Instantaneous spectral analysis (ISA) is a continuous time-frequency analysis technique that provides a frequency spec-trum for each time sample of a seismic trace. ISAachieves bothexcellent time and frequency localization utilizing wavelettransforms to avoid windowing problems that complicateconventional Fourier analysis. Applications of the methodinclude enhanced resolution, improved visualization of strati-graphic features, thickness estimation for thin beds, noise sup-pression, improved spectral balancing, and direct hydrocarbonindication. We have seen four distinct ways in which ISA canhelp in the detection of hydrocarbons: (1) anomalously highattenuation in thick or very unconsolidated gas reservoirs, (2)low-frequency shadows in reservoirs where the thickness isnot sufficient to result in significant attenuation, (3) preferen-tial illumination at the “tuning” frequency which can be dif-ferent for gas or brine-saturated rocks, and (4)frequency-dependent AVO. In this paper, we describe the ISAtechnique, compare it to other spectral decomposition meth-ods, and show some examples of the use of ISA to detect low-frequency shadows beneath gas reservoirs.

The ISA method involves the following steps: (1) decom-pose the seismogram into constituent wavelets using wavelettransform methods such as Mallat’s matching pursuit decom-position, (2) sum the Fourier spectra of the individual waveletsin the time-frequency domain to produce “frequency gathers,”and (3) sort the frequency gathers to produce common (con-stant) frequency cubes, sections, time slices, and horizon slices.The results can be viewed using animation techniques avail-able in commercial interpretation and visualization packages.

Figure 1 shows a synthetic seismic trace and the corre-sponding ISAtime-frequency analysis. The time-frequency plotshows amplitude spectra for each time sample. We refer tothis kind of plot as a “frequency gather.” The first arrival onthe synthetic seismogram results from an isolated reflector. Thefrequency spectrum is the spectrum of the wavelet. Note thatthe duration of the spectrum is identical to the duration of thearrival in the time domain as opposed to Fourier-based meth-ods in which the time duration is equal to the window length.The second event is a composite of two events of differing cen-ter frequency arriving precisely at the same time. The fre-quency spectrum indicates a low-frequency arrival spreadover time and a higher-frequency arrival that is more local-ized in time. The third event is caused by two interferingarrivals of the same frequency. Although the presence of twoarrivals is not immediately apparent on the seismogram, thetime-frequency decomposition clearly shows two distinctarrivals. The fourth event is a composite of four waveformsarriving at two distinct times evident on the time-frequencyanalysis. The final event consists of three arrivals of the samefrequency that are very closely spaced in time. The three dis-tinct arrivals cannot be resolved at low frequencies, but theseparation is clearly evident on the time-frequency analysisat high frequency. It is apparent that ISA provides a usefulrepresentation of the information contained in a seismic trace.

Spectral decomposition methods. Various techniques havebeen utilized in time-frequency analysis. Traditionally, theFast Fourier transform (FFT) and discrete Fourier transform

(DFT) have been applied (DFT having the advantages ofgreater speed and not mandating transform lengths that area power of 2 as required by FFT). Both techniques have lim-ited vertical resolution because the seismogram must be win-dowed. The spectral energy is distributed in time over thelength of the window, thereby limiting resolution. If the timewindow is too short, the spectrum is convolved with the trans-fer function of the window, and frequency localization is lost(i.e., the frequency spectrum is smeared). This can be mitigatedto some extent by tapering the window, but it is obviouslypreferable to avoid windowing altogether. Another disad-vantage of a short window is that side lobes of arrivals appearas distinct events in the time-frequency analysis. If the timewindow is lengthened to improve frequency resolution, mul-

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Instantaneous spectral analysis: Detection of low-frequency shadows associated with hydrocarbonsJOHN P. CASTAGNA and SHENGJIE SUN, University of Oklahoma, Norman, U.S.ROBERT W. SIEGFRIED, Gas Technology Institute, Des Plaines, Illinois, U.S.

