Time-variant spectral whitening - book
As a waveform travels deeper into the earth, its higher frequencies are attenuated more than its lower frequencies are. The inverse Q-filtering, or Q-compensation, is one way to correct for such losses (Kjartansson, 1979; Ecevitoglu and Costain, 1988; Dillon, 1991). The term Q is the seismic quality factor and is a measure of seismic attenuation (see Chapter 14 for more details).
The correction process can be implemented as follows: Apply a series of narrow band-pass filters to a seismic trace. A low-frequency component of the trace will have a lower decay rate than does an intermediate-frequency component. Likewise, an intermediate-frequency component will have a lower decay rate than does a high-frequency component. A series of gain functions can be designed to describe the decay rates within each frequency band. One way to accomplish that is to compute the envelope of each band-pass-filtered trace. The inverse of each of these “envelope gain functions” is applied to each frequency band, and the results are summed. This sum trace is the output of such a time-variant process — the time-variant spectral-whitening (TVSW) process. The number of filter bands, the width of each band, and the overall bandwidth are the TVSW parameters. These parameters must be prescribed for each application.
The TVSW process corrects for attenuation effects and partially deconvolves the seismic wavelet. Spiking deconvolution not only compresses the wavelet but also attenuates reverberations. In contrast, TVSW mainly compresses the wavelet without significantly changing the ringy character of the data. In practice, TVSW might do a better job of flattening, or “whitening,” the amplitude spectrum than is the case for conventional deconvolution. This property can help us deal with broadband data that have a large dynamic range.
Spectral flattening also is achievable with other frequency-domain approaches. For example, spiking deconvolution can be formulated in the frequency domain. In addition, the method of zero-phase frequency-domain deconvolution flattens the amplitude spectrum without touching the phase. Zero-phase frequency-domain deconvolution designed to attain a result equivalent to TVSW requires that the input trace be partitioned into small time gates. When zero-phase frequency-domain deconvolution is performed over multiple time gates along the trace, the result is roughly equivalent to TVSW.
- Kjartansson, E., 1979, Constant Q, wave propagation and attenuation: Journal of Geophysical Research, 84, 4737-4748.
- Ecevitoglu, B. G., and J. K. Costain, 1988, New look at body wave dispersion: 58th Annual International Meeting, SEG, Expanded Abstracts, 1043-1045.
- Dillon, P. B., 1991, Q and upward extension of VSP data through the energy flux theorem: First Break, 9, no. 6, 289-298.
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Also in this chapter
- Prediction-error filters
- Water reverberations
- Gap deconvolution of a mixed-delay wavelet
- Prediction distance
- Model-driven predictive deconvolution
- Convolutional model in the frequency domain
- Model-based deconvolution
- Surface-consistent deconvolution
- Interactive earth-digital processing
- Appendix K: Exercises