Watch videos of Presentations from SEG 2017
Using Attributes to Interpret the Environment of Deposition - A Video Course. Taught by Kurt Marfurt, Rocky Roden, and ChingWen Chen
Dr. Kurt Marfurt and Dr. Tom Smith featured in the July edition of AOGR on Machine Learning and Multi-Attribute Analysis
Rocky Roden and Ching Wen Chen in May edition of First Break - Interpretation of DHI Characteristics using Machine Learning
Seismic interpretation and machine learning by Rocky Roden and Deborah Sacrey, GeoExPro, December 2016

LOGIN / MY PROFILE | LOG OUT

Acceleration of Phase

Acceleration of Phase

Attribute Description:  This measures the rate of change for instantaneous phase

Interpretation Use:  This attribute may provide value by

  • Accentuating bedding differences
  • Higher Resolution may have a higher noise level due to differentiation
  • May reflect some elastic properties of resolvable beds

Recommended Colorbar:

Since the distribution of this data is gaussian with spikes at the extremes, a normal seismic colorbar can be used to visualize this attribute – it will show a contrast between “Peaks” and “Troughs” and highlight them against the central data.   A phase colorbar with common colors at peak and trough zones can also be used and will emphasize the central data more.

In this example, we are using a black – white – red colorbar.

Colorbar and amplitude spectrum of Acceleration of Phase

Colorbar and amplitude spectrum of Acceleration of Phase

Note that a phase based colorbar could also be used.

Example:

Vertical display of Acceleration of Phase

Vertical display of Acceleration of Phase

Time slice of Acceleration of Phase

Time slice of Acceleration of Phase

Frequency Spectrum

Frequency Spectrum

Computation:  Acceleration of Phase (cycles/sec/sec) is based on instantaneous acceleration. The time derivative of instantaneous frequency/wavenumber, by definition, gives the instantaneous acceleration. This can be computed both from instantaneous frequency/wavenumber and from time averaged instantaneous frequency/wavenumber. It is obvious that the time derivative of instantaneous frequency/wavenumber will accentuate the local frequency/wavenumber jumps, and hence will make the thin bed indicators more prominent. It should also indicate to some degree the effect of absorption by showing the frequency/wavenumber dispersion of seismic signals going through unconsolidated or quickly deposited layers:

Acceleration of Phase - 5.png

The corresponding value in depth domain is calculated as the second derivative of phase with respect to depth:

Acceleration of Phase - 6.png

References:

  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization:  Society of Exploration Geophysicists, Geophysical Developments #11.
  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.

Amplitude -> Curvedness, Shape Index, Shape Dome, Shape Bowl, Shape Ridge, Shape Saddle, and Shape Valley

Amplitude -> Curvedness, Shape Index, Shape Dome, Shape Bowl, Shape Ridge, Shape Saddle, and Shape Valley

Attribute Description:  The E_pos and E_neg attributes can be combined to generate components of curvature that describe deformation and shapes.  The Curvedness attribute is a measure of total deformation, whereas the Shape Index attribute provides a qualitative description of the local morphology (AASPI Documentation).

Interpretation Use:  The Curvedness attribute describes the deformation of rocks. The Shape Index attribute gives a qualitative description of shapes. The Shape Dome, Shape Bowl, Shape Ridge, Shape saddle, and Shape Valley are attributes extracted from the Shape Index attribute and these attributes depict the local morphology of the rocks in terms of a dome, bowl, ridge, saddle, and valley shapes (Chopra and Marfurt, 2007). Attribute results are better analyzed in plan view or draped over a horizon displays.

Recommended color palette:  For the Curvedness attribute, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. For the Shape Index attribute, a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.   

Figure 1. Color bar examples of seismic amplitude (a) and output attribute: curvedness (b) and shape index (c).

Figure 1. Color bar examples of seismic amplitude (a) and output attribute: curvedness (b) and shape index (c).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: curvedness(b) and shape index (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: curvedness(b) and shape index (c).

Recommended color palette:  For the Shape Dome, Shape Bowl, Shape Ridge, Shape Saddle, and Shape Valley attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.

Figure 3. Color bar examples of output attributes: shape dome (a), shape bowl (b), shape ridge (c), shape saddle (d) and shape valley (e).

Figure 3. Color bar examples of output attributes: shape dome (a), shape bowl (b), shape ridge (c), shape saddle (d) and shape valley (e).

Examples:

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: shape dome (b), shape bowl (c), shape ridge (d), shape saddle (e), and shape valley (f).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: shape dome (b), shape bowl (c), shape ridge (d), shape saddle (e), and shape valley (f).

Computation: The Curvedness, Shape Index, Shape Dome, Shape Bowl, Shape Ridge, Shape Saddle, and Shape Valley attributes are computed using this workflow:

Input data → Estimates of reflector dip → Estimates of coherent energy gradients through measurements of similarity → Curvature amplitude data

The process can be summarized as follows (AASPI documentation):

  1. Take the Inline Dip, Crossline Dip, and seismic amplitude volumes as input data (refer to Filter Dip Components -> Inline Dip, Crossline Dip, and Confidence attribute description section)
  2. Run a similarity computation by...
    1. Computing the covariance matrix from the complex trace signal
    2. Decomposing the covariance matrix into its corresponding ktheigenvalue and vtheigenvectors. Then normalize the eigenvectors to be unit length
    3. Estimating a scaled principal component, where the scale is the inner product or correlation of the eigenvector
  3. Generate estimates of inline and crossline coherent energy gradient
  4. Compute E_pos and E_neg (refer to Curvatures -> Amplitude -> E_max, E_min, E_pos, and E_neg attributes description section). Where Curvedness and Shape Index attributes are (AASPI Documentation),

Note that the Shape Index attribute range varies between -1.0 and +1.0, with a bowl-shaped indicated by -1.0, a valley shaped indicated by -0.5, a saddle shape indicated by 0.0, a ridge shape indicated by +0.5, and dome shape indicated by +1.0 (AASPI documentation).

Figure 5. Definition of 3D shapes based on their Shape Index (s), E_pos and E_neg attribute responses (after AASPI documentation).

Figure 5. Definition of 3D shapes based on their Shape Index (s), E_pos and E_neg attribute responses (after AASPI documentation).

To generate the Shape Dome, Shape Bowl, Shape Ridge, Shape Saddle, and Shape Valley attributes only requires to multiply the Curvedness attribute at every point in the volume by a filtered version of the Shape Index attribute (shown in Figure 4) that passes a shape component of interest.

Figure 6. Filters used to extract shape information from Shape Index attribute (after Chopra and Marfurt, 2007).

Figure 6. Filters used to extract shape information from Shape Index attribute (after Chopra and Marfurt, 2007).

References

Amplitude -> E_gauss, E_mean, E_max_azimuth, and E_min_azimuth

Amplitude -> E_gauss, E_mean, E_max_azimuth, and E_min_azimuth

Attribute Description:  The E_gauss, E_mean, E_max_azimuth and E_min_azimuth are additional curvature attributes that can be used to complement geologic information extracted from E_pos and E_neg attributes (refer to Curvatures -> Amplitude -> E_max, E_min, E_pos, and E_neg attributes description section).

