this section are processes that can correct for the time-varying effects of attenuation by spectral flattening. A deterministic alternative to compensate for frequency-dependent attenuation is provided by inverse Q filtering.
The amplitude spectrum of the inverse Q filter is given by (Appendix B.9)
where ω is the angular frequency component associated with the input trace and τ is the time variable associated with the output trace from inverse Q filtering. The phase spectrum of the inverse Q filter usually is assumed to be minimum-phase, which can be computed by taking the Hilbert transform of the amplitude spectrum (Appendix B.4).
Application of the inverse Q filter requires knowledge of the attenuation factor Q, which usually is assumed to be constant. A compilation of laboratory measurements of Q for some rock samples is given by Table 2-28.
Note from Table 2-28 that most measurements have been made at extremely high frequencies compared to the typical frequency band of seismic waves. Nevertheless, by assuming frequency-independent Q factor , these measurements can still be considered useful. Also note that the Q factor can vary significantly for limestone, sandstone, and shale rock samples of different origin.
The inverse of the amplitude spectrum defined by equation (48) can be used to obtain a quantitative measure of attenuation. In terms of frequency f, wave velocity v and depth z = vτ, the inverse is
To determine how far in depth the wave has to travel before its amplitude reduces to, say, one-tenth of its amplitude at the surface z = 0, rewrite equation (49) as follows:
|Rock Type||Attenuation Constant, Q||Frequency Range (Hz)|
Note that the smaller the Q factor, the lower the velocity and the higher the frequency, the shallower the depth at which the wave amplitude decays to a fraction of the wave amplitude at z = 0. Table 2-29 lists the z values for a frequency of 30 Hz and a velocity of 3000 m/s for a range of Q values.
|Q Factor||Depth in m|
Note that the smaller the Q factor the shallower the depth at which the amplitude drops to the specified value of one-tenth of the original value at the surface z = 0. For very large Q values, that is, for small attenuation, the amplitude reduction to the specified value does not take place until the wave reaches very large depths beyond the exploration objectives.
Unfortunately, there is no reliable way to estimate the attenuation factor Q directly from seismic data. At best, inverse Q filtering can be applied to post- or prestack data (Appendix B.9) using a range of constant Q factors to create a Q panel, much like a filter panel (The 1-D Fourier transform). The factor that yields the flattest spectrum in combination with other signal processing applications — deconvolution and time-variant spectral whitening, is chosen as the optimum Q value.
- Time-variant deconvolution
- Time-variant spectral whitening
- Frequency-domain deconvolution
- Deconvolution strategies
- The problem of nonstationarity
- Kjartansson, 1979, Kjartansson, E., 1979, Constant Q-wave propagation and attenuation: J. Geophys. Res., 84, 4737–4748.
- Waters, 1981, Waters, K. H., 1981, Reflection seismology: Second edition, John Wiley & Sons.