Zeng & Backus 2005 Part 1

GEOPHYSICS, VOL. 70, NO. 3 (MAY-JUNE 2005); P. C7–C15, 11 FIGS.10.1190/1.1925740

Interpretive advantages of 90◦◦-phase wavelets: Part 1 — Modeling

Hongliu Zeng1 and Milo M. Backus1

ABSTRACT

We discuss, in a two-part article, the benefits of 90◦-phase wavelets in stratigraphic and lithologic interpre-tation of seismically thin beds. In Part 1, seismic modelsof Ricker wavelets with selected phases are constructedto assess interpretability of composite waveforms in in-creasingly complex geologic settings. Although supe-rior for single surface and thick-layer interpretation,zero-phase seismic data are not optimal for interpret-ing beds thinner than a wavelength because their an-tisymmetric thin-bed responses tie to the reflectivityseries rather than to impedance logs. Nonsymmetri-cal wavelets (e.g., minimum-phase wavelets) are gener-ally not recommended for interpretation because theirasymmetric composite waveforms have large side lobes.Integrated zero-phase traces are also less desirable be-cause they lose high-frequency components in the inte-gration process. However, the application of 90◦-phasedata consistently improves seismic interpretability. Theunique symmetry of 90◦-phase thin-bed response elim-inates the dual polarity of thin-bed responses, result-ing in better imagery of thin-bed geometry, impedanceprofiles, lithology, and stratigraphy. Less amplitude dis-tortion and less stratigraphy-independent, thin-bed in-terference lead to more accurate acoustic impedance es-timation from amplitude data and a better tie of seismictraces to lithology-indicative wireline logs. Field dataapplications are presented in part 2 of this article.

INTRODUCTION

Zero-phase wavelets have long been considered the bestinterpretive wavelets for seismic interpretation. Wood (1982)outlines properties of zero-phase wavelets. Wood (1982) andYilmaz (2001) address the principles of wavelet processingto replace a source wavelet with a zero-phase, interpretivewavelet without altering its amplitude spectrum. Brown

Manuscript received by the Editor June 4, 2004; revised manuscript received September 8, 2004; published online May 20, 2005.1The University of Texas at Austin, Bureau of Economic Geology, Austin, Texas 78758-4445. E-mail: [email protected];

[email protected]© 2005 Society of Exploration Geophysicists. All rights reserved.

(1991) summarizes many advantages of the zero-phasewavelet, including wavelet symmetry, minimal ambiguity ofwavelet shape in correlation, center and maximum amplitudeof the wavelet coinciding in time with reflection interface,and best resolution among wavelets with the same amplitudespectrum.

The assumption under which zero-phase wavelets aredeemed superior to other wavelets is that the reflection mustcome from a single interface, or seismic data must haveenough resolution to resolve individual reflections from thetop and bottom of a bed. Sengbush et al. (1961), Widess(1973), and Meckel and Nath (1977) analyze the issue of inter-face resolution of zero-phase wavelets and conclude that theresolution limit for zero-phase wavelets is about one-quarterof the dominant wavelength (λ/4). Below λ/4, the top andbottom of the bed can no longer be picked correctly in time,and reflections from the two surfaces interfere, resulting incomposite waveforms. From λ/4 to λ, the top and bottom arebetter resolved, but composite waveforms still differ criticallyfrom the zero-phase wavelet.

Most petroleum reservoirs are small in vertical dimension(<λ), and many are below seismic resolution (<λ/4). For in-terbedded thin reservoirs, we cannot resolve top and bottominterfaces for accurate bed geometry, and we regularly evenhave difficulties locating interfaces from interference-plaguedcomposite waveforms. As a result, zero-phase wavelets, de-signed for optimal mapping of single interfaces, become obso-lete. We are therefore more interested in detecting the entirethin bed as a composite event, which requires recondition-ing seismic data with a different wavelet. Studies by Sicking(1982), Zeng et al. (1996, 2003), Zeng (2003), and Zeng andHentz (2004) suggest that 90◦-phase wavelets are a betterchoice for seismic thin-bed interpretation. A 90◦-phase seis-mic volume is not only better for mapping reservoir geom-etry in an interbedded sandstone/shale section (Zeng et al.,1996; Zeng and Hentz, 2004), but it also better fits the geolo-gists’ view of the subsurface by mimicking the impedance log(Sicking, 1982).

These two companion papers discuss the interpretive ad-vantages of 90◦-phase wavelets. Part 1 covers the seismic

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modeling of simple and complex geologic models for basicprinciples and benefits of 90◦-phase wavelets. Part 2 (Zengand Backus, 2005) deals with field data applications that il-lustrate the value of 90◦-phase data in lithology and porosityprediction, reservoir geometry mapping, stratigraphic correla-tion, and the study of depositional geomorphology.

