instantaneous spectral attributes to detect channels
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GEOPHYSICS, VOL. 72, NO. 2 �MARCH-APRIL 2007�; P. P23–P31, 12 FIGS.10.1190/1.2428268
nstantaneous spectral attributes to detect channels
ianlei Liu1 and Kurt J. Marfurt2
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ABSTRACT
Channels filled with porous rock and encased in a nonpo-rous matrix constitute one of the more important stratigraphicexploration plays. Although attributes such as coherence canbe used to map channel width, they are relatively insensitiveto channel thickness. In contrast, spectral decomposition canbe used to map subtle changes in channel thickness. The peakspectral frequency derived by using the short-window, dis-crete Fourier transform �SWDFT� is an excellent tool formapping such changes along an interpreted horizon. Weshow that by use of instantaneous spectral attributes, we cangenerate equivalent maps for complete seismic volumes. Be-cause we are often interested in mapping high-reflectivitychannels encased in a lower-reflectivity matrix, we find that acomposite plot of the peak frequency and the above-averagepeak amplitude accentuates highly tuned channels. Finally,by generating a composite volume using peak frequency,peak amplitude, and coherence, we can establish not only thechannel thickness, but also its width. We demonstrate the val-ue of such 3D volumetric estimates through application to �1�a marine survey acquired over Tertiary channels from theGulf of Mexico and �2� a land data survey acquired over Pale-ozoic channels from the Central Basin Platform, west Texas,United States. The channels in both marine and land surveyscan be highlighted through composite-volume analysis.
INTRODUCTION
Channels filled with porous rock and encased in a nonporous ma-rix constitute one of the more important stratigraphic explorationlays. However, detailed mapping of channels has a much broadermpact. By using modern and paleoanalogues, mapping channelselps us to understand the paleodepositional environment and there-y permits us to interpret less obvious prospective areas such as fans
Manuscript received by the Editor February 23, 2006; revised manuscript r1Formerly University of Houston, Center for Applied Geosciences and E
exas. E-mail: [email protected] of Houston, Department of Geosciences, Houston, Texas. E-m2007 Society of Exploration Geophysicists.All rights reserved.
P23
nd levees. By mapping the width, tortuosity, and spatial relation-hip of meandering channels, avulsions, and braided streams, amongther features, geomorphologists are able to infer channel depth anduid velocity during the time of formation and thus better determinehether the fill is sand or shale prone.Seismic coherence and other edge-sensitive attributes �Bahorich
nd Farmer, 1995; Luo et al., 2003� are among the most populareans of mapping channel boundaries. Although these attributes
an easily detect channel edges, they cannot indicate the channel’shickness. In addition, as channels become very thin �well belowne-quarter wavelength�, their waveform becomes constant, suchhat coherence measures based on waveform shape cannot see thehannel at all �Chopra and Marfurt, 2006�.
From its inception, spectral decomposition has also been used toighlight channels �Partyka et al., 1999; Peyton et al., 1998�. Thepectral-decomposition images are complementary to coherencend edge-detection attribute images in that they are sensitive tohannel thickness rather than to lateral changes in seismic waveformr amplitude. Initially, spectral-decomposition analysis was doneithin a fixed-sized analysis window �by using a short-window, dis-
rete Fourier transform �SWDFT�� following a picked stratigraphicorizon, thereby generating a suite of constant-frequency spectral-mplitude maps. Most commonly, the interpreter animates throughhe maps and chooses the ones whose spatial pattern corresponds to aeasonable geologic model. Specifically, there is a strong correlationetween channel thickness and spectral amplitude �Laughlin et al.,002�.
Widess �1973� shows that for thin beds below the tuning thick-ess, the composite seismic amplitude decreases linearly with thick-ess. Chuang and Lawton �1995� generalizes this work to a frequen-y spectrum and observed that the peak frequency slightly increasess the layer thickness decreases. Marfurt and Kirlin �2001� exploithis observation and apply it to an SWDFT over the Pliocene-leistocene deposits of the Mississippi River in the Gulf of Mexico.hey found that the frequency corresponding to the peak spectral
September 28, 2006; published online January 22, 2007.ouston, Texas; presently Chevron Energy Technology Company, Houston,
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mplitude is an excellent means of summarizing the informationontent of the full spectrum: a low-peak frequency corresponds tohick channels, and a high-peak frequency corresponds to thin chan-els.
