Lithological constraints from seismic waveforms: application to the opal-A to opal-CT transitionMohammad Maysami1 and Felix J. Herrmann, University of British Columbia

SUMMARY

In this paper, we present a new method for seismic wave-form characterization whose aim is threefold, namely (i) ex-traction of detailed information on the sharpness of transitionsin the subsurface from seismic waveforms, (ii) reflector mod-eling, based on binary-mixture and percolation theory, and (iii)establishment of well-seismic ties, through parameterizationsof our waveform and critical reflector model. We test thismethodology on the opal-A (Amorphous) to opal-CT (Cristo-balite/Tridymite) transition imaged in a migrated section ofNorth Sea field data West of the Shetlands.

INTRODUCTION

Integrating information observed from seismic waveforms tothe fine details of unconformities in the subsurface has been achallenge for many years. Not only does this integration re-quire a thorough understanding of the (rock)physics responsi-ble for the unconformity but one also needs to come up withparameterizations, and subsequent nonlinear inversion method-ologies, that capture the essence of what constitutes reflectivityand what reflector information translates into the waveforms.In this paper, we address all of these issues by presenting (i)a parametric representation for reflectors and waveforms thatborrows from earlier work by the authors (see e.g. Herrmannet al., 2001; Herrmann, 2005, and the references therein), (ii)a new nonlinear estimation method for the reflector parame-ters, and (iii) a rock-physical model that generates transitions,consistent with the parametric model proposed by Herrmannand Bernabe (2004) to model mineralogical phase transitionsat upper-mantle discontinuities.

It is well-known that seismic waves reflect specularly at re-gions where the elastic moduli (not necessary the density aswe show below) vary significantly at the length scale propor-tional to the seismic wavelength. Motivated by empirical stud-ies on sedimentary records, Herrmann et al. (2001) made theargument that reflectors can be considered as algebraic singu-larities, parameterized by a scaling exponent determining thetransition sharpness. It was shown that this parameterizationcorresponds to a generalization of zero- or first-order disconti-nuities towards fractional-order scale-invariant transitions, in-dexed by a scale exponent. Ever since this generalization, sci-entists, including the authors, have been making efforts to ex-tract these exponents from seismic data (Herrmann et al., 2001;Herrmann, 2001, 2005). These efforts are hampered by the factthat scale exponents are typically found by conducting a mul-tiscale analysis, which is appropriate for broadband data suchas sedimentary records. Unfortunately, migrated data is band-width limited and this explains the challenge. In this paper,we present the latest developments in the estimation of thisimportant parameter through a detecting-estimation technique

1Currently at Stanford University

during which the waveforms are first located, then segmentedand subsequently inverted with a nonlinear procedure.

With the estimated sharpness information, interesting ques-tions arise, such as what is the geological and rock-physicalsignificance of this parameterization by scale exponents? In-deed, as reported by Herrmann et al. (2001), there are indica-tions that the proposed scale exponent correlates with changesalong clinoforms that are consistent with changes in the depo-sitional environment. Albeit insightful, in this paper we aimto answer the question at the rock-physical level by studyingthe behavior of binary mixtures (e.g. opal-A to opal-CT) asa function of their volume fractions. At this point one maybe tempted to resort to traditional Hashin-Shtrikman (HS) orReuss-Voigt bounds to constrain the elastic moduli (see e.g.Hashin and Shtrikman, 1962). Unfortunately, these modelsare smooth as a function of the volume fraction, which wewill consider as a proxi for depth. This smoothness excludessharp transitions and hence specular reflections. To counterthis unsatisfactory aspect of equivalent medium theory, Her-rmann and Bernabe (2004) proposed an alternative model, whe-re the elastic moduli are predicted to change drastically as soonas a critical volume fraction for the stiffer of the two materialsof the binary mixture is reached. At this critical threshold, ac-cording to this model, the stiffer of the two materials connectsand the overall strength of the macroscopic rock matrix willdepend on the size of the connected cluster of the strong ma-terial. As this cluster grows with a scale exponent, a cusp-likesingularity is created (Herrmann and Bernabe, 2004).

In this paper, we will use this percolation model to explain theunconformity associated with the opal-A to opal-CT transitionobserved throughout a migrated section near the Shetland is-lands. Constrained by core data from a well intersecting thisseismic line, we built a reflector model that generates a seismicwaveform similar to the one observed from the data and hencewe are providing a well tie.

