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Computers & Geosciences 32 (2006) 681–695

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Wavelet extractor: A Bayesian well-tie and waveletextraction program$

James Gunninga,�, Michael E. Glinskyb

aCSIRO Division of Petroleum Resources, Bayview Avenue, Clayton 3150, Vic., AustraliabBHP Billiton Petroleum, 1360 Post Oak Boulevard Suite 150, Houston, Texas 77056, USA

Received 8 March 2005; received in revised form 3 October 2005; accepted 11 October 2005

Abstract

We introduce a new open-source toolkit for the well-tie or wavelet extraction problem of estimating seismic wavelets

from seismic data, time-to-depth information, and well-log suites. The wavelet extraction model is formulated as a

Bayesian inverse problem, and the software will simultaneously estimate wavelet coefficients, other parameters associated

with uncertainty in the time-to-depth mapping, positioning errors in the seismic imaging, and useful amplitude-variation-

with-offset (AVO) related parameters in multi-stack extractions. It is capable of multi-well, multi-stack extractions, and

uses continuous seismic data-cube interpolation to cope with the problem of arbitrary well paths. Velocity constraints in

the form of checkshot data, interpreted markers, and sonic logs are integrated in a natural way.

The Bayesian formulation allows computation of full posterior uncertainties of the model parameters, and the important

problem of the uncertain wavelet span is addressed uses a multi-model posterior developed from Bayesian model selection

theory.

The wavelet extraction tool is distributed as part of the Delivery seismic inversion toolkit. A simple log and seismic

viewing tool is included in the distribution. The code is written in Java, and thus platform independent, but the Seismic

Unix (SU) data model makes the inversion particularly suited to Unix/Linux environments. It is a natural companion piece

of software to Delivery, having the capacity to produce maximum likelihood wavelet and noise estimates, but will also be

of significant utility to practitioners wanting to produce wavelet estimates for other inversion codes or purposes. The

generation of full parameter uncertainties is a crucial function for workers wishing to investigate questions of wavelet

stability before proceeding to more advanced inversion studies.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Wavelet extraction; Well-tie; Bayesian; Seismic; Signature; Inversion; Open-source

e front matter r 2005 Elsevier Ltd. All rights reserved

geo.2005.10.001

ble from server at http://www.petroleum.csiro.au.

ing author. Tel.: +61395458396;

8380.

ess: [email protected] (J. Gunning).

1. Introduction

The common procedure of modeling post-stack,post-migrated seismic data as a simple convolutionof subsurface ‘reflectivity’ with a band-limitedwavelet, and the use of this model as the basis ofvarious probabilistic inversion algorithms has been

.

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ARTICLE IN PRESSJ. Gunning, M.E. Glinsky / Computers & Geosciences 32 (2006) 681–695682

the subject of some very vigorous—and occasionallyheated—debates in the last 20 years or so. It is notour intention to rehearse the many and diversearguments at length in this article: trenchant viewsboth in support and opposition to this model havebeen expressed at length in the literature (see, e.g.,the series of vigorous exchanges followingZiolkowski (1991)). Many detractors believe thatthis approach places excessive importance onstatistical machinery at the expense of physicalprinciples; a perfectly reasonable objection in caseswhere important experimental or modeling issuesare oversimplified and the statistical analysis isdisproportionately ornate. While these criticismscan have considerable weight, our view is thatinverse modeling in geophysics must always dealwith uncertainty and noise, and the communityseems to have come to the practical conclusion thatgood modeling can dispense with neither solidphysics nor sensible statistics.

There is a perfectly adequate theoretical justifica-tion for the convolutional model, as long asabsorption and reflections are sufficiently weak,and the seismic processing preserves amplitudes.Closely related assumptions must also hold for theimaging/migration process to be meaningful; theseare usually based on ray-tracing and the Bornapproximation. The migrated images will then bepictures of the true in situ subsurface reflectivity—albeit bandlimited by an implied filter whichembodies the wavelet we seek to extract. Sincemodern inversion/migration methods are closelyrelated to Bayesian updates for the subsurfacevelocity model based on common image gathers(Tarantola, 1984; Gouveia and Scales, 1998; Jinet al., 1992; Lambare et al., 1992, 2003), it would bedesirable if well-log data were directly incorporatedinto the migration formula. This is not commonlydone, for reasons that may relate primarily to thesegregation of workflows, but also technical diffi-culties. There remains a practical need for waveletextraction tools operating on data produced by themigration workflow.

Much of the skepticism about convolutionalmodels has arisen from the observation thatwavelets extracted from well-ties often seen tovary considerably over the scale of a survey. Thisin itself does not seem to us a sufficient argumentto dismiss the model as unserviceable. Firstly, a hostof reasons associated with inadequacies in theseismic processing may create such effects. Forexample, sometimes there is heavy attenuation from

patchy shallow reefs which is imperfectly compen-sated for. Secondly, it is rarely—if ever—demon-strated that the difference in wavelets extracted atdifferent parts of the survey is statistically signifi-cant. As before, an ideal solution would involveunifying the imaging problem with the well-tieproblem for each new well, so imaged amplitudesare automatically constrained to log information.But until this is commonplace, independent partieswill be responsible for the seismic processing andwell-tie workflows, so the well-tie work has toproceed with the ‘best case’ processed seismic dataat hand. In this situation, errors in the imaging haveto be absorbed in modeling noise, and the practi-tioner should at least attempt to discern if thewavelets extracted at different wells are statisticallydifferent.

