Wavelets, Well Ties, And the Search

GEOPHYSICS, VOL.63, NO. 1 (JANUARY-FEBRUARY 1998); P. 297-313, 19 FIGS.

Wavelets, well ties, and the searchfor subtle stratigraphic traps

Anton Ziolkowski*, John R. Underhill*,and Rodney G.K. Johnston*

ABSTRACTWe examine the conventional methodology for ty-

ing wells to processed seismic data and show why thismethodology fails to allow for reliable interpretationof the seismic data for stratigraphy. We demonstratean alternative methodology that makes the tie with-out the use of synthetic seismograms, but at the priceof measuring the seismic source signature, the cost ofsuch measurements being about 1 % of data acquisi-tion costs. The essence of the well tie is (1) to iden-tify geological and seismic interfaces from the logs andcore, (2) to measure the one-way traveltime to theseinterfaces using downhole geophones, and (3) to usethe polarity information from (1) and the timing infor-mation from (2) to identify the horizons on the zero-phase processed seismic data. Conventional process-ing of seismic data usually causes the wavelet to varyfrom trace to trace, and conventional wavelet extrac-tion at a well using the normal-incidence reflection coef-ficients relies on a convolutional relationship betweenthese coefficients and the processed data that has nobasis in the physics of the problem. Each new well in-troduces a new wavelet and poses a new problem

INTRODUCTION

Interpretation of seismic reflection data for stratigraphy re-quires precise knowledge of the wavelet (Anstey, 1978). Con-ventionally, "the wavelet" that "ties" seismic data to the stratig-raphy known from a well is determined from both the seismicdata and the well logs (e.g. White, 1980). This approach hasproblems that have not been addressed properly.

Conventionally, the well-tie problem is this. The seismic dataare processed to a "known" phase, typically zero phase. Thismeans the phase of the seismic wavelet that was assumed to be

how should the zero-phasing filter be derived betweenwells?

Our methodology consists of three steps: (1) deter-mination of the wavelet consisting of all known convo-lutional effects before any processing using measure-ments of the source time function made during dataacquisition, (2) compression of this wavelet to the short-est zero-phase wavelet within the bandwidth available,and (3) elimination of uncontrolled distortions to thewavelet in subsequent processing. This method is illus-trated with data from the prospective Jurassic successionin the Moray Firth rift arm of the North Sea in which wehave identified, for the first time on seismic data, a majorregional unconconformity that cuts out more than 20 Maof geological time. This method offers two major benefitsover the conventional approach. First, all lateral varia-tions in the processed seismic data are caused by the ge-ology. Second, events on the processed seismic data maybe identified from well logs simply by their polarity andtiming. It follows that events can then be followed on theseismic data from one well to another with confidence,the seismic data can be interpreted for stratigraphy, andsubtle stratigraphic traps may be identified.

in the data has been removed. At the well position, the seismicdata should then "tie," that is, strongly correlate, with the syn-thetic seismogram calculated by convolving the primary reflec-tivity series with an appropriate zero-phase wavelet. Normallythere is a mis-tie, and the synthetic seismogram is then adjust-ed to make the tie. This adjustment has three components: atime shift, or linear phase component in the frequency domain,an adjustment to the amplitude spectrum of the wavelet, and anadjustment to the phase spectrum of the wavelet (e.g. Nymanet al., 1987; Richard and Brac, 1988). This approach essen-tially admits that the so-called zero-phase seismic data are not

Publish on "Geophysics on Line" April 22, 1997. Manuscript received by the Editor November 2, 1995; revised manuscript received January 15,1997.*Department of Geology and Geophysics, University of Edinburgh, Grant Institute, Kings Building, West Mains Road, Edinburgh EH9 3JW, UnitedKingdom. E-mail: [email protected]; [email protected]; [email protected]. 1998 Society of Exploration Geophysicists. All rights reserved.

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zero-phase, as pointed out in Nyman et al. (1987). An almostperfect tie can be made at any well with this approach. How-ever, If such a "tie" is made at one well, and a marker horizonis followed from this well to the next well, there is normally amis-tie of, typically, 30 ms at the next well. A new wavelet mustbe found to "tie" the seismic data at the new well. It followsthat "the wavelet" is unknown unless there is a well and theinterpretation of the seismic data for stratigraphy between thewells or beyond well control can be unreliable. Poggiagliolmiand Allred (1994) have developed a method that forces theseismic data to tie with the same zero-phase wavelet at everygiven well, but requires not only time shifts of the syntheticseismograms, but also a space-adaptive operator that changesthe amplitude spectrum and phase spectrum of all traces in theseismic data.

There are two problems with these approaches. First, thetime-shifts of the synthetic seismograms ignore the constraintsimposed by the check-shot data at each well. Second, sincethe wavelet that makes the tie is, in principle, known only atthe well, any interpretation of the seismic data for stratigra-phy between the wells or beyond well control is, in principle,unreliable.