Interpretation Use:  The E_gauss, E_mean, E_max_azimuth, and E_min_azimuth attributes offer additional measurements of volumetric curvature. They can be used to support the stratigraphic features, faults, and fractures that were identified by E_pos and E_neg attributes (Chopra and Marfurt, 2007). Attribute results are better analyzed in plan view or draped over a horizon displays.

Recommended color palette:  For the E_gauss and E_max_azimuth attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.       

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_gauss (b) and e_max_azimuth (c).

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_gauss (b) and e_max_azimuth (c).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_gauss (b) and e_max_azimuth (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_gauss (b) and e_max_azimuth (c).

Recommended color palette:  For the E_mean and E_min_azimuth attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: e_mean (b) and e_min_azimuth (c).

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: e_mean (b) and e_min_azimuth (c).

Examples:

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: e_mean (b) and e_min_azimuth (c).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: e_mean (b) and e_min_azimuth (c).

Computation: The E_gauss, E_mean, E_max_azimuth, and E_min_azimuth attributes are computed using this workflow:

Input data → Estimates of reflector dip → Estimates of coherent energy gradients through measurements of similarity → Curvature amplitude data

The process can be summarized as follows (AASPI documentation):

  1. ake the Inline Dip, Crossline Dip, and seismic amplitude volumes as input data (refer to Filter Dip Components -> Inline Dip, Crossline Dip, and Confidence attribute description section)
  2. Run a similarity computation (refer to Similarity attributes description section)
    1. Computing the covariance matrix from the complex trace signal
    2. Decomposing the covariance matrix into its corresponding kth eigenvalue and vth eigenvectors. Then normalize the eigenvectors to be unit length
    3. Estimating a scaled principal component, where the scale is the inner product or correlation of the eigenvector
  3. Generate estimates of inline and crossline coherent energy gradient
  4. Use the inline and crossline energy gradients to solve the coefficients of the quadratic surface:
e_gauss - 06.png

where,

e_gauss - 07.png
e_gauss - 08.png

and the azimuth of the maximum and minimum curvatures are respectively (Chopra and Marfurt, 2007):

e_gauss - 09.png

References

Amplitude -> E_max, E_min, E_pos, and E_neg

Amplitude -> E_gauss, E_mean, E_max_azimuth, and E_min_azimuth

Attribute Description:  The E_pos and E_neg attributes are measures of the most-positive and most-negative curvature anomalies, whereas the E_max and E_min attributes correspond to the maximum and minimum curvature anomalies.

Interpretation Use:  The E_pos, E_neg, E_max, and E_min attributes correspond to the maximum and minimum curvature anomalies. These attributes capture geologic information in the form of the second derivative of the amplitude behavior of seismic data. They can be used to identify stratigraphic features, faults, and fractures (Chopra and Marfurt, 2007; Chopra and Marfurt, 2013). Attribute results are better analyzed in plan view or draped over a horizon displays.

Recommended color palette:  For the E_pos and E_neg attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds. 

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_pos (b) and e_neg (c).

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_pos (b) and e_neg (c).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_pos (b) and e_neg (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_pos (b) and e_neg (c).

Recommended color palette:  For the E_max and E_min attributes, a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds. 

Figure 3. Color bar examples of seismic amplitude (a) and output attribute: e_max (b) and e_min (c).

Figure 3. Color bar examples of seismic amplitude (a) and output attribute: e_max (b) and e_min (c).

Examples:

Figure 4. Time slice displays of seismic amplitude (a) and output attribute: e_max (b) and e_min (c).

Figure 4. Time slice displays of seismic amplitude (a) and output attribute: e_max (b) and e_min (c).

Computation: The E_max, E_min, E_pos, and E_neg attributes use seismic amplitude data (time or depth domain). Unlike to structural curvature attributes that measure the changes in the dip in m/m or ft/ft, the amplitude curvature attributes to measure the dip in mV2/m or mV2/ft (assuming the amplitude is measured in mV [milli-volts]). Figure 1 shows the definition of 3D shapes expressed as a function of the most-positive curvature (E_pos) and the most-negative curvature (E_neg):

Figure 5. Definition of 3D shapes based on their E_pos and E_neg attribute responses (after AASPI documentation).

Figure 5. Definition of 3D shapes based on their E_pos and E_neg attribute responses (after AASPI documentation).

Then, the computation process uses this workflow:

Input data → Estimates of reflector dip → Estimates of coherent energy gradients through measurements of similarity → Curvature amplitude data

It can be summarized as follows (AASPI documentation):

  1. Take the Inline Dip, Crossline Dip, and seismic amplitude volumes as input data (refer to Filter Dip Components → Inline Dip, Crossline Dip, and Confidence attribute description section)
  2. Run a similarity computation (refer to Similarity attributes description section).
    1. Computing the covariance matrix from the complex trace signal
    2. Decomposing the covariance matrix into its corresponding kth eigenvalue and vth eigenvectors. Then normalize the eigenvectors to be unit length
    3. Estimating a scaled principal component, where the scale is the inner product or correlation of the eigenvector
  3. Generate estimates of inline and crossline coherent energy gradient
  4. Use the inline and crossline energy gradients to solve the coefficients of the quadratic surface:
E_Max - 06.png
E_Max - 08.png

and,

  1. Then, the E_max and E_min attributes are generated using the following relationships (AASPI documentation):
E_Max - 10.png

References

  • AASPI documentation, http://mcee.ou.edu/aaspi/documentation/Volumetric_Attributes-curvature3d.pdf
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 110 – 111.
  • Chopra, S. and K. J. Marfurt, 2013, Structural curvature versus amplitude curvature: The Leading Edge, 32, 178 – 184.

Amplitude -> E_pos_strike and E_neg_strike

Amplitude -> E_pos_strike and E_neg_strike

Attribute Description:  The E_pos_strike and E_neg_strike attributes are additional curvature attributes that can be used to complement geologic information extracted from E_pos and E_neg attributes (refer to Curvatures -> Amplitude -> E_max, E_min, E_pos, and E_neg attributes description section).

Interpretation Use:  The E_pos_strike and E_neg_strike are estimates of the component curvature projected along the strike direction of a tangent plane to the surface. These additional curvature attributes can be used to identify stratigraphic features, faults, and fractures (Chopra and Marfurt, 2007). Attribute results are better analyzed in plan view or draped over a horizon displays.

Recommended color palette:  For the E_pos_strike and E_neg_strike attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_pos_strike (b) and e_neg_strike (c).

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: e_pos_strike (b) and e_neg_strike (c).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_pos_strike (b) and e_neg_strike (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: e_pos_strike (b) and e_neg_strike (c).