Figure 1. Synthetic waveform with transient arrivals (black seismogram),constituent wavelets (color coded by center frequency), and time-frequency analysis (red represents high amplitude). At any given time, thespectrum of the seismogram is the weighted superposition of the waveletspectra. Since the exact wavelets summing to form the seismogram areknown, exact spectra versus time may be computed.

Figure 2. Comparison of true spectra and ISA for a synthetic waveform.

tiple events in the window will introduce notches that dom-inate the spectrum. Long windows thus make it very difficultto ascertain the spectral properties of individual events. Thisis one reason why Q-attenuation over short intervals is diffi-cult to measure quantitatively using Fourier-based techniques.

Recently, the maximum entropy method (MEM) has beenused for spectral decomposition. This technique can achieveexcellent frequency resolution but can be unreliable if the sig-nal violates the assumptions of the method or if the windowis too short. The major disadvantage, in our experience, is thatit seems to be unstable especially for less-than-expert users.Note in the example presented in this article that, even for thefirst arrival, there are two peak frequencies.

Wavelet transforms decompose a seismogram into con-stituent wavelets. The superposition principle tells us that thefrequency spectrum of a seismogram is the sum of the fre-quency spectra of the wavelets that sum to produce that seis-mogram. At any given time, the frequency spectrum is thesuperposition of weighted wavelet spectra in the vicinity ofthat time sample. In a somewhat overlooked paper publishedin GEOPHYSICS in 1995, Chakraborty and Okaya showed howwavelet transforms can be used in time-frequency analysis.They utilized matching pursuit decomposition to producehigh-resolution time-frequency analyses. However, our expe-rience with this method suggests that it introduces artifactsinto the time-frequency analysis manifested as high-amplitudebursts at a given time over a wide frequency band or at a given

frequency over a long time interval, thereby superimposinga cross-hatch pattern on the time-frequency analysis. This isa consequence of computational shortcuts designed specifi-cally for rapid computation made possible by selecting a par-ticular wavelet dictionary. We prefer to select the waveletdictionary to better capture the features of the seismogramwhile selecting parameters judiciously and avoiding as manycross-correlation operations as possible to achieve reasonablecomputation time. Thus, wavelet-transform based instanta-neous spectral analysis (ISA) can be done accurately withacceptable speed while simultaneously achieving excellenttime and frequency resolution.

Comparison of methods. When comparing different spectraldecomposition techniques, it is useful to have “the rightanswer” as a guide. When dealing with synthetic data, con-structed as a superposition of wavelets, the “true” time-fre-quency analysis is readily calculated as the sum of the spectraof the known wavelets. As shown in Figure 2, the ISA tech-nique does not yield the true spectrum precisely becausewavelet decomposition is not unique in very much the sameway that seismic inversion is not unique. This fact can be usedto prove that time-frequency analysis itself is not unique.Many different time-frequency decompositions can resultfrom the same seismogram and, conversely, can be inversetransformed to produce the same seismogram. This then leadsto the question: How can a particular frequency-decomposi-

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Figure 3. Comparison of ISA and time-frequency decomposition obtainedwith FFT using a 200-ms window. Notice the poor vertical resolution andspectral distortion (ribs and notches) caused by FFT windowing.

Figure 4. Comparison of ISA and time-frequency analysis obtained withDFT using a short time window. Although DFT with a short window hasexcellent vertical resolution, the frequency spectrum has been smoothed byconvolution with the spectrum of the window and false events are associ-ated with side lobes of transient arrivals.

Figure 5. Comparison of ISA and time-frequency analysis obtained withMEM using a long time window. Although MEM at times has superbfrequency resolution, vertical resolution is limited by the window lengthnecessary for reliable spectral determination.

Figure 6. A frequency gather (left) and a common frequency section(right) obtained by sorting many frequency gathers according tofrequency. The common frequency gathers can be thought of as instanta-neous amplitude at a given frequency.

trace and wavelets

tion be judged as better than another if there is no uniqueanswer? In our opinion, the important question is not whichdecomposition is right or wrong, but which captures the essen-tial features important in interpretation. We believe it is impor-tant that the following criteria are met:

1) The sum of the time-frequency analysis over frequencyshould approximate the instantaneous amplitude of theseismic trace.