It is the author’s experience that commercialwavelet extraction codes do not proceed from anexplicit probabilistic approach to wavelet extrac-tion, and thus are not capable of producingstatistical output. Most appear to implement areasonable least-squares optimization of a modelmisfit function, but produce only maximum like-lihood (ML) estimates (with no associated uncer-tainty measures), and often only cosmeticallyfiltered versions of these. In addition, there are anumber of parameters associated with the time-to-depth mapping (henceforth called ‘stretch-and-squeeze’), multi-stack and incidence angle (AVO)effects, and imaging errors that ought in principle tobe jointly optimized along with the wavelet para-meters to improve the well-tie. To the authorsknowledge, control of such parameters are notavailable in commercial codes. Many of thecommercial programs quote the fine work ofWalden and White (1998) in their pedigree.These algorithms are entirely spectrally based,which make them very fast and well suited to(probably) impatient users. However, the formula-tion is not explicitly probabilistic, and the spectralmethods will no longer hold once extra modelingparameters are introduced which will move the well-log in space or time. More recently, a paper byBuland and Omre (2003) presents a model verymuch in the same spirit as that we advocate, but nocode is supplied. Some notable differences inmodeling priority exist between our work and thatof Buland. We consider the problem of theunknown wavelet span as very important,and devote considerable effort to modeling this.Conversely, Buland puts some focus on correlated

ARTICLE IN PRESSJ. Gunning, M.E. Glinsky / Computers & Geosciences 32 (2006) 681–695 683

noise models and assumes the wavelet span tobe known apriori. It is our experience that thewavelet span, treated as an inferable parameter,couples strongly with the noise level, and is unlikelyto be well known in a practical situation. We alsoprefer a more agnostic approach to modeling theeffective seismic noise, since this is a complexmixture of forward modeling errors, processingerrors, and actual independent instrumental noise.It seems unlikely that such a process would beGaussian, and still less susceptible of detailedmodeling of the two-point Gaussian correlationfunction.

We have no objection in principle to theapplication of cosmetic post-extraction filtering inorder to improve the apparent symmetry oraesthetic appeal of extracted wavelets, but feel thatthese requirements would be better embedded insome kind of prior distribution in a Bayesianapproach. Similarly, the relationship between thesonic log and any time-to-depth informationderived from independent sources (like a checkshot)seems most naturally expressed with a Bayesianlikelihood function.

In summary, we feel that the simple convolutionalmodel is likely to linger on indefinitely as thestandard model for performing probabilistic orstochastic inversion. It is then crucially impor-tant to ensure that inversions are run with waveletsand noise parameters optimally derived fromwell data, checkshot information and the sameseismic information. There is then a considerableneed for quality software for performing well-tiesusing a full probabilistic model for the wave-let coefficients, span, time-to-depth parameters,and other related parameters that may be re-quired for inversion or other quantitative inter-pretations.

We present just such a model and its softwareimplementation details in this paper. Section 2presents the Bayesian model specification, andsuitable choices for the priors. The forward model,likelihoods and selection of algorithms is discussedin Section 3, with the final posterior distributiondescribed in Section 4. Questions of statisticalsignificance are addressed in Section 5. Detailsabout the code and input/output data forms arediscussed in Section 6, but most detail is relegated tothe electronic appendices 2 and 3 (see Appendix A).A variety of examples are presented in Section 7,and some final conclusions are then offered inSection 8.

2. Problem definition

2.1. Preliminaries

The wavelet extraction problem is primarily oneof estimating a wavelet from well-log data, imagedseismic data, and a time-to-depth relation thatapproximately maps the well-log onto the seismicdata in time. We will use the following forwardmodel in the problem, with notation and motiva-tions developed in the remainder of this section. Theobserved seismic Sobs is a convolution of thereflectivity r with a wavelet w plus noise en

Sobsðxþ Dx; yþ Dy; tþ DtRÞ

¼ rðx; y; tjsÞ � wþ en, ð1Þ

taking into account any lateral positioning (Dx;Dy)and registration error DtR of the seismic data withrespect to the well coordinates. The well-log dataare mapped onto the time axis using time-to-depthparameters s.

A few remarks about the various terms of thisequation are required. The imaged seismic data Sobs

are likely to have been processed to a particular(often zero) phase, which involves estimation of theactual source signature (gleaned from seafloor orsalt-top reflections, or perhaps even explicit model-ing of the physics of the impulsive source), and asubsequent removal of this by deconvolution andbandpass filtering. The processed data are thencharacterized by an ‘effective’ wavelet, which isusually more compact and symmetrical than theactual source signature. This processing clearlydepends on the early time source signature, sosubsequent dispersion and characteristics of theamplitude gain control (and possible inverse-Qfiltering) may also make the ‘effective’ wavelet atlonger times rather different than the fixed waveletestablished by the processing. Since the waveletappropriate for inversion work will be that applic-able to a time window centered on the inversiontarget, it may be very different in frequency content,phase and amplitude from that applicable to earliertimes. Thus, we begin with the assumption that theuser has relatively weak prejudices about thewavelet shape, and any more definite knowledgecan be integrated into the well-tie problem at a laterstage via appropriate prior terms.