PHASE CONTROL OF WAVELET SHAPE

In the frequency domain, a wavelet can be uniquelyexpressed by amplitude and phase spectra. The amplitudespectrum determines the energy distribution of different fre-quency components of the seismic signals; the phase spectrumdisplays phase lags of respective signals in the amplitude spec-trum. Assuming the same amplitude spectrum, a linear-phasechange leads to a time shift of a wavelet; a constant-phase shiftalters wavelet shape. By modifying the slope of the linear-phase shift and constant-phase shift, an infinite number ofwavelets can be created from the same amplitude spectrum(e.g., Yilmaz, 2001).

Figure 1 examines the constant phase shift of a Ricker wave-let obtained by applying the Hilbert transform (Bracewell,1999). Starting with a symmetric, zero-phase Ricker wavelet,a 90◦-phase shift converts a zero-phase wavelet to an anti-symmetric wavelet, whereas a 180◦-phase shift reverses its po-larity. A 270◦-phase shift reverses polarity while making thewavelet antisymmetric again. Eventually, a 360◦-phase shiftmoves the wavelet back to itself. As far as arrival time is con-cerned, the maximum amplitudes of zero-, 180◦-, and 360◦-phase wavelets are aligned at t = 0, whereas zero crossingsof 90◦- and 270◦-phase wavelets are pointed at t = 0. Otherintermediate changes of constant phase lead to nonsymmet-rical wavelets (e.g., 30◦, 60◦, 120◦, 150◦, 210◦, 240◦, 300◦, and330◦ phase, Figure 1) that have no distinctive amplitude char-acteristics at t = 0. In particular, the 90◦-phase wavelet isequivalent to the quadrature trace (Taner and Sheriff, 1977)if the original wavelet is zero phase.

Other important seismic phase characteristics, such as min-imum phase, mixed phase, and maximum phase, are causedby nonlinear phase distributions and are not discussed here indetail. A zero-phase wavelet can be converted to a minimum-phase or maximum-phase wavelet having the same amplitudespectrum by applying spectral factorization (e.g., Claerbout,1976).

The interpretive quality of a wavelet is also related to therelative size of its side lobes. Smaller side lobes mean less am-biguity, fewer interferences, and higher accuracy in interpre-tation. In our synthetic experiments, we exclusively use theRicker wavelet, which is characterized by a main- to side-lobeamplitude ratio of 2.24. For field seismic data, the size of side

Figure 1. Wavelet phase control over wavelet shape and sym-metry. Ricker wavelets of nonzero-phase lags are created byapplying the Hilbert transform to the Ricker wavelet.

lobes is controlled by seismic bandwidth (Yilmaz, 2001). Thewider the bandwidth, the smaller the side lobes and the greaterthe ratio. Ignoring side lobes beyond the first side lobe, a 2.35-octave (5/20/60/75) bandpass wavelet is equivalent to a 40-HzRicker wavelet in terms of the main- to side-lobe amplituderatio, a condition easily satisfied for most modern seismic data.Therefore, observations from Ricker wavelet models shouldbe applicable for seismic interpretation of reasonably widebandwidth field data.

WEDGE MODEL

To assess the merits of 90◦-phase wavelets in seismic thin-bed interpretation, we first performed a simple model studyof an isolated thinning bed. The purpose was to establish ba-sic thin-bed waveform characteristics of selected wavelets andtheir interpretive implications. A two-bed interference modeland a geologically more realistic interbedded thin-bed modelare discussed in the following sections.

Model design

The fundamental model consists of a wedge of materialencased in acoustically dissimilar material (Figure 2). Thetop and bottom of the wedge show opposite reflection po-larities, a realistic and typical situation in most, if not all,stratigraphic profiles. We assume that sandstone wedge ma-terial is lower in acoustic impedance (AI) than the surround-ing material (shale) to mimic the petrophysical profile ofthe shallow Gulf of Mexico and other areas. The reflectioncoefficient R, calculated from the impedance profile, is neg-ative at the top and positive at the bottom with the samemagnitude. The model covers a thickness range from seismi-cally thin (<λ/4) to marginally thick (λ/4 − λ) and seismicallythick (>λ).