Consider the history of hydrocarbon exploration in theNorth Sea. In more than three decades of exploration, over70% of all the oil and gas has been found to be reservoiredin Jurassic sediments in structural traps (Spencer et al., 1996),most notably those created during the Late Jurassic rift episodeincluding the Brent, Ninian and Statfjord fields. Over 90% ofthese reserves were discovered in the 1970s in the most obviousseismically defined traps (Spencer et al., 1996). Even thoughcomparatively few readily identifiable major structures havebeen found subsequently, attention has turned only recentlyto the deliberate search for subtle stratigraphic traps. Of thosefound so far, none appears to have been the result of a deliber-ate search. The Miller Field is a case in point: it is essentially aLate Jurassic stratigraphic trap in an intrabasinal location andwas discovered only because of the drilling of a minor struc-tural expression at the base Cretaceous level (Rooksby, 1991;Garland, 1993).

For the deliberate search for stratigraphic traps to be suc-cessful, it is essential that the data be acquired and processedwith this as a major goal, rather than merely as a secondarypossibility following the determination of structure. Determi-nation of structure from seismic data is essentially a search fora velocity model that allows the seismic data to be migrated todetermine a sharp image. The velocities are found by lining upseismograms, using the rate of change of traveltime with offset.This is a first-order effect. In contrast, determination of stratig-raphy from seismic data is essentially the correlation of seismicreflection amplitudes with changes in acoustic impedance inthe subsurface. This is a second-order effect and requires moreprecision.

We argue that this correlation requires the absolute arrivaltimes of the reflections at a well to be known. In the presenta-tion of our argument, we found it necessary to examine what ismeant by "a good well tie," "the seismic wavelet," "the convo-lutional model of the seismogram," the effects of viscoelasticabsorption on wave propagation, and the role of conventionalpredictive deconvolution in seismic data processing. It is shownthat conventional processing forces the wavelet to vary uncon-trollably from trace to trace.

In our approach, the wavelet is determined from measure-ments made during acquisition, and then is compressed to theshortest possible wavelet within the bandwidth available. Sub-sequent processing of the data is constrained to introduce nouncontrolled distortions to the wavelet. In particular, predic-tive deconvolution with a predictive gap less than the length ofthe wavelet is eliminated.

The argument is illustrated with data from a seismic surveytying exploration wells that targeted the prospective Jurassicreservoir interval in the Inner Moray Firth, at the western endof one of the three main rift arms in the North Sea Basin.

THE CONVOLUTIONAL MODEL OF THE SEISMOGRAM

Consider the modeling of seismic reflection data in a com-puter, using, for example, a finite-difference scheme in threedimensions. Every element in the earth is parameterized byits density and elastic constants. Absorption can be includedby allowing the stress to depend linearly on the strain rate aswell as on the strain itself. Each element in the model earthwould than be linearly viscoelastic. The source and receiversmust also be characterized.

The receivers have the known, measured, characteristics ofhydrophone or geophone arrays, together with the known re-sponses of the associated recording systems. Over the band-width of interest, and at the signal levels we measure, thesedevices are linear, and the effects of their responses are strictlyconvolutional. It is possible that there are severely attenuatingzones beneath the geophones. These can be characterized us-ing linear viscoelasticity, as described above. Such zones are, inany case, part of the earth, and not part of the receivers.

The input to the model is the source time function of thesource in the right units. For vibroseis vibrators, this is theforce applied by the vibrator plate to the surface of the earth,and is not necessarily the same for each vibrator (Baeten andZiolkowski, 1990). For the dynamite source it is the radial dis-placement of the cavity generated by the dynamite explosion.For an air gun, it is the displacement of the air-gun bubble.

The signal generated by the computer program at a receiveris the convolution of the source time function with the earthimpulse response. This response is the pressure or particle ve-locity that would be generated at the receiver if the sourcetime function were an impulse. The impulse response at eachreceiver is different, because the travel paths vary from receiverto receiver. At a given receiver the impulse response is the sumof many different arrivals, each of which has traveled a differ-ent path. The viscoelastic absorption of energy along each ofthese paths is different. As a result, the effects of absorptionare different for each arrival in the seismogram and are there-fore not convolutional. Therefore, they cannot, in principle, beremoved by deconvolution.

In the recorded seismogram there are three convolutionalcomponents: the source time function, the receiver response,and the response of the recording system. These compo-nents may be removed by deconvolution, provided they areknown. In real seismic data, the characteristics of the receiversand recording systems are known with great precision. Thesource time function can be measured for every major source(Ziolkowski, 1991) and the cost, for marine seismic data,is on the order of 1% of the data acquisition cost (Hones,1996).

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Wavelets, Well-Ties and Stratigraphy 299

THE WAVELET

What is "the wavelet"? It is impossible to give a simple an-swer to this question. One reason we find a different waveletat every well is because conventional seismic data processingcan introduce uncontrolled distortions to the wavelet. Anotheris that conventional methodology (White, 1980) requires thesimple normal-incidence synthetic seismogram to "match" theseismic data after stack and migration after stack. However,there is no theory to support this requirement. Therefore, thematch tells us nothing about the meaning of this "wavelet," andtherefore nothing about the stratigraphy. These claims are dis-cussed further below. At this point we introduce our approach.

