What happens to a wavelet when its amplitude spectrum is changed while its zero-phase character is preserved? To begin, consider the wavelet in Figure 1.1-21 (summed trace 1) resulting from superposition of two very low-frequency components. Then, add increasingly higher frequency components to the Fourier synthesis (summed traces 2 through 5). Note that the wavelet in the time domain is compressed as the frequency bandwidth (the range of frequencies summed) is increased. Ultimately, if all the frequencies in the inverse Fourier transformation are included, then the resulting wavelet becomes a spike, as seen in Figure 1.1-22 (summed trace 6). Therefore, a spike is characterized as the in-phase synthesis of all frequencies from zero to the Nyquist. For all frequencies, the amplitude spectrum of a spike is unity, while its phase spectrum is zero.
Figure 1.1-23 shows five zero-phase wavelets, synthesized as shown in Figure 1.1-21. Note that all of them have band-limited amplitude spectra. A zero-phase band-limited wavelet can be used to filter a seismic trace. The output trace contains only those frequencies that make up the wavelet used in filtering. The time-domain representation of the wavelet is the filter operator. The individual time samples of this operator are the filter coefficients. The process described here is zero-phase frequency filtering, since it does not modify the phase spectrum of the input trace, but merely band-limits its amplitude spectrum.
Frequency-domain filtering involves multiplying the amplitude spectrum of the input seismic trace by that of the filter operator. The procedure is described in Figure 1.1-24. On the other hand, the filtering process in the time domain involves convolving the filter operator with the input time series. Figure 1.1-25 is a description of the filter design and its time-domain application. The frequency- and time-domain formulations of the filtering process (Figures 1.1-24 and 1.1-25) are based on the following important concept in time series analysis: Convolution in the time domain is equivalent to multiplication in the frequency domain. Similarly, convolution in the frequency domain is equivalent to multiplication in the time domain.
Frequency filtering can be in the form of band-pass, band-reject, high-pass (low-cut), or low-pass (high-cut) filters. All of these filters are based on the same principle — construction of a zero-phase wavelet with an amplitude spectrum that meets one of the four specifications.
Band-pass filtering is used most commonly, because a seismic trace typically contains some low-frequency noise, such as ground roll, and some high-frequency ambient noise. The usable seismic reflection energy usually is confined to a bandwidth of approximately 10 to 70 Hz, with a dominant frequency around 30 Hz.
Band-pass filtering is performed at various stages in data processing. If necessary, it can be performed before deconvolution to suppress remaining ground-roll energy and high-frequency ambient noise that otherwise would contaminate signal autocorrelation. Narrow band-pass filtering may be necessary before crosscorrelating traces in a CMP gather with a pilot trace for use in estimating residual statics shifts. Band-pass filtering also can be performed before computing crosscorrelations during construction of the velocity spectrum for improved velocity picking (velocity analysis). Finally, it is a standard practice to apply a time-variant band-pass filter to stacked data (basic data processing sequence).
- Analog versus digital signal
- Frequency aliasing
- Phase considerations
- Time-domain operations
- Crosscorrelation and autocorrelation
- Vibroseis correlation
- Practical aspects of frequency filtering
- Bandwidth and vertical resolution
- Time-variant filtering