Synthetics and amplitude tuning curves

A zero-phase Ricker wavelet synthetic section of the wedgemodel is shown in Figure 3. The polarity convention of thezero-phase data adopted in this paper is that a negative re-flection coefficient corresponding to a decrease in acousticimpedance is displayed as a trough. The composite wave-forms of the wedge show dramatic changes with thickness.We observe symmetrical waveforms whose centers (maxi-mum amplitudes) correspond to the top and bottom of the

Figure 2. Wedge AI model of sandstone encased in thick shale.Assuming constant AI in sandstone and shale, R is negative atthe top and positive at the bottom, with the same magnitude;λ denotes seismic wavelength.


Advantages of 90◦◦-phase Wavelets: Part 1 C9

sandstone when the sandstone is thicker than a wavelength(λ). However, for seismically thin beds (<λ/4) and marginallythick beds (λ/4 − λ), reflection amplitudes are composite seis-mic responses that mix reflections from the top and the bottom(Figure 3). In this situation, the observed waveform is suchthat the bed corresponds to an antisymmetric, trough-then-peak couplet (Widess, 1973). When the bed is thinner than3λ/4, the center of the bed is aligned at the seismic zerocrossing.

A 90◦-phase Ricker wavelet seismic model is displayed inFigure 4. For a 90◦-phase wavelet, the polarity convention ischosen such that a negative-then-positive reflection coefficientset corresponding to a thin bed with acoustic impedance lowerthan in the host rock is displayed as a trough. This is equiva-lent to applying a 90◦-phase shift to a zero-phase wavelet andthen reversing the polarity. In this model, when the sandstoneis thicker than λ, we see antisymmetric waveforms whose zerocrossings align with the top and bottom of the sandstone. Forseismically thin beds and marginally thick beds, reflection am-plitudes are composite seismic responses (Figure 4, <λ). Theobserved waveform becomes symmetric so that the bed cor-responds to a seismic trough event. When the bed is thinnerthan 3λ/4, the center of the bed is aligned at the center of thetrough (maximum negative amplitude).

If we measure composite amplitude for the zero-phasemodel and maximum negative amplitudes for the 90◦-phasemodel, comparable thin-bed amplitude tuning curves can beseen (Figure 5). In both models, we can approximately calcu-late actual bed thickness by measuring peak-to-trough trav-eltime (zero-phase case, Figure 3) or traveltime between zerocrossings of the main lobe (90◦-phase case, Figure 4) if the bedis thicker than λ/4 or by linearly linking amplitude to the ac-tual thickness if the bed is thinner than λ/4. As a result, theapplication of 90◦-phase wavelets does not reduce interfaceresolution or detection power of seismic amplitude.

Interpretive advantages of 90◦-phase seismic data

By comparing the zero-phase seismic model (Figure 3) andthe 90◦-phase model (Figure 4), we observe that 90◦-phasedata are more desirable for geologic interpretation of thinbeds. Of course, these observations are restricted to noise-freedata having simple wavelets of small, Ricker-like side lobes.

First, the 90◦-phase model has easier-to-interpret wave-forms and simpler interpretive rules. In the zero-phase model(Figure 3), the seismic response to a thin bed is a trough-then-peak couplet. The top of the bed corresponds to a mainseismic trough, whereas the bottom of the bed can be corre-lated to a main seismic peak. When the bed thins to belowλ/4, a time deviation between reflection interfaces and maxi-mum seismic energy (trough or peak) occurs. To use seismicamplitude for geometric and lithologic interpretation, we maychoose to deal with dual polarities (trough and peak) at thesame time. However, doing so hampers an interpreter’s abil-ity to identify the trough/peak couplet associated with a giventhin bed, potentially causing confusion. To use trough or peakamplitude alone may create a greater error when thin-bed in-terference of other geologic units is involved (as discussed inthe following section). On the other hand, in the 90◦-phasemodel (Figure 4), the seismic response to a thin bed is a singlemain trough. The center of the bed lines up with the maximum

Figure 3. Zero-phase Ricker seismic model. Despite the sym-metric shape of the Ricker wavelet, composite waveforms aresymmetric only in seismically thick beds (>λ). Antisymmet-ric seismic responses dominate when a bed thins. If the bedis thinner than λ/4, deviation between reflection interfacesand seismic trough–peak measurements (indicated by dashedlines) occurs. Neither polarity nor amplitudes in asymmetricwaveforms match wedge geometry (lithology).

Figure 4. Ninety-degree Ricker seismic model. Regardless ofantisymmetric wavelet, composite waveforms become sym-metric when thickness is less than λ. Although not very accu-rate when thickness is less than λ/4, the seismic trough (nega-tive amplitudes) matches wedge geometry, with zero crossingscorresponding to the boundaries of the wedge.

