Spectral decomposition provides anovel means of utilizing seismic dataand the discrete Fourier transform(DFT) for imaging and mapping tem-poral bed thickness and geologic dis-continuities over large 3-D seismicsurveys. By transforming the seismicdata into the frequency domain via theDFT, the amplitude spectra delineatetemporal bed thickness variabilitywhile the phase spectra indicate lateralgeologic discontinuities. This technol-ogy has delineated stratigraphic set-tings (such as channel sands andstructural settings involving complexfault systems) in 3-D surveys.

Widess pioneered a widely usedmethod for quantifying thin-bed thick-ness in 1973. Because it uses peak-to-trough time separation in conjunctionwith amplitude, this method dependson careful processing to establish thecorrect wavelet phase and true trace-to-trace amplitudes. Although similarin context, the spectral method pro-posed here uses a more robust phase-independent amplitude spectrum andis designed for examining thin-bedresponses over large 3-D surveys.

The concept behind spectraldecomposition is that a reflection froma thin bed has a characteristic expres-sion in the frequency domain that isindicative of the temporal bed thick-ness. For example, a simple homoge-neous thin bed introduces apredictable and periodic sequence ofnotches into the amplitude spectrumof the composite reflection (Figure 1).The seismic wavelet, however, typi-cally spans multiple subsurface lay-ers—not just one simple thin bed. Thislayered system results in a complextuned reflection that has a unique fre-quency domain expression.

The amplitude spectrum interfer-ence pattern from a tuned reflectiondefines the relationship betweenacoustic properties of the individualbeds that comprise the reflection.Amplitude spectra delineate thin-bedvariability via spectral notching pat-terns, which are related to local rockmass variability. Likewise, phase spec-tra respond to lateral discontinuitiesvia local phase instability. Together,the amplitude- and phase-relatedinterference phenomena allow inter-preters to quickly and efficiently quan-

tify and map local rock mass variabil-ity within large 3-D surveys.

The difference in frequencyresponse between a long-window anda short-window amplitude spectrumis significant. The transform from along trace approximates the spectrumof the wavelet (Figure 2), but the trans-form from a short trace comprises awavelet overprint and a local inter-ference pattern representing theacoustic properties and thickness ofthe geologic layers spanned by thewindow (Figure 3).

With a few exceptions (e.g.,cyclothems and sabkhas), long analy-sis windows encompass many geo-logic variations that statisticallyrandomize interference patterns ofindividual thin beds. The resultinglong-window reflectivity spectraappear white or flat. This is the premisebehind multiple suppression viadeconvolution. Given a large enoughwindow, the geologic stacking of indi-vidual thin layers can be consideredrandom. The convolution of a sourcewavelet with a random geologic sec-

0000 THE LEADING EDGE MARCH 1999 MARCH 1999 THE LEADING EDGE 353

Interpretational applications of spectraldecomposition in reservoir characterization

GREG PARTYKA, JAMES GRIDLEY, and JOHN LOPEZ, Amoco E&P Technology Group, Tulsa, Oklahoma, U.S.

Figure 1. Thin-bed spectral imaging.

Figure 2. Long-window spectral decomposition and its relationship to theconvolutional model. A long temporal window samples random geologythat commonly exhibits a white (flat) amplitude spectrum.

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tion creates an amplitude spectrumthat resembles the wavelet.

The response from a short windowis dependent on the acoustic proper-ties and thicknesses of the layersspanned by the window. The shorterthe window, the less random the sam-pled geology. The amplitude spectrumno longer approximates just thewavelet but rather the wavelet pluslocal layering. In such small windows,the geology acts as a local filter on the

reflecting wavelet, thereby attenuat-ing its spectrum. The resulting ampli-tude spectrum is not white andrepresents the interference patternwithin the window.

The short-window phase spectrumis also useful in mapping local rockmass characteristics. Because phase issensitive to subtle perturbations in theseismic character, it is ideal for detect-ing lateral acoustic discontinuities. Ifthe rock mass within the window is lat-

erally stable, its phase response willlikewise be stable. If a lateral discon-tinuity occurs, the phase responsebecomes unstable across that discon-tinuity. Once the rock mass stabilizeson the other side of the discontinuity,the phase response likewise stabilizes.