Computation: The E_pos_strike and E_neg_attributes strike are computed using this workflow:

Input data → Estimates of reflector dip → Estimates of coherent energy gradients through measurements of similarity → Curvature amplitude data

The process can be summarized as follows (AASPI documentation):

  1. Take the Inline Dip, Crossline Dip, and seismic amplitude volumes as input data (refer to Filter Dip Components -> Inline Dip, Crossline Dip, and Confidence attribute description section)
  2. Run a similarity computation (refer to Similarity attributes description section)
    1. Computing the covariance matrix from the complex trace signal
    2. Decomposing the covariance matrix into its corresponding kth eigenvalue and vth eigenvectors. Then normalize the eigenvectors to be unit length
    3. Estimating a scaled principal component, where the scale is the inner product or correlation of the eigenvector
  3. Generate estimates of inline and crossline coherent energy gradient
  4. Use the inline and crossline energy gradients to solve the coefficients of the quadratic surface:
E_pos - 03.png

References

Attenuation

Attenuation

Attribute Description:  Attenuation attempts to represent the rate of absorption of the seismic energy as it goes through the earth.  The absorption, transmission and reflection of sound directly relate to rock type and its impedance value as reflected by P and S wave velocity and density. These are effected by conditions such as porosity, saturation and pore pressure so it can be useful in reservoir characterization.

Interpretation Use:  This attribute may provide value by

  • Localizing Lithology types
  • Providing an indication of pore pressure, fluid saturation and porosity

Recommended Colorbar: 

In this example, we are using “Average Energy”, a spectral based colorbar. This attribute puts out all positive values, in this example from 0 to 462. The brighter colors represent the higher rates of attenuation while the darker blues and greens represent the lower rates of attenuation. As seen in the example below, the deeper rocks attenuate the sound less, which makes sense as they are more compacted, and have a higher velocity than the shallower rocks. The spectrum reflects the large number of “0” values

Attenuation - 1.png

Example:

Vertical Seismic Section using Attenuation

Vertical Seismic Section using Attenuation

Time slice of Attenuation

Time slice of Attenuation

Frequency Spectrum for Attenuation


Frequency Spectrum for Attenuation

Computation: 

Attenuation is computed by dividing the first derivative of envelope by smoothed frequency:

Attenuation - 5.png

The smoothed frequency is used due to the presence of spikes in normal instantaneous frequency.

References:

  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization:  Society of Exploration Geophysicists, Geophysical Developments #11.
  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.

Banded Attributes Overview

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,

Bandwidth

Bandwidth

Attribute Description:  Instantaneous bandwidth is a statistical measure of the seismic wavelet, but it relates to various physical conditions, representing seismic data bandwidth sample by sample. It is one of the high-resolution character correlators.  It shows overall effects of absorption and seismic character changes.

Interpretation Use:  This attribute may provide value because it:

  • Represents the seismic data bandwidth sample by sample.
  • Is a high-resolution character correlator.
  • Shows overall effects of absorption and seismic character changes.

Recommended Colorbar: 

Since the distribution of this data is Gaussian in nature, a standard seismic colorbar can work.

In this example, we are using a yellow-red-black-white-black—blue-cyan colorbar which highlight the higher values in the histogram so that they stand out between the bright red and blue values. 

Amplitude spectrum of Bandwidth

Amplitude spectrum of Bandwidth

Example: 

Vertical display of Bandwidth

Vertical display of Bandwidth

Time slice of Bandwidth

Time slice of Bandwidth

Frequency Spectrum

Frequency Spectrum

Computation:  Bandwidth (octaves) is based on Barnes’ (1993) suggestion that instantaneous bandwidth can be computed as:

Bandwidth - 5.png
chrome

Where ė(t) is the time or depth derivative of the envelope.

The variance with respect to the mean frequency/wavenumber (standard deviation) indicates the width of power spectral density distribution over a band of frequencies; hence we can use it as an indication of the spectral bandwidth. This equation measures the absolute value of the rate of change of envelope amplitude. Instantaneous bandwidth is a statistical measure of the seismic wavelet, but it relates to various physical conditions.
 
References:

  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization:  Society of Exploration Geophysicists, Geophysical Developments #11.
  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.

CMP -> Bandwidth, Reconstructed Data, Mean Frequency, Peak Frequency, Peak Magnitude, Peak Magnitude Above Average, Peak Phase, Modeled, Residual, Roughness, Range Trimmed Mean Magnitude, Slope

CMP -> Bandwidth, Reconstructed Data, Mean Frequency, Peak Frequency, Peak Magnitude, Peak Magnitude Above Average, Peak Phase, Modeled, Residual, Roughness, Range Trimmed Mean Magnitude, Slope

Attribute Description: 

The statistical summary attributes (i.e., Bandwidth, Reconstructed Data, Mean Frequency, Peak Frequency, Peak Magnitude, Peak Magnitude Above Average, Peak Phase, Modeled, Residual, Roughness, Range Trimmed Mean Magnitude, Slope) generated by the Continuous Wavelet Transform (CWT) can also help in the interpretation of anomalies associated with reservoirs or other zones of interest (AASPI Documentation; Zhang, 2010).

Interpretation Use: 

The statistical summary attributes can be also useful for providing more information sensitive to stratigraphy or reservoir physical properties (Chopra and Marfurt, 2007). Attribute results can be analyzed in different ways, from a plan view, vertical transects, or draped over a horizon display.

Recommended color palette: 

For the statistical summary attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different hues. In the examples below, the hues were set to light blue and yellow to better highlight geologic features. Or even, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of output attributes: bandwidth (a), mean frequency (b), peak frequency (c), modeled (d), and reconstructed (e).

Figure 1. Color bar examples of output attributes: bandwidth (a), mean frequency (b), peak frequency (c), modeled (d), and reconstructed (e).

Figure 2. Color bar examples of output attributes: peak magnitude (a), peak magnitude above average (b), range trimmed mean magnitude (c), residual (d), and roughness (e).

Figure 2. Color bar examples of output attributes: peak magnitude (a), peak magnitude above average (b), range trimmed mean magnitude (c), residual (d), and roughness (e).

Examples:

Figure 3. Time slice displays of seismic amplitude and output and output attributes: bandwidth (b), mean frequency (c), peak frequency (d), modeled (e), and reconstructed (f).

Figure 3. Time slice displays of seismic amplitude and output and output attributes: bandwidth (b), mean frequency (c), peak frequency (d), modeled (e), and reconstructed (f).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: peak magnitude (b), peak magnitude above average (c), range trimmed mean magnitude (d), residual (e), and roughness (f).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: peak magnitude (b), peak magnitude above average (c), range trimmed mean magnitude (d), residual (e), and roughness (f).

Recommended color palette: 

For the Peak Phase attribute, a cyclic color scheme is suggested. In this color palette, the hues wrap around so that the red follows purple. A specific color is assigned to different phase ranges, so then the display can be used to infer the continuity of seismic events. We suggest using the histogram of values to guide setting color value thresholds.

Figure 5. Color bar examples of seismic amplitude (a) and output attribute: peak phase (b).

Figure 5. Color bar examples of seismic amplitude (a) and output attribute: peak phase (b).