Figure 5. Comparison of amplitude tuning curves for zero- and90◦-phase seismic models. Two models exhibit similar tuningcurves with the same tuning thickness and similar apparentthicknesses measured from traveltime.


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negative amplitude (<3λ/4) or with the center of the compos-ite trough waveform (3λ/4 − λ). Although not very accurateif the thickness is less than λ/4, the top and bottom of the bedcorrespond to seismic zero crossings. If side lobes are ignored,seismic polarity (negative amplitudes, in this case) becomes aunique indicator of the thin bed. A simple highlight of (neg-ative) amplitudes is adequate for depicting geometry of thethin bed. A 90◦-phase shift converts a surface (reflectivity)-responsive signal (zero phase) into a more layer-responsivesignal (90◦ phase). This conversion greatly reduces ambiguityin associating seismic events to thin beds, simplifying seismicinterpretation rules.

Second, the 90◦-phase model is a better representationof impedance profile and lithology-indicative wireline logs.When the bed is thinner than λ, we hope to tie the seis-mic waveform directly to the impedance profile for the bestpossible correlation between seismic traces and stratigraphicarchitecture. If impedance is linked to lithology, as in our geo-logic model (Figure 2), we also hope to tie seismic waveformsto lithology-indicative wireline logs (e.g., gamma-ray andspontaneous potential logs in a sandstone/shale sequence). Ina zero-phase model, however, the same lithology of the sameimpedance can be tied to opposite polarities. The correlationbetween impedance curves in the model and seismic traces istypically very poor. The 90◦-phase model corrects the problem(Figure 4). Seismic responses are symmetric to the sandstonebed of symmetric impedance curves, making the main seismicevent (a trough, in this case) coincide with the geologicallydefined sandstone bed. As a result, seismic polarity (negativeamplitudes, in this case) is tied uniquely to lithology, and thecorrelation between impedance curves and seismic traces isgreatly increased. This improvement should make a seismicsection look more like a geologic section, encouraging moregeologists to be interested in seismic interpretation.

Minimum phase and integrated zero phase

Minimum phase has been widely discussed in the litera-ture. We apply the minimum-phase equivalent of a Rickerwavelet to generate a seismic wedge model (Figure 6). Com-paring zero-phase and 90◦-phase models (Figures 3 and 4),we find the biggest difference in seismic waveform is theloss of symmetry. With a nonsymmetrical wavelet, the am-

Figure 6. Minimum-phase seismic model. The minimum-phasewavelet is converted from a Ricker wavelet of the same am-plitude spectrum as in Figures 3 and 4 by applying a spectralfactorization. The minimum-phase model is characterized byasymmetric waveforms with large side lobes that cause ambi-guity in interpreting geologic surfaces and lithology.

plitude difference in main lobes and side lobes is reduced.Large side lobes make it more difficult to identify geo-logic surfaces (in this case, top and bottom of sandstone inFigure 6). Also, amplitudes from surfaces with the same mag-nitude of reflectivity are typically not identical. As a result, it isconsiderably more challenging to link amplitudes to thicknessand lithology. Maximum- and mixed-phase data have simi-lar problems. Although minimum-phase and maximum-phasewavelets may have advantages if geologic interference is de-rived mainly from one direction (below and above the layer,respectively) (Zeng et al., 1996), nonsymmetrical wavelets areless desirable interpretive wavelets and generally should beavoided.

Another common approach is to integrate zero-phase seis-mic traces for an estimate of thin-bed AI. A zero-phase seis-mic trace is an estimate of reflectivity series; reflectivity canbe approximated by the derivative of AI. Integrating the esti-mated reflectivity series (Figure 3) results in an estimated AIprofile (Figure 7). However, the integration process reducesdominant and high-end frequencies of seismic traces, therebyreducing seismic resolution. Integrated traces are character-ized by longer main lobes (at 0 − λ/2) and side lobes (at anythickness) compared with the 90◦-phase data (Figure 4) thatpreserve the same frequency components of zero-phase seis-mic data. As a result, integrated zero-phase data are less de-sirable for seismic interpretation.

TWO-BED INTERFERENCE MODEL

We further examine thin-bed interference patterns by mod-eling two low-AI thin sandstone beds inserted in a high-AIshale host (Figure 8). To simplify the discussion, thicknesses ofthe thin beds are assumed equal and fixed (λ/4, Figure 8a,b).Zero-phase (Figure 8a) and 90◦-phase (Figure 8b) seismicmodels demonstrate more complex reflection patterns thando their single-wedge counterparts (Figures 3 and 4, respec-tively).