Our approach is (1) to determine the wavelet before anyprocessing, using measurements of the source wavefield madeduring data acquisition, (2) to compress this wavelet to theshortest possible wavelet within the bandwidth available, and(3) to ensure that the subsequent processing introduces no un-known distortions to the wavelet. With this approach, it is pos-sible to say what the wavelet is. It involves measurement of thesource signature. This step is indispensable and we consider itto be a small price to pay to get precise well ties, and to be ableto interpret stratigraphy reliably between wells.

The convolutional wavelet is the combination of all convolu-tional effects in the shot record. For a point source and point re-ceiver it is the source time function convolved with the receiverresponse and the recording instrument response. In principle,the surface of the earth is a boundary to the earth and shouldnot be considered part of the source or receiver. Until we havean exact solution to the removal of free-surface effects, how-ever, source ghosts should be included where applicable (airguns and dynamite), and also receiver ghosts, where applica-ble (marine surveys). Marine source arrays and vibrator ar-rays are directional, as are the receiver groups. In theory, thesedirectional effects can be removed in common-receiver gath-ers for source effects (assuming shot-to-shot repeatability) andin common-shot gathers for receiver effects (e.g., Baeten andZiolkowski, 1990, Chap. 7). In practice, it may be sufficient formost purposes simply to take an average angle of incidence, say20, to allow for the nonvertical incidence of the raypath fromthe source to the target and back to the receiver. This raypathis illustrated in Figure 1 for a marine example. The resulting

wavelet, shown in Figure 2 and calculated using source mea-surements recorded through the same instruments as the seis-mic reflection data, defines the bandwidth of the data: the datacannot have a bandwith any greater than the bandwidth of thiswavelet.

The method used to calculate the wavelet from measure-ments is discussed in detail in Ziolkowski and Johnston (1997).The source wavefield of the air-gun array was measured foreach shot using near-field hydrophones. Using the method ofZiolkowski et al. (1982) and Parkes et al. (1984), the measure-ments were used to decompose the wavefield into the individ-ual spherical waves generated by the oscillating bubbles emit-ted by the air guns. The spherical waves were then superposedin the frequency domain, taking into account the geometry ofthe array and the free surface, to compute a far-field signa-ture at 20 to the vertical. A receiver ghost was added in thefrequency domain, taking into account the 20 angle of inci-dence and the measured depth of the streamer. A reflectioncoefficient of -1 was assumed at the water surface.

The wavelet thus obtained, as shown in Figure 2, containsbubble pulses and is very long. The bubble pulses do not appearto be very significant when the full frequency bandwidth isincluded. At the target the highest frequency in the data maybe 40 Hz. Figure 3 shows the wavelet of Figure 2 filtered witha minimum-phase filter with a high-cut frequency of 40 Hz,and the primary-to-bubble ratio is reduced by a factor of 2.

20 degree wavelet: P/B ratio = 16.4Amplitude x ]0 3

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FIG. 2. The 20 wavelet.

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source depth = 6m `(,

I----receiver depth = 7m

20 degree wavelet with 40Hz high-cut: P/B ratio = 8.2Amplitude x 103

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TARGET ZONE 1.00

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FIG. 3. The wavelet of Figure 2 after low-pass filtering withFIG. 1. Geometry for calculation of the wavelet.

40 Hz high cut.

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The wavelet clearly needs to be shortened to obtain maximumresolution within the bandwidth available.

Figure 4 shows the construction of a signature deconvolu-tion filter that makes the known long unfiltered convolutionalwavelet of Figure 2 into a short minimum-phase wavelet withessentially the same bandwidth. The application of the sig-nature deconvolution filter to the data makes the wavelet inthe data short, thus increasing the resolution. We now havea known short wavelet, Figure 4b, in the data. If the waveletchanges, for instance because of changes in the source signaturedetected through continuous monitoring during data acquisi-tion, we must keep the desired short wavelet, Figure 4b, fixedand design a new filter, Figure 4c, to shape this new long waveletto the same short desired wavelet. This will ensure that, aftersignature deconvolution, the same known wavelet, Figure 4b,exists throughout the data.

a) Input farfield waveletAmplitude x 10 3

-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.500.600.700.800.901.00

Figure 5 shows the process of Figure 4 in the frequency do-main. The air-gun bubble pulses seen in Figure 4a exhibit thecharacteristic peaks and notches in the amplitude spectrum asshown in Figure 5a. The tuned air-gun array reduces the bub-ble pulses somewhat, but not completely. Figure 5b shows theamplitude spectrum of the desired wavelet: it is designed tobe essentially the same as Figure 5a, but is smooth. The am-plitude spectrum of the filter, Figure 5c, that shapes the calcu-lated wavelet into the desired wavelet is essentially flat over thebandwidth of the wavelet, but with superimposed notches andpeaks to compensate for the corresponding peaks and notchesin the spectrum, Figure 5a, of the wavelet. The result of multi-plying Figure 5a by Figure Sc is Figure 5d, which is very similarto the desired spectrum, Figure 5b. Because the filter spectrum,Figure Sc is essentially flat, the signal-to-random noise level es-sentially is unchanged.

