# Illustrations of spectra

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Series Geophysical References Series Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing Enders A. Robinson and Sven Treitel 7 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Let us now illustrate the spectra of different dipoles. We start with the unit spike: the coefficient 1 at time index 0. In terms of a two-length causal wavelet, the unit spike is (1, 0). The Fourier transform of (1, 0) is 1 for all $\omega$ . Hence, the amplitude spectrum of the unit spike is equal to one for all frequencies or, in other words, the amplitude spectrum is white. If we write 1 = 1 + 0 i, we see that the phase-lag spectrum of the unit spike is $-{\rm {tan}}^{-1}\left({0/1}\right)$ , which is zero for all $\omega$ . In other words, the unit spike has the minimum-phase spectrum, which is equal to zero for all frequencies.

If we shift the unit spike by one time unit, we obtain the delayed spike (0, 1). Its Fourier transform is

 {\begin{aligned}{\rm {\ exp\ }}\left(-i\omega \right)={\rm {\ cos\ }}\omega -i{\rm {\ sin\ }}\omega .\end{aligned}} (28)

Hence, the amplitude spectrum of the delayed spike is equal to one for all frequencies. The phase-lag spectrum is

 {\begin{aligned}&-{\rm {tan}}^{-1}\left({\frac {-{\rm {\ sin\ }}\omega }{{\rm {\ cos\ }}\omega }}\right)={\rm {tan}}^{-1}\left({\rm {\ tan\ }}\omega \right)=\omega .\end{aligned}} (29)

The delayed spike has a maximum-phase-lag spectrum, which is equal to the frequency $\omega$ for all frequencies in the Nyquist range. Figure 5 shows the amplitude spectrum (dashed line) that these two wavelets have in common. The minimum-phase spectrum is the $0^{\rm {o}}$ line through the origin, and the maximum-phase-lag spectrum is the ${4}{5}^{o}$ line through the origin. The sum of the two phase curves is also the ${4}{5}^{o}$ line. Figure 5.  Spectra for the dipole made up of the minimum-delay wavelet (1, 0) and the maximum-delay wavelet (0, 1).

We look next at the minimum-delay wavelet (1, 0.5) and the corresponding maximum-delay wavelet (0.5, 1). Figure 6 shows the common-amplitude spectrum (dashed line) of these two wavelets. The amplitude spectrum is a hill-shaped curve with its peak at frequency 0. In other words, the amplitude spectrum is that of a low-pass filter. We see that the minimum-phase-lag spectrum (heavy line) and the maximum-phase-lag spectrum (medium-light line) add up to a ${4}{5}^{o}$ line (light line). Note that the two phase curves in Figure 6 are closer together than are the phase lines in Figure 5. Figure 6.  Spectra for the dipole made up of the minimum-delay wavelet (1, 0.5) and the maximum-delay wavelet (0.5, 1).

Let us review the terminology we introduced at the beginning of this chapter. As we know, there is phase lead and there is phase lag, with one being the negative of the other. In particular, the phase-lag spectrum is the negative of the phase-lead spectrum. Either one can be called the phase spectrum. According to the electrical engineering convention (which we follow), the exponential in the direct Fourier transform carries a negative sign. For that reason, we choose to let the word phase refer to the phase lag and not to the phase lead. Electrical engineers often say phase for phase lag, so we are in good company. For example, electrical engineers say minimum phase for minimum-phase lag, as we do.

Let us look at the minimum-delay wavelet (1, 0.9) and the corresponding maximum-delay wavelet (0.9, 1). Figure 7 shows the common-amplitude spectrum (dashed line) of these two wavelets. The amplitude spectrum is a hill-shaped curve with its peak at frequency 0. The amplitude spectrum is close to zero at the Nyquist frequencies $\pm \pi$ . The amplitude spectrum is that of a low-pass filter. We see that the minimum-phase spectrum (thick line) and the maximum-phase spectrum (medium-thick line) almost intersect with the line of slope 0.5. Because ${\rm {tan}}^{-{l}}\left({0.5}\right)=26.6^{\circ }$ , we call this line the $26.6^{\circ }$ line. The two phase spectra add up to a $45^{\circ }$ line (thin line). Note that the phase curves in Figure 7 are closer together than are the phase curves in Figure 6. Figure 7.  Spectra for the dipole made up of the minimum-delay wavelet (1, 0.9) and the maximum-delay wavelet (0.9, 1).

Let us look at the equal-delay wavelet (1, 1) and its corresponding reverse wavelet (1, 1), which is of course the same wavelet. Figure 8 shows their spectra. By definition, an equal-delay wavelet is both a minimum-delay wavelet and a maximum-delay wavelet. The hill-shaped amplitude spectrum touches zero at frequencies $\pm \pi$ . The phase spectrum is the $26.6^{\circ }$ line (thick line) through the origin. This line added to itself gives the $45^{\circ }$ line (thin line). Figure 8.  Spectra for the dipole made up of the equal-delay wavelet (1, 1) and the (same) equal-delay wavelet (1, 1).

Let us look at the minimum-delay wavelet (1, –0.5) and the corresponding maximum-delay wavelet (–0.5, 1). Figure 9 shows the common-amplitude spectrum (dashed line) of these two wavelets. The amplitude spectrum is a valley-shaped curve with its trough at frequency 0. In other words, the amplitude spectrum is that of a high-pass filter. We see that the minimum-phase spectrum (thick line) and the maximum-phase spectrum (medium-thick line) add up to a $45^{\circ }$ line (thin line). Note that the two phase curves in Figure 9 are farther apart than are the two phase curves in Figure 6. Figure 9.  Spectra for the dipole made up of the minimum-delay wavelet (1, –0.5) and the maximum-delay wavelet (–0.5, 1).

Let us look at the minimum-delay wavelet (1, –0.9) and the corresponding maximum-delay wavelet (–0.9, 1). Figure 10 shows the common-amplitude spectrum (dashed line) of these two wavelets. It is a V-shaped curve with its trough at frequency 0. The amplitude spectrum is that of a high-pass filter. The minimum-phase spectrum (thick line) and the maximum-phase spectrum (medium line) almost intersect with two 26.6° lines, one through $\pi {/2}$ on the vertical axis and the other through $-\pi {/2}$ on the vertical axis. The two phase spectra add up to a $45^{\circ }$ line (thin line). Figure 10.  Spectra for the dipole made up of the minimum-delay wavelet (1, –0.9) and the maximum-delay wavelet (–0.9, 1).

Let us look at the equal-delay wavelet (1, –1) and the corresponding reverse wavelet (–1, 1) (Figure 11). By definition, an equal-delay wavelet is both a minimum-delay wavelet and a maximum-delay wavelet. The common V-shaped amplitude spectrum touches 0 at frequency 0. The figure shows two $26.6^{\circ }$ lines, one through $\pi /2$ on the vertical axis and the other through $-\pi /2$ on the vertical axis. The minimum-delay phase spectrum is made up of the left side of the $26.6^{\circ }$ line that goes through $\pi /2$ and the right side of the 26.6° line that goes through $-\pi /2$ . The opposite is true for the maximum-phase spectrum. These two phase spectra add up to the $45^{\circ }$ line (thin line). Figure 11.  Spectra for the dipole made up of the equal-delay wavelet (1, –1) and the equal-delay wavelet (–1, 1).