# Instantaneous attributes - book

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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 12 http://dx.doi.org/10.1190/1.9781560801610 9781560801481 SEG Online Store

Instantaneous attributes, which are based on the Hilbert transform, were introduced into electrical engineering by Dennis Gabor (Gabor, 1946[1]), who invented the prototype hologram. John S. Farnbach (Farnbach, 1975[2]) introduced into earthquake seismology several attributes derived from complex-signal analysis. Nigel Anstey authored two widely circulated booklets privately published by Seiscom in 1972 and 1973. These booklets made Anstey the pioneer in appreciating the geologic significance of instantaneous attributes. Tury Taner and colleagues (Taner et al., 1979[3]) added to the attention that exploration geophysicists paid to seismic attributes.

In the original sense, an instantaneous attribute is a quantity derived from what is called complex-signal analysis or complex-trace analysis. A conventional seismic trace can be viewed as the real component of a complex trace. We now will see how the complex trace is obtained.

Let ${\displaystyle x_{n}}$ be the given real signal, which usually is the seismic trace. Complex-signal analysis involves obtaining the complex signal

 {\displaystyle {\begin{aligned}z_{n}=x_{n}+iy_{n}.\end{aligned}}} (1)

In this representation, the imaginary part ${\displaystyle y_{n}}$, called the quadrature signal, is obtained by computing the Hilbert transform of ${\displaystyle x_{n}}$. The Hilbert transform and some of its properties are described in Appendix L.

It can be shown that the energy spectrum of ${\displaystyle x_{n}}$ and the energy spectrum of ${\displaystyle y_{n}}$ are equal. In other words, if ${\displaystyle X\left(\omega \right)}$ is the Fourier transform of ${\displaystyle x_{n}}$ and ${\displaystyle {\rm {y}}\left(\omega \right)}$ is the Fourier transform of ${\displaystyle y_{n}}$, then

 {\displaystyle {\begin{aligned}{|}Y\left(\omega \right){|}^{2}={\ |}X\left(\omega \right){|}^{2}.\end{aligned}}} (2)

From this result, it follows that the total energy of a signal and its Hilbert transform are also equal.

The complex signal can be written in polar form as

 {\displaystyle {\begin{aligned}z_{n}=A_{n}e^{i{\theta }_{n}}{\ =}A_{n}{\rm {\ cos\ }}{\theta }_{n}+iA_{n}{\rm {\ sin\ }}{\theta }_{n},\end{aligned}}} (3)

where

 {\displaystyle {\begin{aligned}A_{n}={+}{\sqrt {x_{n}^{2}+y_{n}^{2}}}{\;\;\;\;\theta }_{n}={\rm {tan}}^{-1}{\frac {y_{n}}{x_{n}}}.\end{aligned}}} (4)

The positive quantity ${\displaystyle A_{n}}$ is called the instantaneous amplitude at time instant n. The instantaneous amplitude ${\displaystyle A_{n}}$ is the amplitude of the complex signal. The instantaneous amplitude can be visualized as forming the envelope of the real seismic trace. Note that instantaneous amplitude also is called envelope amplitude. Either designation can be used.

In Chapter 6, we defined phase lead and phase lag. Phase lag is defined as the negative of the phase lead. For example, the unit-delay filter can be represented as ${\displaystyle Z=e^{i\left(-\omega \right)}=e^{-i\left(\omega \right)}}$. It has phase lead ${\displaystyle -\omega }$, which is negative for positive frequencies. It has phase lag ${\displaystyle \omega }$, which is positive for positive frequencies. The unit-advance filter can be represented as ${\displaystyle Z^{-{\rm {l}}}=e^{i\left(\omega \right)}=e^{-i\left(-\omega \right)}}$. It has phase lead ${\displaystyle \omega }$, which is positive for positive frequencies. It has phase lag ${\displaystyle -\omega }$, which is negative for positive frequencies. Because realizable filters involve delay and not advance, we prefer to use the phase lag instead of the phase lead. However, when it comes to instantaneous attributes, the choice is invariably phase lead.

The angle ${\displaystyle {\theta }_{n}}$ in equation 4 is called the instantaneous phase lead at time instant n. The instantaneous angular frequency ${\displaystyle {\omega }_{n}}$ at time instant n is defined as the numerical derivative of the phase lead with respect to time. Let the sampling time increment be ${\displaystyle \Delta t=1}$ so that the Nyquist angular frequency is ${\displaystyle {\omega }_{N}=\pi }$ Various expressions for the numerical derivative of the phase lead are available. One that gives a good approximation should be used. The simplest approximation (but by no means the best approximation) for the numerical derivative is

 {\displaystyle {\begin{aligned}{\omega }_{n}={\frac {\Delta \theta }{\Delta t}}={\frac {{\theta }_{n+1}-{\theta }_{n}}{1}}={\theta }_{n+1}-{\theta }_{n}.\end{aligned}}} (5)

A complex trace permits separation of instantaneous amplitude information from phase-lead information, and it also allows us to compute the instantaneous frequency. These and other related quantities usually are displayed in a color-coded manner, thereby allowing an interpreter to see their interrelationships in time and space.