Wedge model response. Spectraldecomposition and thin-bed tuningphenomena can be illustrated by a sim-ple wedge model (Figure 4a). The tem-poral response consists of tworeflectivity spikes of equal but oppo-site magnitude. The top of the wedgeis marked by a negative reflection coef-ficient and the bottom by a positivereflection coefficient. The wedge thick-ens from 0 ms on the left to 50 ms onthe right. Filtering the reflectivitymodel (using an 8-10-40-50 HzOrmsby filter) illustrates the tuningeffects brought on with a change inthickness (Figure 4b). The top and bot-tom reflections are resolved at largerthicknesses but blend into one reflec-tion as the wedge thins.

A short-window amplitude spec-trum was computed for each reflec-tivity trace. These are plotted withfrequency as the vertical axis (Figure4c). The temporal thickness of thewedge determines the period of thenotches in the amplitude spectrumwith respect to frequency (Figure 5a).Pf = 1/t, where Pf = period of notchesin the amplitude spectrum withrespect to frequency (Hz), and t = thin-bed thickness (s).

Another viewpoint illustrates thatthe value of the frequency componentdetermines the period of the notchesin the amplitude spectrum withrespect to thin-bed thickness (Figure5b). Pt = 1/f, where Pt = period ofnotches in the amplitude spectrumwith respect to temporal thickness (s),and f = discrete Fourier frequency.

Even a relatively low-frequencycomponent such as 10 Hz quantifiesthin-bed variability.

The wedge model (Figures 4 and5) illustrates the application of thisapproach to a very simple two-reflec-tor reflectivity model. Increasing thecomplexity of the reflectivity modelwill in turn complicate the interfer-ence pattern.

The tuning cube. Amoco’s mostcommon approach to characterizereservoirs using spectral decomposi-tion is via the “zone-of-interest tuningcube” (Figure 6). The interpreter startsby mapping the temporal and verticalbounds of the zone-of-interest. Ashorttemporal window about this zone is

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Figure 3. Short-window spectral decomposition and its relationship to theconvolutional model. A short temporal window samples ordered (nonran-dom) geology that tunes the amplitude spectrum.

Figure 4. Blocky wedge model. (a) Reflectivity; (b) filtered reflectivity; (c)spectral amplitudes.

a)

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transformed from the time domaininto the frequency domain. The result-ing “tuning cube” can be viewed incross-section or plan view (commonfrequency slices).

The frequency slice is typicallymore useful because it allows inter-preters to visualize thin-bed interfer-ence patterns in plan view from whichhe or she can, drawing on experience,

identify textures and patterns indica-tive of geologic processes. Amplitudeor phase versus frequency behav-ior/tuning is fully expressed by ani-mating through the entire frequencyrange (i.e., through all frequencyslices).

Tuning cube maps of the earth’ssubsurface are in many ways analo-gous to satellite imaging maps of the

earth’s surface. Both methods takeadvantage of frequency subbands tomap lateral variability in surface prop-erties.

Removing the wavelet overprint. Thetuning cube has three components:thin-bed interference, wavelet over-print, and noise (Figure 7). Since thegeologic response is the most inter-esting component for the interpreter,it is prudent to balance the waveletamplitude without degrading the geo-logic information. This reduces thetuning cube to thin-bed interferenceand noise.

Common spectral balancing tech-niques rely on sparse invariant sta-tionary statistics. If we assume that thegeologic tuning varies considerablyalong any flattened horizon, then webalance the wavelet spectrum byequalizing each frequency slice accord-ing to its average amplitude (Figure 8).After whitening to minimize thewavelet effect, the tuning cube retainstwo main components: thin-bed inter-ference and noise.

In frequency-slice form, thin-bedinterference appears as coherentamplitude variations. Random noisespeckles the interference pattern in asimilar fashion to poor quality televi-sion reception. At dominant frequen-cies, the relatively high signal-to-noise

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Figure 6. Zone-of-interest tuningcube.

Figure 5. Thin-bed tuning of amplitudes versus frequency (a) with respectto frequency and (b) with respect to thin-bed thickness.

a)

b)

Figure 7. Prior to balancing the spectrum, the tuning cube consists of thin-bed interference, the seismic wavelet, and random noise.

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ratio (SNR) results in clear pictures ofthin-bed tuning. Movement awayfrom dominant frequency causes theSNR to degrade. At frequenciesbeyond usable bandwidth, the poorSNR results in a noise map.