Our paper is organized as follows. First, we present the para-metric reflector model, followed by a parameterization-esti-mation technique for waveform data. Next, we briefly intro-duce the percolation model and discuss the type of seismicresponse it generates. We conclude by using this model, inconjunction with the parametric estimation, to constrain theopal-A to opal-CT transition from well and seismic data.

SEISMIC WAVEFORM CHARACTERIZATION

In this section, we present our parametric waveform inversionmethod consisting of a detection, segmentation, and an estima-tion stage.

Waveform parameterization: As proposed by Herrmann et al.(2001); Herrmann (2001), transitions in the Earth’s subsurfacecan be modeled by algebraic singularities, parameterized by

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Figure 1: Generalized transition model with scale exponentsvarying from zero- to first-order. (a) Zero- (blue), first- (red),and fractional-order transitions. (b) Corresponding seismicwaveforms, yielded by the convolution of the reflectivity witha Ricker wavelet. Notice how the reflected waveform changesfrom the Ricker wavelet for the zero-order transition to the in-tegral of the Ricker wavelet for the first-order discontinuity.

a scale exponent. As can be seen from Figure 1, the corre-sponding waveforms yielded by the linear convolution modelchange significantly as a function of the singularity order. Forthe zero-order transition, we observe the Ricker wavelet itself,which is consistent with the fact that the reflectivity for thattransition is given by the delta Dirac distribution, while for thefirst-order discontinuity time-integrated waveform is observed.

In all generality according to this transition model, we can con-sider a seismic trace as a superposition of waveforms generatedby different fractional-order transitions. Mathematically, thissuperposition can, in the Fourier domain, be written in the fol-lowing algebraic form (Blu and Unser, 2003):

s(ω) =X

i

ciϕθi(ω), (1)

with ϕθ (ω) = ( jω)−α/2+φ (− jω)−α/2−φ e−(σ2ω2)

2 e− jωτ , -where α is related to the singularity order, σ is the bandwidth,φ the phase rotation, and τ the location of the waveform (seealso Herrmann, 2005, for further details). According to Equa-tion 1, the seismic response is made out of waveforms whichfor each i are parameterized with θi = {τi,σi,αi,φi}. Becausethe seismic trace is assumed to be linearly related to the reflec-tivity, the attributes of this parameterization, in particular thescale exponent α , are directly related to the order of the tran-sition. Moreover, this analytic model lends itself for nonlinearinversion—i.e., find the sets of parameters θi given s(t).

Multiscale detection: Seismic traces as defined in Equation 1are challenging because they violate most of the assumptionsthat underlay methods such as deconvolution. This is becausethe waveforms in Equation 1 are allowed to vary in a non-stationary fashion, which for a single source function, corre-sponds to an Earth whose transitions differ from ordinary stepfunctions. The latter would justify deconvolution approachessuch as spiky deconvolution to locate the reflectors. Instead,

we opt to use maxima along the Wavelet Transform ModulusMaxima Lines (WTMMLs) (Mallat, 1997). We write the com-plex wavelet transform of a seismic trace s as a convolutionproduct

W s(t,σ) =`s∗ ψσ

´(t), (2)

where ψσ (t) = 1√σ

ψ∗(−tσ

), and σ ≥ 0 is the scale of waveletψ . Here, the symbol ∗ denotes the complex conjugate. Follow-ing Mallat (1997), we define a WTMML as a connected curve,σ(t), in the time-scale plane (t,σ), along which all points aremodulus maxima. Singularities (events) are detected by find-ing the maximum point along each modulus maxima line (seeFig. 2(a)). These points yield approximate estimates for thescale (= bandwidth) and location of the reflection events. Aninitial approximation for the phase can be calculated from thecomplex wavelet coefficients as well. The advantage of thismethod is that very few assumptions are made regarding thetype of reflected waveforms, making this approach a relativelyrobust event detector.

Segmentation: The result of the detection stage is a set oflocations, scales, and phases— i.e., {(τ(n),σ (n),φ (n)) | n =1 · · ·N}, where N is the number of detected maxima. We ex-tract the nth detected waveform by multiplying the seismictrace by a window function centered at τ(n) and with a supportproportional to σ (n) (see Fig. 2(b)). The output of this proce-dure consists of N signals with ‘isolated’ events s(n)(t). Eventhough this segmentation procedure is somewhat arbitrary, wefound this method to perform reasonably well for cases whereinter-event distances are not too small. Sub-wavelength detailsare not extracted and are left to the next estimation stage.

