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Banded Attributes Overview

In the Paradise Attributes there is a group of attributes that we refer to as Banded attributes.   They are based on the Envelope and Instantaneous Phase attributes that are part of the original instantaneous attributes as derived by Turhan Taner.

Background information

Just a quick review of some of the foundational concepts to base our discussion of the Banded attributes calculated in Paradise. For a more detailed explanation, please refer to the help document labeled “Complex Trace Attributes Overview”

This paper will illustrate Paradise instantaneous attributes on a portion of a seismic trace in the Stratton Field 3D Seismic Survey, a widely-distributed and accessible dataset available from the Bureau of Economic Geology, University of Texas.  Specifically, the trace samples in this time window are from line 79, trace 89, sample 601 to 801 (time 1.2 to 1.6 by .002s). 

  The trace that all displays will be based on.

The trace that all displays will be based on.

Envelope

The envelope is calculated using the vector for the Real and Hilbert (imaginary) traces. We call e(t) the envelope signal. The complex time signal c(t) combines real amplitude signal a(t) and its Hilbert transform noted as H(a(t)) but shortened to H(t).

Banded Attributes Overview - 2.png

Now the complex signal is a vector with one axis pointing in the real direction and the other axis pointing in the Hilbert transform direction.  The amplitude and Hilbert transform are Cartesian coordinates, X and Y, of the vector respectively.

At any instant of time the length of the vector is the Euclidean distance

Banded Attributes Overview - 3.png

We call e(t) the envelope signal.

  This figure illustrates a(t), H(t) and e(t).  Envelope is red, amplitude is blue and Hilbert transform is green.

This figure illustrates a(t), H(t) and e(t).  Envelope is red, amplitude is blue and Hilbert transform is green.

Phase

The phase signal φ(t) is found as the angle of the vector for the Euclidean distance.

Banded Attributes Overview - 5.png
  Phase φ(t) is illustrated below amplitude a(t) to correlate phase discontinuities with the period of the local reflection wavelet. Period is measured peak-to-peak or trough-to-trough and is reciprocal of frequency. Notice here that another way to estimate period is to measure time between phase discontinuities which is at every other zero crossing of amplitude.

Phase φ(t) is illustrated below amplitude a(t) to correlate phase discontinuities with the period of the local reflection wavelet. Period is measured peak-to-peak or trough-to-trough and is reciprocal of frequency. Notice here that another way to estimate period is to measure time between phase discontinuities which is at every other zero crossing of amplitude.

Envelope and Phase Breaks
 

The group of banded attributes are based on picked properties of a signal using either the envelope or phase attributes.  The phase breaks and envelope breaks attributes represent minima that are picked from their respective attribute signals.

Phase Breaks

The first of these, phase breaks, is computed in several steps.  First, smooth phase is computed as

Banded Attributes Overview - 7.png

where b(t) is a boxcar function with length of T samples and height 1/T.

Phase breaks are computed by convolving a 3-sample Hanning smoother han(t) with the Hilbert transform of the difference between the phase and the smoothed phase.

Banded Attributes Overview - 8.png

Energy bands on phase breaks starts by picking times of peaks of the phase breaks signal.  Here is a single peak pick of the phase break signal pb(t) with an overbrace to represent a peak pick.

Banded Attributes Overview - 9.png

The pick is a vector of two elements – pick time and phase break value. The peak is a point on the phase break curve where the first derivative is zero and the second derivative is positive. A picking level controls the number of picks. It sets the threshold of phase break picking as a number of decibels below the largest peak value of phase break of the signal. Below the threshold, phase breaks simply are not picked.  When the decibel range is set to zero, the threshold is ignored and all phase breaks are picked.

The ordered set of selected phase break picks is

Banded Attributes Overview - 10.png

which represents the list of all phase break picks on the phase break signal.  That is, the set Banded Attributes Overview - 14.png is the ordered set Banded Attributes Overview - 15.png given that Banded Attributes Overview - 16.png are for all Banded Attributes Overview - 18.png in pb(t). The number of picks is the count.

Banded Attributes Overview - 11.png

From this set of peak picks, an envelope spike signal of pick times is constructed with Kronecker delta functions.

Banded Attributes Overview - 21.png

The overbrace notation here represents a picked peak signal and is the time of the i-th pick.

Banded Attributes Overview - 22.png

This result is the phase breaks attribute.

  Phase breaks signal pb(t) marks phase discontinuities by picking peaks from a difference signal constructed by subtracting the phase signal φ(t) from the smoothed phase signal  .