Even with careful tapering, spectral decomposition using theWDFT has window effects �Cohen, 1995�. For this reason, Casta-na et al. �2003� looked at alternative methods for time-frequencyecomposition based on wavelet transforms to compute what isommonly called instantaneous spectral attributes �ISA�. Much ofhis work has been used to identify channels �e.g., Sinha et al., 2005;
atos et al., 2005�, but to our knowledge, little has been publishedn the use of peak-frequency volumes based on these techniques, al-hough Liu et al. �2004� showed how the ISApeak frequency signifi-antly increases with decreasing layer thickness.
Precompute complexseismic wavelets
Read seismic trace
Generate complex seismic trace
Set residual = original complex traceSet spectrum = 0.0
Calculate envelope of residual
Pick the time position of strongest envelo
Least-squares fit complex wavelets to resi
Subtract complex wavelets from previous residcompute new residual
Add complex spectrum of subtracted wavelets to pre
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More traces?
Output complex spectraof all traces
igure 1. A flowchart for a wavelet-based spectral-decompositionatching-pursuit technique. Typically we precompute a complex w
ntervals of 0.5 Hz.
In this paper, we show how spectral decomposition can be used toenerate complete volumes of both peak spectral frequency and themplitude at that peak spectral frequency. We begin by reviewingow we compute ISAcomponents by using a matching-pursuit algo-ithm. Because we are interested in lateral changes in tuning associ-ted with channels, rather than absolute spectral amplitudes, we in-roduce the peak spectral amplitude above the average spectral am-litude at each analysis point. Next, we show how these algorithmsehave on simple synthetics. We then show how the peak frequencynd the amplitude of the peak frequency can be effectively coren-ered through the use of a 2D color map �or palette�. By generating aomposite of this 2D color map with a gray scale, we can also coren-er coherence. Finally, we apply this workflow to two channels sys-ems: the first seen in a marine survey over Tertiary channels ac-
quired in the Gulf of Mexico and the second seenin a land survey over Paleozoic channels acquiredin the Central Basin Platform, west Texas, UnitedStates.
ALGORITHM DESCRIPTION
We decompose the data by using a matching-pursuit wavelet-based spectral-decomposition al-gorithm shown in Figure 1 and presented by Liuand Marfurt �2005�. The details of the algorithmdescription can be found in Appendix A. We be-gin by calculating the instantaneous envelope andfrequency for each input trace. We then identifykey seismic events by picking a suite of envelopepeaks that fall above a user-specified percent-age of the largest peak in the current �residual�trace. We have found that this implementationconverges faster and provides a more-balancedspectrum of interfering thin beds than the alterna-tive “greedy” matching-pursuit implementation�Mallat and Zhang, 1993� that fits the wavelethaving the largest envelope, one at a time. Thepeak frequency of the wavelet can be approxi-mately calculated by the instantaneous frequencyof the residual trace at the envelope peak �Robert-son and Nogami, 1984�. The amplitudes andphases of each of the selected wavelets are com-puted together by using a simple least-squares al-gorithm. Each picked event has a correspondingwavelet. We compute the complex spectrum ofthe modeled trace by simply adding the complexspectrum of each constituent wavelet. This pro-cess is repeated until the residual falls below a de-sired threshold. In spite of our original work usingthe popular Morlet wavelets �Liu and Marfurt,2005�, we have found that a Ricker wavelet basisfunction proposed by Liu et al. �2004� better fitsseismic data that tend to be biased toward thelower frequencies.
To better illustrate the flow shown in Figure 1,we apply it to a seismic line extracted from a 3D
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urvey acquired over the Central Basin Platform. The amplitude andime of the chosen waveletsare displayed in Figure 2d. At each itera-ion, we generate the corresponding seismicavelets and add them to the previously modeledata �Figure 2a�, compute a new data residualunmodeled data� �Figure 2b�, and accumulatehe complex spectrum, of which we display the0-Hz component in Figure 2c.
