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Spectral Attributes: Program spec_cwt

Attribute-Assisted Seismic Processing and Interpretation - 8 December 2015 Page 1

COMPUTING SPECTRAL COMPONENTS USING THE CONTINUOUS WAVELET TRANSFORM – PROGRAM spec_cwt

Alternative Spectral Decomposition Algorithms Spectral decomposition methods can be divided into three classes: those that use quadratic forms, those that use linear forms and those that use atom decomposition. Quadratic forms are based on Wigner-Ville distribution, which can be easily designed but lose the phase component of the data and therefore cannot be used in reconstruction. Linear forms are based on short time Fourier transform (STFT), and include the S transform and the continuous wavelet transform (or CWT) as in program spec_cwt. Program spec_clssa is a nonlinear implementation of the short time Fourier transform. Atomic decomposition reconstructs the signal by using small “atom-sized” signals (in our case wavelets), such as matching pursuit (program spec_cmp) and the Hilbert-Huang transform.

spec_cwt computation flow chart While there is only one input file to program spec_cwt, many output files can easily fill your disk drives. Detailed spectral analysis will typically be done about a reservoir or other zone of interest, such that the input data volume may be windowed. If the data have not previously been spectrally whitened, it may be beneficial to reconstruct the spectrally balanced amplitude. In either case, it is best to ask for spectrally balanced spectral components and statistical measures, since this will suppress the effect of the dominant frequency of the source wavelet, leaving the tuning effects of the geology in place.

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Computing spectral components To begin, click the Attributes calculation tab in the AASPI-util window and select program spec_cwt:

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Program spec_cwt performs spectral decomposition by using continuous wavelet transform method. The following window appears:

First, enter the (1) name of the Seismic Input (*.H) file you wish to decompose, as well as a (2) Unique Project Name and (3) Suffix as you have done for other AASPI programs. The decomposition will generate date between (4) f1 and (5) f4 at (6) increments of Δf. f1 ,f2, f3 and f4 define the corner frequencies of an Ormsby filter used in spectral balancing and data reconstruction. Program spec_cwt computes spectral (7) magnitude, (8) phase, and (9) voice components that can be output for further analysis.

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`

Xx Overview of the Continuous Wavelet Transform (CWT)

As formally introduced by Grossmann and Morlet (1984), a function 𝝍(𝒕) ∈ 𝓛𝟐(𝕽) with zero mean is called

a wavelet if it has finite energy concentrated in time and satisfies some well-established conditions. A family of

wavelet functions can be obtained from a basis (or “mother”) wavelet 𝝍(𝒕) by scaling it by s and shifting by u:

𝝍𝒖,𝒔(𝒕) =𝟏

√𝒔𝝍 (

𝒕 − 𝒖

𝒔) = 𝝍𝒔(𝒕 − 𝒖); 𝒘𝒉𝒆𝒓𝒆 𝝍𝒔(𝒕) = 𝝍 (

𝒕

𝒔) (A1)

We know from the Fourier transform scaling property (e.g. Mallat, 2009), that if

𝜳(𝝎)is the Fourier transform of the wavelet 𝝍(𝒕), then the Fourier transform of the same wavelet scaled by s,

𝝍 (𝒕

𝒔) , is given by |𝒔|𝜳(𝒔𝝎). Therefore, if we compress the wavelet by increasing s, its spectrum will dilate and the

peak frequency will shift to a higher value. In contrast, if we dilate a wavelet by decreasing s, its spectrum will

compress and the peak frequency will shift to a lower value. Therefore, by varying the scaling factor s, such a

wavelet family can represent broad-band spectra. Recall that the wavelet must have zero mean; this causes no

serious problem for seismic amplitude data, but will for various inversion products such as acoustic impedance and

gamma ray volumes which have a DC component.

