Phase rotation

Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing

Series	Geophysical References Series
Title	Digital Imaging and Deconvolution: The ABCs of Seismic Exploration and Processing
Author	Enders A. Robinson and Sven Treitel
Chapter	13
DOI	http://dx.doi.org/10.1190/1.9781560801610
ISBN	9781560801481
Store	SEG Online Store

I speak without exaggeration that I have constructed three thousand different theories in connection with the electric light. Yet in only two cases did my experiments prove the truth of my theory.

-Thomas A. Edison

Chapter 12 introduced the concept of instantaneous attributes. Instantaneous means that we consider properties that are associated with the trace at one instant of discrete time n. A trace (which is a real-valued signal) can be considered to be the real part of a complex trace

${\begin{aligned}z_{n}=x_{n}+iy_{n}.\end{aligned}}$

(1)

In this representation, the trace $x_{n}$ can be called the in-phase signal, and the imaginary part $y_{n}$ is called the quadrature signal. The quadrature signal is the Hilbert transform of the in-phase signal. The complex trace can be written as a vector, which in polar form is

${\begin{aligned}z_{n}=A_{n}e^{i{\theta }_{n}}=A_{n}{\rm {\ cos\ }}{\theta }_{n}+i\ A_{n}{\rm {\ sin\ }}{\theta }_{n},\end{aligned}}$

(2)

where

$x_{n}=A_{n}{\rm {\ cos\ }}{\theta }_{n}{\ ,\ }y_{n}=A_{n}{\rm {\ sin\ }}{\theta }_{n},$

${\begin{aligned}A_{n}=+{\sqrt {x_{n}^{2}+y_{n}^{2}}}{,\ }{\theta }_{n}={\rm {tan}}^{-1}{\frac {y_{n}}{X_{n}}}.\end{aligned}}$

(3)

The length $A_{n}$ is called the instantaneous amplitude, and the angle ${\theta }_{n}$ is called the instantaneous phase lead. For the rest of this chapter, we shall simply say phase instead of phase lead.

What is phase rotation? Phase rotation refers to rotation of the complex-trace vector. Phase rotation for a specified phase angle is done as follows: A rotation of $90^{\circ }\pi /2$ transforms the complex trace, as given by equation 2 of Chapter 12, into the rotated complex trace

${\begin{aligned}z_{n}e^{i\pi {/2}}=A_{n}e^{i\left({\theta }_{n}+{\frac {\pi }{2}}\right)}=A_{n}{\rm {\ cos\ }}\left({\theta }_{n}+{\frac {\pi }{2}}\right)+iA_{n}{\rm {\ sin\ }}\left({\theta }_{n}+{\frac {\pi }{2}}\right)\end{aligned}}$

(4)

$=-A_{n}{\rm {\ sin\ }}{\theta }_{n}+iA_{n}{\rm {\ cos\ }}{\theta }_{n}=u_{n}+iv_{n}.$

Figure 1.  A complex-trace z and the rotated complex trace w for a rotation angle of $90^{\circ }$

We recognize the real part, namely $u_{n}=-A_{n}{\rm {\ sin\ }}{\theta }_{n}$, as the negative of the quadrature component (Figure 1).

A $180^{\circ }$ rotation gives the factor $e^{i\pi }=-{\rm {1}}$. Thus, the rotated complex trace is $-z_{n}$, which is out of phase with the unrotated complex trace $z_{n}$. In other words, a $180^{\circ }$ rotation is equivalent to multiplying the trace values by –1. A positive phase shift pulls an event up in time (i.e., a positive phase shift produces an advance), whereas a negative phase shift pushes an event down in time (i.e., a negative phase shift produces a delay).

Vibroseis data often require phase shifting. A vibroseis trace can be $90^{\circ }$ out of phase with other traces recorded in the same area. A $90^{\circ }$ rotation is required to correct this situation. However, the polarity must be checked carefully, because a $-90^{\circ }$ correction might be needed instead of one of $+90^{\circ }$.

However, why limit ourselves to rotations of $90^{\circ }$, $-90^{\circ }$, and $180^{\circ }$? Suppose that we want to phase-shift (i.e., phase-rotate) the complex trace by the phase angle $\phi$ We form the complex trace (after first converting $\phi$ to radians)

${\begin{aligned}w=z_{n}e_{i}^{\phi }=u_{n}+iv_{n}=A_{n}e^{i(\theta _{n}+\phi )}=A_{n}\cos(\theta _{n}+\phi )+iA_{n}\;\sin(\theta _{n}+\phi ).\end{aligned}}$

(5)

Figure 2.  A complex-trace z and the rotated complex trace w for a rotation angle of $\phi$.

Then the real signal $u_{n}=A_{n}{\rm {\ cos\ }}\left({\theta }_{n}+\ \phi \right)$ is the required phase-rotated version of the real signal $x_{n}$ (Figure 2). A rotation of $\phi$ leaves the instantaneous amplitude $A_{n}$ intact but changes the instantaneous phase from ${\theta }_{n}$ to ${\theta }_{n}+\phi$.