Interference patterns

For interpretive purposes, the simplest way to character-ize interference patterns is to study seismic amplitude ver-sus distance between thin beds. For the zero-phase model(Figure 8a), peak (0◦P), trough (0◦T), and trough-to-peakcomposite amplitude (0◦TP) can be measured for upper and

Figure 7. Integrated version of zero-phase seismic model(Figure 3). Although the model visually resembles the 90◦-phase seismic model (Figure 4), it is characterized by a fattermain trough and larger side lobes (peak).



lower sandstones, respectively (Figure 8c,d). For the 90◦-phase model (Figure 8b), trough amplitude (90◦T) is a suit-able measurement (Figure 8c,d). For comparison, all ampli-tudes in the zero-phase and 90◦-phase models are normalizedto respective single-bed responses. Four interference patternsare observed as a function of the shale wedge thickness:

1) Interference free. If the shale is thicker than 3λ/4, thereis little amplitude change compared with single-bed re-sponses. If the shale bed is thicker than λ, amplitudes haveno changes and single-bed tuning curves (Figure 5) apply.

2) Constructive interference. If the shale ranges roughly fromλ/4 to 3λ/4, amplitudes are above the normal level.

3) Destructive interference. If the shale ranges from λ/8 toλ/4, amplitudes are below the normal level.

4) Nontraceable. If the shale is thinner than λ/8, compos-ite waveforms for the two sandstones merge. No sepa-rate trough-to-peak couplets (zero phase) or troughs (90◦

phase) can be recognized for individual sandstones in thevertical section, although horizontal imaging by slicing a3D volume is still possible.

To further demonstrate the influence of interference pat-terns on stratigraphic and impedance interpretation, somevariably spaced sandstone patches are added to a singlesandstone model to simulate representative constructive, de-structive, and nontraceable interference patterns identified inFigure 8 (Figure 9).

Bed tracing and stratigraphic interpretation

In the zero-phase model (Figure 8a), both upper and lowersandstone beds can be traced as trough-then-peak couplets ifthe beds are adequately separated (shale wedge >λ/4) despiteconstructive interference. If sandstone beds are close (shalewedge ranging from λ/8 to λ/4), however, uneven destructiveinterferences of troughs and peaks severely damage the wave-form symmetry, and tracing of the reflection couplets may be-come confusing (Figures 8a and 9a). If sandstone beds are tooclose to each other (shale <λ/8), two couplets are reduced toone, and bed tracing of individual sandstones in a vertical sec-tion becomes impossible. If we attempt to trace continuoussandstone in the model in destructive and nontraceable inter-ference ranges by slicing against a seismic reference (here, thegenerally continuous peak or trough), polarity reversals mayoccur (Figure 9a). These false polarity reversals are particu-larly harmful for stratigraphic analysis by creating pseudoseis-mic events that would be wrongly interpreted as additionaldepositional units or stratigraphic relationships. As a result,we potentially see more depositional bodies and erosional sur-faces than actually exist or are resolved.

In the 90◦-phase model (Figures 8b and 9b), waveform sym-metry as observed in the single wedge model (Figure 4) isnot lost. As a result, delineation of the sandstones is muchsimpler. A trace of troughs recovers the accurate geometryof sandstone beds. Polarity reversals are rare and occur onlyin the nontraceable interference range, ensuring qualitativelycorrect horizontal amplitude patterns for closely spaced sand-stone beds. This accuracy offers a clear advantage in the studyof seismic stratigraphy and geomorphology, as shown in ex-amples in part 2 of this article (Zeng and Backus, 2005).

Amplitude distortion and impedance estimation

Seismic interference causes amplitude distortions in bothmodels, although with different magnitudes and potential in-terpretation pitfalls. In the zero-phase model (Figures 8a and

Figure 8. Interference patterns of two approaching thin bedsencasing shales. AI is assigned the same as it is in Figure 2.Beds are equally thick at λ/4. (a) Zero-phase synthetic sec-tion. (b) 90◦-phase synthetic section. (c) Amplitude versusthickness of the shale wedge for the upper sandstone. (d) Am-plitude versus thickness of the shale wedge for the lower sand-stone: 0◦T = trough (negative) amplitude in the zero-phasedata, 0◦P = peak (positive) amplitude in the zero-phase data,0◦TP = trough-to-peak composite amplitude in the zero-phasedata, and 90◦T = trough amplitude in the 90◦-phase data.Four interference patterns are identified as interference free(>3λ/4), constructive interference (λ/4 − 3λ/4), destructiveinterference (λ/8 − λ/4), and nontraceable (<λ/8). If am-plitudes are traced from troughs and peaks separately, con-structive and destructive interferences for zero-phase data aredifferent for the two beds and are therefore location (strati-graphic position) dependent.


