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**Chapter 4**

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### Minimum-phase signal

### A cosine signal with two impulses

### Amplitude spectrum of the two impulses

### A quadratic chirp signal

### A cosine signal with a shutdown period

### A localized modulating function

### Windowed Fourier transform

### Windowed Fourier transform

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Minimum-phase signal
A cosine signal with two impulses
Amplitude spectrum of the two impulses
A quadratic chirp signal
A cosine signal with a shutdown period
A localized modulating function
Windowed Fourier transform
Windowed Fourier transform (Cont'd)
Windowed Fourier transform (Cont'd)
Windowed Fourier transform
Windowed Fourier transform (Cont'd)
Windowed Fourier transform (Cont'd)
Windowed Fourier transform
Windowed Fourier transform (Cont'd)
Synthetic data
Acquisition geometry
Amplitude spectrum of the wavelet transform
Wigner-Ville transform
Choi-Williams distribution
Uncertainty relation

Images - Chapter 4
(a) The minimum-phase signal, (b) the mixed-phase signal, (c) the maximum-phase signal, and (d) the zero-phase signal.

It is a common practice in petroleum seismology to classify signals in rather broad terms according to their frequency content. We here present a classification of seismic signals as a function of their phase spectrum. A classification as a function of bandwidth will be introduced in the next chapter.

The term *phase* in seismic signals is often referred to as minimum phase, maximum phase, mixed phase, or zero phase. Each of these terms refers to the characteristics of the signal shape and starting location in the data. This figure shows four signals, each of a different phase. Each type of phase is described below:

(i) Minimum phase: The minimum-phase signal, shown in (a), is described as a front-loaded signal. This means that the energy in the signal is concentrated in the front of the pulse. The signal is not symmetrical. The phase of this signal will vary for each frequency component of the signal. Note that the convolution between two minimum-phase signals is always minimum phase. However, convolution between one minimum-phase and one nonminimum-phase signal does not produce a minimum phase signal.

For a group of signals with the same amplitude spectra, the minimum-phase signal will have the smallest phase shift at all frequencies, cause the least time delay, have the most front-loaded energy distribution, and have the largest time-zero sample value.

(ii) Mixed phase: The mixed-phase signal, shown in (b), is described as a signal with its energy concentrated in the center of the pulse. It can be divided into minimum-phase and maximum-phase signals. The signal is usually not symmetrical. The phase of this signal will vary for each frequency component of the signal.

(iii) Maximum phase: The maximum-phase signal, shown in Figure (c), is described as an end-loaded signal. This means that the energy in the signal is concentrated toward the end of the pulse. The signal is not symmetrical. The phase of this signal will vary for each frequency component of the signal. The characteristics of the maximum-phase signal are the opposite of the minimum-phase signal.

(iv) Zero phase: The zero-phase signal, shown in (d), is symmetrical and centered on zero time. The zero-phase signal has the shortest duration and largest peak amplitude of any

signal with the same amplitude spectrum. These characteristics make it the most desirable of all the signals because of its resolution capability. The phase of the zero-phase signal is zero for all frequency components contained within the signal.

Images - Chapter 4
(a) A cosine signal with two impulses (in the time domain) and (b) its amplitude spectrum.

From 0 to 600 ms, we have a standard stationary signal which corresponds to the sum of two cosine waves, and at 600 ms we have a sharp peak in amplitude followed by a drop. After the sharp peak in amplitude, we return to our stationary behavior of two cosine waves until we reach 800 ms, where another sharp peak in amplitude occurs and we return to a standard stationary signal again. Just as in the analogy of the restaurant noise used earlier to introduce nonstationary signals, the cosine waves represent the classical restaurant background noise of music and conversations, whereas the impulses represent the accidental dropping of a ceramic plate on the floor. So the signal here is effectively nonstationary.

Let us now look at the Fourier transform of this signal. Note that the Fourier transform detects the two cosine wave signals which correspond to the two spikes in (b). And what about the impulses? By replotting the Fourier transform at a different scale, we can now see the superposition of frequencies corresponding to the two impulses. So the Fourier transform does not provide clear information about the two impulses because of a scale problem.

Images - Chapter 4
(a) The amplitude spectrum of the two impulses and (b) the amplitude spectrum of a signal. Notice that we have truncated the amplitude spectrum in (b) to see the small-scale features.