Examples:

Figure 6. Vertical transect views of seismic amplitude (a) and output attribute: peak phase (b).

Figure 6. Vertical transect views of seismic amplitude (a) and output attribute: peak phase (b).

Computation:

The statistical summary attributes are additional outputs of the spectral decomposition based on Complex Matching Pursuit (refer to Spectral Decomp-> CMP -> Spectral Magnitude, Spectral Phase, and Spectral Voice Componentsattributes description section).  Prior to computing these summary attributes, the amplitude volume (time or depth domain) is spectrally whitened to account for changes in the source wavelet with depth and a non-flat source spectrum. Thereafter, the output volume shows a relatively flat spectrum bound by two tails (see Figure 2). Following this behavior, the statistical summary attributes are generated.

Figure 7. The representative spectrum of a balanced seismic amplitude volume (after Zhang, 2010).

Figure 7. The representative spectrum of a balanced seismic amplitude volume (after Zhang, 2010).

References

  • AASPI documentation, http://mcee.ou.edu/aaspi/documentation/Spectral_Attributes-spec_cmp.pdf
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 123 – 151.
  • Zhang, K., 2010, Seismic attribute analysis of unconventional reservoirs, and stratigraphic features: Ph.D. Thesis, University of Oklahoma.

CMP -> Spectral Magnitude, Spectral Phase, and Spectral Voice Components

CMP -> Spectral Magnitude, Spectral Phase, and Spectral Voice Components

Attribute Description: 

The spectral components generated by the Complex Matching Pursuit (CMP) are complex time-frequency signals. As a result, the Spectral Magnitude, Spectral Phase, and Spectral Voice attributes are generated at each time-frequency sample. The Spectral Magnitude attribute represents the energy that correlates with the seismic signal (e.g., similar to Envelop), the Spectral Phase attribute denotes the phase rotation between the seismic trace and the modeled Ricker (or Morlet) wavelets, and the Spectral Voice attribute corresponds to the real component of complex spectrum (e.g., Amplitude) and can be used to enhance the ability of coherence, curvature, or other AASPI structural attribute computations at uncovering thin or small structural and stratigraphic details.

Interpretation Use: 

The Spectral Magnitude and Spectral Voice attributes can provide detailed subtle stratigraphic information about a reservoir or other zone of interest. Also, these attributes can help at examining geologic features in the form of spectral components. Additionally, the Spectral Phase can provide insights into discontinuity features as well as onlap, offlap, and erosional unconformities (Chopra and Marfurt, 2017). Attribute results can be analyzed in different ways, from a plan view, vertical transects, or draped over a horizon display.

Recommended color palette: 

For the Spectral Voice attributes a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. Or any color scheme that works well with normal seismic amplitude data. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: spectral voice 20 Hz(b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: spectral voice 20 Hz(b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attribute: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Figure 2. Time slice displays of seismic amplitude (a) and output attribute: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Recommended color palette: 

For the Spectral Magnitude attributes a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. We suggest using the histogram of values to guide setting color value thresholds.

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Examples:

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Recommended color palette: 

For the Spectral Phase attribute, a cyclic color scheme is suggested. In this color palette, the hues wrap around so that the red follows purple. A specific color is assigned to different phase ranges, so then the display can be used to infer the continuity of seismic events. We suggest using the histogram of values to guide setting color value thresholds.

Figure 5. Color bar examples of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Figure 5. Color bar examples of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Examples:

Figure 6. Vertical transect views of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Figure 6. Vertical transect views of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

The Spectral Magnitude, Spectral Phase, and Spectral Voice attributes are computed using small modeled Ricker or Morlet wavelets. Prior to computing the spectral decomposition, the amplitude volume (time or depth domain) is spectrally whitened to account for changes in the source wavelet with depth and a non-flat source spectrum. The spectral decomposition based on CMP technique is outlined in Liu and Marfurt (2017).

The Magnitude Spectral (Eq. 1) and Phase (Eq. 2) Spectral attributes are computed as follows:

CMP - 08.png

where v(t,f) and vH(t,f) are the real and imaginary part of the complex spectrum (Figure 1). Note that the Phase Spectral attribute ranges between -180° and +180°. Then, the Voice Spectral attribute is given by Voice(t,f) = v(t,f) (Chopra and Marfurt, 2016).

References

  • AASPI documentation, http://mcee.ou.edu/aaspi/documentation/Spectral_Attributes-spec_cmp.pdf
  • Aarre, V., t. N. A. Al Dayyni, S. L. Mahmoud, A. B. S. Clark, B. Toelle, O. V. Vejbaek, and G. White, 2012, Seismic detection of subtle faults and fractures: Oilfields Review Summer, 24, 28 – 43.
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 123 – 151.
  • Chopra, S. and K. J. Marfurt, 2016, Spectral decomposition and spectral balancing of seismic data: The Leading Edge, 35, 176 – 179.
  • Liu, J. and K. J. Marfurt, 2007, Instantaneous spectral attributes to detect channels: Geophysics, 72, P23 – P31.

Coherent Energy, Total Energy, Energy Ratio Similarity, Outer Product Similarity and Sobel Filter Similarity

Coherent Energy, Total Energy, Energy Ratio Similarity, Outer Product Similarity and Sobel Filter Similarity

Attribute Description: Similarity is a measurement between traces based on the coherence of the waveforms. Some forms of similarities may be more sensitive to amplitudes than others while the results of the energy ratio volume are designed to not be sensitive to the amplitude changes.

Interpretation Use: The similarity attribute identifies lateral changes in waveforms such as stratigraphic features and faults/fractures. This attribute highlights the abrupt boundaries or discontinuities in the waveform caused by geological features. The linearity is better seen on the time/depth or horizon slices. Coherent Energy and Total Energy are two volumes to show regional energy trends within the data.

Recommended Color Palette: A colorbar where one end of the color is black and is gradationally transitioned to white on the other end, is recommended for energy ratio similarity display. The idea is to highlight the discontinuity by the dark color and makes the coherent background white. Gray and purple are the common intermediate colors. Coherent energy and total energy can be displayed with the gradational color bar, or alternatively, add red-yellow gradational color to emphasize the energy trend.

coherent energy - 01.png
coherent energy - 02.png
coherent energy - 03.png

Example:
 
a)

b)

coherent energy - 05.png

c)

coherent energy - 06.png

d)

coherent energy - 07.png

e)

Figure 1: a) Coherent Energy b) Total Energy c) Energy Ratio Similarity d) Outer Product Similarity e) Sobel Filter Similarity. Coherent Energy and Total Energy images look very similar, whereas Energy Ratio, Outer-Product and Sobel Similarity provides similarity features but the noise level can vary.

Figure 1: a) Coherent Energy b) Total Energy c) Energy Ratio Similarity d) Outer Product Similarity e) Sobel Filter Similarity. Coherent Energy and Total Energy images look very similar, whereas Energy Ratio, Outer-Product and Sobel Similarity provides similarity features but the noise level can vary.