Deviated wells impose the problem of notknowing precisely the rock properties in the regionabove and below a current point in the well, since


the log is oblique. Given that a common procedurewould be to krige the log values out into the near-well region, and assuming the transverse correlationrange used in such kriging would be fairly long, afirst approximation is to assume no lateral varia-tions in rock properties away from the well, so theeffective properties producing the seismic signal arethose read from the log at the appropriate zðtÞ. Thewavelet extraction could, in principle, model errorsin the reflectivity calculation, which could thenabsorb errors associated which this long-correlationkriging assumption. Another way to think aboutthis approximation is the leading term in a ‘near-vertical’ expansion. The seismic amplitudes to usewill be those interpolated from the seismic cube atthe appropriate ðx; y; tðzÞÞ. In this approximation achange in the time-to-depth mapping will result in anew t0ðzÞ, but not a new ðx; yÞ. Extraction of theseamplitudes for each possible realization of thetime-to-depth parameters must be computationallyefficient.

Wavelet extraction for multiple wells will betreated as though the modeling parameters at eachwell are independent (e.g., time-to-depth para-meters). The likelihood function will be a productof the likelihood over all wells. The wavelet will bemodeled as transversely invariant.

The wavelet is parametrized by a suitable set ofcoefficients over an uncertain span and will benaturally tapered. Determination of the waveletspan is intrinsic to the extraction problem. Thetime-to-depth mapping from, e.g., checkshot datawill be in general not exact, since measurementerrors in first-break times in this kind of data maynot be negligible. We allow subtle deviations fromthe initial mapping (‘stretch-and-squeeze’ effects) soas to improve the quality of the well-tie, and theextent of these deviations will be controlled by asuitable prior.

Other parameters that may be desirable to modelare (1) time registration errors in the seismic: theseare small time errors that may systematicallytranslate the seismic data in time, and (2) positioning

errors: errors that may systematically translate theseismic data transversely in space. The latter inparticular are useful in modeling the effects ofmigration errors when using post-migration seismicdata. Since the wavelet extraction may be performedwith data from several wells, the positioning andregistration errors may be modeled as either (a)independent at each location, which assumes thatthe migration error is different in widely separated

parts of the survey, or (b) synchronized betweenwells, which may be an appropriate assumption ifthe wells are close.

The extraction is also expected to cope withmulti-stack data. Here the synthetic seismic differsin each stack because of the different angle used inthe linearized Zoeppritz equations. Because ofanisotropy effects, this angle is not perfectly known,and a compensation error for this angle is anotherparameter which is desirable to estimate in thewavelet extraction. Users may also believe that thewavelets associated with different stacks may bedifferent, but related, for various reasons associatedwith the dispersion and variable traveltimes ofdifferent stacks. It is desirable to build in thecapacity to permit stretch and scale relationsbetween wavelets associated with different stacks.

Finally, the extraction must produce usefulestimates of the size of the seismic noise, which isdefined as the error signal at the well-tie for eachstack.

2.2. Wavelet parameterization

2.2.1. Basic parameters

Let the wavelet wðawÞ be parametrized by a set ofcoefficients aw, with suitable prior pðawÞ. LikeBuland and Omre (2003), we use a widely dispersedGaussian of mean zero for pðawÞ. The wavelet isparameterized in terms of a set of equispacedsamples i ¼ �M ; . . . ;N (kW in total), spaced atthe Nyquist rate associated with the seismic bandedge (typically about dt ¼ 1=ð4f peakÞ). The first andlast samples must be zero, and the samples for thewavelet at the seismic data rate (e.g., 2,4ms) aregenerated by cubic splines with zero-derivativeendpoint conditions. See Fig. 1. Note there arefewer fundamental parameters than seismic sam-ples. This parameterization enforces sensible band-width constraints and the necessary tapering.

Cubic splines are a linear mapping, so thecoefficients at the seismic scale aS are related tothe coarse underlying coefficients aW linearly. Givena maximum precursor and coda length, the twoindices M ;N then define a (usually small) set ofwavelet models with variable spans and centering.

For two-stack ties, the default is to assume thesame wavelet is used in the forward model for allstacks. However, users may believe the stacks mightlegitimately differ in amplitude and frequencycontent (far-offset stacks usually have about 10%less resolution than the near stack). We allow the

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-0.4

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0.2

0.4

0.6

0.8

1

Fig. 1. Parameterization of wavelet in terms of coefficients at

Nyquist spacing associated with band edge (black boxes), and

resulting coefficients generated at seismic time-sampling rate

(circles) by cubic interpolation. Cubic splines enforce zero

derivatives at edges for smooth tapering.

J. Gunning, M.E. Glinsky / Computers & Geosciences 32 (2006) 681–695 685

additional possibility of ‘stretching and scaling’ thefar wavelet (two additional parameters added ontoaw) from the near wavelet to model this situation.The priors for the additional ‘stretch and scale’factors are taken to be Gaussian, most commonlywith means close to 1 and narrow standarddeviations.

2.2.2. Wavelet phase constraints

Sensible bandwidth and tapering constraints arebuilt into the wavelet parameterization. Addition-ally, users often believe that the wavelet ought toexhibit some particular phase characteristic, e.g.,zero or constant phase. Since the wavelet phase f isobtained directly from a Fourier transform of thewavelet ~w ¼ F ðwðawÞÞ as fi ¼ tan�1ðIð ~wiÞ=Rð ~wiÞÞ,for frequencies indexed i, a suitable phase-con-straint contribution to the prior may be written as

pðawÞ� exp �X

i

ðfiðawÞ � f̄Þ2=2s2phase

!,

where the sum is over the power band of the seismicspectrum (the central frequencies containing about90% of the seismic energy). Here f̄ is a target phase,either user-specified, or computed as hfiðawÞi if aconstant (floating) phase constraint is desired. Aproblem with this formulation is that the branchcuts in the arctan function cause discontinuities inthe prior. To avoid this, we use the form

pphaseðawÞ� exp �X

i

ðDð ~wi; f̄Þ2=2s2phaseÞ

!,

where Dð ~wi; f̄Þ is the shortest distance from thepoint ~wi to the (one-sided) ray at angle f̄ headingout from the origin of the complex ~w plane. Thisformulation has no discontinuities in the errormeasure at branch cuts.