b) Desired minimum-phase waveletAmplitude x 103

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II

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Time (s) ' I I ' Time (s)0.00 0.10 0.20 0.30 0.40 0.50 0.60

C) Optimum lag shaping filter (lag=84ms; q=0.99; coeffs w/o dt) d) Actual minimum-phase wavelet realisedAmplitude Amplitude x 103

180.00160.00140.00120.00100.0080.0060.0040.0020.000.00

-20.00-40.00-60.00-80.00

' ' ' ' ' ' ' ' Time (s)0.00 0.10 0.20 0.30 0.40 0.50 0.60

-0.50-0.40-0.30-0.20-0.100.000.100.200.300.400.500.600,700.800.901.00

I Time (s)0.00 0.10 0.20 0.30 0.40 0.50 0.60

FIG. 4. Signature deconvolution: (a) the wavelet of Figure 2; (b) the delayed desired minimum-phase wavelet; (c) the signaturedeconvolution filter; (d) the convolution of the wavelet (a) with the filter (c).

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Optimum lag shaping filter Actual smooth spectrum realised

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d)Amplitude10.00 rr


Thus the application of the signature deconvolution filter,designed in this way, increases the resolution, by shorteningthe wavelet, but without decreasing the signal-to-noise ratio.

performing the signature deconvolution step before predictiveTHE ROLE OF PREDICTIVE DECONVOLUTION deconvolution is critical. If the wavelet is not known, as is usu-

all the case this critical sten cannot be done and the waveletPredictive deconvolution is required in conventional seis-

mic data processing to attenuate multiples generated by thefree surface (Peacock and Treitel, 1969; Taner et al., 1995).The predictive gap must be shorter than the multiple period.In the case of marine seismic data processing, the predictivegap must be shorter than the two-way traveltime in the water.If the wavelet is shorter than this gap it is unaffected by thepredictive deconvolution operator. In other words, if the orig-inal long wavelet is compressed to a known wavelet, shorterthan the predictive gap, before predictive deconvolution is

a) Input farfeld spectrumAmplitude

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applied, the wavelet will be the same after predictive deconvo-lution.

If the wavelet is known to be long, as in marine surveys,

will be altered by the predictive deconvolution operator, asdiscussed below.

TRUE -AMPLITUDE RECOVERY

The seismic reflection data are not stationary. That is, thestatistical properties of the data are not independent of thetime origin. In particular, there are no data before the shot isfired, the biggest amplitudes are usually the first arrivals, theamplitude and bandwidth decreases with time, and there are

0.00 20.00 40.00 60.00 80100 100.00 120.00' Freq (Hz) 0.00 20.00 40.00 60.00 80.00 100.00 120.00 Freq (Hz)

Fre Hz) ' ' ' ' ' Fr (Hz) 20.00 40.00 60.00 80.00 100.00 120.00 q

( 0.00 20.00 40.00 60.00 80.00 100.00 120.00 ( )FIG. 5. Frequency-domain equivalent of Figure 4: (a) amplitude spectrum of 4(a); (b) amplitude spectrum of 4(b), which is asmoothly interpolated version of (a); (c) amplitude spectrum of 4(c) which is, on average, flat in the bandwidth of the wavelet;(d) amplitude spectrum of 4(d).

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302 Ziolkowski et al.

no data after recording has stopped. A stationary time seriesis infinitely long. Predictive deconvolution uses a prediction-error filter (Peacock and Treitel, 1969; Taner et al., 1995) toremove the so-called "predictable part" of the data (the mul-tiples) from the seismogram, revealing the so-called "unpre-dictable part" (the primaries). The "predictable part" is esti-mated from the autocorrelation of the data, which also containsthe primaries. Predictive deconvolution assumes that there areno accidental correlations between primary reflections that cancontribute to the autocorrelation of the data. If the data wereonly a small part of an infinitely long stationary time series, theprediction-error filter could be computed from the autocorre-lation function estimated from any suitably long segment in thetime series, by definition, including the segment containing theprimary reflections we are trying to reveal. Since that segmentis the only segment available, the stationarity assumption iscritical to the predictive deconvolution process. This argumentis discussed in more detail in Ziolkowski (1984, Chap. 5).

Since seismic data clearly are not stationary, the predictivedeconvolution step is preceded by a step, usually known as"true-amplitude recovery," that applies a time-dependent gainto the data to make the data look more as if they are seg-ments of stationary time series. Typically, this gain functionis t" where t is time and n is approximately 2. This gain cor-rection introduces a time-dependent distortion to the wavelet:that is, after application of this gain correction the waveletvaries down each trace and the convolutional model is violated(Ziolkowski, 1984, Chap. 2). The only gain correction that doesnot violate the convolutional model is an expontial gain, e"`,typically about 12 dB/s.

Neither t" nor e"` make the data stationary. There is no "true-amplitude recovery" correction that makes the data stationary.If we want to keep track of the wavelet, we should use theexponential gain; if we are more concerned about stationaritythan wavelet distortion, it may be better to use t". These twooperations are equivalent only at one time instant.