Computation:

Energy ratio similarity is the ratio of the coherent energy over the total energy of the input data within the analysis window. Coherence energy here is estimated based on eigenstructure methodology by Gersztenkorn and Marfurt (1999). The first step is to capture the trace information by first constructing a covariance matrix centered around the analysis point d. The matrix contains the cross-correlation value from the data matrix within the analysis window. The covariance matrix can be written as

coherent energy - 09.png

C: M by M square covariance matrix
M: number of traces in the analysis window
λk: the kth eigenvalue
ν(k): the kth eigenvector

Then the eigenstructure coherence was derived by calculating the eigenvalues of the covariance matrix. The assumption is that the first eigenvector is sufficient to represent the coherent energy of all traces

coherent energy - 10.png

J, referring to the total number of eigenvalue-eigenvector pairs, equals to M. Ceigen is the eigenstructure coherence based on the first eigenvalue (λ1).

Instead of calculating use the first eigenvalue to represent the coherent energy, Chopra and Marfurt (2007) offer a closely related method to calculate energy ratio similarity defined as

coherent energy - 11.png

Where Coherence energy (Kcoh) is the sum of the energy of the corresponding weighted PC-filtered traces.

coherent energy - 12.png

and

coherent energy - 13.png

dH= Traces after Hilbert transform
d(j)pc= Filtered jth principal component centered at the point (t0,xn,yn)

The Total Energy (Ktot) is the sum of the energy of the weighted analytic traces used to compute the covariance matrix:

coherent energy - 14.png

On the other hand, Marfurt et al. (1999) define Outer Product Similarity to be the ratio of the energy of the average trace to the average of the energies of each of the traces.

The calculation adopts the idea from Luo et al. (1996) which normalized Sobel filter by dividing the inline and crossline component by the energy.

Reference

  • AASPI Document: http://mcee.ou.edu/aaspi/documentation/Volumetric_Attributes-similarity3d.pdf
  • Gersztenkorn, A., and K. J. Marfurt, 1999, Eigenstructure-based coherence computations as an aid to 3-D structural and stratigraphic mapping: Geophysics, 64, no. 5, 1468-1479, http://dx.doi.org/10.1190/1.1444651
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11
  • Luo, Y., W. G. Higgs, and W. S. Kowalik, 1996, Edge detection and stratigraphic analysis using 3-D seismic data: 66th Annual International Meeting, SEG, Expanded Abstracts, 324–327

Complex Trace Attribute overview

Complex Trace Attribute overview

Complex Trace Attribute overview

There are a multitude of seismic attributes in the interpretation world today. A large group of these are referred to as single trace attributes as the computation involved in computing that attribute involves only one trace. The bulk of these attributes are based off of 5 basic attributes developed by M.Turhan Tanner, Sven Treitel, and Bob Sheriff that are referred to has the complex signal attributes or Hilbert transform attributes. These five attributes are:

  • Amplitude or Real Trace
  • Hilbert or Imaginary Trace
  • Envelope
  • Instantaneous Phase
  • Instantaneous Frequency

This document will go through these 5 attributes in an overview setting and specific information can be found in the trace specific documentation included with this series.

For this explanation, a single, final stack trace will be used to illustrate these traces. This same trace will be used in all the individual attribute documentation. This comes from the Stratton field survey which is a publicly available 3D survey and is extracted from line 79, trace 89 which ties to Well # 9 which contains a VSP and full series of logs. This is a subset of the trace from 1.2 seconds to 1.6 seconds and includes two tops, C3 and F1 for reference.

This is the amplitude or real trace.

This is the amplitude or real trace.

The complex trace consists of two parts mathematically, the real (or amplitude) part and the imaginary (or Hilbert trace) part. The complex time signal c(t) combines real amplitude signal a(t) and its Hilbert transform noted as iH(a(t)) but usually shortened to H(t).

complex trace attribute overview - 2.png

In multi-dimensional space, the real trace is shown in a vertical display with the imaginary trace perpendicular to it at the zero axis in a horizontal display. With this understanding the complex signal is a vector with one axis pointing in the real direction and the other axis pointing in the imaginary or Hilbert transform direction. The amplitude and Hilbert transform are Cartesian coordinates, X and Y, of the vector respectively. This vector can also be expressed in polar form with a vector length and angle with respect to the amplitude (X) axis.

At any instant of time the length of the vector is the Euclidean distance as indicated by the double brackets:

complex trace attribute overview - 3.png

We call e(t) the envelope signal.

complex trace attribute overview - 4.png

Shown are the real trace (blue), imaginary trace (green) and the envelope trace (red)

From these attributes, we can calculate out two basic measurements – the instantaneous Phase and instantaneous Frequency. Instantaneous refers to the fact that these are sample by sample calculations made down the traces, a single trace at a time.

The phase signal Φ(t) in Cartesian coordinate form is defind as

complex trace attribute overview - 5.png

The phase signal Φ(t) in polar form is defined as

complex trace attribute overview - 6.png
complex trace attribute overview - 7.png

The phase signal Φ(t) changes with time so we estimate the instantaneous frequency as the change of phase with time. This is the first derivative of phase over time. There are many ways to calculate derivatives, but the one often used is the following.

complex trace attribute overview - 8.png

where square brackets indicate square of Euclidean distance. This is the instantaneous frequency attribute. 

Here is a display of frequency ƒ(t) in blue, smoothed frequency ƒ˜(t) in green and thin bed tb(t) in red. All three are included here as the thin bed is the difference between the frequency and the smoothed frequency and it makes sense to show all three.

complex trace attribute overview - 9.png

Note that these five attributes form the foundation for most of the rest of the single trace attributes.

 

References:

  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.

CWT -> Peak Frequency, Peak Magnitude, Peak Phase, Peak Magnitude Above Average, Roughness, Range Trimmed Mean Magnitude, and Slope

CWT -> Peak Frequency, Peak Magnitude, Peak Phase, Peak Magnitude Above Average, Roughness, Range Trimmed Mean Magnitude, and Slope

Attribute Description: 

The statistical summary attributes (i.e., Bandwidth, Reconstructed Data, Mean Frequency, Peak Frequency, Peak Magnitude, Peak Magnitude Above Average, Peak Phase, Modeled, Residual, Roughness, Range Trimmed Mean Magnitude, Slope) generated by the Continuous Wavelet Transform (CWT) can also help in the interpretation of anomalies associated with reservoirs or other zones of interest (AASPI Documentation; Zhang, 2010).

Interpretation Use: 

The statistical summary attributes can be also useful for providing more information sensitive to stratigraphy or reservoir physical properties (Chopra and Marfurt, 2007). Attribute results can be analyzed in different ways, from a plan view, vertical transects, or draped over a horizon display.

Recommended color palette: 

For the statistical summary attributes a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different hues. In the examples below, the hues were set to light blue and yellow to better highlight geologic features. Or even, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of output attributes: bandwidth (a), bandwidth extension (b), mean frequency (c), peak frequency (d), and inverse reconstructed (e).

