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GCPrevious HitPhaseNext Hit Residules Can Help Define Channels*

 

Marcilio Matos1,2 and Kurt J. Marfurt2

 

Search and Discovery Article #41140 (2013)

Posted June 30, 2013

 

*Adapted from the Geophysical Corner column, prepared by the author, in AAPG Explorer, March, 2013, and entitled “Out of Previous HitPhaseNext Hit Doesn’t Mean Out of Luck”.
Editor of Geophysical Corner is Satinder Chopra ([email protected]). Managing Editor of AAPG Explorer is Vern Stefanic

 

1Signal Processing Research, Training and Consulting, Norman, Oklahoma ([email protected])
2University of Oklahoma, Norman, Oklahoma

 

General Statement

Interpreters use Previous HitphaseNext Hit each time they design a wavelet to tie Previous HitseismicNext Hit data to a well log synthetic. A 0-degree Previous HitphaseNext Hit wavelet is symmetric with a positive peak, while a 180-degree Previous HitphaseNext Hit wavelet is symmetric with a negative trough. Given a 0-degree Previous HitphaseNext Hit source wavelet, thin beds give rise to ±90-degree Previous HitphaseNext Hit wavelets.

Mathematicians define Previous HitphaseNext Hit using a “complex” trace, which is simply a pair of traces:

1) The first trace is the measured Previous HitseismicNext Hit data, and forms the “real” part of the complex trace.

2) The second trace is the Hilbert transform of the measured data, and forms the imaginary part of the complex trace.

Note in Figure 1a that when the real part of the trace is positive, the imaginary part is a minus-to-plus zero crossing. In contrast, when the real part of the data is a minus-to-plus zero crossing, the Hilbert transform is a trough. This latter phenomenon allows us to use the “instantaneous” Hilbert transform to generate an amplitude map of a thin bed that was previously picked on the well log as zero crossing of the measured (or real) data.

Now let’s map both parts of the complex trace on the same plot. As you may remember from high school algebra, the real part is plotted against the x-axis and the imaginary part against the y-axis. We plot the same 100 ms (50 samples) of data “parametrically” on the complex plane.

Note in Figure 1b that the waveform progresses counterclockwise from sample to sample. We map this progression using the Previous HitphaseNext Hit between the imaginary and real parts. If we use the arctangent to compute the Previous HitphaseNext Hit, we encounter a 360-degree discontinuity each time we cross ±180 degrees (Figure 1c). Note how peaks and troughs in Figure 1a appear at 0 degrees and ±180 degrees in Figure 1c.

Now, if we computed the Previous HitphaseNext Hit by hand, we would obtain the much more continuous Previous HitphaseNext Hit shown in Figure1d, which is an “unwrapped” version of Figure 1c, and in this unwrapped image, note there is still a discontinuity at t=850 ms; however, this discontinuity is associated with waveform interference (geology) and not mathematics. Such discontinuities form the basis of the “thin-bed indicator” instantaneous attribute introduced 30 years ago.

The above discussion illustrates the concept of Previous HitphaseNext Hit unwrapping and discontinuities based on the complex trace used in instantaneous attributes. A more precise analysis can be obtained by applying the same process to spectral components of the Previous HitseismicNext Hit data. Spectral decomposition is a well-established interpretation technique. The Previous HitseismicNext Hit data are decomposed into a suite of spectral components, say at intervals of five Hz.

Most commonly we use spectral magnitude components to map thin bed tuning, while some workers use them to estimate Previous HitseismicNext Hit attenuation, 1/Q. The Previous HitphaseNext Hit components are less commonly used, but often delineate subtle faults. Here, we will show how the identification of discontinuities in the unwrapped instantaneous Previous HitphaseNext Hit discussed above can be extended to unwrapped Previous HitphaseNext Hit of spectral components.