Apparent polarity is the polarity of the real trace at the instantaneous amplitude peak. In other words, apparent polarity is defined as the sign of ${\displaystyle x_{n}}$ when ${\displaystyle A_{n}}$ has a local maximum. Apparent polarity can help interpreters to identify gas accumulations. Lateral variations in all these displays can help to localize stratigraphic changes. Seismic traces representing wave motion can be transformed into corresponding traces representing a particular set of attributes, which then are available for further analysis.

What does instantaneous amplitude portray? Instantaneous amplitude is a robust, smoothed, polarity-independent measure of the energy in a trace at a given time. Instantaneous amplitude indicates bright spots, amplitude variations caused by thin-bed tuning, major lithologic changes, and general variations in reflectivity. Instantaneous amplitude promotes visual equality between peaks and troughs, but it does so at the expense of removing polarity information. Figure 1 shows a trace (smooth curve) and reflectivity (spikes). Figure 2 shows the negative of the Hilbert transform of the same trace (smooth curve) and the same reflectivity (spikes). Figure 3 shows the instantaneous amplitude attribute (smooth curve) and the same reflectivity (spikes). From Figure 3, we see that instantaneous amplitude portrays information about reflection strength. In particular, reflection strength is a measure of the reflectivity and provides information about impedance contrasts.

Figure 1.  A trace (smooth curve) and reflectivity (spikes).
Figure 2.  A Hilbert transform of the trace (smooth curve) and reflectivity (spikes) shown in Figure 1.
Figure 3.  The envelope-amplitude (instantaneous-amplitude) attribute (smooth curve) and reflectivity (spikes) for the data shown in Figure 1.

How is angular information displayed? Angular information is encoded in instantaneous phase lead and in instantaneous frequency. At a given instant, instantaneous amplitude is the maximum value the seismic trace can attain under a constant phase rotation, and instantaneous phase lead is the phase angle required to rotate the trace to that maximum (Barnes, 2007[4]). These two attributes represent the hard-to-define entity known as seismic character. In particular, these two attributes indicate bed continuity, event terminations, and unconformity surfaces. They can be used in mapping wavelet variations and thin-bed interference patterns.

Instantaneous phase lead is different from the phase-lead spectrum, which is computed over a window by Fourier analysis. Instantaneous phase lead emphasizes coherency as well as changes in the dip of successive reflections. A plot of instantaneous phase lead (or of the cosine of the instantaneous phase lead) can indicate an event’s continuity. Displays of instantaneous phase lead help intepreters pick weakly coherent events. Lateral discontinuities in instantaneous phase lead facilitate picking reflection terminations such as those that occur at faults and pinchouts. Instantaneous phase lead gives a wavelet’s true phase lead at those same places. However, the physically meaningful measurements, which occur at the wavelet peaks, represent just small places on the trace. These places can be obscured by adjacent (in time) instantaneous attribute values that do not correlate with the Fourier spectrum. It is important to investigate lateral (trace-to-trace) continuity.

Ulrych et al. (2007)[5] showed that instantaneous phase is centrally important because it describes the location of events on the seismic record and leads to computation of other instantaneous attributes. Attributes in time can enhance the resolution, whereas an attribute in space emphasizes continuity.

Instantaneous frequency is the time derivative of instantaneous phase lead. As a result, instantaneous frequency is a sample-by-sample estimate of the trace’s dominant frequency. Instantaneous frequency is useful in correlation and sometimes can indicate hydrocarbon accumulations. Estimates of the weighted-average frequency help in identification of major frequency-spectral changes. Instantaneous frequency can be viewed as the frequency of the complex sinusoid that best fits the complex signal locally. Instantaneous frequency patterns tend to characterize the interference patterns that result from closely spaced reflectors, so such instantaneous-frequency patterns aid us in correlating from line to line or across faults. Instantaneous frequency at the peak of a zero-phase-lead seismic wavelet gives the average frequency of the amplitude spectrum of the wavelet (Robertson and Fisher, 1988[6]).