Figure 1. Color bar examples of output attributes: bandwidth (a), bandwidth extension (b), mean frequency (c), peak frequency (d), and inverse reconstructed (e).

Figure 2. Color bar examples of output attributes: peak magnitude (a), peak magnitude above average (b), range trimmed mean magnitude (c), roughness (d), and slope (e).

Figure 2. Color bar examples of output attributes: peak magnitude (a), peak magnitude above average (b), range trimmed mean magnitude (c), roughness (d), and slope (e).

Examples:

Figure 3. Time slice displays of seismic amplitude (a) and output attributes: bandwidth (b), bandwidth extension (c), mean frequency (d), peak frequency (e), and inverse reconstructed (f).

Figure 3. Time slice displays of seismic amplitude (a) and output attributes: bandwidth (b), bandwidth extension (c), mean frequency (d), peak frequency (e), and inverse reconstructed (f).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: peak magnitude (b), peak magnitude above average (c), range trimmed mean magnitude (d), roughness (e), and slope (f).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: peak magnitude (b), peak magnitude above average (c), range trimmed mean magnitude (d), roughness (e), and slope (f).

Recommended color palette: 

For the Peak Phase attribute, a cyclic color scheme is suggested. In this color palette, the hues wrap around so that the red follows purple. A specific color is assigned to different phase ranges, so then the display can be used to infer the continuity of seismic events. We suggest using the histogram of values to guide setting color value thresholds.

Figure 5. Color bar examples of seismic amplitude (a) and output attribute: peak phase (b).

Figure 5. Color bar examples of seismic amplitude (a) and output attribute: peak phase (b).

Examples:

Figure 6. Vertical transect views of seismic amplitude (a) and output attribute: peak phase (b).

Figure 6. Vertical transect views of seismic amplitude (a) and output attribute: peak phase (b).

Computation: The statistical summary attributes are additional outputs of the spectral decomposition based on Complex Matching Pursuit (refer to Spectral Decomp-> CWT -> Spectral Magnitude, Spectral Phase, Spectral Voice Components, and Spectral Shape (Ridge) attributes description section).  Prior to computing these summary attributes, the amplitude volume (time or depth domain) is spectrally whitened to account for changes in the source wavelet with depth and a non-flat source spectrum. Thereafter, the output volume shows a relatively flat spectrum bound by two tails (see Figure 2). Following this behavior, the statistical summary attributes are generated.

CWT - 07.png

References

  • AASPI documentation, http://mcee.ou.edu/aaspi/documentation/Spectral_Attributes-spec_cwt.pdf
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 123 – 151.
  • Zhang, K., 2010, Seismic attribute analysis of unconventional reservoirs, and stratigraphic features: PhD Thesis, University of Oklahoma.

CWT -> Spectral Magnitude, Spectral Phase, Spectral Voice, and Spectral Shape (Ridge) Attribute Components

CWT -> Spectral Magnitude, Spectral Phase, Spectral Voice, and Spectral Shape (Ridge) Attribute Components

Attribute Description: 

The spectral components generated by the Continuous Wavelet Transform (CWT) are complex time-frequency signals. This type of transform has Morlet wavelets that are proportional to the center frequency. So then, the narrow-band ringing and broad-band impulsive reflections are better positioned in time (Chopra and Marfurt, 2007). The Spectral Magnitude, Spectral Phase, and Spectral Voice attributes are generated at each time-frequency sample. The Spectral Magnitude attribute represents the energy that correlates with the seismic signal (e.g., similar to Envelop), The Spectral Phase attribute denotes the phase rotation between the seismic trace and the modeled Morlet wavelets, the Spectral Voice attribute is the real component of the complex spectrum (e.g., Amplitude) and can be used to enhance the ability of coherence, curvature, or other AASPI structural attribute computations at uncovering thin or small structural and stratigraphic details. On the other hand, the Shape (Ridge) attribute can be used to enhance seismic resolution and reconstructing the seismic trace using broader band wavelets (Davogustto et al., 2013).

Interpretation Use: 

The Spectral Magnitude and Spectral Voice attributes can provide detailed subtle stratigraphic information about a reservoir or other zone of interest. Also, these attributes can help at examining geologic features in the form of spectral components. Additionally, the Spectral Phase can provide insights into discontinuity features as well as onlap, offlap, and erosional unconformities. On the other hand, the Spectral Shape (Ridge) attribute can be used to improve the geometry interpretation of geologic features (Chopra and Marfurt, 2007). Attribute results can be analyzed in different ways, from a plan view, vertical transects, or draped over a horizon display.

Recommended color palette: 

For the Spectral Voice attribute, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. Or any color scheme that works well with normal seismic amplitude data. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Figure 1. Color bar examples of seismic amplitude (a) and output attributes: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Examples:

Figure 2. Time slice displays of seismic amplitude (a) and output attribute: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Figure 2. Time slice displays of seismic amplitude (a) and output attribute: spectral voice 20 Hz (b), spectral voice 32 Hz (c), and spectral voice 44 Hz (d).

Recommended color palette: 

For the Spectral Magnitude attribute, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. We suggest using the histogram of values to guide setting color value thresholds.

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Figure 3. Color bar examples of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Examples:

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Figure 4. Time slice displays of seismic amplitude (a) and output attributes: spectral magnitude 20 Hz (b), spectral magnitude 32 Hz (c), and spectral magnitude 44 Hz (d).

Recommended color palette: 

For the Spectral Shape (Ridge) attribute, a divergent color scheme is suggested. The midpoint color is white to emphasize the progression outward two different dark hues. In the examples below, the dark hues were set to blue and red to better highlight geologic features. We suggest using the histogram of values to guide setting color value thresholds.

Figure 5. Color bar examples of seismic amplitude (a) and output attributes: spectral ridge 20 Hz (b), spectral ridge 32 Hz (c), and spectral ridge 44 Hz (d).

Figure 5. Color bar examples of seismic amplitude (a) and output attributes: spectral ridge 20 Hz (b), spectral ridge 32 Hz (c), and spectral ridge 44 Hz (d).

Examples:

Figure 6. Color bar examples for seismic amplitude (a) and output attributes: spectral ridge 20 Hz (b), spectral ridge 32 Hz (c), and spectral ridge 44 Hz (d).

Figure 6. Color bar examples for seismic amplitude (a) and output attributes: spectral ridge 20 Hz (b), spectral ridge 32 Hz (c), and spectral ridge 44 Hz (d).

Recommended color palette: 

For the Spectral Phase attribute, a cyclic color scheme is suggested. In this color palette, the hues wrap around so that the red follows purple. A specific color is assigned to different phase ranges, so then the display can be used to infer the continuity of seismic events. We suggest using the histogram of values to guide setting color value thresholds.

Figure 7. Color bar examples of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Figure 7. Color bar examples of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Examples:

Figure 8. Vertical transect views of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Figure 8. Vertical transect views of seismic amplitude (a) and output attributes: spectral phase 20 Hz (b), spectral phase 32 Hz (c), and spectral phase 44 Hz (d).