2) The sum of the time-frequency analysis over time shouldapproximate the spectrum of the seismic trace.

3) Distinct seismic events should appear as distinct events onthe time-frequency analysis. In other words, the vertical res-olution of the time frequency analysis should be compara-ble to the seismogram. The time duration of an event onthe time-frequency analysis should not differ from the timeduration on the seismogram.

4) Side lobes of events on the seismogram should not appearas separate events on the time-frequency analysis.

5) The amplitude spectrum of an isolated event should beundistorted. The spectrum should not be convolved withthe spectrum of the window function.

6) There should be no spectral notches related to the time sep-aration of resolvable events.

A sure sign of improper spectral decomposition is a largedc (zero frequency) component for given events on data thathave had low-cut filters applied and should have little low-frequency energy. This is usually caused by spectral smear dueto windowing.

By definition, the ISA technique is designed to mathe-matically conform to criteria (1) and (2). As the ISA methodinvolves no windowing of the seismogram it also does wellat meeting criteria (3) to (6). As can be seen in Figure 2, ISAdoes a good job of capturing the essential features of the truetime-frequency spectra. One caveat: the more appropriate theselection of the wavelet dictionary, the better the time-fre-quency representation. An inappropriate wavelet dictionarywill cause the ISA method to fail in the sense that criteria (3)and (4) will not be achieved.

Figure 3 compares ISA and FFT spectra for the same syn-thetic trace discussed previously. The FFT was calculated witha 200-ms window. Notice that the FFT spectral energy is spreadout over the window length. When multiple arrivals occur inthe same window, severe spectral notches appear as well as“ribs” between events. The location of the notches and peri-odicity of the ribs depend entirely on the location of events intime and tell little about the spectral characteristics of indi-vidual reflectors. Certainly, closely spaced arrivals cannot beresolved in time on the FFT time-frequency analysis. Theirspacing in time can only be inferred from location of spectralnotches—this being a strongly model dependent inference. Thetemporal resolution of the FFT can be improved by reducingthe window length, the cost being loss of frequency resolu-tion and smearing of the spectrum (Figure 4). In this case, DFTwas used rather than FFT because the window length was nota power of 2. At first glance, DFT appears to have resolutioncomparable to ISA. Part of this improvement is illusory, how-ever, as false events (that can be associated with side lobes of

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Figure 7. Broad-band migrated stacked section for offshore Tertiary clasticsection. Troughs are blue, and peaks are red. The reservoir (arrow) is aclassic bright spot (low-impedance gas sands with a characteristic leadingtrough). No shadowing beneath the reservoir is apparent. Timing linesrepresent 20 ms.

Figure 8. 10-Hz common frequency section corresponding to the broad-band section in Figure 7. Significant low-frequency energy occurs beneaththe reservoir but is absent elsewhere. Timing lines are 20 ms.

Figure 9. 30-Hz common frequency section corresponding to the broad-band section in Figure 7. The low-frequency shadow in Figure 8 hasdisappeared. Events immediately below the reservoir appear somewhatattenuated. Timing lines are 20 ms.

Figure 10. Broad-band seismic section from NW Shelf of Australia. Gassands are pink and brine sands are blue.

arrivals) are evident at low frequencies. Furthermore, it canbe seen that frequency resolution has been lost and the spec-tra have been spread out over a broader frequency band thanis actually present in the data.

Superb frequency resolution can be achieved with MEM(Figure 5); however, the method becomes unstable when shortwindows are used. Thus, in practice, the temporal resolutionthat can be achieved is relatively poor.