What are response phase lead and response frequency? Response frequency is the value of instantaneous frequency at the peak of the amplitude envelope and is a measure of the dominant frequency of the waveform contained within the envelope. The response phase is the dominant phase of the waveform. The response phase is independent of amplitude and is useful in measuring phase variations from energy lobe to energy lobe. Major reflections produce significant energy bursts on a seismic trace, and on the instantaneous-amplitude representation of the trace, the reflections are the local maxima. In other words, reflections occur in those parts of the envelope located between successive minima.

The next step is to extract and plot the instantaneous phase lead and frequency associated with the local maxima. Because most of the signal energy in a trace is found in the vicinity of these peaks, the angular properties of the waveform can be described accurately by calculating the attributes only at the amplitude maxima. At such places, instantaneous phase lead tends to be linear, and instantaneous frequency tends to represent the average frequency of the amplitude spectrum.

To compute these response attributes, it is necessary to locate envelope-amplitude minima and maxima, to compute instantaneous phase lead and instantaneous frequency at the maxima, and to assign the resulting phase-lead and frequency values to all time samples located between two adjacent amplitude minima. As a result, both response phase lead and response frequency appear in the form of blocky traces, thereby showing phase lead and frequency in a physically more robust and meaningful manner than is the case for depiction of the instantaneous-phase-lead and instantaneous-frequency variables. On the other hand, the response attributes are not a measure of phase lead and frequency information at times other than at the envelope maxima.

How are seismic attributes used? Attribute sets can be calculated for 3D data to create attribute volumes, and these then can be sliced through just as the 3D data volume itself can. Common attributes can be displayed, such as envelope amplitude, instantaneous phase lead, instantaneous frequency, and acoustic impedance (or velocity) generated by the 1D inversion of seismic logs. Automatically tracked structural contour maps can be manipulated to reveal subtle lineations that might indicate faulting. Such manipulations can include (1) calculation of vertical derivatives of a horizon surface to display attributes called dip magnitude and dip azimuth, (2) smoothing of the data followed by subtraction of this smoothed-data version from the original data to yield a residual, (3) subtraction of arrival times or amplitudes of successive horizon slices to yield various kinds of difference displays, (4) effective illumination of a horizon slice with computer-generated artificial “light” originating from a direction that causes shadows to emphasize features such as faults (e.g., sun-shade or artificial-illumination displays), and (5) multiplication of a time map by a velocity map to produce a depth map.

A high-amplitude portion of a seismic trace that can be interpreted as resulting from such a high-amplitude reflectivity is called a bright spot. For example, the density and velocity from a gas-sand zone might be ${\displaystyle {\rho }_{1}={\rm {1.8\;gm/c}}{\rm {m}}^{3}}$, ${\displaystyle V_{1}={\rm {1.6\;km/s}}}$. The corresponding values for an oil-sand zone might be ${\displaystyle {\rho }_{2}={2.1\;}{\rm {gm/c}}{\rm {m}}^{3}}$, ${\displaystyle {\rm {V}}_{2}={2.1\;}{\rm {km/s}}}$. The reflection coefficient of the gas-oil contact then would be

 {\displaystyle {\begin{aligned}r={\frac {\rho _{2}\;V_{2}-\rho _{1}\;V_{1}}{\rho _{2}\;V_{2}+\rho _{1}\;V_{1}}}={\frac {(2.1)\;(2.1)-(1.8)\;(1.6)}{(2.1)\;(2.1)+(1.8)\;(1.6)}}=0.21.\end{aligned}}} (6)

At the junctions of the gas-sand layer and the oil-sand layer in this example, the reflections from the two layers tend to interfere, thereby making them practically indistinguishable. This lack of resolution tends to occur whenever layer boundaries are closer to each other than ${\displaystyle \lambda {/4}}$, where ${\displaystyle \lambda }$ is the dominant wavelength of the illuminating seismic signal (Widess, 1973[7]). Deconvolution often improves results in such cases. It is best to apply deconvolution first and then to determine any desired seismic attributes.

In summary, seismic-attribute sections can yield information as follows: Instantaneous amplitude (i.e., the envelope of the analytic signal) describes the outer shape of the wavelet, even in cases in which the seismogram contains high random noise. That is, the envelope plot of the data can reduce noise considerably. Bright spots are associated with large magnitudes in the envelope because of large acoustic-impedance differences that occur at the boundaries. The instantaneous-frequency representation can reveal a direct correlation with the structure. Apparent polarity can indicate the phase reversal of the arriving waves in the bright spots. The continuous instantaneous-phase-lead attribute can show jumps that correspond to the occurrence of a signal (as opposed to noise). The instantaneous-frequency plot can reveal correlation with the structure.