Conventional seismic data can be improved by adequate phase corrections. For example, phase distortion imposed on the received seismic signal by the data-acquisition system must be accounted for and corrected. Correct determination of the phase in seismic data processing is important. Here, we summarize the important results.

In matching two surveys, one might tend to think that the first survey is either in phase or out of phase with the second survey. In other words, one tends to think in terms of either positive polarity (corresponding to a phase rotation of $0^{\circ }$) or negative polarity (corresponding to a phase rotation of $180^{\circ }$). However, the dichotomy of either normal polarity or reverse polarity is not adequate for matching the wide variation of phase shifts that occurs in recording and processing. Accurate phase corrections are vital when one is dealing with synthetic logs. Seismic records show phase shifts other than $180^{\circ }$ for different configurations of recording instruments. As a result, seismic sections from different surveys usually do not tie adequately. A simple reversal of polarity on one section can improve the tie at some levels, but it usually makes other ties worse. Proper phase matching is vital.

Seismic data processing is an enterprise that requires both initiative and resourcefulness. It employs the methodologies of two disciplines. One is geophysics, in which we are concerned with the physics of wave motion. The other is petrophysics, in which we are concerned with the physics of the rock layers in our mapping of petroleum reservoirs. Successfully linking geophysics and petrophysics is fundamentally important. Certainly, this integration is a requirement in all oil exploration surveys today.

When we are dealing with the geophysics of wave propagation, we must conduct the processing under the minimum-phase assumption. Minimum phase is a characteristic of many physical systems involved in the wave-propagation process. The minimum-phase assumption is in keeping with the physical action of the earth as the seismic signal travels through the various strata. In addition, recording instruments usually are minimum phase, with a linear phase shift added in case of a constant time delay. Deconvolution depends on the minimum-phase assumption in its task of increasing seismic resolution and removing multiple reflections.

However, at some point in the data-processing scheme, the physics of wave propagation must be abandoned, and our attention must turn to petrophysics. The final seismic section must reveal the rock layers in their greatest detail. In this phase of processing, the minimum-phase component should be removed to yield a seismic section made up of symmetric wavelets. The symmetric wavelet is better suited for stratigraphic resolution and yields better synthetic logs. Several methods of wavelet estimation are found in Osman and Robinson (1996).

A minimum-phase wavelet is a causal wavelet that has its energy more concentrated toward the front than does any other causal wavelet with the same autocorrelation as that of the minimum-phase wavelet (Appendix N). The actual amount of phase shift for a typical seismic minimum-phase wavelet is close to $90^{\circ }$ in many cases because such a minimum-phase wavelet tends to have a dipolelike structure. That is, the typical minimum-phase wavelet tends to oscillate from one polarity to the other polarity, with nearly equal amplitudes. In comparison, a zero-phase (or 180°-phase) wavelet is completely symmetric about the central peak.

Let us illustrate the phase rotation of a wavelet. We will construct the wavelet in a special way. We start with a band-limited low-pass real symmetric frequency spectrum $S\left(\omega \right)$, which is called the envelope spectrum. The inverse Fourier transform of this envelope spectrum gives a real symmetric wavelet $s_{n}$, which is called the envelope wavelet. Now we use the envelope wavelet $s_{n}$ to modulate a carrier signal. The carrier signal is the high-frequency sinusoidal signal ${\rm {\ cos\ }}{\omega }_{0}n$. We assume that the carrier frequency ${\omega }_{0}$ is positive and lies above the pass-band of the envelope spectrum. The required signal is the modulated wavelet $x_{n}=s_{n}{\rm {\ cos\ }}{\omega }_{0}n$. The frequency spectrum of the modulated wavelet is

${\begin{aligned}X\left(\omega \right)=\sum {x_{n}}e^{-i\omega n}=\sum {s_{n}}e^{-i\omega n}{\rm {\ cos\ }}{\omega }_{0}n={\frac {1}{2}}\sum {s_{n}}e^{-i\omega n}\left(e^{i{\omega }_{\rm {o}}n}+e^{-i{\omega }_{0}n}\right)\\\;\;\;\;\;\;\;\;\;={\frac {1}{2}}\sum {s_{n}}e^{-i\left(\omega -{\omega }_{\rm {o}}\right)n}+{\frac {1}{2}}\sum {s_{n}}e^{-j\left(\omega +{\omega }_{0}\right)n}={\frac {1}{2}}S\left(\omega -{\omega }_{0}\right)+{\frac {1}{2}}S\left(\omega +{\omega }_{0}\right),\end{aligned}}$

(6)

where each summation is over all values of n. In other words, the frequency spectrum of the modulated wavelet is the sum of one-half the envelope spectrum shifted to the right by the amount of the carrier frequency plus one-half the envelope spectrum shifted to the left by the amount of the carrier frequency.