The difficulty with the Fourier transform in this example is that cosine wave signals are well localized in the Fourier domain, whereas impulses are spread over all the frequencies. In other words, we need to sum all frequencies so that constructive and destructive interferences can allow us to reconstruct simple signals such as impulses. As we discussed earlier in this chapter, the criterion for selecting cosine and sine functions as the basis of the functions of the Fourier representation is that cosine and sine functions are a set of simple functions which can be used to describe complex functions. There are exceptions to this statement: representing an impulse by combining an infinite number of cosine and sine functions is not an effective way of representing such a simple signal.

In general, if a signal contains sudden changes (e.g., discontinuities), the high frequencies relative to these changes are detected by the Fourier transform, but their contributions are spread all over the Fourier domain because the modulating function exp(*i ω t*) is not limited in a specific interval. The Fourier analysis is therefore more efficient in the study of signals that do not suffer sudden variations with time (or, if the sudden variations exit, they must exist at all times); i.e., they must be stationary signals.

Images - Chapter 4
(a) A quadratic chirp signal (in the time domain) and (b) its amplitude spectrum.

The plot of this signal is shown here, along with its amplitude spectrum. We can see that the Fourier transform suggests that the quadratic chirp signal has most of its energy at zero frequency; but in actuallity it does not. If we divide this signal into small time intervals, we can recognize that the frequencies of cosine waves vary from one time window to another. Actually, the frequencies of this signal have a quadratic growth with time. Therefore the quadratic chirp signal is a nonstationary signal, because in a stationary signal, the frequency contents is relatively uniform with time. So to properly represent the quadratic chirp signal, we simultaneously need to do so in frequency and time.

Images - Chapter 4
(a) A cosine signal with a shutdown period (in the time domain) and (b) its amplitude spectrum.

Note that if we are just concerned with the steps of forward and inverse Fourier transform, without any processing (like filtering some unwanted features) or interpretation between these two steps, the scale problem pointed out here will not be an issue. However, in petroleum seismology, we invoke the Fourier representation for identifying features of signals which are easily detectable in the time domain and for processing to remove unwanted energy or just to speed up computations. In all these cases, we need to have the best possible Fourier representation of signals. Unfortunately, such representation is possible only in the time-frequency domain for nonstationary signals.

Images - Chapter 4
Illustration of a localized modulating function: (a) the modulating function used in the Fourier transform and (b) the localized modulating function used in the windowed Fourier transform. The auxiliary function used in (b) is the Gaussian window. The jet colorscale is also used here [going from blue (minimum value) to red (maximum value)].

Images - Chapter 4
The magnitude of the windowed Fourier transform of a signal using (a) a Gaussian window of width 16 ms, (b) a Gaussian window of width 32 ms, (c) a Gaussian window of width 64 ms, (d) a Gaussian window of width 128 ms, (e) a Gaussian window of width 256 ms, and (f) a Gaussian window of width 512 ms. The jet colorscale is also used here [going from blue (minimum value) to red (maximum value)].

Let us now look at examples of WFT representations. We will use the examples of cosine waves with impulses, cosine waves with shutdown time, and quadratic chirp signals. Let us start with the example of cosine waves with impulses. This signal consists of the sum of two cosine waves and two impulses. This figure shows the WFT of this signal for various window widths [16 ms (i.e., 8 samples), 32 ms (i.e., 16 samples), 64 ms (32 samples), 128 ms (64 samples), 256 ms (128 samples), and 512 ms (256 samples)]. We have used the Gaussian window for the signal g(t) (i.e., g(t) = exp [- a x^{2}] for a convenient value a> 0) for the computations of the WFT in these figures and actually for all the WFT figures in this chapter. The following observations can be made from these figures:

- In (a) we see that two impulses were detected with a good localization in time. The two cosine waves have also been detected, but the localization of their frequencies is not accurate.
- In (b) we have improved the localization of the two cosine frequencies, but they are still not clearly distinguishable.
- In (c) and (d) we have a localization of the two impulses in time and a good localization of the cosine wave frequencies. Notice that if the two impulses were close together, we would not be able to distinguish them because a small time spread is still visible around the locations of these impulses.
- In (e) and (f), we can see that as the window g(t) get wider, the WFT representation converges toward the classical Fourier transform representation.

Images - Chapter 4 The magnitude of the windowed Fourier transform of a signal using (a) a Gaussian window of width 32 ms, (b) a Gaussian window of width 64 ms, (c) a Gaussian window of width 128 ms, (d) a Gaussian window of width 256 ms, (e) a Gaussian window of width 512 ms, and (f) a Gaussian window of width 1024 ms. The jet colorscale is also used here [going from blue (minimum value) to red (maximum value)].