2.2.3. Wavelet timing

In many instances, users also believe that the peakof the wavelet response should occur at zero time, sotiming errors will appear explicitly in the time-registration parameters, rather than being absorbedinto a displaced wavelet. Very often, a zero-phaseconstraint is too strong a condition to impose on thewavelet to achieve the relatively simple effect ofaligning the peak arrival, since it imposes require-ments of strong symmetry as well. This peak-arrivalrequirement can be built into the prior with theadditional term

ppeak-arrivalðawÞ

� expð�ðtpeakðawÞ � t̂peakÞ2=2s2peakÞ,

where t̂peak and speak are user-specified numbers,and tpeakðawÞ is the peak time of the wavelet inferredfrom the cubic spline and analytical minimization/maximization. Clearly the peak time is only a piece-wise continuous function of the wavelet coefficients,so we advise the use of this constraint only where anobvious major peak appears in the unconstrainedextraction. If not, the optimizer’s Newton schemesare likely to fail.

2.3. Time-to-depth constraints

2.3.1. Checkshots and markers

The primary constraint on time to depth is aseries of checkshots, which produce data pairsfzðcÞi ; t

ðcÞi g with associated time uncertainty sðcÞt;i ,

stemming primarily from the detection uncertaintyin the first arrival time. The depths are measured(well-path) lengths, but convertible to true depthsfrom the well survey, and we will assume no error inthis conversion. Such pairs can often be sparse, e.g.,500 ft spacings, and will not necessarily coincidewith natural formation boundaries.

Markers are major points picked from the seismictrace and identified with events in the logs. Theyform data triples fz

ðmÞi ;Dt

ðmÞi ;sðmÞDt;ig which are depths

zðmÞ and relative timing errors DtðmÞi with respect to

the time-depth curve associated with the linearlyinterpolated checkshot. The picking error is sðmÞDt;i.These can obviously be converted to triples


fzðmÞi ; tðmÞi ;sðmÞt;i g, which are the same fundamental

type of data as the checkshot. Here

tðmÞi ¼ tinterpolated-from-checkshotðz

ðmÞi Þ þ Dt

ðmÞi

and sðmÞt;i ¼ sðmÞDt;i. This formula is used to convertmarker data to checkshot-like data, and is notintended to imply or induce any correlation betweenthe two sources of information in the prior.

We use the vector s ¼ ftðcÞi ; t

ðmÞi g (combining

checkshots and marker deviations) as a suitableparameterization of the required freedom in thetime-to-depth mapping. We assume that the errorsat each marker or checkshot position are indepen-dent and normal, so the prior for the time-to-depthcomponents becomes

pðtiÞ� exp �ðt� t

ðc;mÞi Þ

2

2s2t;i

!.

The complete time prior distribution pðsÞ ¼Q

i pðtiÞ

is truncated to preserve time ordering by subtractinga large penalty from the log-prior for any state s

that is non-monotonic.

2.3.2. Registration and positioning errors

A further possibility for error in the time-to-depthmapping is that the true seismic is systematicallyshifted in time by a small offset. Such a registrationerror DtR is modeled with a Gaussian prior, andindependently for each stack. Similarly, a simplelateral positioning error Drp ¼ fDx;Dyg is used tomodel any migration/imaging error that may havemis-located the seismic data with respect to thewell position. This too is modeled by a Gaussianprior pðDrpÞ� expð�ðDx� DxiÞ

2=2s2Dx;iÞ expð�ðDy�

DyiÞ2=2s2Dy;iÞ for well i. Each seismic minicube

(corresponding to each well) will have an indepen-dent error, but this can (optionally) be taken as thesame for each stack at the well. Positioning errorsfor closely spaced wells can be mapped onto thesame parameter. We write DrR ¼ fDtR;i;Dxi;Dyig asthe full model vector required for these registrationand positioning errors.

2.4. Log data

For computational efficiency during the extrac-tion, the sonic and density log data are firstsegmented into chunks whose thickness ought notto exceed lB �

16f BE, where f BE is the upper band-

edge frequency of the seismic. We use the aggrega-tive methods of Hawkins and ten Krooden (1978)

(also Hawkins, 2001) to perform the blocking, basedon the p-wave impedance. The effective propertiesof the blocked log are then computed using Backusaveraging (volume weighted arithmetic densityaverage plus harmonic moduli averages (Mavkoet al., 1998)) yielding the triples Dwell ¼ fvp; vs;rg forthe sequence of upscaled layers. The reflectivities rin Eq. (1) are computed at these upscaled-layerboundaries. The shear velocity may be computedfrom approximate regressions if required (a near-stack wavelet extraction will be relatively insensitiveto it anyway). This blocking procedure is justified bythe fact that the convolutional response of veryfinely layered systems is exactly the same as theconvolutional response of the Backus-averagedupscaled system, providing the Backus average isdone to ‘first order’ in any deviations of the logsfrom the mean block values.

The raw log data and its upscaled values can beexpected to have some intrinsic error, whichultimately appears as an error er of the reflectivityvalues computed at the interfaces using the linear-ized Zoeppritz equations. The nature of errors inwell-logging is complex in general. A small whitenoise component always exists, but the more seriouserrors are likely to be systematic or spatiallyclustered, like invasion effects or tool-contactproblems. For this reason, we make no attempt tomodel the logging errors using a naive model, andexpect the logs to be carefully edited before use.