ZERO-PHASING

At some point we should like to make the wavelet zero phase,to enable the interpreter to line up known acoustic interfaceswith peaks or troughs in the seismic data. Zero-phasing canbe done before or after predictive deconvolution. After theexponential gain correction, the wavelet is w(t), and is relatedto the original desired wavelet d(t) as

w(t) = ecet d(t). (1)

After application of the zero-phasing filter, each seismogramin the data is now the earth impulse response (with exponentialgain applied) convolved with this wavelet, with its peak exactlyat the arrival time of the event. The polarity is our choice. In thisexample, the polarity is chosen to give a black peak in the seis-mic data to correspond to a step increase in acoustic impedancein the well. This is SEG normal polarity (Sheriff and Geldart,1995, 183). If it is required that a black peak should corre-spond to a step decrease in acoustic impedance, for instanceto show gas, the polarity of the data should be reversed afterzero-phasing.

It should be noted that the zero-phasing step can be com-bined with the signature deconvolution step. Instead of theminimum-phase wavelet d(t), the wavelet e-"` wo (t) is used asthe desired wavelet. A different signature operator would bedesigned and applied, causing this wavelet to be present in thedata after signature deconvolution. After the exponential gaincorrection, the wavelet in the data would then be wo (t) and nofurther operations to the wavelet would be required.

THE EFFECT OF CONVENTIONAL PROCESSINGON THE WAVELET

Consider what happens to the wavelet with conventionalprocessing. First the conventional t" process of "true ampli-tude recovery" applies a distortion to the wavelet that variesalong each trace of the data, thus violating the convolutionalmodel. Next, predictive deconvolution is applied with a gapthat is shorter than the wavelet. The predictive deconvolutionoperator is derived from the autocorrelation of the trace,which is different for every trace in the gather in the 1-D case(Peacock and Treitel, 1969) and different for every gatherin the 2-D case (Taner et al., 1995). This is partly becausethe so-called "predictable part" of the trace, including thewavelet and the multiples, is expected to vary with offset,but it is also because the so-called "unpredictable part," theresponse of the earth, is not white (Fokkema and Ziolkowski,1987) and contributes to the autocorrelation function in away that must vary with offset and from gather to gather. Theoperator is therefore different for every trace of the data.After predictive deconvolution there is therefore a differentwavelet on every trace of the data. Even if the quality controlin data acquisition ensured perfect repeatability of the wavelet

Amplitude x 103 Zero phase wavelet

The wavelet w(t) has a Fourier transform W(f), where f isfrequency and

W(f) = lW(.f)I exp{iow(.f)}, (2)in which W (f) I is its amplitude spectrum and cw (f) is itsphase spectrum. The zero-phasing filter is simply the multi-plication of the Fourier transform of the data by the functionexp{-i0.(f)}. The result is to produce a wavelet wo (t) in thedata that has the Fourier transform

Wo(.f) = IW(f). (3)

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This is the shortest possible wavelet within the bandwidth ofthe data (Berkhout, 1984, Chap. 1) and is illustrated in Figure 6.

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FIG. 6. Zero-phase equivalent of 4(b).

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from trace to trace, the predictive deconvolution step woulddestroy it. There is, therefore, no such thing as "the wavelet"after conventional prestack predictive deconvolution.

The data are stacked. Then, after stack, predictive decon-volution is often applied again, usually with different param-eters, again with a different operator on every trace, thus en-suring that there are further differences in the wavelet fromtrace to trace. The conventional use of predictive deconvolu-tion thus guarantees that lateral variations in the data cannot beattributed solely to lateral variations in the geology, thus ren-dering the data unreliable for the interpretation of stratigraphy.

THE WELL TIE BY CHARACTER-MATCHING AND THECONVOLUTIONAL MODEL

The goal of seismic data processing is to render the seismicdata interpretable. The interpreter attempts to correlate re-flections seen in the data with known acoustic impedance con-trasts in the earth. The processed seismic reflection data arepresented to the interpreter as if they contain primary normal-incidence reflections only, with all unwanted multiples, refrac-tions, diffractions, surface waves, compression-to-shear conver-sions, etc. having been removed successfully by the processing.

The convolutional model described above for each recordedtrace of a shot gather is valid because of the linearity of theviscoelastic earth model and the linear response of both thereceiver and the recording system. After the shot records havebeen combined in conventional processing to produce stackedsections, the resulting seismic traces are not the convolution ofsome physically meaningful wavelet with the primary reflectioncoefficient series.

If, nevertheless, this convolution is assumed (e.g., White,1980; Nyman et al., 1987; Richard and Brac, 1988; Poggiagliolmiand Allred, 1994), each well introduces a new wavelet and anew question is now posedhow should the zero-phasing fil-ter be derived between wells? This approach does not solve aproblem; it simply creates a new one.

PRECISE SYNTHETIC SEISMOGRAMS

Precise synthetic shot records could in principle be com-puted properly using finite differences, as outlined above, pro-vided a completely faithful 3-D anisotropic viscoelastic modelof the earth existed, and provided all receiver and source effectswere included, as discussed above. To calculate these syntheticshot records would require much more knowledge of the earththan can be obtained from a single well and also would requireknowledge of the source that is usually not available. If all therequired knowledge were available, the synthetic shot recordswould resemble the recorded data. Then there would be a rea-sonable match to the processed data only if the synthetic shotrecords were processed in the same way as the real data.