General statement

Figures

Example

Conclusion

 

 

 

 

 

 

 

 

 

General statement

Figures

Example

Conclusion

 

 

 

 

 

 

 

 

General statement

Figures

Example

Conclusion

 

 

 

 

 

 

 

 

Figure Captions

Figure 1. (a) The “complex” trace composed of the original measured trace, d(t), (the real part, in red) and its Hilbert transform, dH(t), (the imaginary part, in blue) extracted from the survey shown in Figure 2 and Figure 3. The envelope and its reverse are plotted in orange. Note how it “envelopes” the real and the imaginary trace (and indeed any Previous HitphaseNext Hit-rotated version of the trace). (b) The complex trace plotted parametrically against time on a complex plot. Each time sample can also be represented in polar coordinates as a magnitude and Previous HitphaseNext Hit, with Previous HitphaseNext Hit being measured counterclockwise from the real axis. (c) The wrapped Previous HitphaseNext Hit computed as φ=ATAN2[dH(t),d(t)]. The definition of the arctangent gives rise to discontinuities at ±1800. (d) The unwrapped Previous HitphaseNext Hit, retaining only discontinuities associated with waveform interference (geology and crossing noise).

Figure 2. (a) A time slice at t=842 ms through a data volume acquired over Stratton Field, south Texas. Block arrow indicates a channel that gives rise to an amplitude anomaly. (b) The same time slice co-rendered with a vertical slice through the corresponding spectral Previous HitphaseNext Hit residue volume. Previous HitSeismicNext Hit data courtesy of the University of Texas Bureau of Economic Geology.

Figure 3. Vertical slices along line AA’ shown in the previous image through (a) the Previous HitseismicNext Hit amplitude, (b) the instantaneous Previous HitphaseNext Hit, (c) the Previous HitphaseNext Hit residue, and (d) the co-rendered Previous HitphaseNext Hit residue and Previous HitseismicNext Hit amplitude volumes. The instantaneous Previous HitphaseNext Hit is plotted using a cyclical color bar. The Previous HitphaseNext Hit residues are color coded by the magnitude and frequency of the spectral components at which they occur.

Example

Let us illustrate the use of such discontinuities by applying them to the well-studied Stratton Field data volume acquired over a south Texas fluvial-deltaic system by the University of Texas Bureau of Economic Geology. In our Stratton Field example, thin channels give rise to tuning effects and subtle amplitude anomalies as shown in Figure 2a. While we can detect the channel system on time and horizon slices, they are difficult to see on vertical slices through the Previous HitseismicNext Hit amplitude data (Figure 3a). Determining the thickness of the channel on the Previous HitseismicNext Hit amplitude image is even more difficult. The corresponding slice through the instantaneous Previous HitphaseNext Hit volume (Figure 3b) shows a subtle change, but again does not help delineate the channel.

One approach to improving this image is to unwrap the instantaneous Previous HitphaseNext Hit volume (as we did in Figure 1d) and compute its vertical derivative, thereby highlighting Previous HitphaseNext Hit discontinuities due to waveform interference (in this case geology). Our approach is based on the computation of Previous HitphaseNext Hit residues of spectral components computed at five Hz intervals, which provides not only an image of waveform interference, but also a measure of our confidence in the interference pattern (provided by the corresponding spectral magnitude) and the frequency component at which it occurs.

Figure 3c shows this computation, where the hue component of color corresponds to the frequency of the discontinuity and the intensity or brightness to its strength. A block arrow clearly delineates the top and bottom of the channel. Figure 3d co-renders the Previous HitphaseNext Hit residue image with the original Previous HitseismicNext Hit amplitude using 50 percent opacity.

Conclusion

Thin meandering channels are often visible on amplitude time slices (Figure 2). Previous HitPhaseNext Hit residues add the third dimension. In the April AAPG Explorer Geophysical Corner (Search and Discovery Article #41141), our colleagues will show how Previous HitphaseTop residues provide a powerful tool for geobody extraction and interpretation.

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