In addition, the spectrum of each wavelet in the family maintains a constant ratio between its peak frequency

and the corresponding bandwidth. Once a wavelet family is chosen, then the Continuous Wavelet Transform, CWT,

of a function 𝒇(𝒕) ∈ 𝓛𝟐(𝕽) at time u and scale s is defined as

𝑪𝑾𝑻𝒈(𝒖, 𝒔) =𝟏

√𝒔∫ 𝒈(𝒕)

+∞

−∞𝝍∗ (

𝒕−𝒖

𝒔) 𝒅𝒕, (A2)

where the superscript * indicates the complex conjugate.

The CWT can be interpreted as the cross-correlation between the signal and the wavelet, shifted by u with

scale s. Because the wavelet 𝝍(𝒕) has zero average, we can rewrite equation A2 as a convolution operation:

𝑪𝑾𝑻[𝒈(𝒖, 𝒔)] = ∫ 𝒈(𝒕)𝟏

√𝒔

+∞

−∞

𝝍∗ (𝒕 − 𝒖

𝒔) 𝒅𝒕

= 𝒈(𝒖) ∗ 𝝍𝒔̅̅̅̅ (𝒖),

(A3)

where

𝝍𝒔̅̅̅̅ (𝒕) =

𝟏

√𝒔𝝍∗ (−

𝒕

𝒔). (A4)

Therefore, the CWT algorithm can be implemented by simply convolving, in the time or frequency domain,

the seismic trace with a reversed and scaled wavelet represented by equation A3. Because the spectrum of 𝝍𝒔(𝒕)

resembles a bandpass filter, then CWT can also be interpreted as a multiple bandpass filtering operation, where the

signal 𝒈(𝒕) is filtered by a bank of filters created from a single mother wavelet scaled by several factors s. The scale

range should be chosen by taking into account the similarity of the scaled wavelets with the seismic trace, and by

verifying if the frequency bandwidth of the scaled wavelets filter-bank covers the seismic trace spectrum. The

energy representation of the CWT coefficients is called scalogram.

Wavelet analysis is very popular on different scientific areas because of its versatility to compare signals from

different natural phenomena with a desired or modeled basic wavelet function. In the seismic exploration area, we

can model zero-offset seismic traces by convolving the earth’s reflectivity series with a seismic wavelet. Therefore,

CWT mother wavelets (such as Ricker wavelets) that better represent the seismic wavelet produce better focused

time-frequency spectra. Morlet et al. (1982) showed that a complex sinusoid, with frequency fc, modulated by a

Gaussian function, with variance σ2, resembles seismic wavelets very well when scaled by s:

2

2/1

2

2/1exp2expexp2exp

1)(

s

tf

s

tfi

s

f

s

t

s

tfi

st b

cb

cs

(A5)

In the frequency domain, the Morlet wavelet represented by equation A5 (Teolis,1998), is a bandpass filter

centered at fc/s Hz with half-bandwidth of fb/s Hz. Figure A1 shows the Morlet wavelet and its spectrum for

different scales s when fc=1 Hz and fb=1 Hz. In theory, the spectrum of the Morlet wavelet is not zero for zero

frequency, and cannot be used in the CWT. In practice, most seismic data are missing the lowest frequencies, such

that the resulting small amount of aliasing can be tolerated when computing the CWT.

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Figure A1. (Top) A suite of four complex Morlet wavelets centered about 10, 20, 40, and 60 Hz with the real part in blue

and the imaginary part in green. (Bottom) Their corresponding magnitude spectra at 10 (blue), 20 Hz (green), 40 Hz

(red), and 60 Hz (cyan). By construction the bandwidth increases proportionately to the center frequency.

Since the Morlet wavelet is a complex function, the CWT spectral components are also complex. Figure A2 shows

the CWT of a seismic trace. The CWT magnitude represents the amount of energy that correlates with the trace, while the

CWT phase represents the phase rotation between the seismic trace and the Morlet wavelet at each instant of time.