Because the carrier frequency ${\omega }_{0}$ lies to the right of the passband of the envelope spectrum, it follows that the shifted envelope spectrum $Z\left(\omega \right)=S\left(\omega -\ {\omega }_{0}\right)$ vanishes for negative values of frequency $\omega$. Hence, $Z\left(\omega \right)$ is the frequency spectrum of an analytic signal Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_n} . The inverse Fourier transform of $Z\left(\omega \right)$ gives $z_{n}=s_{n}e^{i{\omega }_{0}n}$. The real part of $z_{n}$ is the modulated wavelet $x_{n}=s_{n}{\rm {\ cos\ }}{\omega }_{0}n$, and the imaginary part $y_{n}=s_{n}{\rm {\ sin\ }}{\omega }_{0}n$ is the Hilbert transform of the modulated wavelet. We see that the envelope wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle {{\rm s}}_n} gives the instantaneous amplitude and that ${\omega }_{0}n$ gives the instantaneous phase.

Let us look at an example. The fast Fourier transform (FFT) uses the same number of sample points in both the time and frequency domains. Let the number of sample points be $N={64}$ (note that N is a power of 2). Suppose that the sampling interval is dt = 0.004 ms, in which case the Nyquist frequency is $f_{n}={1/}\left(2dt\right)={125}$. The Nyquist range is $2f_{n}={250Hz}$. The frequency sampling interval is $df={250/64=3.90625}$ Hz. Let the envelope spectrum be a rectangular spectrum of amplitude 1 centered at the origin, with a total bandwidth equal to one-eighth of the Nyquist range. The Nyquist range is ${250=64}df$, so one-eighth of the Nyquist range is 8 df, and the bandwidth is 8 $df={31.25}$ Hz. One-half of the bandwidth is 4 $df={15.625}$ Hz. The envelope spectrum is equal to 1 within the passband from the lower frequency –15.625 Hz to the upper frequency 15.625 Hz, and it is equal to 0 outside this passband (Figure 3b). The inverse Fourier transform of the envelope spectrum gives a symmetric envelope wavelet (Figure 3a). In fact, the envelope is called a positive envelope because its major peak is positive.

Figure 3.  (a) The symmetric envelope wavelet and (b) the spectrum of the symmetric envelope wavelet.

Figure 4.  (a) The symmetric modulated wavelet and (b) the spectrum of the symmetric modulated wavelet.

We use the envelope wavelet to modulate a sinusoidal carrier signal of frequency Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_0={7}df={27.3438}} Hz. The lower frequency of the positive-frequency passband of the modulated spectrum is $f_{\rm {1}}={3}df={11.7188}$ Hz, and the upper frequency is $f_{2}={\rm {1l}}df=42.9688$ Hz. The bandwidth is $f_{2}-f_{1}={8\ }df={31.25}$ Hz. Because of symmetry, the negative-frequency passband has the same bandwidth (Figure 4b). If we take the inverse Fourier transform of the modulated spectrum, we obtain the symmetric modulated wavelet (Figure 4a).

Figure 5.  (a) Symmetric modulated wavelet after a phase rotation of $\phi =90^{\circ }$. (b) The corresponding (approximate) minimum-phase wavelet. (c) One signal plotted on top of the other for comparison.

Figure 5a shows the symmetric modulated wavelet after a phase rotation of $\phi =90^{\circ }$. Figure 5b shows the corresponding (approximate) minimum-phase wavelet. The minimum-phase wavelet is approximate because a true minimum-phase wavelet would have no precursor before the large positive peak that appears immediately after time 0.10. Figure 5c shows one signal plotted on top of the other for comparison. This figure illustrates the point of this set of plots: In this case, the symmetric modulated wavelet rotated by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 90^\circ} is a reasonable approximation of the minimum-phase wavelet. The phase-rotation process is a two-way street. That is, in this case, a rotation of $-90^{\circ }$ would convert the minimum-phase wavelet into the symmetric wavelet. It follows that by means of an appropriate phase rotation, the minimumphase wavelets on a seismic trace can be converted approximately into the desired symmetric wavelets.

Figure 6.  A dipping event is seen across the record, but each trace is a 5° rotation of the trace above.

Let us now change the subject. Look at the seismic section shown in Figure 6. A dipping event runs smoothly across the section. Looking at the event from bottom to top, the wavelets appear to be traveling. However, each wavelet is only a $5^{\circ }$ phase-rotated version of the one above it. The wavelets are not traveling at all; instead, the phase goes through a $360^{\circ }$ change across the record. The top trace is identical to the bottom trace.

If two surveys do not tie, the interpreter might try reversing the polarity on one of them. That might improve the fit at one place but degrade it at another. Such a procedure usually is not warranted because phase differences between instruments tend to be something like $40^{\circ }$ or $60^{\circ }$, depending on configuration. Different field equipment usually has intermediate phase differences that are smaller than $180^{\circ }$.

Sawtooth effects on synthetic logs can result from phase shifts. Two sets of seismic data at a common tie point but with different instruments can produce synthetic logs that have improper phase characteristics. A constant phase shift applied to one of the synthetic logs can result in a modified log that has a better phase appearance.

Bishop and Nunns (1994) described algorithms for computing the amplitude, time, and phase differences at each intersection among a series of 2D seismic lines. They gave an iterative least-squares technique for deriving optimal mis-tie corrections for each line. They also included a necessary modification of the least-squares technique for nonlinear phase data.

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Also in this chapter

• Alternate approaches to phase rotation
• Band-limiting root approximation
• Appendix N: The energy-delay theorem

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