3. Forward model and likelihood

The forward model for the seismic is the usualconvolutional model of Eq. (1). We suppressnotational baggage denoting a particular well andstack. The true reflectivity is that computed fromthe well-log projected onto the seismic time axisrðx; y; tjsÞ (which is a function of the currentparameters time-to-depth map parameters s).

The reflection coefficients r are computed fromthe blocked log properties, using the linearizedZoeppritz form for the p–p reflection expanded toOðy2Þ (top of p. 63, Mavko et al., 1998):

RppðBÞ ¼1

2

Drrþ

Dvp

vp

� �

þ By2Dvp

2vp

�2v2s ðDr=rþ 2Dvs=vsÞ

v2p

!, ð2Þ

with notation r ¼ 12ðr1 þ r2Þ, Dr ¼ r2 � r1; etc., for

the wave entering layer 2 from layer 1 above. Here,


the factor B is introduced to model a situationwhere the angle y for a given stack obtained fromthe Dix equation is uncertain. The stack has user-specified average stack velocity V st, event-time T st

and stack range ½X st;min;X st;max�. Assuming uniformweighting, the mean-square stack offset is

hX 2sti ¼

ðX 3st;max � X 3

st;minÞ

3ðX st;max � X st;minÞ, (3)

from which the y2 at a given interface is computedas

y2 ¼v2p;1

V4stT

2st=hX

2sti

. (4)

Due to anisotropy and other effects related tobackground AVO rotation (Castagna and Backus,1993), the angle may not be perfectly known, so B isintroduced as an additional parameter, typicallyGaussian with means close to unity and smallvariance: pðBÞ� expð�ðB� B̄Þ2=2s2BÞ. The B para-meter can be independent for each stack if multiplestacks are used.

The reflection Rpp is computed at all depths z

where the blocked log has segment boundaries, andthe properties used in its calculation are the segmentproperties from Backus averaging, etc. The reflec-tion is then projected onto the time axis using thecurrent time to depth map and four-point Lagrangeinterpolation. The latter approximates the reflectionby placing four suitably weighted spikes on the fournearest seismic sampling points to the projectedreflection time.

3.1. Contributions to the likelihood

The optimization problem aims at obtaining asynthetic seismic close to the observed data for allwells and stacks, whilst maintaining a crediblevelocity model and wavelet shape. The priorregulates the wavelet shape, so there are contribu-tions to the likelihood from the seismic noise andinterval velocities. We address these in turn.

3.1.1. Seismic noise pdf parametrization

Physical seismic noise is largely a result of smallwaves with the same spectral character as the mainsignal (multiple reflections, etc.). However, the noiseprocess we model, en, is a mixture of this real noiseand complex modeling errors, much of the latteroriginated in the seismic processing. We take this asa random signal with distribution PN ðenjanÞ, the

meta-parameters an (e.g., noise level, covarianceterms, etc.) having prior pðanÞ.

Probably the simplest approach to the meta-parameters in the noise modeling is to avoid theissue of the noise correlations as much as possibleby subsampling. It is trivial to show that if arandom process has, e.g., a Ricker-2 power spec-trum (i.e., �f 2 expð�ðf =f peakÞ

2Þ), then 95% of the

spectral energy in the process can be captured bysampling at times

DTs ¼ 0:253=f peak, (5)

where f peak is the peak energy in the spectrum (andoften about half of the bandwidth). Most practicalseismic spectra will yield similar results. To keep themeta-parameter noise description as simple as isreasonable, we choose to model the prior distribu-tion of the noise for stack j as Gaussian, with Nj

independent samples computed at this sampling-rate, mean zero, and overall variance s2n;j . Since thevariance is unknown, it must be given a suitableprior. Gelman et al. (1995) suggest a non-informa-tive Jeffrey’s prior (PN ðsn;jÞ�1=sn;j) for dispersionparameters of this type, so the overall noiselikelihood + prior will look like

PN ðen;rnÞ�PNðenjrnÞpðrnÞ

�Y

stacks j

1

sNjþ1n;j

expð�e2n=2s2n;jÞ. ð6Þ

Correlations between the noise level on closelyspaced stacks may be significant, so we assume theuser will perform multi-stack ties only on well-separated stacks, so the priors for each stack aresensibly independent.

3.1.2. Interval velocities

Any particular state of the time depth parametervector s corresponds to a set of interval velocitiesVint between checkshot points. It is desirable forthese to not differ too seriously from an upscaledversion of the sonic log velocities. If Vint;log are thecorresponding (Backus-upscaled) interval velocitiesfrom the logs (which we treat as observables), weuse the likelihood term

pðVint;logjsÞ� expð�ðVintðsÞ � Vint;logÞ2=2s2Vint

Þ,

where sVintis a tolerable interval velocity mismatch

specified by the user. Typically acceptable velocitymismatches may be of the order of 5% or so,allowing for anisotropy effects, dispersion, model-ing errors, etc.


Even in the case that the interval velocityconstraints are weak or disabled, a large penaltyterm is introduced to force monotonicity in thecheckshot points, since the Gaussian priors willallow an unphysical non-monotonic time-to-depthmap.