Goupillaud et al. (1984) showed that the CWT preserves the signal energy and is invertible, such that the signal can

be reconstructed from the CWT coefficients as a convolution along the scales plus an integration along time,

𝒈(𝒕) =𝟐

𝑪𝝍

𝓡𝓮 ∫ ∫ 𝑪𝑾𝑻𝒈(𝒖, 𝒔)𝝍𝒔(𝒕 − 𝒖)

+∞

−∞

+∞

𝟎

𝒅𝒖𝒅𝒔

𝒔𝟐

=𝟐

𝑪𝝍𝓡𝓮 ∫ [𝑪𝑾𝑻𝒈(. , 𝒔) ∗ 𝝍𝒔(𝒕)]

+∞

𝟎

𝒅𝒔

𝒔𝟐,

(A6)

where the constant 𝑪𝝍 is given by

𝑪𝝍 = ∫|𝚿(𝝎)|𝟐

𝝎𝒅𝝎

+∞

𝟎. (A7)

Perfect reconstruction is achieved for the continuous case, when, theoretically, g(t), u and s are infinitely dense. In

practice, we need to sample sufficiently the CWT by the scale, s, to allow a good inverse CWT reconstruction in equation

A6 (Goupillaud et al.,1984; Li and Ulrych, 1999).

Figure 2 shows the CWT of a seismic trace and the corresponding inverse CWT, or ICWT, reconstruction using

equation A6.

Figure A2. (a) Seismic trace and corresponding CWT(b) magnitude and (c) phase computed using a Morlet mother

wavelet with center frequency fc=1.0 Hz and half-bandwidth fb=1.0. (c) CWT phase. (d) The inverse CWT (e) the error in

reconstruction.

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Examples of Morlet Wavelets As described in the gray theory box, the “mother” wavelet is defined by a center frequency, fc, and a half-bandwidth, fb. Other members of the wavelet family are scaled and shifted versions of the mother wavelet. Figures 1-3 show representative wavelets constructed from a broad band, moderate band, and narrow band mother wavelets. The choice of which wavelet to use depends on the application. Liner et al. (2004) used narrow band wavelets in estimate spectral magnitude discontinuities based on a Hölder transform. Singleton et al. (2006) also used narrow band wavelets in their Q estimation work.

(a)

(b)

(c)

(d)

Figure 1. A suite of Morlet wavelets plotted with center frequencies of (a) 10 Hz, (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The (program spec_cwt default) mother wavelet is defined with center frequency fc=1.0 Hz and half-bandwidth fb=1.0 Hz (variance σ2=1.00 s2) giving rise to moderate temporal and frequency resolution.

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(a)

(b)

(c)

(d)

Figure 2. A suite of Morlet wavelets plotted with center frequencies (a) 10 Hz, (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The broad band mother wavelet in this example is defined with center frequency fc=1.0 Hz and half-bandwidth fb=1.5 Hz (variance σ2=0.44 s2), giving rise to a low temporal resolution, but higher frequency resolution.

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(a)

(b)

(c)

(d)

Figure 3. A suite of Morlet wavelets plotted with center frequencies (a) 10 Hz, (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The narrow band mother wavelet is defined with center frequency fc=1.0 Hz and half-bandwidth fb=0.5 Hz (variance σ2=2.23 s2), giving rise to higher temporal resolution, but lower frequency resolution. Spectral Magnitude, Phase, and Voices

Spectral decomposition generates a suite of spectral magnitude, phase, and voice component at each time-frequency sample. Though not often used in seismic interpretation, the concept of the spectral voice provides a particularly clear means of illustrating both the mechanics and the value of spectral decomposition. The information content of voices is clearly understood when one listens to a Mozart opera, where each of the Soprano, Alto, Baritone, and Base performers often sing different words in harmony.Figure 4 shows a seismic trace, along with it time-frequency spectral magnitude, phase, and voice components. A vertical slice through the 20 Hz spectral

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voice component is equivalent to applying a narrow band filter centered about 20 Hz to the data.