Low-frequency shadows. Since the inception of bright spottechnology in the 1960s, low-frequency shadows beneathamplitude anomalies have been used as a substantiating

hydrocarbon indicator. These shadows are often attributed byexplorationists to abnormally high attenuation in gas-filledreservoirs. However, it is often difficult to explain observedshadows under thin reservoirs where there is insufficienttravel path through absorbing gas reservoir to justify theobserved shift of spectral energy from high to low frequen-cies. At the 1996 SEG/EAGE Summer Research Workshop,Dan Ebrom summarized at least 10 mechanisms that canexplain these low-frequency shadows. Other than intrinsicattenuation, we think that one or more of the following mech-anisms for introducing low-frequency shadows may be atwork at any given time: stacking in of locally converted shear-waves and peg-leg multiples; NMO stretch of far-offset infor-mation; improper moveout correction and consequent loss of

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Figure 11. (a) 10-Hz common frequency section corresponding to thebroad-band seismic section in Figure 10. The low-frequency shadowbeneath the lower gas sand is the strongest event at 10 Hz. (b) 20-Hzcommon frequency section corresponding to the broad-band seismic sec-tion in Figure 10. The low-frequency shadow beneath the lower gas sandis now weaker than the overlying gas sands. (c) 30-Hz common frequencysection corresponding to the broad-band seismic section in Figure 10. Thelow-frequency shadow beneath the lower gas sand is now gone and thegas sands are the strongest events on the section.

b)

a)

c)

Figure 12. Broad-band seismic section for offshore Gulf of Mexico brightspot. The top of reservoir is the trough at 2500 ms at trace 190. Low-frequency shadowing is not particularly apparent on the broad-band data.

high frequencies upon stacking; and time-varying deconvo-lution where the gas bright spot is in the design window.

In the examples which follow, the frequency gathers aresorted into common frequency cubes, sections, and horizonslices (Figure 6). Each common frequency display is thus thespectral amplitude for that frequency versus time.

The first example is a bright spot from the Gulf of Mexico(Figure 7). The reservoir has a characteristic leading-trough(blue) on the broadband seismic data and is slightly brighterthan nearby events. Figure 8 shows the corresponding ISAsec-tion at 10 Hz. The reservoir is anomalously bright at this fre-quency, but what is most intriguing is the zone of abnormally

strong low-frequency energy beneath the reservoir. At 30 Hz(Figure 9) the reservoir is clearly defined, though less anom-alous in amplitude, and the energy under the reservoir appar-ent at 10 Hz is gone.

The next example (Figure 10) from the NW Shelf ofAustralia exhibits two distinct gas reservoirs. At 10 Hz (Figure11a) the brightest event on the section is beneath the deepergas pay. We believe this to be a low-frequency shadow. At 20Hz, the gas reservoirs are brighter than the shadow which stillpersists. The shadow has completely disappeared at 30 Hz(Figure 11c).

Returning to the Gulf of Mexico, a weak amplitude anom-aly can be seen at the crest of the structure in Figure 12. Figure13 shows the relative variations in event amplitudes as fre-quency increases. At 8 Hz the strongest event is a low-fre-quency shadow under the reservoir which extends to thereservoir limits (Figure 13a). At 12 Hz, the top reservoir-sandreflection is now strong and the shadow persists (Figure 13b).At 20 Hz the shadow is gone (Figure 13c).

Viewing frequency-dependent effects in map view is alsovery revealing. Figure 14 shows frequency-dependent hori-zon slices at the top of the reservoir (left) and for a 50-ms timewindow immediately beneath the reservoir. The reservoirdimensions are outlined by the dashed contour. At 6 Hz(Figure 14a) the reservoir amplitude is not particularly bright.The deeper window shows a strong shadow under the reser-

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Figure 13. (a) 8-Hz common frequency section corresponding to thebroad-band seismic section in Figure 12. The low-frequency shadow justbeneath the reservoir is the strongest event on the section. The reservoirextends from xline 165 to 200. (b) 12-Hz common frequency sectioncorresponding to the broad-band seismic section in Figure 12. The low-frequency shadow and the reservoir have comparable amplitude, but theextent of the shadow better defines the reservoir dimensions (from xline165 to 200). (c) 20-Hz common frequency section corresponding to thebroad-band seismic section in Figure 12. The low-frequency shadow iscompletely attenuated.