DETERMINISTIC WELL-TIE METHODOLOGY

The best match we can hope to get by the simple convolu-tion of a wavelet with the primary normal-incidence reflectioncoefficients obtained from a limited part of the well is for thetiming and polarity of the principal reflections.

Suppose we have measured the signature and have followedthe recommended processing scheme described above, makingsure that no uncontrolled distortions have been applied to thewavelet. The wavelet in the data is some known, short, zero-

phase wavelet wo(t), as shown in Figure 6. Both the polarity andthe arrival time of events in the real data are now the same asthe normal-incidence reflection coefficient series converted totwo-way traveltime using the measured check-shot data at thewell locations. With this processing scheme known interfacesin the wells may therefore be correlated with reflections in theseismic data without synthetic seismograms. We illustrate thismethod with data from the Inner Moray Firth of the North Sea.

THE STRATIGRAPHIC SETTING OF THE INNER MORAYFIRTH BASIN

The Jurassic stratigraphy of the Inner Moray Firth Basin iswell known from nearby onshore outcrops, drilling, and seismicexploration (e.g., Sykes, 1975; Andrews et al., 1990; Underhill,1991a and b; Stephen et al., 1993; Wignall and Pickering,1993; Davies et al., 1996). Regional uplift during the Tertiary(Thomson and Underhill, 1993; Hillis et al., 1994) has left theJurassic relatively shallow in the Inner Moray Firth, and theseismic resolution of Jurassic stratigraphy is therefore gener-ally much better here than elsewhere in the North Sea.

Seismic stratigraphic studies, based on the recognition andcorrelation of surfaces defined by reflector terminations, en-able the Jurassic to be subdivided into two main megase-quences Jl and J2 (Underhill, 1991a and b). The boundarybetween the two megasequences is marked by regional onlaponto a strong reflector that corresponds to a unit known lo-cally as the Alness Spiculite. Analysis of well logs enables theJl megasequence to be subdivided further into genetic strati-graphic sequences through the identification and correlationof through-going marine shale horizons, or maximum flood-ing surfaces, as shown in Figures 7 and 8 (Stephen et al., 1993,Partington et al., 1993).

A regional correlation of these events has led to the identi-fication of the Mid-Cimmerian Unconformity that allows theJl seismic megasequence to be subdivided into two parts J1aand Jib (of Underhill, 1991a and b). This unconformity is asignificant, but subtle, gentle surface of truncation and onlap,which records an episode of thermal doming and subsequentdeflation of the North Sea area (Underhill and Partington, 1993and 1994). The best sequence-stratigraphic definition of the topof the Jl megasequence and of the Mid-Cimmerian Uncon-formity occurs in the central areas of the Inner Moray Firth(Stephen et al., 1993).

RESULTS OF THE NEW WELL-TIE METHODOLOGY

The methodology we propose provides two separate benefitsover the conventional approach: (1) accurate stratigraphic cor-relation between wells and (2) detection of subtle stratigraphictruncations. We illustrate these benefits with examples from aspeculative well-tie seismic survey shot in October 1992 by theSeismograph Service Limited (SSL) Seisventurer in the InnerMoray Firth of the North Sea. The layout of the survey is shownin Figure 9. The source wavefield was measured according tothe method in Ziolkowski et al. (1982) and Parkes et al. (1984),and the seismic reflection data were processed to zero phaseaccording to our method, described above. Wells 12/22-2 and12/22-3 are connected by a seismic line, as shown in Figure 9,and lie in the area where the top of the J1 megasequences trun-cated (Figure 7) and hence where the Mid-Cimmerian Uncon-formity might best be resolved.

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Accurate correlation of stratigraphy between wells

Figure 10 shows part of the well completion log from 12/22-2.At a depth of 4205 It bkb (below kelly bushing), 4123 ft ss (subsea), there is a sharp decrease in the gamma-ray log (extremeleft) and a sharp increase in the sonic velocity log (extremeright) that marks the top of the J1 megasequence of Underhill(1991a and b). This is the top of the Alness Spiculite, a poroussandstone containing oil, but with very poor permeability, andtherefore it is not currently a producible reservoir. The sharpincrease in sonic velocity indicates that the top of this sand-stone should provide a clear seismic reflection with a positivereflection coefficient for the measured pressure.

The sonic log is calibrated with a check-shot survey, partof which is shown in Figure 11, in which a geophone is putat known interfaces in the well and measures the downgoingseismic wave from a seismic source just below the sea surface.The vertical traveltime of the sound wave from sea level to thegeophone placed at the top of the Alness Spiculite is measured0.5390 s, corrected for offset, datum, picking errors, etc. Two-way time is thus 1078 ms.

In his mind's eye the interpreter has the following idea ofthe reflection process, as illustrated in Figure 12. The sharpincrease in acoustic impedance gives a reflection, and the peakof the zero-phase seismic wavelet occurs exactly at the two-way traveltime to the interface. Figure 13 shows the relevantportion of the seismic line at its intersection with 12/22-2. Aclear black reflection can be seen in the zero-phase data at1078 ms, centered on the top Alness Spiculite event.