Nonlinear parametric inversion: To complete the charac-terization, we need to estimate the attributes of the individualwaveforms—i.e., we need to estimate θ (n) for each n = 1 · · ·Ngiven s(n)(t). For this purpose, we solve N nonlinear least-squares problems

θ(n) = arg min

θ∈Θ

‚‚‚s(n)−M[θ ]‚‚‚2

2with n = 1 · · ·N, (3)

where M[θ ] = { fθ : θ ∈ Θ} is the collection of all feasiblewaveforms—i.e., θ = [τ,σ ,α,φ ]∈Θ with Θ the set of feasibleparameters. The set of estimated parameters θ (n) signifies theattributes of the nth event. We solve the above optimizationproblem with a quasi-Newton method (BFGS, Nocedal andWright, 1999; Kelley, 1999). In Figure 2, we show an exampleof our waveform characterization method.

RELATION TO ROCK PHYSICS

At first glance, our expose so far may leave the reader with theimpression that we are merely interested in some mathematicaltechnique without an obvious link to the (rock) physical pro-cesses that may be responsible for transitions other than zero-or first-order discontinuities. One can argue that the formermodels a major unconformity where two different rock typesare in welded contact, whereas the second could arguably beused to model some transitional region, which leaves the frac-tional transitions without an obvious physical interpretation.

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Figure 2: Seismic waveform characterization for a syntheticseismic trace. (top) Magnitude complex wavelet coefficientswith warm colors corresponding to large magnitudes. Thevertical and horizontal axes show scale and location, respec-tively. Dark blue lines show WTMML and white circles iden-tify events. (middle) Segmentation of individual events (solidwaveform) by using a window function (dashed line with samecolor) (bottom) Comparison between the reconstructed trace,formed by superposition of all characterized events, with theoriginal seismic trace.

The site-percolation model: Variations in both the densityof mass and bulk modulus determine the reflection of acous-tic waves. Even though these properties appear on an equalfooting in the acoustic impedance, their physical significanceis very different. For bi-compositional mixtures, for instance,the density is trivially related to the volume fractions of thetwo rock components. This property does not hold for the bulkmodulus, which depends on the microscopic arrangement ofthe two rock constituents. Upper- and lower bounds, such asVoigt and Reuss, are used to ‘sandwich’ all possible values formodulus for all possible arrangements. Unfortunately, this sortof formulation is smooth in the volume fraction—i.e., thesebounds are smooth as observed in Figure 3 dashed lines. Thissmoothness implies a smooth depth dependence for volumefractions that increase linearly with depth and hence will notgenerate a specular reflection.

By bringing in the notion of connectivity, arguably importantfor elastic moduli, Herrmann and Bernabe (2004) proposed touse ideas from percolation theory, where this concept plays acentral role. According to this model, the elastic moduli ofa binary mixture change suddenly when the volume fraction,p of the stronger of the two materials connects. We illustratethis in Figure 3(a), where a connected cluster of strong (black)material is formed at some critical volume fraction. At that

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Figure 3: Schematic site-percolation model. The percolationthreshold pc = p(zc), which corresponds to a critical depth zc,is denoted by a vertical dashed line. (top) Illustration of co-existence region. Black and white ellipsoids show inclusionsof strong and weak materials, respectively. At critical volumefraction pc (dashed vertical lines), strong material inclusionspercolate and an infinite connected cluster is formed. (bottom)Corresponding compressional wave velocity (solid line) with asingularity at pc as a function of volume fraction. It is boundedwith Reuss and Voigt averages shown in dashed lines. Thesingularity order was taken to be β = 0.41.

point, an ∞-cluster of strong inclusions is formed. The volumefraction of the cluster for p ≥ pc is given as p∗ = p ( p−pc

1−pc)β ,

where β is an exponent that depends on the type of percolation(see Herrmann and Bernabe (2004) for more details). Conse-quently, the bulk modulus, and hence the acoustic wavespeed,will have a β -order discontinuity (see Figure 3(b)). Accordingto this model, the mixture’s wavespeed for p < pc (and hencefor shallower depth) follows the lower Reuss bound, to ’cuspup’ to the upper Voigt bound for p ≥ pc—i.e., a β -order re-flector emerges at a depth that corresponds to pc. This cuspreflects and hence can be used to explain seismically observedtransitions. However, since the exponents β predicted by per-colation theory depend on several factors over which we haveno control in this setting, we will have to use seismic imagesto estimate this exponent.