Figure 4. A representative trace (line 150, cdp 74) showing the seismic amplitude in wiggle display, and the time frequency magnitude, phase, phase corrected for two-way travel time, and voice components. The output phase from program spec_cwt is corrected for the two-way travel time (i.e. the phase at each sample has been shifted by -2πft ) to better represent the phase of the reflectors rather than the two way traveltime. Note the phase residues (discontinuities in phase) are readily apparent in the correct image. The voice, v, is a simple function of the magnitude,m, and phase, ϕ, given byv(t,f)=m(t,f)exp[-iφ(t,f)].The sum of the voices reconstructs the original trace, d(t). Note the rapid variation in phase laterally with frequency at t=0.8 s. Such rapid changes require sampling with a fine frequency increment (∆f=2 Hz) in order to obtain accurate estimates of the phase residue.

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Figure 5. Vertical slices along line 150 through the spectral voice components at (a) 10 Hz , (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz obtained by cross-correlating the seismic amplitude data with the wavelets shown in Figure 1. These images clearly demonstrate that the CWT is equivalent to filtering the seismic data with a suite of relatively narrow band filter banks.

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Figure 6. Vertical slices along line 150 through the spectral magnitude components at a) 10 Hz , (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The mother wavelet has the default half-bandwidth of fb=1.0 Hz, resulting in a moderately localized wavelet, giving moderate temporal resolution.

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Figure 7. Vertical slices along line 150 through the spectral magnitude components at a) 10 Hz , (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The mother wavelet had a half-bandwidth of fb=1.5 Hz, resulting in a narrow wavelets shown in Figure 2, giving higher temporal resolution.

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Figure 8. Vertical slices along line 150 through the spectral phase components at 10, 20, 50, and 100 Hz. In this conventional CWT image, the phases include the 2πft phase delay associated with the position along the time axis. These phase are the onesused internally in programspec_cwtneeded to reconstruct the original or spectrally balanced trace.

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Figure 9. Vertical slices along line 150 through the spectral phase components at a) 10 Hz , (b) 20 Hz, (c) 50 Hz, and (d) 100 Hz. The output phase from program spec_cwt is corrected for the two-way travel time (i.e. the phase at each sample has been shifted by -2πft ) to better represent the phase of the reflectors rather than the two way traveltime. These phases will also be the ones used in program complex_stratal_slice and complex_spectra_pca prior to use in our future Q estimation applications.

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Figure 10. Spectral magnitude components for the same trace shown in the previous figure, showing the effect of the bandwidth of the mother wavelet, for half-bandwidths (a) fb=1.5 , (b) fb=1.0, and (c) fb=0.67 Hz. The center frequency is fc=1.0 Hz in all three cases. Note that the broader band wavelets provide greater temporal resolution, in (a) but less frequency resolution, with the magnitude varying smoothly across the frequency axis. In contrast, the narrow band wavelets in (c) provide greater frequency resolution, but less temporal resolution, with the magnitude varying more smoothly along the vertical axis. For this reason the right most narrow band choice may be better suited for spectral balancing and Q estimation, but not for interpretation of the spectral response of individual reflectors. Spectral Balancing and Data Reconstruction Programs spec_cmp and spec_cwtmay be better suited for spectral balancing. At present, the average spectrum is computed for the entire survey, and then a single time-variant spectral balancing operator is applied to each trace. Trace-by-trace spectral balancing can be dangerous, where notches in the spectra are not statistically averaged out, such that one can remove geologic features of interest. Future developments may include computing such spectral balancing within fairly large multitrace windows, or using low order polynomials to fit the spectra. To invoke spectra balancing and data reconstruction, click the (1) Want inverse CWT reconstructed data? and (2) the Spectrally balance output? buttons. The (3) Spectral balancing factor, α, and (4) Bluing exponent, β, are described in the gray box below. The half-bandwidth value fb, (5) the Morlet mothe wavelet bandwidth, also impacts the quality of the spectral balancing, with narrow-band wavelets providing greater spectral resolution:

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The impact of alterative values of half-bandwidth fb are shown in Figure 11.

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Figure 11. Input seismic amplitude data (a) before and (b-d) after spectral balancing using a value of α=0.01 (1 percent) and Morlet wavelet half-bandwidth values of (b) fb=1.0 Hz, (c) fb=1.5 Hz, and (d) fb=0.67 Hz. The narrow band Morlet wavelet used in (d) allows for the greatest amount of spectral balancing. Ideally, noise should be eliminating using structure-oriented filtering (program sof3d) prior to spectral balancing.