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9a), measured amplitude is a function of stratigraphic posi-tion. For the upper sandstone in Figure 8c, the seismic trough,which corresponds roughly to the top of the sandstone, un-dergoes minimal amplitude distortion. With shale wedge thin-ning, the trough amplitude (Figure 8c, 0◦T) decreases slightly(−1.34 dB, λ − λ/8). However, the seismic peak, which tiesroughly to the base of the upper sandstone, suffers muchmore severe distortion (Figure 8c, 0◦P). Amplitude first in-creases significantly (1.15 dB in λ − λ/4), then decreases rad-ically (−10.92 dB in λ/4 − λ/8). For the lower sandstone(Figure 8d), the interference pattern is just the opposite: thetrough amplitude (Figure 8d, 0◦T) is distorted much morethan the peak amplitude (Figure 8d, 0◦P). Figure 9c illustratessimilar relationships. This dependency of amplitude on strati-graphic position is obviously not desirable for interpretation,especially if we want to link amplitude to impedance contrast,lithology, and stratigraphy. To reduce amplitude distortionand eliminate amplitude dependency on stratigraphy, we canreplace the measurement of trough or peak amplitude with

Figure 9. Typical amplitude distortion of a thin bed caused byone other thin bed nearby at different interference patternsas defined in Figure 8. (a) Zero-phase synthetic section. (b)90◦-phase synthetic section. (c) Amplitude distortion at non-traceable, destructive, and constructive interference patterns.Amplitudes are measured at surfaces made by linearly inter-polating between maximum troughs or peaks where only onesandstone exists. Potential for polarity reversal in a horizontalamplitude display is seen easily in the zero-phase data.

composite amplitude (Figure 8c,d, 0◦TP, from 0.52 to −4.14dB). However, this approach would be difficult to apply infield data, in which trough–peak couplets may be tricky topick in closely spaced thin beds without dense well control.The 90◦-phase data are less plagued by amplitude distortionproblems. The interference pattern is the same for both upperand lower beds (Figure 8b), so there is no stratigraphic depen-dency of amplitude. Amplitude distortion is mild (0.22 dB inλ − λ/2 and −3.47 dB in λ/2 − λ/8) and remains the same forboth sandstone beds (Figures 8c, 90◦T, and 8d, 90◦T). As a re-sult, 90◦-phase data are generally better correlated to acousticimpedance.

Note that in this model, thin beds are fixed in thickness.In fact, the seismic interference pattern should change withbed thickness. However, general relationships observed in thismodel should hold if two beds are roughly equally thick andtotal thickness does not exceed 3λ/4. Conclusions should alsoapply to cases of more than two thin beds, with minor modifi-cations from additional modeling.

REALISTIC MODEL OF INTERBEDDEDSANDSTONES AND SHALES

Now we expand seismic modeling to a geologically re-alistic case. Mapped from well data, multiple interbeddedsandstones and shales are modeled with geologically reason-able thickness and impedance distributions. Isolated beds andclosely spaced beds are included, with variable magnitudeof reflectivity at each depositional surface. Impedance is as-sumed to be an indicator of lithology (low AI as sandstoneand high AI as shale), although more ambiguous situationsmay occur in field data applications.

Geologic and impedance models

Zeng et al. (1996) have built a fine-scale Miocene geologicmodel based on depositional facies analysis and facies-guidedproperty mapping in the Powderhorn field, Calhoun County,Texas. More than 100 wells were used to constrain thicknessand clay-content (Cclay) mapping of 14 subsurface sandy de-positional units (labeled 1–14, Figure 10a). A stack of thesesandy depositional units and interbedded shale units composea 3D model. The model is geologically realistic in the sensethat all rock properties mapped (thickness, Cclay, porosity, AI,etc.) honor the facies interpretation of fluvial and microtidalshore-zone systems in the formation. In this 200-ms (300 m,assuming an average velocity of 3000 m/s) section, most of thebeds are less than 15 ms (<22 m) thick, well within the range ofseismic interference with a typical seismic wavelet in the area(3λ/4 = 28 m at 40 Hz dominant frequency). Vertical varia-tion of rock properties within beds (or in transitional zones)was not modeled.

Seismic models and interpretation

Assuming low-impedance sandstones in high-impedanceshales and a linear relationship between sandstone AI andshale AI with Cclay (Han et al., 1986), AI distribution is ascaled version of the Cclay model (Figure 10a). For simplic-ity, the low-frequency, vertical trend of AI is not modeled. Byconvolving 40-Hz, zero-phase, and 90◦-phase Ricker wavelets



with the normal-incidence reflectivity model calculated fromthe acoustic impedance model, we obtain respective seismicmodels (Figure 10b,c).