The synthetic seismogram was obtained by convolving thereflection coefficient series (obtained from the density and

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FIG. 9. Map of the Inner Moray Firth showing the 1992 well-tiesurvey.

sonic logs) with the same zero-phase wavelet, shown inFigure 6. For the best visual comparison with the seismic data,the synthetic seismogram was filtered with the same time-varying band-pass filter and displayed with the same AGC andplotting parameters as the seismic data. the synthetic seismo-gram shows that the timing and polarity of the seismic reflec-tion data are correct. As discussed above, we do not expectthere to be an especially good character match between thesynthetic and real seismic data, because the processed seismicdata are not equal to a convolution of the wavelet with thenormal-incidence reflection coefficient series.

Note that the synthetic seismogram is not required to identifythe events on the seismic section: it simply corroborates theidentification already made from the well completion logs andthe check-shot survey.

Figure 14 shows the complete seismic line linking 12/22-2 and12/22-3. The top Alness Spiculite reflection can be followedconfidently between these two wells as a prominent black pos-itive reflection. Many other events (not marked) can be fol-lowed on the seismic section with equal confidence. This canbe checked at 12/22-3.

Figure 15 shows part of the well completion log from 12/22-3. The top of the Alness Spiculite is seen here at a depth of6250 It bkb (6168 It ss), where there is a sharp decrease againin the gamma-ray log and a sharp increase in the sonic velocitylog. The relevant portion of the check-shot survey for this wellis shown in Figure 16, in which the traveltime of the soundwave from sea level to the geophone placed at the top of theAlness Spiculite is measured to be 0.7210 s, giving a two-waytraveltime of 1442 ms.

Figure 17 shows the relevant portion of the same zero-phaseprocessed seismic line at its intersection with 12/22-3. The topof the Alness Spiculite can be seen as the clear black reflectionat 1442 ms, thus confirming the interpretation of the seismicdata.

Detection of subtle stratigraphic truncations

An application of this methodology to the interpretationof seismic data has also enabled a subtle stratigraphic trun-cation, the Mid-Cimmerian Unconformity, to be detected andmapped. The existence of this truncation had been recognizedpreviously through a stratigraphic and sedimentological analy-sis of over 2000 North Sea biostratigraphically calibrated elec-trical well logs (Underhill and Partington, 1993 and 1994).Figure 7 shows the truncation in the Inner Moray Firth.

Regional well correlation panels across the basin indicatethat the Mid-Cimmerian Unconformity is marked by suddentruncation below and onlap above (Figures 7 and 8). Further-more they demonstrate that the most pronounced truncationoccurs in different places from the most pronounced onlap. Theseismic line connecting wells 12/22-2 and 12/22-3 (Figure 14)corresponds to the area in which truncation, but no onlap, isnoted from well correlations. Interpretation of the line follow-ing accurate well tie of a Lower Jurassic marker bed, The Lady'sWalk member, indicates where the truncation occurs betweenthe wells. This demonstrates the power of iteration betweenthe electrical well-log correlations, the well ties, and the seis-mic data to define the point of truncation more precisely.

The most pronounced onlap seen from the electrical well-logcorrelations is in more southerly regions. With our seismic data

tea Inner M ray Firth

C 12/22-3

Line 1 D1 R12i22-2

12/27-1 Line D4-D5

NE Scotian , UKMercator projection

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acquisition and processing methodology, we can now identifyit on seismic data. Figure 18 shows part of the well completionlog from 12/27-1 in which part of the well has actually beencored, including the Mid-Cimmerian Unconformity identifiedat 3566 ft ss. Figure 19 shows a portion of a seismic line fromthe same survey, intersecting 12/27-1. The unconformity is indi-cated by the arrows; the onlap of reflectors onto it from above,and truncation of reflectors against it from below can be seenclearly. Although this is a major regional event, cutting outmore than 20 Ma of geological time, this is the first time thisunconformity has been convincingly demonstrated on seismicdata in this part of the North Sea. Its successful identificationon this 2-D survey suggests that subtle truncations and onlaps

are likely to be readily identifiable in this and other basins ifour approach is adopted with 3-D data.

CONCLUSIONS

Compared with the spectacular success in finding structuralhydrocarbon traps; the success in finding subtle stratigraphictraps is poor. In the North Sea, for example, any success has oc-curred by accident rather than by design. The poor success rateis a consequence of the conventional approach to the problemof tying wells to seismic data. Now that few structural traps re-main to be drilled, the deliberate search for subtle stratigraphictraps is of paramount importance for extending the life of thebasin. Seismic methodology has not kept pace with this need.

FIG. 10. A portion of the well completion log for 12/22-2.