If the seismic data have not been previouslypectrally balanced, it is common practice to doo within the spectral-decomposition algorithm.ollowing Partyka et al. �1999�, we assume theeology to be random over a suite of neighboringime slices, giving rise to an underlying “white”eflectivity spectrum. We then balance the spec-rum within a user-defined bandwidth as shown inigure 3. Spectral balancing accounts for changes
n the source wavelet with depth. The averagepectrum over the entire survey at a given timeefore balancing is shown in Figure 3a. The bal-nced average spectrum is demonstrated in Fig-re 3b. If we define the average spectrum as�� f��, which is a function of frequency f , and itsaximum as �max, we estimate a noise level as a
raction � of the peak spectral amplitude. Then weescale each spectral component by 1/���� f��
� ·�max� thereby obtaining a “balanced” aver-ge spectrum. Some workers find that balancinghe median rather than the mean spectrum pro-ides better results in the presence of bright spotsG. A. Partyka, personal communication, 2005�.or our two examples, the mean and median spec-
ra are nearly identical. Such spectral balancing ismportant in extracting tuning effects �such as theeak spectral frequency� of the geology, in orderot to be biased toward the spectral behavior ofhe seismic wavelet.
Once balanced, we can animate through timer horizon slices of discrete spectral components,nterpret selected volumes of discrete spectralomponents, or generate composite volumes ofeak frequency, peak amplitude, and coherencettributes. Peak amplitude is the maximum mag-itude of the amplitude spectrum. And peak fre-uency is the frequency at the peak amplitude.
SYNTHETIC MODEL
A simple 1D synthetic seismic trace is shownn Figure 4a. The source wavelets are Morletavelets with peak frequency at 10 Hz, 30 Hz,
nd 50 Hz. The Morlet wavelets with 10-Hz peakrequency have 0° phase. The Morlet waveletsith 50-Hz peak frequency have 90° phase. Thehases of Morlet wavelets with 30-Hz peak fre-uency are 0° and 45°. Figure 4b demonstrates
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he final time-frequency distribution plot of Figure 4a. The time ex-ension of the time-frequency distribution is constrained by the en-
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ion 4 iterations 16 iterations
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of the matching-pursuit algorithm described in Figure 1 applied to athe Central Basin Platform. Columns represent algorithm results af-d 16th iteration of �a� modeled data, �b� residual �unmodeled� data,
nent of the modeled data, and �d� the wavelet location and envelopeThe frequency and phase of the modeled data are not displayed. Datagton Resources.
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lancing to account for changes in the source wavelet with depth �re-mponent by 1/���� f�� + � ·�max��. �a� The average spectrum beforeerage spectrum after balancing.
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elope of matched wavelets. We can easily see that the low-frequen-y wavelet has a longer time spreading, whereas the high-frequencyavelet has a shorter time extension, but has greater bandwidth.
2D COLOR CHART
Many seismic attributes are only meaningful when put in the con-ext of a second, independent, attribute. For example, a measure-
ent of reflector azimuth is meaningless if the dip magnitude of theeflector is flat. Similarly, a measure of wavelet phase is meaninglessf its amplitude falls below the signal-to-noise level. In this paper, thealue of the peak spectral frequency has meaning only if that peakies significantly above the average spectrum. We use a 2D color
ap that combines hue and lightness to represent peak frequencynd above-average peak amplitude, respectively �Figure 5a�, andwo idealized spectra �Figure 5b and c�, one that is high amplitude,ighly peaked at a high frequency, and one that is lower amplitude,atter, and peaked at a lower frequency. Because the spectrumhown in Figure 5b has a relatively higher peak frequency andbove-average peak amplitude, it is represented by the bright orangeolor shown in Figure 5a. In contrast, the spectrum shown in Figurec has a lower peak frequency and only an average peak amplitudend is represented by the dark green color shown in Figure 5a.
To display coherence, we turn to the concept of composite dis-lays discussed by Chopra �2001� and Lin et al. �2003�. In this paper,f the coherence is above a threshold, we display the peak frequencynd amplitude through the use of the 2D color table displayed in Fig-re 5a. If the coherence falls below a threshold, we display the coher-nce against a 1D gray scale. In this manner, coherence defines thedges �and thus width� of the channels, and the peak frequency de-nes the relative thickness.
EXAMPLES FROM FIELD DATA
marine survey over Tertiary channels:outh Marsh Island, Gulf of Mexico
Conventional analysis through animation of spectral componentss described by Partyka et al. �1999� works very well when appliedo a horizon. This methodology breaks down, however, when ana-yzing volumes of seismic data. Generation of 80 output volumes at-Hz increments between 10 and 90 Hz quickly fills all the diskpace available. The computational effort of spectral decompositions greatly outweighed by the shear amount of output data. Evenhough we can reduce the output volumes by sampling every 10 Hz,t is still awkward to simultaneously deal with nine common-fre-uency volumes. For this reason, we propose initially generatingnly the peak frequency and peak amplitude volumes by time-fre-uency decomposition, and combining them with coherence, there-y providing an image that can be used to rapidly identify features oftratigraphic interest. If appropriate, the individual spectral compo-ents can be regenerated and examined either along constrainedones of interest or for a constrained range of frequencies �such asone for reservoir illumination by Fahmy et al., 2005�. A compositeolume of peak frequency, peak amplitude, and coherence is createdn the following examples.