For seismically thin (<λ/4), sandy stratigraphic units en-cased in thick shale (e.g., 1–3, 9, and 12 in Figure 10b,c), im-provement can be seen in impedance and lithology profilingbut not in geometry (thickness) interpretation with 90◦-phasedata. Amplitudes in 90◦-phase data (Figure 10c) clearly tieto geologically defined impedance layers or lithology (trough

Figure 10. Clay content (Cclay), AI, and seismic models in ageologically realistic setting; (a) Cclay and AI section; Cclayand AI are linearly related, with low AI indicating low Cclay(sandstone) and high AI denoting high Cclay (shale). (b) 40-Hz, zero-phase Ricker synthetics. (c) 40-Hz, 90◦-phase Rickersynthetics. A, B, and C refer to closely spaced thin beds (in-dividual beds thinner than λ/4) with destructive interferencepattern (Figures 8 and 9). D denotes a marginally thick bed(λ/4 − λ). E and F are seismically thick beds (total thick-ness >λ). Numbers 1–4, 8, 10–11, and 13 are fluvial facies;5, 9, and 12 are wave-dominated delta facies; and 6, 7, and14 are barrier/bar-lagoon facies. Dashed lines are geologicallydefined unit boundaries and impedance surfaces.

for lower AI/sandstone and peak for higher AI/shale), whichis not the case when zero-phase data (Figure 10b) are used.However, trough-to-peak couplets of these thin beds in zero-phase data can be traced reliably as long as the stratigraphicarchitecture is known. Picking zero crossings in the 90◦-phasedata produces similar thickness estimation.

For seismically thin (<λ/4), sandy stratigraphic units in-terbedded with thin shales (e.g., 6–7 in A, 5–7 in B, and 9–10in C, Figure 10), 90◦-phase data are a better choice. Similarto observations in two-bed interference models (Figures 8 and9), closely spaced thin beds are difficult to trace in zero-phasedata because of the loss of amplitude symmetry of trough–peak couplets. This difficulty can be avoided, however, by fol-lowing seismic troughs in 90◦-phase data. Amplitude distor-tion is also greater and is stratigraphy dependent in the zero-phase data, typically leading to a poorer estimate of acousticimpedance.

For marginally thick sandstones (λ/4 − λ), 90◦-phase datastill show a better fit to the lithologic profile, thanks to thesymmetry of the seismic response (e.g., D in 14; compare Fig-ure 10b and 10c).

For seismically thick (collective thickness >λ) units (e.g., Ein 5–7, F in 13–14, Figure 10b,c), both data sets show separatereflections corresponding to the top and bottom of the beds.Although traveltime can be measured for unit geometry, am-plitude and polarity are not a direct indicator of bed geometry.However, zero-phase data are better because the waveform issymmetric to the bed boundaries.

Crosscorrelation between AI profileand seismic amplitude

Quantitative comparison between zero-phase and 90◦-phase data can be achieved by crosscorrelating between theAI model and respective seismic models (Figure 11). For 40-Hz synthetics, 90◦-phase amplitude traces show the best cor-relation to the AI model without time shift (correlation co-efficient ρ = 0.73, Figure 11b). Zero-phase amplitude tracesare poorly tied to the AI model at zero time shift (ρ = 0.01,

Figure 11. Crosscorrelation functions between the AI model(Figure 10a) and seismic models of selected frequencies andphases. Compared with zero-phase cases, 90◦-phase data withdominant frequencies of 20, 40, and 80 Hz are all better corre-lated to the AI model, even if the zero-phase data are properlyshifted in time. The AI logs (A, B, C) and thickness × AI logs(D, E, F) are treated separately to show the influence of thick-ness tuning.


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Figure 11b), but the correlation improves dramatically (ρ =0.66) when the zero-phase model is shifted λ/4 downward sothat the seismic troughs are best tied to sandstones in AI logs.However, overall correlation is still poorer than that in 90◦-phase data results. Crosscorrelation functions from 20- and80-Hz models exhibit similar trends. The quality of correla-tion, however, deteriorates for both zero- and 90◦-phase data.The 40-Hz model has the best correlation because most thinbeds in the model are in the range of constructive interfer-ence for the 40-Hz data but are in the range of destructiveinterference for the 20-Hz data and are the seismically thickbeds for the 80-Hz data (refer to Figure 8c,d). Increased am-plitude distortion in the 20-Hz data and the greater number ofresolved surfaces in the 80-Hz data reduce AI correlation tothe amplitude. A more precise treatment considers thicknesstuning. Reducing data frequency places more beds into aseismically thin (<λ/4) category, with enhanced (linear)amplitude-thickness correlation (Figure 5). As a result, replac-ing AI logs with thickness-weighted AI (thickness × AI) logsimproves crosscorrelation in 20- and 40-Hz data (Figure 11d,e)but reduces correlation in the 80-Hz case (Figure 11f).