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The arithmetic of spectral balancing and bluing

Flattened spectra are obtained by balancing the power. The power of the jth trace is simply the spectral

magnitude squared:

),(),(2

ftaftP jj . (B1)

This spectral magnitude is averaged over all traces j=1,…,J and a 2K+1 sample vertical analysis window to

obtain the average power for each time slice t

K

Kk

J

j

avg ftktPKJ

ftP1

),()12(

1),(

. (B2)

The peak of the average power spectrum at time t is defined as

),()( ftPMAXtP avgf

peak . (B3)

With these definitions and a prewhitening value of ε=0.02 (2%) the flattened magnitude spectrum is

computed as

),()(),(

)(),(

21

ftatPftP

tPfta j

peakavg

peakflat

j

. (B4)

Traditionally, the goal of seismic processing was to produce a flat spectrum. However, Neep (2007) and

others built on earlier “colored inversion” work that showed the reflectivity spectrum derived from well logs

behaves as fβ where 0.0 < β < 0.4. A more general spectral bluing filter is then

),()(),(

)(),(

21

ftaftPftP

tPfta j

peakavg

peakblue

j

. (B5)

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Figure 12. Vertical slices through (a) original data, (b) reconstructed data with without spectral balancing, (c) difference between original and reconstructed data, and (d) reconstructed data with spectral balancing using a 0.5 s window and 4% of the peak magnitude in the balancing formula. No blueing has been applied. Note that the scale of (d) differs from (a)-(c) since the magnitude of both low and high frequencies have been increased. Also note that (d) has low frequency noise that was diminished with a bluing factor of β=0.3 in the previous figure.

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Figure 13. Vertical slices through spectrally balanced amplitude section with a bluing value of (a) β=0.0 resulting in a flat spectrum, and (b) β=0.3 such that the average spectrum behaves as f0.3 . Corresponding average time-frequency spectra with (c) β=0.0 and (d) β=0.3 . Phase Residues In the last GUI image note (4) the Want phase residue attributes?Option. The phase residue is a measure of disconuities in the phase component of the seismic data. Examination of Figures 4 and 9 will reveal discontinuities in the phase. These discontinuities are referred to as “phase residues” (see gray box below) that are often associated with abrupt changes in geologic deposition. Similar spectral discontinuities in the spectral magnitude components form the basis of the “Spice” algorithm developed by Liner et al. (2006).

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The Phase Residue

Ghiglia and Pritt (1998) provide an excellent survey of 2D phase-unwrapping techniques and

show how a complex residue theorem based on vector calculus can be applied to the phase-

unwrapping problem. They use a rectangular integration path aligned with the t and f axes. We

choose a smaller diamond-shaped integration path about each sample (j∆t, k∆f) given by

2

),(),(

2

),(),(

2

),(),(

2

),(),(

1111

1111

kjkjkjkj

kjkjkjkj

jk

ftftWftftW

ftftWftftWI

(C1)

where ψ is the phase and W is a wrapping operator that produces an output that falls between

±π.If the integral Ijk in equation C1 is nonzero, there are inconsistent phase points, which

Ghiglia and Pritt (1998) call “phase residues”. The figure below shows how the residue is

calculated for a small portion of a typical wrapped time-frequency phase matrix.

Figure B1. Diamond-shape integration paths used in computing the phase residue. Area A is

continuous, with a phase residue = 0.0, while area B is discontinuous with a phase residue =

1.0 .

Matos et al. (2011) find phase attributes are sensitive to the same kinds of stratigraphic

discontinuities seen by analyzing the magnitude component of time-frequency distribution

using wavelet transforms and the continuous wavelet transform. Because phase is often a more

accurate seismic measure than magnitude, it holds significant promise in mapping stratigraphic

unconformities.

Accurate computation of phase residues requires relatively fine sampling across frequency.