4. Forming the posterior

For a given wavelet span, the Bayesian posteriorfor all the unknowns will then be

Pðaw; s;DrR;B; rnÞ

�pðrnÞpðBÞpðawÞpphaseðawÞppeak-arrivalðawÞ

�Y

wells i

pðsiÞpðDriÞ ðpriorÞ

�Y

wells i

pðVint;logjsiÞ ðlikelihoodsÞ

�Y

wells istacks j

PN ðSobs;ijðxþ Dxi; yþ Dyi; tþ DtR;iÞ

� ðrijðsiÞ � wðawÞÞjrnÞ. ð7Þ

The maximum a posteriori (MAP) point of thismodel is then a suitable estimate of the full set ofmodel parameters, and the wavelet can be extractedfrom this vector. This point is found by minimizingthe negative log posterior using either Gauss–Newton or standard Broyden–Fletcher–Goldfarb–Shanno (BFGS) methods (Nocedal and Wright,1999; Koontz and Weiss, 1982). The optimizer bydefault is started at the prior-mean values, withtypical scales set to carefully chosen mixtures of theprior standard deviations or other suitable scales.The parameter uncertainties are approximated bythe covariance matrix formed from the quadraticapproximation to the posterior at the MAP point,as per Appendix 1 (see Appendix A).

In some cases, especially those where registrationor positioning terms exist in the model, the posteriorsurface is multi-modal. The code can then employ aglobal optimization algorithm formed by sortinglocal optimization solutions started from dispersedstarting points, at the user’s request. The startingpoints are distributed in a hypercube formedfrom the registration/positioning parameters. Thismethod is naturally more expensive to run thanthe default. An illustration of this accompaniesExample 7.2.

Where lateral positioning errors are suspectedand modeled explicitly, the MAP parametersobtained in the optimization may not be especially

meaningful if the noise levels are high. Better ‘fits’ tothe tie can be obtained through pure chance aseasily as through correct diagnosis of a misposition-ing, and users have to beware of this subtlestatistical possibility. A more detailed discussioncan be found in Appendix 4 (see Appendix A). Weurge the use of this facility with caution.

4.1. The uncertain span problem

From the user-specified maximum precursor andcoda times, a set of candidate wavelet models withindices M ;N can be constructed which all lie withinthe acceptable bracket. These can be simplyenumerated in a loop. The posterior space is thenthe joint space of models and continuous para-meters, and each model is of different dimensions.We treat the wavelet span problem as a model-selection problem where we seek the most likelywavelet model k (from among these k ¼ 1; . . . ;Nm

models) given the data D (D ¼ fSobs;Dwellg). Thesemodels are assumed to have equal prior weight. TheMAP wavelet model measured by the marginal

likelihood of the model k

PðkjDÞ�

ZPðaw; s;DrR;B;rnÞ

�daw ds dDrR dBdrn ð8Þ

is an appropriate measure to determine the MLwavelet. For linear models, it is well known that theoverall model probability computed from thisrelation (and the associated Bayes factors formedby quotients of these probabilities when comparingmodels) includes a strong tendency to penalizemodels that fit only marginally better than simplermodels. A simple approximation to the integral isthe standard Bayesian information criterion (BIC)penalty (Raftery, 1996), which adds a term12np logðndÞ to the negative log-posterior, nd beingthe number of independent data points (which willbe the number of near-Nyquist samples of the noiseSobs � Ssynth when the mismatch trace is digitizedover the time-interval of interest), and np is thenumber of parameters in the wavelet. We do not usethe BIC directly, but evaluate integral (8) using theLaplace approximation (Raftery, 1996), basedon the numerical posterior covariance matrix ~Cobtained in Appendix 1 (see Appendix A).

Users are sometimes confused as to why longwavelets yield better ties than short ones, and howto choose the length. Use of the formal marginal

ARTICLE IN PRESS

1Hoschek, W., The CERN colt Java library. http://tilde-

hoschek.home.cern.ch/�hoschek/colt/index.htm.2Gunning, J., 2003. Delivery website: follow links from

http://www.petroleum.csiro.au.3Cohen, J.K., Stockwell, Jr., J., 1998. CWP/SU: Seismic Unix

Release 35: a free package for seismic research and processing.

Center for Wave Phenomena, Colorado School of Mines,

http://timna.mines.edu/cwpcodes.


model likelihood immediately solves this problem:the better ties are invariably not statisticallysignificant. The Bayes factor concentrates themarginal likelihood on the wavelet span of moststatistical significance.

Readers unfamiliar with model-selection pro-blems should be aware that the choice of the priorstandard deviations for the wavelet coefficients is nolonger benign when there are different size modelsto compare. If the prior standard deviations arechosen too large, the model-selection probabilitiessuffer from the Lindley Paradox (Dennison et al.,2002) and the posterior probability always falls onthe shortest wavelet. To prevent this, the character-istic scale in the prior is chosen about three timeslarger than the RMS seismic amplitude divided by atypical reflection coefficient.

The code computes a normalized list of allposterior wavelet mode probabilities, so if the userrequests realizations of the wavelet from the fullposterior, these are sampled from the joint modeland continuous-coefficient space. Very often, theposterior probability is concentrated strongly on aparticular span.

5. Questions of statistical significance

Even with a sophisticated well-tie code such asthat described in this paper, wavelet extraction isnot a trivial workflow. In many cases, the mainreflections in the data do not seem to correspondvery well to major reflections in the well-logs, andwell-matched synthetics can be produced only byswitching on many degrees of freedom, such aspositioning, registration errors, and highly flexiblecheckshot points, with narrow time gates for theextraction. In such situations, the tie may well betotally spurious. A useful flag for such a situation iswhen the posterior probability lands almost entirelyon the shortest wavelet, and also when the quality ofthe tie degrades rapidly with increasing time gatewidth.