DISCUSSION

Seismic modeling suggests that different wavelet phases canbe designed for different geologic objectives. Neither the zerophase nor the 90◦ phase is universally better, and neither isbetter in some important cases. Whenever we are dealing witha well-resolved interface reflection, all published advantagesof the zero-phase wavelets hold true. Also, if we are interpret-ing a major unconformity or a crisp fault interface reflection,zero-phase wavelets have advantages, although a good casecan be made for use of a minimum-phase or maximum-phasewavelet under certain conditions (e.g., Zeng et al., 1996). Inall cases dealing with seismically thin beds, 90◦-phase waveletshave advantages.

In this work we have examined low-AI sandstones (rela-tive to shale) only. Low-AI sandstones are typical among shal-low formations with minor compaction, especially marine sed-iments in the Gulf of Mexico and elsewhere. However, othertypes of sandstone–shale acoustic relationships also have sig-nificant impact on the seismic behavior of any given wavelet.In hard-rock onshore sediments or deeply buried marine sed-iments, high-AI sandstones (relative to shale) are more repre-sentative. Compared with the low-AI sandstone relationship,there is a reversal in AI contrast and resultant seismic polar-ity. Therefore, observations made from our low-AI sandstonemodels should hold true if reversed polarities are applied toseismic models.

In seismic interpretation of thin reservoirs, another relevantissue is how to characterize a bed bounded above and belowby acoustically dissimilar rocks or an acoustically transitionalbed. All discussions so far have assumed equal-magnitude re-flectivity at the top and bottom of a thin bed, implicatingblocky AI layering without vertical AI variation in the hostrocks and within the unit. Many times, vertical changes of AIin the host rocks or within the thin bed occur because of depo-sitional architecture and resultant grain-size trend variations.We have observed significant waveform and phase changescaused by AI variations in host rocks and within thin beds inmodel and field seismic data. Further investigation is needed

to evaluate how the seismic-phase property influences charac-terization of such thin beds.

CONCLUSIONS

Despite its superiority in single-interface and thick-bedinterpretation, the zero-phase wavelet is less suitable thanthe 90◦-phase wavelets for interpretation of seismically thin(<λ/4) and marginally thick (λ/4 − λ) beds. Disadvantagesinclude a poor match of seismic traces to AI logs and detri-mental seismic interference in a thinly interbedded environ-ment. Minimum-phase and other nonsymmetrical waveletsare also not recommended because of ambiguity caused bylarge, asymmetric side lobes.

Seismic modeling of a thinning bed reveals that a 90◦-phaseseismic trace tends to be a good estimate of relative AI log.The application of 90◦-phase wavelets leads to a better tie be-tween seismic events and impedance/lithology layering whilemaintaining an amplitude-tuning relationship similar to thatof zero-phase data. Integration of seismic traces can convertzero-phase data to fit the impedance profile, but the processenhances low-frequency energy and reduces seismic resolu-tion.

Seismic interference patterns in a model of converging thinbeds further illustrate that a 90◦-phase shift reduces polarityreversals and amplitude distortion in thin-bed imagery com-pared with zero-phase data. The 90◦-phase data also eliminatedependency of amplitude on stratigraphy. As a result, accu-racy and resolution of impedance, lithology, and stratigraphiccorrelation can be improved.

Finally, 90◦-phase seismic models of an interbedded sand-stone/shale model demonstrate consistent improvement instratigraphic correlation and lithologic mapping over zero-phase data in a geologically realistic setting. Crosscorrelationbetween the AI model and the seismic amplitude section ishigher for 90◦-phase data than for zero-phase data for a widerange of data frequencies.

ACKNOWLEDGMENTS

The authors thank S. Fomel and X. Janson at the Bureau ofEconomic Geology, the Geophysics associate editor, reviewerW. T. Wood, and two anonymous reviewers for their helpfulcomments and technical input. The manuscript was edited byL. Dieterich before submission. Graphics were drafted withthe assistance of J. Ames. Partial support for this publica-tion was received from the John A. and Katherine G. JacksonSchool of Geosciences and the Geology Foundation of TheUniversity of Texas at Austin. Published by permission of theDirector, Bureau of Economic Geology.

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