We suggest setting the computational ∆f=1 Hz. Unfortunately, if you do this and choose to

output these many components as well, you may fill up your disk drive. To analyze individual

components we suggest a coarser output ∆f=5 Hz for subsequent loading into your

interpretation workstation.

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Figure 14 shows the phase residues corresponding to the phase components shown in Figure 9. Corresponding to line 125 of the Boonsville data volume.

Figure 14. (a) The phase residues corresponding to the same vertical slice shown in Figure 9. (b) The same phase residues co-rendered with the original seismic amplitude data. Example 1: Mapping incised channels using phase residues This example comes from Davogustto et al.’s (2012) analysis of an incised Red Fork valley in the Anadarko Basin of Oklahoma. There were 660 wells within the seismic survey that provided detailed analysis that fell below seismic resolution.

a) b)

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Figure 15. The Watonga data volume have been reprocessed by CGG-Veritas as part of a larger “mega-merge” survey using 2010 technology and the increased migration aperture afforded by the adjacent surveys. The arbitrary line connects a suite of wells, several of which were described by Peyton et al. (1998). (After Davogustto et al., 2012).

Figure 16. The shale filled Stage 5 southern incised valley is easily identified on the seismic data. Unfortunately, the more northern sand-filled incised valleys to the north are "invisible" on the vertical seismic although we can see them on a time slice on the flattened data. This same phenomenon occurs on the Mississippian section in the TX-OK panhandles. Blue picks are the Inola lm and the pink picks are the Pink lm. The squares represent the well tops for each formation on each well. (After Davogustto et al., 2012).

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Figure 17. Phase residue magnitude on the same vertical section. Now we see anomalies that look like channels or in this case incision stages. Cyan is stage 1 & 2, purple is stages 3 & 4 and green is stage 5. I am still showing the tops from the Pink and the Inola limestones. (After Davogustto et al., 2012).

Figure 18. Using the tops from the logs Davogustto proceeds to interpret each of the anomalies. He is able to extract surfaces for each one of the stages from the attribute and use these to build the geological model. (After Davogustto et al., 2012).

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Figure 19. Time-structure map of the top Red fork computed by picking the conventional seismic amplitude volume. (After Davogustto et al., 2012).

Figure 20. The resulting geomodel from the seismic interpretation. On the right is the regional red fork which corresponds to delta plains and marshes. Note each one of the stages of incision of the valley. The advantage of this is that it simplifies building the geocellular model or the geological model and that each one of the "environments" can

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be modeled using different techniques and different net to gross or porosity schemes. (After Davogustto et al., 2012). Statistical measures of the spectrum Programs spec_cmp, spec_cwt, and spec_clssa provide several statistical measures of the spectrum that can be used in addition to or in place of the full 4D spectral components. The peak spectral magnitude, peak spectral frequency, and peak spectral phase are easy to understand. You obtain these by placing a checkmark in front of Want peak attributes? If you place a checkmark in front of Want spec mag cmpt?or Want spec phase cmpt?, you will obtain each of the spectral components that range between f1 and f4 with the desired Frequency Increment. For interpretation of the components on most interpretation workstations, it may be easier to load these components separately. If you place a check mark in of (9) Store cmpts as 4D cubes?, you obtain spectral gathers that are ordered with the time axis running fastest, followed by the frequency axis, (such that the first two indices represent a time-frequency distribution) followed by the crossline number axis (inlines) followed by the inline axis no. (crosslines). The 4D volumes will have the following names for this job:

If you ask for spectral components not to be stored as a 4D cube the constant-frequency 3D spectral magnitude and spectral phase volumes will have the frequency value encoded in the file name:

There are several additional controls under the Extended tab:

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Since the amount of output can be quite large, it may be useful to run spec_cwt on only a limited range of (1) inlines and (2) crosslines. You will probably want to experiment with these parameters a bit to calibrate them for the kind of data your encounter. It is reasonable to expect that most surveys of a similar vintage from the Gulf of Mexico will have similar spectral ranges and signal-to-noise ratios. Likewise, similar surveys acquired in the Permian Basin of west Texas will be similar to each other. In order to simplify parameter choices, will be similar to each other. In order to simplify parameter choices, you can then cat the AASPI “.parms” file to examine its content or set it to your desired default parameters:

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The file in your home directory will always take precedence over the one in the ${AASPIHOME}/scripts directory. As in all the AASPI GUIs, click Execute to run the program. The end of your run should looks something like the following:

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Now, let us plot some of the results. Since we did not choose to store the spectral magnitude and phase components as a 4D cubes, we have several 3D volumes we can plot separately. Plotting the same time slice as in all the other examples, and setting Allpos=y in our SEP Viewer GUI for the strictly positive magnitude, the spec_mag_cwt_d_10.H (the 10 Hz magnitude component) file looks like this:

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Figure 21.A time slice through the 10 Hz spectral magnitude component. While the spec_mag_cwt_d_80.H (the 80 Hz magnitude component) file looks like this:

Figure 22.A time slice through the 80 Hz spectral magnitude component. The phase components will range from -1800 to +1800, so set Allpos=n and choose a cyclical color bar to plot spec_phase_cwt_0_10.H (the 10 Hz phase component):

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Figure 23.A time slice through the 10 Hz spectral phase component. and spec_phase_boonsville_0__80.H (the 80 Hz phase component):

Figure 24.A time slice through the 80 Hz spectral magnitude component. Plotting Spectral Components

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We provide a simple graphical interface to quality control the spectral components. Many commercial workstation software products now provide excellent interactive visualization of 4D volumes (t, x, y, and typically offset h, but in our case frequency, f). To generate 4D volumes rather than a suite of 3D volumes, simply place (1) a checkmark in front of the Store spectral cmpts as 4D cubes option:

To plot the results, we can use the plot_4D_spectral_componentscan be found under the Display tools tab:

Spectral Attributes: Program spec_cwt

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Previously, I had computed spectral components for the Boonsville survey and stored them as a 4D volume (t,f,line_no,cdp_no) in a file spec_mag_4d_cwt_d_cwt.H

Spectral Attributes: Program spec_cwt

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The upper selection bar allows me to plot a constant frequency section or a constant timeslice. I have chosen the 1.1s time slice. I choose the energy.sep color bar and obtain these slices of different frequencies: First, 35 Hz component at 1.1s:

Spectral Attributes: Program spec_cwt

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Figure 25.A time slice through the 10 Hz spectral magnitude component.

Then, 50 Hz component at 1.1s:

Figure 26.A time slice through the 50 Hz spectral magnitude component.

Spectral Attributes: Program spec_cwt

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The upper selection bar allows me to plot a constant frequency section or a constant time slice. I have chosen the 40 Hz component. I choose the energy.sep color bar and obtain these slices of different frequencies:

Spectral Attributes: Program spec_cwt

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First, 0.9s time slice of 40 Hz component:

Figure 27. A time slice at t=0.090 s through the 40 Hz spectral magnitude component. Then, 1.0 s time slice of 40 Hz component:

Figure 28. A time slice at t=1.000 s through the 40 Hz spectral magnitude component.

Spectral Attributes: Program spec_cwt

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I can generate similar plots of the spec_phase_4d_cwt_cwt_d.H 4D volume:

These parameters provide the phase component at 50 Hz:

Spectral Attributes: Program spec_cwt

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Figure 29. A time slice at t=1.000 s through the 40 Hz spectral phase component. Plotting three spectral components against RGB Landmark, OpenDtect, Transform, and other commercial workstations do an excellent job of plotting spectral components against the red, green, and blue color channels. We can show a similar capability using our crude tool rgbplot, which can be found under the Display tools tab. The quality of rgbplot is a direct function of the number of colors used. The above-mentioned commercial implementations use 256 colors for each of R, G, and B, giving a total of (28)3 or 224 (24-bit) or over 16 million colors. Trying to emulate RGB with only 256 colors gives only 6 shades of R, G, and B, giving rise to images that have very limited color depth.