A CASE STUDY: OPAL-A TO OPAL-CT TRANSITION

To test whether the above model can be used to explain seismi-cally observed waveforms, we submit a seismic subsection ofthe Faeroe-Shetland basin West of the Shetlands to our wave-form characterization and we zoom in onto the opal-A to opal-CT transition that also contains a well constraint (Figure 4).Core samples taken from this borehole indicate that the strongevent at t = 2.9 s is diagenetic and corresponds to an opal-A toopal-CT transition (Davies et al., 2001; Davies and Cartwright,2002). Moreover, the samples were found to change in vol-ume fraction for opal-CT from 10% at the top of the transitionzone to 76% below the unconformity (Davies and Cartwright,

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2002). If we assume a linear gradient for the volume fractionbetween these two values, we can use our site-percolation tomodel the shape of the transition as a function of depth. Sinceno sonic well data is available at this depth, we take represen-tative values for the bulk modulus for both opal types from theliterature (Guerin, 2000, logs measured in hole 904A).

At this point, we are in principle in a position to tie seismic to awaveform generated from our percolation model. Two piecesof the puzzle are still missing, namely the seismic source func-tion and the order of discontinuity β . We estimate the seismicsource function by taking the average over the estimated scaleexponents from the sea-bottom reflector, which we assume tobe close to zero-order. This procedure allows us to correctfor the order of the seismic wavelet in additional to applyingthe unit shift between the order of the reflectivity and the bulkmodulus. The second piece of the puzzle is solved by averag-ing the estimated exponents along the opal-A to opal-CT tran-sition, which after correction for the source leads to an estimateof α = 0.79. This estimate is close to the value of β = 0.81,which leads to the best trial-and-error match between the seis-mic in Figure 4 and our synthetic trace. We compute this traceusing a convolution model, where the volume fraction of opal-CT is a proxy for depth or time. Figure 5 contains plots forthe density, which is smooth, the compressional wavespeed,including the bounds, its derivative and the reflection coeffi-cients. It is clear, that the wavespeed has a singularity at thecritical depth.

DISCUSSION AND CONCLUSIONS

This paper is a first attempt towards a combination of sophis-ticated data inversion techniques with a rock-physical modelthat predicts the fine-structure of transitions as a function ofthe volume fraction for the stronger of the two components ofa binary-rock mixture. Application of this methodology to theopal-A to opal-CT transition gives us the following insights.First, detailed information on the nature of transitions is re-flected into waveforms. Second, this information can be char-acterized by first detecting the events by the multiscale wavelettransform, followed by a segmentation into individual wave-forms and a subsequent parametric inversion. Third, repeated

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Figure 5: Modeled opal-A to opal-CT transition. Propertiesfor opal-A are taken to be ρ = 1713.90 kg/m3 and Vp =1889.65 m/s. For opal-CT, density and P-wave velocity areassumed to be ρ = 2006.06 kg/m3 and Vp = 2237.71 m/s,respectively. The density (top left) and P-wave velocity (topright) profiles are plotted as a function of volume fraction ofopal-CT. The velocity is bounded by Reuss and Voigt averagesand shows a critical point. This singularity is clearly visiblefrom the derivative of the velocity (bottom left), and can alsobe observed in the reflection coefficients (bottom right).

application of this procedure to a seismic section leads to anattribute map where reflectors are overlaid with colored dots,which localize the positions of the transitions and whose colorindicates their order. Fourth, with the estimate for the orderparameter, a model for the transition can be built, yielding awaveform close to the seismically imaged waveform associ-ated with the opal transition. This latest finding is particu-larly exciting since it gives an indication that the percolationmodel for the bulk modulus can be used to explain the detailsof waveforms associated with the opal transition. This meansthat sharp seismic reflectors can emerge even in cases wherethe composition of the rock seemingly changes smoothly. Theonset of connectivity amongst the stronger of two mixing ma-terials as function of the volume fraction is responsible for thissharp transition. This finding is new and encouraging and willhelp us to constrain unconformities from seismic and to bet-ter understand the processes responsible for transitions in theEarth’s subsurface. In addition, our work may give us comple-mentary information since the scale exponent associated withpercolation depends on factors such as the type of mixing. Thisimplies that our attribute maps may provide complementaryinformation on changes along major unconformities.

Acknowledgments: The authors thank Chevron company forfinancially supporting the grant in aid ChaRM. We would alsolike to thank Veritas DGC for providing the dataset and ad-ditional information. Finally, we would like to thank DaveWilkinson and Yves Bernabe for fruitful discussions. Thiswork was in part financially supported by NSERC DiscoveryGrant 22R81254 and CRD Grant DNOISE 334810-05 of F. J.H.

Lithology constraints from seismic waveforms

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