Computation:

Spectral decomposition can help at illuminating subtle geologic features that are below the resolution of the full frequency seismic data. This method consists of separating seismic signals into different frequency components:

Figure 9. Sketch showing an optical analog of spectral decomposition where light (i.e., seismic volume) is decomposed into its spectral components.

Figure 9. Sketch showing an optical analog of spectral decomposition where light (i.e., seismic volume) is decomposed into its spectral components.

The Spectral Magnitude, Spectral Phase, Spectral Voice, and Spectral Shape (Ridge) attributes are computed using modeled Morlet wavelets. The input amplitude volume (time or depth) is spectrally whitened to account for changes in the source wavelet with depth and a non-flat source spectrum prior to applying the spectral decomposition. The steps involved in computing CWT spectral decomposition are outlined in ASSPI documentation.

The Magnitude Spectral (Eq. 1) and Phase (Eq. 2) Spectral attributes are given by:

CWT - 10.png

where v(t,f) and vH(t,f) are the real and imaginary part of the complex spectrum (Figure 1). Note that the Phase Spectral attribute ranges between -180° and +180°. Then, the Voice Spectral attribute is given by Voice(t,f) = v(t,f) (Chopra and Marfurt, 2016).

References

  • AASPI documentation, http://mcee.ou.edu/aaspi/documentation/Spectral_Attributes-spec_cwt.pdf
  • Aarre, V., t. N. A. Al Dayyni, S. L. Mahmoud, A. B. S. Clark, B. Toelle, O. V. Vejbaek, and G. White, 2012, Seismic detection of subtle faults and fractures: Oilfields Review Summer, 24, 28 – 43.
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 123 – 151.
  • Chopra, S. and K. J. Marfurt, 2016, Spectral decomposition and spectral balancing of seismic data: The Leading Edge, 35, 176 – 179.
  • Davogustto, O., M. C. de Matos, C. Cabarcas, T. Dao, K. J. Marfurt, 2013, Resolving subtle stratigraphic features using spectral ridges and phase residues: Interpretation, 1, SA93 – SA108.

Dip Magnitude and Dip Azimuth

Dip Magnitude and Dip Azimuth

Attributes Description: The Dip Magnitude and Dip Azimuth attributes are estimates of the descent and direction of seismic reflectors.

 

Interpretation Use: The Dip Magnitude and Dip Azimuth attributes enable the mapping of subtle trace-to-trace variations on reflection character. Also, a composite display of dip magnitude and azimuth can help to emphasize more relevant geologic information (Chopra and Marfurt, 2007). Attribute results are better analyzed in plan view or draped over a horizon display.

Recommended color palette: For the Dip Magnitude attribute, a grayscale gradient color scheme is suggested. The color progression could begin with white (to highlight useful geological features) and finish with black (to denote shadow areas), or vice-versa. For the Dip Azimuth, a rainbow color scheme is suggested. A specific color is assigned to different azimuth ranges, so then the display can be used to infer the dip direction of deformed rocks. We suggest using the histogram of values to guide setting color value thresholds.

Figure 1. Color bar examples of seismic amplitude (a) and output attribute: dip magnitude (b) and dip azimuth (c).

Figure 1. Color bar examples of seismic amplitude (a) and output attribute: dip magnitude (b) and dip azimuth (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: dip magnitude (b) and dip azimuth (c).

Figure 2. Time slice displays of seismic amplitude (a) and output attributes: dip magnitude (b) and dip azimuth (c).

Computation: The Dip Magnitude and Dip Azimuth use seismic amplitude data (time or depth domain) as input. The attributes are computed from the estimates of dip angle along the inline and crossline directions in a 3D survey (refer to Filter Dip Components -> Inline Dip, Crossline Dip, and Confidence attributes description section). To avoid smearing of faults, fractures, and other discontinuities, the algorithm uses a multi-window moving around an analysis point (Luo et al., 2002). The inline and crossline dip estimates are done using the window with the maximum coherency and are related to the estimated true dip magnitude and dip azimuth by these simpel geometric relationships (Marfurt, 2006):

Dip Magnitude - 07.png

The dip magnitude is greater than or equal to apparent dip pairs θx and θy. The dip azimuth is measured from the inline 3D seismic survey direction. A coordinate rotation is applied when the inline acquisition is not aligned with the North. The apparent dip pairs θx and θy are related to the estimated true dip and azimuth by the simple geometric relationships (Marfurt et al., 1998)

Note that in the presence of noisy input data, the following structure-oriented filters can be also applied as an optional step: i) Lower-Upper-Middle (LUM), ii) Multistage Median-based Modified Trimmed-Mean (MSMTM), iii) Alpha-Trimmed Mean, and iv) Mean. The Alpha-Trimmed Mean and Mean filters work best in areas where the data has random incoherent noise but might misestimate the amplitude. Also, the Alpha-Trimmed Mean filter is relatively insensitive to spikes in the data. Refer to Filter Dip Components -> Inline Dip, Crossline Dip, and Confidence attributes description section for a description of how these filters are computed.

References

  • AASPI Documentation, http://mcee.ou.edu/aaspi/documentation/Volumetric_Attributes-filter_dip_components.pdf
  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization: SEG Geophysical development series, 11, 27 – 43.
  • Marfurt, K., 2006, Robust estimates of 3D reflector dip and azimuth: Geophysics, 71, P29 – P40.
  • Marfurt, K., R. L. Kirlin, S. L. Farmer, and M. S. Bahorich, 1998, 3-D seismic attributes using a semblance-based coherency algorithm: Geophysics, 63, 1150 – 1165.

Envelope

Envelope

Attribute Description: The Envelope attribute is one of the early complex trace attributes generated by Turhan Tanner, and Sven Treitel.  It forms the basis for many of the other single trace attributes used today.  It is the length of the vector between the Real and Imaginary trace (Hilbert Transform).

This figure illustrates the real (amplitude), Hilbert transform, and Envelope traces.  The Envelope is red, the amplitude is blue, and Hilbert transform is green. 

This figure illustrates the real (amplitude), Hilbert transform, and Envelope traces.  The Envelope is red, the amplitude is blue, and Hilbert transform is green. 

Interpretation Use:  This attribute may provide value by:

  • Representing the layer based reflectivity
  • Acting as a DHI indicator
  • Highlighting thin bed tuning effects
  • Showing boundaries of sequences and depositional environments
  • Major lithologic changes
  • Showing layer effects over boundary effects

Recommended Colorbar: 

Since the distribution of this data is all positive, highlighting the very high values with unique colors, and using a gradational colorbar for the rest seem to highlight anomalous areas.

Example colorbar and amplitude spectrum

Example colorbar and amplitude spectrum

Example

Vertical display of Envelope

Vertical display of Envelope

Time slice of Envelope

Time slice of Envelope

Frequency Spectrum

Frequency Spectrum

Computation: The formula is comprised of the square root of the sum of the squares of the real and imaginary traces, or the Euclidian distance:

Envelope - 6.png

For a more detailed explanation of Envelope and other complex attributes, please read “Complex Trace Attributes” in this help section.