Invoking rgbplot gives the following GUI:

Spectral Attributes: Program spec_cwt

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I have chosen three input attributes: the 20, 40, and 60 Hz spectral coherence components generated by program spec_cwt. In the resulting image features with lower frequencies appear as red or orange, while features with higher frequencies appear as purple or blue.

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Figure 30. RGB plot of 20, 40, and 60 Hz spectral magnitude components using 256 colors (6 levels of R, G, and B).

Figure 31. RGB plot of 20, 40, and 60 Hz spectral magnitude components using 4096 colors (16 levels of R, G, and B).

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Figure 32. RGB plot of 20, 40, and 60 Hz spectral magnitude components using 262,144 colors (32levels of R, G, and B). Computing coherence from different spectral components The amplitude and phase of spectral components is a function of thickness and changes in reflection coefficients. Li and Lu (2014) showed how different these different spectral components give rise to different coherence images. Using commercial software (dGB’s OpenDtect) to display three different coherence volumes against R, G, and B, they generated the following two images of a western China survey showing karst, faults, and channels.

Spectral Attributes: Program spec_cwt

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Figure 33. RGB plot coherence computed from 10, 30, and 50 Hz spectral voice components (filtered data). (After Li and Lu, 2014).

References

Davogustto, O., M. Castro de Matos, and K. J. Marfurt, 2012, Using phase residues to map Red Fork channels: Geophysical Society of Oklahoma City Spring Expo.

Davogustto, O., M. Castro de Matos, C. Cabarcas, T. Doan, and K. J. Marfurt, 2013, Resolving subtle stratigraphic features using spectral ridges and phase residues: Interpretation, 1, SA93-SA108.

Ghiglia, D., and M. D. Pritt, 1998, Two-dimensional phase unwrapping: Theory, algorithms, and software: Wiley-Interscience.

Goupillaud, P., A. Grossman, and J. Morlet, 1984, Cycle-octave and related transforms in seismic signal analysis: Geoexploration, 23, 85-102.

Grossmann, A., and J. Morlet, 1984, Decomposition of Hardy functions into square integrable wavelets of constant shape: SIAM Journal on Mathematical Analysis, 15, 723-736, doi:10.1137/0515056.

Li, X. G., and T. J. Ulrych, 1999, Well log analysis using localized transforms, J. Applied Seismology, 8, No. 3, 243-260.

Li, F. Y., and W. K. Lu, 2014, Coherence attributes at different spectral components: Interpretation, 2.

Liner, C., C.-F. Li, A. Gersztenkorn, and J. Smythe, 2004, SPICE: A new general seismic attribute: 72nd Annual International Meeting of the Society of Exploration Geophysicists, Expanded Abstracts, 433-436.

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Mallat, S., 2009, A wavelet tour of signal processing, 3rd ed.: Academic Press. Mallat, S., 2009, A wavelet tour of signal processing: Academic Press. Matos, M. C., O. Davogustto, K. Zhang, and K. J. Marfurt, 2011, Detecting stratigraphic

discontinuities using time‐frequency seismic phase residues: Geophysics, 76, P1–P10.

Morlet, J., G. Arens, E. Fourgeau and D. Giard, 1982, Wave propagation and sampling theory - Part II: Sampling theory and complex waves, Geophysics, 47, 222-236.

Neep, J. P., 2007, Time variant colored inversion and spectral bluing: 69th EAGE Annual Meeting, Extended Abstracts, B009.

Peyton, L., R. Bottjer, and G. Partyka, 1998, Interpretation of incised valleys using new 3D seismic techniques: A case history using spectral decomposition and coherency: The Leading Edge, 17, 1294-1298.

Singleton, S.W., M.T. Taner, and S. Treitel, 2006, Q-estimation using Gabor-Morlet joint time-frequency analysis: 76th Annual International Meeting, SEG, Expanded Abstracts , 1610-1614.

Teolis, A., 1998, Computational Signal Processing with Wavelets, 1st ed: Birkhäuser Boston.

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