High
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igure 4. �a�Asimple 1D synthetic seismic trace that is composed ofifferent Morlet wavelets. �b� Time-frequency distribution plot ofa�.
Low HighPeak frequency (Hz)
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igure 5. �a� A 2D color chart of peak frequency �hue� and peak am-litude above background �lightness� �as shown in Lin et al., 2003�.b and c� Two idealized spectra: �b�A spectrum with a high peak fre-uency and high above-average peak amplitude maps to a bright or-nge color, and �c� a spectrum with a low peak frequency and onlyverage peak amplitude maps to a dark green color.
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Instantaneous spectral attributes P27
Figure 6a shows the time slice of the coherence volume at t1.416 s. White arrows indicate a wide main channel, and yellow
rrows indicate a narrow branch channel. In the time slice throughhe 20-Hz spectral component �Figure 6b�, only the main channelhows up. In contrast, in the time slice through the 60-Hz spectralomponent �Figure 6c�, only the branch channel shows up. This phe-omenon implies that the main channel is thicker than its branch.
Figure 7a shows the time slice of the coherence volume at t1.230 s. The meandering channels are easily interpreted in the co-
Wide channel
High
LowNarrow channel
2 km
Wide channelHigh
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Narrow channel Low
High
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igure 6. Time slices at t = 1.416 s through a survey acquired overertiary channels, South Marsh Island, Gulf of Mexico, showing �a�oherence, �b� the 20-Hz spectral component, and �c� the 60-Hzpectral component. �Horizontal and vertical scales are the same.�
erence volume �white arrows�. Figure 7b demonstrates the com-osite volume of peak frequency, peak amplitude, and coherence ofhe same time. In Figure 7b, white arrows point to the meanderinghannels. Compared to the gray image in Figure 7a, the compositelot �including hue, lightness, and gray� in Figure 7b better distin-uishes the channels from the background. For instance, most of thehannels appear as green. The coherence can detect the discontinuityf seismic events �which means the coherence volume can detect thedge of the channel�, whereas the peak frequency and peak ampli-ude allow the channel to be highlighted in a distinctive color.
land survey over Paleozoic channels: Central Basinlatform, west Texas, United States
We now apply this same technique to older indurated rocks im-ged in a field-data example from west Texas to illustrate the spec-ral-decomposition technique in Figure 2. Figure 8 shows the timelice of seismic volume at t = 1.060 s. The arrows point to Pennsyl-anian channel deposits. Figures 9 and 10 show two time slices at t1.060 s and 1.096 s of peak frequency and above-average peak
mplitude �peak amplitude minus average amplitude value� throughhe use of the 2D color chart shown in Figure 5a. The channels are
High
Low
2 km
a)
b)
igure 7. Time slices at t = 1.230 s through a survey acquired overertiary channels, South Marsh Island, Gulf of Mexico, showing �a�oherence volume and �b� composite volume of peak frequency,eak amplitude, and coherence. �Horizontal and vertical scales arehe same.�
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learly shown in these two time slices. We note that the channel has areen color whereas the background has blue color, implying that thehannel has a higher peak frequency than the background response.n these images, we did not use coherence to identify the edges of the
Pos.
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igure 8. Time slice of the seismic volume at t = 1.060 s through aurvey acquired over the Central Basin Platform. The survey is alsosed in Figure 2. White arrows point to two different channelranches. �Seismic data courtesy of Burlington Resources.�
2 km
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igure 9. Time slice of the composite volume of peak frequency andbove-average peak amplitude at t = 1.060 s. �Magenta arrowsoint to channel.� Colors are as in the 2D color chart in Figure 5a.
hannel, so the color green itself highlights the channels. To view allhe channels in one slice, we flatten the volume along the unconfor-
ity horizon �the blue pick shown in Figure 12a�. Figure 11a showshe phantom-horizon slice 44 ms above the unconformity. The ac-ompanying amplitude spectra �Figure 11b-d� correspond to theoints indicated by the magenta arrows. Figure 11b shows a peakrequency at about 62 Hz and high above-average peak amplitude;he plot points to an area that is thus mapped as bright yellow in Fig-re 11a. Figure 11c shows a peak frequency at about 50 Hz and mod-rate above-average peak amplitude; the plot points to the chan-elthat is mapped as light green in Figure 11a. Figure 11d shows aeak frequency at about 16 Hz and moderate above-average peakmplitude; these characteristics indicate a background response, andhe area is mapped as a light blue in Figure 11a.