References:

  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.

Envelope Bands on Envelope Breaks

Envelope Bands on Envelope Breaks

Attribute Description: The Envelope Breaks trace reflects the minimums in the Envelope.  The Envelope Bands (Energy Bands) give the average energy of the envelope between adjacent pairs of Envelope breaks.  This identifies regions of strong energy. Since this is using Envelope Breaks, the discontinuities are not always as sharp as the phase breaks.

The upper trace is normal amplitude. The lower trace shows Envelope Bands on Envelope Breaks

The upper trace is normal amplitude. The lower trace shows Envelope Bands on Envelope Breaks

Interpretation Use:  This attribute may provide value by:

  • Identifying major units of strong energy and thus weak energy
  • Showing boundaries of sequences and depositional environments
  • Showing discontinuities and faults

Colorbar:

For this example, we use a colorbar that highlights the higher energy zones in the seismic. Since this is based on Envelope Breaks – it tends to be lower frequency by nature. Note the values are positive.

Example colorbar with an amplitude spectrum

Example colorbar with an amplitude spectrum

Example:

Vertical seismic display of Envelope Bands

Vertical seismic display of Envelope Bands

Time slice displaying Envelope Bands

Time slice displaying Envelope Bands

Frequency Spectrum

Frequency Spectrum

Computation: The Energy bands on envelope is similar to energy bands on phase breaks.  This attribute integrates by summing the envelope signal between envelope troughs.

Envelope Bands on Envelope Breaks - 06.png

This result is the envelope bands on envelope.

References:

  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.
  • Tom Smith, 2017, Paradise Instantaneous Attributes

Envelope Bands on Phase Breaks

Envelope Bands on Phase Breaks

Attribute Description:  The Phase Breaks trace reflects the discontinuities in the phase. The Envelope Bands (Energy Bands) give the average energy of the envelope between adjacent pairs of phase breaks. This identifies regions of strong energy.

Top trace is the amplitude trace. Lower Trace is Envelope Bands on Phase Breaks

Top trace is the amplitude trace. Lower Trace is Envelope Bands on Phase Breaks

Interpretation Use:  This attribute may provide value by:

  • Identifying areas of strong energy and thus weak energy
  • Showing boundaries of sequences and depositional environments
  • Showing discontinuities and faults

Recommended Colorbar: Since this is an average of envelope between phase discontinuities, the same colorbar for phase discontinuities or envelope can work. The colors are designed to focus on the higher energy intervals in the seismic. Note the values are positive.

Example Color bar and Amplitude spectrum

Example Color bar and Amplitude spectrum

Example:

Vertical display of Envelope Bands on Phase Breaks

Vertical display of Envelope Bands on Phase Breaks

Time slice of Envelope Bands on Phase Breaks

Time slice of Envelope Bands on Phase Breaks

Frequency Spectrum

Frequency Spectrum

Computation: 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.

Integratee the envelope signal between an adjacent pair of peak pick times amd tp yield a value which will be assigned as a constant to all samples in that range (δ(tj)) and repeat these integration / assingment steps for each park of pick times (2 to N).

Envelope Bands on Phase Breaks - 06.PNG

This result is the energy bands on phase breaks attribute.

For a more detailed explanation of Banded attributes and how they are calculated, read the “Banded Attribute Overview” help document.

References:

  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.
  • Tom Smith, 2017, Paradise Instantaneous Attributes

Envelope Breaks

Envelope Breaks

Attribute Description:  The Envelope Breaks attribute calculates out the minima of the envelope and picks that minima. It helps isolate intervals in the seismic trace and can be useful in extracting block information between each break based on another attribute. A detailed explanation of Banded attributes is described in the Help document “Banded Attribute Overview”.

Interpretation Use: This attribute may provide value by

  • Isolating unit boundaries
  • Defining unit continuity

Recommended Colorbar: 

The data for this attribute is basically the spike that defines the pick location and zero. As seen in the amplitude spectrum, zero dominates the values and the remaining values are all the pick value so only two colors are really used. You just need to set the highest value as the color you want to highlight the picks.

Example colorbar

Example colorbar

Example:

Vertical display of Envelope Breaks

Vertical display of Envelope Breaks

Time slice of Envelope Breaks

Time slice of Envelope Breaks

Frequency Spectrum

Frequency Spectrum

Computation: Envelope breaks are 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.

Envelope Breaks 05.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

Envelope Breaks 06.png

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

Envelope Breaks 07.png

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

Envelope Breaks 08.png

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

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.

Note that since the picks were made on the Envelope of the trace – they don’t always coincide with a trough or peak of the real or amplitude trace. Phase picks do a better job of this. These breaks represent the minima of the energy of the trace as represented by the Envelope and have a different significance for interpretation.

References:

  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.
  • Oliveros, R.B., and B. J. Radovich, 1997, Image-processing display techniques applied to seismic instantaneous attributes on the Gorgon gas field, North West Shelf, Australia; 67th Annual International Meeting, SEG, Expanded Abstracts, 2064-2067

Envelope Second Derivative

Envelope Second Derivative

Attribute Description: Envelope Second Derivative gives a measure of sharpness of the envelope peak, which may be more useful as a principal attribute display. In black and white, it shows all peaks of the envelope, which corresponds to all the reflecting interfaces detectable within the seismic bandwidth.

Interpretation Use:  This attribute may provide value because it:

  • Shows the reflecting interfaces visible within the seismic bandwidth.
  • Shows the sharpness of each event.
  • Indicates changes in the lithology.
  • Shows the large changes of the depositional environment, even though the corresponding envelope amplitude may be small.
  • Is a good representative image of the subsurface within the seismic bandwidth

Recommended Colorbar: 

Since the distribution of this data is Gaussian in nature, a standard seismic colorbar can work.

In this example, we are using a Standard White - Black colorbar giving a standard seismic look and feel. The higher amplitudes show the sharper events in the positive realm while the negative values show those events in the envelope minima. 

Example Colorbar

Example Colorbar

Example:

Vertical display of Envelope Second Derivative

Vertical display of Envelope Second Derivative

Time slice of Envelope Second Derivative

Time slice of Envelope Second Derivative

Frequency Spectrum

Frequency Spectrum

Computation: 

This is the second derivative of the envelope, the envelope being the modulus of the complex function, with the analytic trace defined as the real part of the trace f(t) and the imaginary part of the complex trace g(t):

Envelope Second Derivative - 5.png

The envelope is the modulus of the complex trace function:

Envelope Second Derivative - 6.png

This is the second derivative of the Envelope:

Envelope Second Derivative - 7.png

References:

  • Chopra, S. and K. J. Marfurt, 2007, Seismic attributes for prospect identification and reservoir characterization:  Society of Exploration Geophysicists, Geophysical Developments #11.
  • Taner, M. T., 2001, Seismic attributes:  Canadian Society of Exploration Geophysicists Recorder, 26, no 7.