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The conventional approach suffers from three shortcomings.First, conventional acquisition and processing of seismic reflec-tion data in general guarantees that the convolutional waveletvaries from trace to trace. This follows from the use of predic-tive deconvolution both before and after stack with predictivegaps that are shorter than the wavelet. Second, events on theprocessed seismic data are conventionally identified from welllogs via the construction of a synthetic seismogram that is thebest match, in some sense, to the processed data at the wellposition, and is simply the convolution of "a wavelet" withthe normal-incidence reflection coefficient series. This convo-lutional model has no basis in the physics of the problem, andthe wavelet thus derived is not related to the convolutionalwavelet in the data. Each well introduces a new wavelet andraises a new problem: How should the zero-phasing filter be de-rived between wells? Third, the synthetic seismograms derived

to ensure a good "match" often require time-shifts that are in-consistent with the check-shot data at the wells.

An alternative scheme that has two major benefits is pre-sented. First, all lateral variations in the processed seismic dataare caused by the geology. Second, events on the processedseismic data may be identified from well logs simply by theirpolarity and timing. It follows that events can than be followedon the seismic data from one well to another, with confidence,and the seismic data can be interpreted for stratigraphy. This isdemonstrated with data from the prospective Jurassic succes-sion in the Inner Moray Firth of the North Sea.

The proposed method consists of three steps: (1) determi-nation of all known convolutional effects before any process-ing, using measurements of the source wavefield made duringdata acquisition, (2) compression of this wavelet to the short-est zero-phase wavelet within the bandwidth available, and

FIG. 11. A portion of the calibrated sonic log for 12/22-2.

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(3) elimination of uncontrolled distortions to the wavelet insubsequent processing.

The known convolutional effects are the source time func-tion, the receiver response, and the response of the record-ing system. Absorption is not a convolutional effect in seismicdata and cannot be removed by deconvolution. The receiverresponse and recording system response are normally knownwith great precision. The measurement of the source time func-tion for all major seismic sources is within the capability of mostseismic contractors.

The determination of stratigraphy from seismic data requiresprecise knowledge of the wavelet, which is not obtained withconventional acquisition and processing methods. We proposethat the source time function be measured during data acqui-sition. The cost is on the order of 1% of the cost of seismicdata acquisition. With appropriate data processing the ben-efits are (1) direct identification of seismic events from welllogs on the basis of timing and polarity, (2) correlation of seis-mic events from well to well, and (3) reliable interpretation ofstratigraphy.

Our approach has allowed us to tie all the wells in the In-ner Moray Firth without changing the phase of the wavelet atthe wells and to map the stratigraphy on these lines. The workhas demonstrated that a hitherto undetected subtle, regionalunconformity can be detected on seismic data. The impact of

this approach on the acquisition, processing, and interpreta-tion of 2-D and especially 3-D seismic data in any part of theworld could be very significant and allow the identification ofpreviously undetected stratigraphic hydrocarbon traps.

ACKNOWLEDGMENTS

This paper was presented under a similar title by AntonZiolkowski as the Fall 1994 SEG Distinguished Lecture, andhe is very grateful to the SEG for this honor.

We gratefully acknowledge the cooperation of SeismographService Limited, now Geco-Prakla, for obtaining the data usedin this paper and further for allowing us to use it for researchpurposes. In particular, we thank Ian Cheshire, Director; RexLugg, Manager of the Marine Division; Paul Tollefson, for ob-taining permission for us to carry out the survey from all theoil companies in the Inner Moray Firth; John Connor, PartyChief; Lloyd Peardon, Manager of Research; and Robert Lawsof Research, who also came on board. We are also very gratefulto the Master and crew of the seismic vessel Seisventurer. Wegratefully acknowledge the cooperation with Geco-Prakla inthe joint venture in processing the data, and thank Helen Saalerin particular for the calculation of the reflection coefficients at12/22-2 and 12/22-3.

FIG. 12. The essence of the well tie.FIG. 13. Seismic section of line 1D1R at well 12/22-2, showing

the synthetic seismogram and tie to the seismic data.

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Anton Ziolkowski thanks the Petroleum Science and Tech-nology Institute (PSTI) for funding his chair of Petroleum Geo-science at Edinburgh. The work reported here is based partlyon a study funded by PSTI and Total Oil Marine entitled Doesit make sense to measure the source signature continuously inmarine seismic exploration? which was completed in February1994. Rodney Johnston is now funded under NERC ProjectNo. GR3/9064, which has supported this research since January1994.

John Underhill acknowledges the help of Kevin Stephen andMark Partington in gaining a fuller understanding of the Juras-sic stratigraphy in the Inner Moray Firth Basin, BP, Shell andEsso for the support of research in the basin, and PSTI andNorsk Hydro for continuing support of research in Edinburgh'sseismic workstation facilities.

Finally, we thank the reviewers for their very constructivecomments on our paper.

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FiG. 14. Line 1D1R, processed to zero phase and linking wells 12/22-2 and 12/22-3. The top of the Alness Spiculite is indicated withthe arrows. The vertical scale is milliseconds; the timing lines are 100 ms.

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FIG. 19. A part of seismic line 3D4-D5 tieing with well 12/27-1 and imaging the Mid-Cimmerian Unconformity. The vertical scale isseconds; the timing lines are 100 ms. This line was processed by Geco-Prakla along the lines described in this paper and is reproducedwith their permission.

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Wavelets, Well Ties, And the Search

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