There are ways to address the problem morerigorously, but the full detail is beyond the scope ofthis paper. Roughly speaking, we perform a simplesignificance calculation by computing an ensembleof wavelet extractions, using completely random

seismic data, and compare the noise-level statisticsof this ensemble to the noise level of the actual dataextraction. The random seismic minicubes aregenerated using a FFT method based on the po-wer spectrum SðxÞ ¼ w2

z expð�w2zÞ expð�ðw

2x þ w2

yÞÞ,

where wz is scaled from a Ricker-2 fit to the data’saverage vertical power spectrum, and wx;y are scaledso as to give a lateral correlation length specified bythe user. The CDF of the final minicube is thenmapped to that of the data, so the univariatestatistics of the data minicubes are preserved. Thelatter step is important, as processed seismic is notunivariate Gaussian, and the wavelet extraction issensitive to amplitudes.

This Monte Carlo test can be run in cases wherethe user is suspicious of the meaningfulness of thefit, and generates a CDF of the noise parametersobtained over the Monte Carlo ensemble. Thisshould have no appreciable overlap with the true-extraction noise best estimates in cases of mean-ingful ties. An example is shown in Section 7.3.

6. The code

The wavelet extraction code is written in Java,and based largely on efficient public domain linearalgebra1 and optimization (Koontz and Weiss,1982) libraries, along with the seismic handlinglibraries of its companion software Delivery.2 Itcomprises about 50K lines of source code. Waveletextraction is a relatively light numerical application:simple extractions take seconds and more complexproblems may take minutes.

Users are expected to be able to provide seismicdata in big-endian ‘minicubes’ centered at each well(the seismic resides entirely in RAM), plus log datain ASCII LAS or simple geoEAS format. Check-shots and well surveys in a simple geoEAS are alsorequired. Details of these formats are given inAppendix 2 (see Appendix A).

Outputs are a mixture of seismic SU3 files forwavelets, synthetic-and-true SU seismic pairs, andsimple ASCII formats for the parameter estimationsand uncertainties. A small graphical visualizationtool (extractionViewer) is provided as part ofthe distribution which produces the typicalcross-registered log and seismic displays shownin the examples. Details of the output files are inAppendix 3 (see Appendix A).

http://tilde-hoschek.home.cern.ch/hoschek/colt/index.htm



http://www.petroleum.csiro.au

http://timna.mines.edu/cwpcodes


The code is available for download (Gunning andGlinsky, 2004), [3], under a generic open-sourceagreement. Improvements to the code are welcometo be submitted to the author. A set of simpleexamples is available in the distribution, and arebriefly illustrated here.

7. Examples

7.1. Simple single reflection

The simplest example possible is a set of logs thatproduce a single reflection, so the synthetic tracereflects the wavelet directly. Two synthetic dataminicubes were generated from the logs with zeroand a modest noise level. With the noise-freeminicube, the wavelet extracts perfectly, andthe posterior distribution converges strongly onthe wavelet model just long enough to capture thetrue wavelet energy.

More interesting is the extraction on the noisyminicube, shown in Fig. 2. Here the central check-shot point was deliberately shifted 10ms, and itshares a 10ms uncertainty with all the checkshot

rho sonic Vp V-interval TVD MD synthetic seismictime(ms) time(

Mis-timed wavelet(A) (B

Fig. 2. Example of recovery of wavelet from single reflection with noi

wavelet. (B) Case where phase-constrained prior forces timing error to

displays co-registered time, total vertical depth (TVD), measured depth

shown are (in order) density logs r, sonic vp, V int;log (interval velocity

observed seismic Sobsðxþ Dx; yþ Dy; tþ DtRÞ. Viewing display can tog

points spaced at 500 ft. The naive extracted waveletis clearly offset. A symmetrical wavelet can beforced by switching on a zero-phase constraint, andthe checkshot point then moves about 10ms back-wards, to allow the zero-phase recovered wavelet tostill generate a ‘good’ synthetic.

This extraction is run with the command line

% waveletExtractor WaveletExtraction.xml -- dump-ML-parameters -- dump-ML -synthetics --dump-ML-wavelets --fake-Vs -v 4 -c -NLR

where the main details for the extraction arespecified in the XML file WaveletExtrac-tion.xml. The XML shows how to set up anextraction on a single trace, where the 3D inter-polation is degenerate. The runtime options corre-spond to frequently changed user preferences. Theirmeanings are documented by the executable wa-veletExtractor in SU style self-documentation[2]. Amongst the most important options are(a) those related to the set of wavelet spans (cf.Section 4.1): -c,-l denote respectively a centeredset of wavelets, or the longest only. (b) Log-data

rho sonic Vp V-interval TVD MD synthetic seismicms)

Zero-phase enforced )

sy data. (A) Case where time-to-depth error forces mis-timing of

be absorbed in uncertain time-to-depth map. Viewing program

(MD) scales for MAP (best-estimate) time-to-depth model. Also

from upscaled blocked sonic), synthetic seismic (rðx; y; tjsÞ � w),gle on acoustic impedance, a blocked vp log, slownesses, etc.

ARTICLE INJ. Gunning, M.E. Glinsky / Computers

options: -NLR; assume no reflections outside log(so entire log can be used), --fake-Vs; concoctapproximate fake shear log from p-wave log anddensity.