Beyond the zone-of-interest. The tun-ing cube addresses the tuning prob-lem on a local zone-of-interest scale.Characterizing larger seismic volumesrequires a different approach. Fordecomposition beyond the singlereflectivity package or zone-of-inter-est, we recommend “discrete fre-quency energy cubes” or, withdifferent data organization, the “time-frequency 4-D cube.”

“Discrete frequency energy cubes”(Figure 9) are computed from a singleinput seismic volume into multiplediscrete frequency amplitude andphase volumes. Computation is donevia running window spectral analysiswhich calculates the amplitude orphase spectrum for each sample. Thespectral components are then sortedinto common frequency componentcubes. This method is typically doneonly after scoping the zone-of-interest,horizon-based tuning cube.

For the “time-frequency 4-D cube,”the spectral decomposition is also com-puted using a running windowapproach. The results are sorted intocommon samples with increasing fre-quency. This volume allows the inter-preter to exploit conventionalinterpretive workstation software andnavigate through the volume at anydepth slice for any frequency. The out-put is many times the size of the input,but it allows the interpreter to navigateand visualize in space, time, and fre-quency.

Gulf of Mexico example. A case his-tory from the Gulf of Mexico illustratesthe use of spectral decomposition toimage the Pleistocene-age-equivalentof the modern Mississippi River delta.Tuning cube frequency slices (Figure10) capture the subtleties of inherenttuning and reveal the various deposi-tional features more effectively thanfull-bandwidth energy and phaseextractions (Figure 11). For example,compare the north-south delineationfor channel A. The image from 26 Hz(Figure 10b) is significantly better thanfrom 16 Hz (Figure 10a). On the otherhand, channel B is better imaged by 16Hz than by 26 Hz. Neither channel isadequately delineated by conventionalenergy envelope extraction. Thestrength of the phase component liesin detecting discontinuities. Theresponse in both phase slices (Figure

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Figure 9. Discrete frequency energy cubes.

Figure 8. Removing the wavelet overprint (balancing the spectrum) withoutremoving the reflectivity tuning characteristics.

Time-frequency4-D cube

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10c,d) is stable away from the faultsbut becomes unstable crossing dis-continuities such as faults. These spec-tral phase maps provide sharperdefinition of faults than a conventionalresponse phase map (Figure 11b).

Conclusions. Spectral decompositioncan be a powerful aid to the imagingand mapping of bed thickness andgeologic discontinuities. Real seismicis rarely dominated by simple blocky,resolved reflections. In addition, truegeologic boundaries rarely fall alongfully resolved seismic peaks andtroughs. By transforming the data intothe frequency domain with the dis-crete Fourier transform, short-windowamplitude and phase spectra localize

thin-bed reflections and define bed-thickness variability within complexrock strata. This allows the interpreterto quickly and effectively quantifythin-bed interference and detect sub-tle discontinuities within large 3-D sur-veys.

Suggestions for further reading. TheFourier Transform and its Applicationsby Bracewell (McGraw-Hill, 1965).“Spectral analysis applied to seismicmonitoring of thermal recovery” byDilay and Eastwood (TLE, 1995).“Identification of deltaic facies with 3-D seismic coherency and spectraldecomposition cube” by Lopez et al.(Abstract, Istanbul ‘97). “Interpret-ational aspects of spectral decomposi-

tion” by Partyka and Gridley(Abstract, Istanbul ‘97). “How thin isa thin bed?” by Widess (GEOPHYSICS,1973). LE

Acknowledgments: We thank Amoco EPTG forpermission to publish this article, Chuck Webbfor recognizing the value of this technology atits inception and for providing the positive feed-back that enabled its development, Amoco’sSeismic Coherency and Decomposition team forits ongoing development and calibration of spec-tral methods, and Amoco’s Unix SeismicProcessing Team for providing the technicalframework for algorithm development and test-ing.

Corresponding author: G. Partyka, gapartyka@amoco.com

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Figure 10. Gulf of Mexico (a) 16-Hz energy map, (b) 26-Hz energy map, (c) 16-Hz phase map, (d) 26-Hz phase map.

Figure 11. Gulf of Mexico full-bandwidth (a) conventional energy envelope extraction, (b) conventional responsephase extraction.

a) b)

c) d)

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