Figure 12a shows the seismic section of line AB from Figure 11a,nd the same amplitude spectra as in Figure 11 are shown in Figure2b-d. In the seismic section, we see that the notches in Figure 12bre cause by bed interferences. Figure 12c corresponds to the ampli-ude spectrum of a channel’s response, which shows a relativelyigher peak frequency compared to the background response of Fig-re 12d. Bed interferences can cause frequency notches �which de-end on bed thickness and geometry�; some of these can be individu-lly resolved �as in Figure 12b� but others cannot �Figure 12c�.
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igure 10. Time slice of the composite volume of peak frequencynd above-average peak amplitude at t = 1.096 s. �Magenta and yel-ow arrows point to channels.� Colors are as in the 2D color chart inigure 5a.
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Instantaneous spectral attributes P29
At present, we interpret these images in threesteps. First we use principles of geomorphology,together with modern and paleoanalogues, toidentify stratigraphic features of interest. Second,we calibrate these patterns through conventionalinterpretation of the vertical seismic section, cou-pled with our understanding of the physics ofthin-bed interference phenomena. Finally, we usethe colors to provide a quantitative estimate ofrelative channel thickness, and we use coherenceto provide a qualitative estimate of channelwidth. These tools allow us to unravel strati-graphic features of interest preserved in the geo-logic record.
CONCLUSIONS
We propose a new spectral-decompositionmethod that combines the concept of matching-pursuit and least-squares solution. Through theuse of instantaneous spectral analysis derivedfrom wavelet-based spectral decomposition, wehave extended the concept of using peak spectralfrequency of mapped horizons to full 3D vol-umes. We find that these peak spectral frequen-cies are most useful if modulated by some mea-sure of the corresponding spectral amplitude. Forchannels where we expect lateral changes in thin-bed tuning, we find that the peak spectral ampli-tude above the average spectral amplitude is par-ticularly useful by deemphasizing the appearanceof strong-amplitude flat spectral responses.
Although spectral decomposition is a good in-dicator of channel thickness, coherence and otheredge detectors are good indicators of channelwidth. For this reason, we advocate displayingboth attributes in a composite image. We haveshown the effectiveness of this technique in map-ping Tertiary channels in a marine survey. We findthis technique to be an excellent tool for rapidlymapping channels that may be of importance bothfor prospect evaluation and for quantifying reser-voir heterogeneity. We are encouraged to thinkthat by using these three measures together, wecan develop improved geostatistics or neural net-work flows that, with well control, can help usquantitatively estimate reservoir thickness.
ACKNOWLEDGMENTS
We thank the sponsors of Allied GeophysicalLaboratories at the University of Houston and theCRC Petroleum Research Fund for their supportof this research effort. We give special thanks toCharlotte Sullivan, Fred Hilterman, and JohnCastagna for their helpful feedback. We alsothank Fairfield Industries and Burlington Re-sources for providing the field data. Finally, we
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igure 11. �a� Phantom-horizon slice 44 ms above the unconformity througholume of peak frequency, above-average peak amplitude, and coherence uique described in Figure 5. �b� Amplitude spectrum of the erosional uhich appears as bright yellow. �c� Amplitude spectrum of the Pennsylvanraining the erosional unconformity, which appears as bright green. �d� Amrum of channel matrix, which appears as a lower-amplitude spectrum andlue.
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igure 12. �a� Seismic section of lineAB from Figure 11. �b–d� The same amra as Figure 11. �Blue line indicates unconformity.�
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hank the associate editor of Geophysics and two anonymous re-iewers for their valuable comments and suggestions.