Figure 14. (a) 6-Hz common frequency horizon slice (left) on the top of areservoir (dotted black line) and for a 50-ms window immediately belowthe reservoir base (right). At 6 Hz, two strong events are apparent—oneassociated with the reservoir dimensions (a shadow) and another indeter-minate elongate feature toward the lower left. (b) 14-Hz common fre-quency horizon slice on the top of a reservoir (left) and for a 50-mswindow immediately below the reservoir base. At 14 Hz, the reservoir isnow bright, the low-frequency shadow is gone, and the indeterminateelongate feature toward the lower left remains. (c) 21-Hz common fre-quency horizon slice on the top of a reservoir (left) and for a 50-ms win-dow immediately below the reservoir base. At 21 Hz, the reservoirremains bright, the low-frequency shadow is gone and the indeterminateelongate feature is starting to look like a channel.

b)

a)

c)

b)

a)

c)

voir but also other strong energy to the lower left of the reser-voir. At 14 Hz (Figure 14b) the reservoir is a clear bright spot,the shadow is gone, and the high-frequency energy to the lowerleft persists (indicating that this energy has another origin).At 21 Hz, the energy to the lower left has developed a crispchannel-like character showing that it is a stratigraphicallyolder geologic feature unrelated to the reservoir.

Discussion and conclusions. In this article we have shownthat the ISA method has a much better combination of tem-poral and frequency resolution than conventional spectraldecomposition methods. This enables the use of ISAas a directhydrocarbon indicator. Low-frequency shadows are muchmore apparent on spectrally decomposed data than on broad-band seismic sections.

For every example shown, the shadow was stronger thanthe reservoir reflection at lower frequencies, suggesting thatshadows are not necessarily a simple attenuation phenome-non because low-frequency energy must have been added oramplified by some physical or numerical process. Attenuationalone should simply attenuate higher frequencies, not boostlower frequencies. Furthermore, attenuation should be lesslocalized in time than what we are observing. We believe thatthese shadows are caused by one or more of the mechanismsdescribed by Ebrom.

Conventional amplitude analysis is made at the arbitraryand largely accidental dominant frequency of the seismic dataresulting from a complex interaction of acquisition parame-ters, earth filtering, and data processing. When one sees howamplitudes change with frequency, the inadequacy of “broad-band” amplitude analysis becomes immediately apparent.Explorationists who construct conventional horizon amplitudemaps need to ask themselves if they should continue to gen-erate maps at the accidental dominant frequency of the data

when their prospect may in fact be more anomalous at someother frequency. In our opinion, such broad-band amplitudeanomaly mapping will become obsolete in the not too distantfuture.

It should also be apparent that the old idea of “tuning thick-ness” is also obsolete. Because we can investigate the data atany frequency, there is no single tuning thickness.Explorationists need to think in terms of the “tuning fre-quency” for a given reservoir, not the tuning thickness for agiven seismic data set. This will be discussed in a future arti-cle.

Suggested reading. As a general introduction to spectral decom-position we recommend various papers by Greg Partyka or KurtMarfurt in GEOPHYSICS. For an introduction to wavelet-transformbased spectral decomposition, Chakraborty and Okaya’s 1995GEOPHYSICS paper “Frequency-time decomposition of seismic datausing wavelet-based methods” is a must read. “The low-fre-quency gas shadow on seismic section” by Ebrom is available inthe proceedings for the 1996 SEG/EAGE Summer Workshop onWave Propagation in Rocks. TLE

Acknowledgments: Thanks to the Gas Technology Institute and FusionGeophysical for financial backing in the development and evaluation of ISA.Seismic data were provided by ChevronTexaco. Thanks are also owed to SvenTreitel who doesn’t even know that he gave me the clue I needed to do thisin a conversation relating the Fourier transform to correlation with sine waves.Special thanks are due to my late father John F. Castagna. I didn’t believehim when he told me years ago that geophysicists were doing things all wrongand that we needed to look at things a single frequency at a time. Thanks,Dad, for being so arrogant in suggesting the ridiculous and for teaching meto do the same.

Corresponding author: castagna@ou.edu

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