Figure 1.1-2 shows how a time-dependent signal was synthesized from its frequency components. Consider a signal with a zero-phase spectrum. Figure 1.1-11 shows sinusoids with frequencies ranging from approximately 1 to 32 Hz. All of these sinusoids have zero-phase lag; thus, the peak amplitudes align at t = 0. The time-domain signal on the trace identified by an asterisk in Figure 1.1-11 is synthesized by summing all these sinusoids — a process described by inverse Fourier transform. Such a time-domain signal is called a wavelet. A wavelet usually is considered a transient signal, that is, a signal with a finite duration. It has a start time and an end time, and its energy is confined between these two time positions. The wavelet that was just constructed is symmetric around t = 0 and has a (positive) peak amplitude at t = 0. Such a wavelet is called zero phase. In fact, the wavelet was synthesized using the zero-phase sinusoids of equal peak amplitude.
A zero-phase wavelet is symmetric with respect to zero time and peaks at zero time. Figure 1.1-12 shows the result of applying a linear phase shift to the sinusoids in Figure 1.1-11. Linear phase shift is described by a function that represents a line in the frequency domain: ϕ = αω, where α is the slope constant and ω is the angular frequency, which is the temporal frequency scaled by 2π. The wavelet, identified by an asterisk in Figure 1.1-12, has shifted in time by -0.2 s, but its shape has not changed. Thus, a linear phase shift is equivalent to a constant time shift. The slope of the line describing the phase spectrum is proportional to the time shift.
Figure 1.1-12 The same sinusoidal components as in Figure 1.1-11, but with a −0.2 s constant-time delay. When summed, these sinusoids yield a band-limited symmetric wavelet that is represented by the trace on the right (denoted by an asterisk). This wavelet is the same as that shown in Figure 1.1-11, except that it is shifted in time by −0.2 s. This time shift is related to the linear phase spectrum associated with the summed frequency components.
The wavelet can be shifted by any amount of time simply by changing the slope of the line ϕ = αω, that describes the phase spectrum. Starting with the zero-phase wavelet, Figure 1.1-13 shows the effect of increasing amounts of the linear phase shift on a zero-phase wavelet. Although not shown, by changing the sign of the slope in the phase spectrum, the wavelet can be shifted in the opposite time direction.
If a 90-degree phase shift is applied to each of the sinusoids in Figure 1.1-11, as shown in Figure 1.1-14, then the zero crossings are aligned at t = 0. The result of this summation yields the antisymmetric wavelet shown on the trace identified by an asterisk. Note that the two wavelets in Figures 1.1-11 and 1.1-14 have the same amplitude spectrum because they have been synthesized from the sinusoidal components with the same peak amplitude and frequency. The difference lies in their phase spectra. The wavelet in Figure 1.1-11 has zero-phase spectrum, while that in Figure 1.1-14 has a constant-phase spectrum (+90 degrees). Therefore, the difference in wavelet shape is a result of the difference in their phase spectra.
Figure 1.1-15 shows the effect of various amounts of constant phase shift on a zero-phase wavelet. The 90-degree phase shift converts the zero-phase wavelet an antisymmetric wavelet. The 180-degree phase shift changes the polarity of the zero-phase wavelet. The 270-degree phase shift changes the polarity of the zero-phase wavelet, while converting it to an antisymmetric wavelet. Finally, the 360-degree phase shift retains the shape of the original wavelet. A constant phase shift to changes the shape of a wavelet. In particular, a 90-degree phase shift converts a symmetric wavelet to an antisymmetric wavelet, while a 180-degree phase shift changes its polarity.
Figure 1.1-16 shows a portion of a seismic section with the application of different degrees of constant phase rotation. Note the change in the wavelet character of the significant reflections. This difference in wavelet character has an impact on picking events for interpretation. When comparing displays of two different sections, which may be associated with two different vintages of processing of the same data or two different lines from the same survey, the wavelet character of the reflection event that is being picked must be consistent from one section to another. A common mistake is displaying two sections to be compared with opposite polarity. The polarity convention set by the Society of Exploration Geophysicists is based on a negative water-bottom reflection coefficient, which corresponds to a positive polarity.
So far, two basic phase spectra have been examined — linear and constant phase shifts. We now examine their combined effect. The phase spectrum is defined by a function ϕ = ϕ0 + αω, where ϕ0 is the constant phase shift and α is the slope of the linear phase shift. Figure 1.1-17 shows the result of applying a linear phase shift (as in Figure 1.1-12) plus a 90-degree constant phase shift (as in Figure 1.1-14) to the sinusoids in Figure 1.1-11. The zero-phase wavelet with the same amplitude spectrum as that in Figure 1.1-11 was shifted in time by -0.2 s because of the linear phase shift, and converted to an antisymmetric form because of the constant 90-degree phase shift.
Figure 1.1-15 Starting with the zero-phase wavelet (a), its shape is changed by applying constant phase shifts. A 90-degree phase shift converts the zero-phase wavelet to an antisymmetric wavelet (b), while a 180-degree phase shift reverses its polarity (c). A 270-degree phase shift reverses the polarity, while making the wavelet antisymmetric (d). Finally, a 360-degree phase shift does not modify the wavelet (e).
Other variations in phase spectrum are shown in Figure 1.1-18. The zero-phase wavelet (Figure 1.1-18a) can be modified to different shapes simply by changing the phase spectrum. It can be modified to the extent that it may no longer resemble the original wavelet shape as illustrated by the last example (Figure 1.1-18d). By keeping the amplitude spectrum unchanged, the wavelet shape can be changed by modifying the phase spectrum.
- Analog versus digital signal
- Frequency aliasing
- Time-domain operations
- Crosscorrelation and autocorrelation
- Vibroseis correlation
- Frequency filtering
- Practical aspects of frequency filtering
- Bandwidth and vertical resolution
- Time-variant filtering