APPENDIX A
MATCHING-PURSUIT SPECTRAL ANALYSIS
We begin our analysis by assuming that each seismic time trace,�t�, is band limited and can be represented by a linear combinationf either Ricker or Morlet wavelets:
u�t� = �j
aj · w�t − tj, f j,� j� + Noise, �A-1�
here aj, tj, f j, and � j represent the amplitude, center time, peak fre-uency, and phase of the jth wavelet. We exploit complex attributenalysis and estimate the center time of each candidate wavelet byeaks in the instantaneous envelope. We estimate the average fre-uency favg of the wavelet by the instantaneous frequency at the en-elope peak �called response frequency by Bodine �1984� and wave-et frequency by Taner �2000��. The peak frequency f j shown inquation A-1 can be computed for the Ricker wavelet by
f j = ���/2�favg, �A-2�
nd for the Morlet wavelet by
f j = favg. �A-3�
he temporal behavior of the Ricker wavelet is given by
wR�t, f j� = �1 − 2�2f j2t2�exp�− �2t2f j
2� , �A-4�
nd its spectrum is given by
wR�f , f j� =2
��
f2
f j3 exp−
f2
f j2 . �A-5�
he temporal behavior of the Morlet wavelet is given by
wM�t, f j� = exp�− t2f j2 · ln 2/k� · exp�i2�f jt� , �A-6�
nd its spectrum is given by
wM�f , f j� =��/ln 2
f j· exp�− k ·
�2�f − f j�2
ln 2 · f j2 � ,
�A-7�
here f j is the peak frequency, and k is a constant value that controlshe wavelet breadth. If we use smaller values of k, we will include
ore cycles in the Morlet wavelet. In our process, we choose k0.5.To efficiently solve for both the amplitude and phase of each
avelet, we use the Hilbert transform to form both an analytic datarace
U�t� = u�t� + iuH�t� �A-8�
nd a table of analytic complex Ricker wavelets
W�t, f j� = w�t, f j� + iwH�t, f j� , �A-9�
here w indicates symmetric cosine wavelets and wH indicates anti-
ymmetric sine wavelets. For the table of analytic complex Morletavelets, we use equation A-6 to generate Morlet wavelets.The analytic analogue of equation A-1 then becomes
U�t� = �j
Aj · Wj�t − tj, f j� + Noise, �A-10�
here the amplitude aj in equation A-1 is represented by the magni-ude of the complex amplitude Aj, and the phase � j is represented byhe phase of Aj.
Our objective is to minimize the energy of the residual trace R�t�,efined as the difference between the seismic trace and the matchedavelets
R�t� = U�t� − �j
J
�Aj · Wj�t − tj, f j���2
, �A-11�
here we recall that tj and f j are known and Aj is unknown. To esti-ate all wavelet coefficients in one iteration, we write equation A-11
n matrix form and simply solve the normal equations
A = �WHW + �I�−1WHU , �A-12�
here W = �W�t − t1, f1�,W�t − t2, f2�, . . . ,W�t − tm, fm�� is a vectorf wavelets centered at each envelope peak, A = �A1,A2, . . .Am�T is aector of complex wavelet amplitudes, I is the identity matrix, and �s a small number that makes the solution stable.
In the matching-pursuit approach, we begin by fitting those wave-ets corresponding to the largest wavelets’ envelope first. In our ap-lication, we have chosen to fit those wavelets whose envelopes arereater than 50 % of the value of the largest event of the current tracethe original trace for the first iteration; the current residual for allubsequent residuals�. We then follow the flow chart described inigure 1 until the residual energy is acceptable.Using equation A-5 or A-7 we compute the complex spectrum by
umming the complex spectra of the constituent wavelets
u�t, f� = �j=1
J
Aj · wj�f , f j�env�w�t − tj, f j��ei2�f�t−tj�,
�A-13�
here
env�w�t − tj, f j�� = ��w2�t − tj, f j�� + �wH�t − tj, f j��2�1/2
�A-14�
s the envelope of the complex wavelets. The amplitude spectrum ishus simply the magnitude of equation A-13, and the phase is the an-le between its real and imaginary parts. We assume the waveletpectra to be mathematically supported by the time duration of theeismic wavelets. If we wish to accurately reconstruct the amplitudef the original data from these complex spectra, we weight them by aoxcar function, giving rise to a rather blocky vertical section. Wend that for interpretation purposes, it is more useful to weight theomplex spectra by the envelope of the wavelets. Within this weight-ng window, the spectra must be phase shifted according to the sam-le time delay or advance from the envelope peak �equation A-14�.
B
B
C
C
C
C
CF
L
L
L
L
L
M
M
M
P
P
R
S
T
W
Instantaneous spectral attributes P31
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