Phase breaks signal pb(t) marks phase discontinuities by picking peaks from a difference signal constructed by subtracting the phase signal φ(t) from the smoothed phase signal .

Envelope Breaks
Envelope breaks may also be picked on envelope minimums.  They are computed in an almost an equivalent way to the way to picking peaks on phase breaks.  Following the same steps as before, a single envelope trough pick of the envelope signal e(t) but with an underbrace to represent a trough pick is written.

Banded Attributes Overview - 24.png

The pick is a vector of two elements – pick time and envelope value.  The trough is a point on the envelope curve where the first derivative is zero and the second derivative is negative.  A picking level controls the number of picks.  It sets the threshold of envelope as a number of decibels below the largest peak value of envelope of the signal.  Below the threshold, troughs are not picked.  The threshold is ignored and all envelope troughs are picked when the decibel range is set to zero.

The ordered set of all trough envelope picks is

Banded Attributes Overview - 25.png

which represents the list of all selected trough picks on the envelope signal.  The number of picks

Banded Attributes Overview - 26.png

From this set of trough picks, an envelope spike signal of pick times is constructed with Kronecker delta functions.

Banded Attributes Overview - 27.png

The underbrace notation here represents a picked trough signal of spikes and is the time of the i-th pick.

The envelope breaks signal is computed by convolving a 3-sample Hanning smoother han(t) with the envelope spike signal as

Banded Attributes Overview - 28.png

This result is the envelope breaks attribute.

  The amplitude a(t) and envelope breaks eb(t) attributes. Notice that the envelope breaks coincide with envelope troughs of the envelope signal shown in Figure 8.

The amplitude a(t) and envelope breaks eb(t) attributes. Notice that the envelope breaks coincide with envelope troughs of the envelope signal shown in Figure 8.

Banded Attributes
Banded attributes are basically taking an attribute, envelope for example, and calculating the average energy between either phase or envelope breaks and writing that result between the times of the breaks used to bound the calculation.  If we used phase breaks, then this would give us Energy bands on Phase breaks

Energy bands on phase breaks
The energy bands on phase breaks attribute integrates by summing the envelope signal between adjacent pairs of phase breaks. Where the envelope is small the integrated value is small and the block value assigned to each sample between pairs of phase breaks is small.  If the envelope is large, the block envelope value between phase breaks is large. This type of attribute helps identify regions of strong energy.  The computation steps are as follows.

Assembling all the calculation components in the expression below, we have the following steps: integrate the envelope signal between adjacent pair of peak pick times and to yield a value which will be assigned as a constant to all samples in that range (δ(tj)) and repeat these integration/assignment steps for each pair of pick times (2 to N).

Banded Attributes Overview - 30.png

This result is the energy bands on phase breaks attribute.

  The amplitude a(t) and energy bands on phase breaks ebp(t) attributes.

The amplitude a(t) and energy bands on phase breaks ebp(t) attributes.

Energy Bands on Envelope Breaks
Energy bands on envelope breaks is similar to energy bands on phase breaks.  This attribute integrates by summing the envelope signal between envelope troughs.

Banded Attributes Overview - 32.png

This result is the envelope bands on envelope.

  The amplitude ebe(t) and energy bands on envelope attributes.  Bands coincide with envelope minima as marked by envelope breaks in Figure 15.  Energy bands are integrated values of envelope between adjacent minima.

The amplitude ebe(t) and energy bands on envelope attributes.  Bands coincide with envelope minima as marked by envelope breaks in Figure 15.  Energy bands are integrated values of envelope between adjacent minima.

Other Banded attributes on Phase or Envelope Breaks

The method for calculating the average energy between two phase or envelope breaks and putting that value between them can be used for any attribute and allows the interpreter to see the unit energy based on that attribute in the seismic display. Since the technique is the same, the remaining bands on breaks attributes will not be discussed in detail, but will be available for the interpreter to use as desired.  Note that the Phase Breaks tend to be more consistent and follow the wavelet closer so most of any new attributes that are generated will probably use Phase Breaks.

Some of these that may be introduced in the future could include:

  • Sweetness Bands on Phase breaks
  • Relative Acoustic Impedance Bands on Phase Breaks
  • Q Bands on Phase Breaks
  • Bandwidth Bands on Phase Breaks
  • Attenuation Bands on Phase Breaks
  • Thin Bed Bands on Phase Breaks
Banded Attributes Overview - 34.png

References

Appendix – Mathematical Notations

  • Constants, variables and subscripts are lower case.
  • Vectors are lower case bold.
  • Sets are upper case bold.
  • Sets of real, imaginary, complex and integer: ℝ,