Standard graphical views of the MAP extractionare obtained with

% extractionViewer MLgraphing-file.MOST_LIKELY.well_log.txt

and the most likely wavelet with (on a little-endianmachine)

% cat MLwavelet.MOST_LIKELY.su j suswap-bytes jsuxgraph

-0.02 -0.02

0

0.02

Most likely

0

0.02

-1000 0

rho sonic Vp V-interval TVD MD synthetic seismictime(ms) ti

mode number0 1 2 3 4 5 6

time

(s)

time

(s)

(A)

(C) (D)

(B

Fig. 3. Example of typical dual-well extraction for ‘Bonnie’ wells: (A)

between interval velocities and sonic logs. (C) Maximum likelihood (ML

Wavelet realizations from full posterior (Section 4.1) showing small post

a section of Bonnie-House log.

7.2. Standard dual-well extraction

PRESS& Geosciences 32 (2006) 681–695 691

Here we illustrate a typical, realistic dual wellextraction for two widely separated wells ‘Bonnie-House’ and ‘Bonnie-Hut’. The checkshots are ofhigh quality (bar a few points), so time-to-deptherrors are confined to registration effects only.We illustrate here the peak-arrival constraintprior (cf. Section 2.2.3), which forces the waveletpeak amplitude to arrive at t ¼ 0� 1ms. Multi-start global optimization was used. A Monte Carlostudy of the extraction here shows the extractionto be significant with very high probability. See,Fig. 3.

1000

rho sonic Vp V-interval TVD MD synthetic seismicme(ms)

(E)

)

Tie at ‘Bonnie-House’, (B) ‘Bonnie-Hut’. Note close agreement

) wavelet for each model-span, with ML wavelet solid curve. (D)

erior uncertainty. (E) Example of acoustic impedance blocking on

ARTICLE IN PRESS

-0.05

0

0.05

-0.05 0 0. 05

532

534

536

538

540

542

-0.05

0

0.05

-0.05 0 0.05

t_B

onni

eHou

se

t_BonnieHut

(A) (B)

Fig. 4. Example of basins of attractions for two registration parameters for Bonnie-House and Bonnie-Hut. Grayscale represents MAP

negative log-posterior value obtained starting from values on axes. All other parameters are started at their prior means. (A) Basins for

problem without peak-arrival constraints. (B) Case with peak-arrival constraint �1ms (--constrain-peak-arrival 0 1). The

darkest shade is the global minimum. Note how peak-arrival term in prior greatly enlarges basin of attraction of desired global optimum,

without affecting overall MAP value.

J. Gunning, M.E. Glinsky / Computers & Geosciences 32 (2006) 681–695692

The vertical wells ‘Bonnie-House’ and ‘Bonnie-Hut’ have relatively good quality checkshots. Forthis reason, time-registration effects were the onlyunknown parameters included in the time-to-depthmapping. This kind of problem creates an obviousnon-uniqueness in the inverse problem if the waveletis long enough to offset itself in time by the sameoffset as the registration error. The prior in theregistration error in principle should remove thenon-uniqueness of the global minimum in this case,but a variety of poorer local minima do appear inthe problem, in which the local optimizer is easilytrapped.

This problem illustrates a typical interactiveworkflow. (1) The extraction is performed with nophase or timing constraints, often for a single longwavelet (the -l option), on each well at a time. Theresulting wavelets are then examined to see if thepeak response (assuming the expectation of asymmetric-looking wavelet) is systematically shiftedin time. Very bad ties may be noted, and the guiltywells excluded. (2) Large systematic offsets can thenbe directly entered into the XML file for each well(in consultation with the imaging specialist!),possibly allowing registration timing errors aroundthe shift. (3) The individual extractions can then bererun, confirming that major timing problems havebeen removed. (4) A variety of joint-well extractionsmay then be attempted, perhaps turning on phase orpeak-arrival constraints.

In principle, the manual setting of offsets issomething that can be automated as part of a

(expensive) global optimization routine (availableusing the --multi-start heuristic), but usersusually want to examine the well-ties on anindividual basis before proceeding to joint extrac-tions. It is often the case that imaging around a wellis suspicious for various reasons—which translatesinto poor ties, so the best decision may be to omitthe well altogether, rather than lump it into whatmay otherwise be a good joint extraction.

The actual extraction here is run with

% waveletExtractor BonnieHousePlusBonnieHut.xml --fake-Vs --dump-ML-parameters--dump-ML-synthetics --dump-ML-wave-lets -v 4 [--constrain-peak-arrival 0 1]-c --multi-start -NLR

An illustration of the complexity of the basins ofattraction is shown in Fig. 4.

7.3. AVO extraction for ‘Bitters’ well

Here we illustrate a typical single-well extractionon log data that has a strong AVO character. Thetwo stacks are at about 5� and 30�. The seismic herewas generated synthetically, with a large level ofwhite noise added to the reflection sequence beforeconvolution. The extraction is then of poorishquality, but still statistically significant at aboutthe 10% level, even though the most likely waveletcorresponds to the shortest model. The same

ARTICLE IN PRESS

rho sonic Vp V-interval TVD MD near-stacksynthetic

near-stackseismic

near-stackseismic

far-stacksynthetic

time (ms)

1.45

1.50

1.55

1.60

1.65

01 02 03 04 0

time

(s)

angle (degrees)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0 2 4 6 8 100 2 4 6 8 10

-0.02

-0.01

0

0.01

0.02

-0.5 0 0.5 1.0

mode numbermode number

time

(s)

(A)

(C) (D) (E)

(B)

Fig. 5. (A) AVO character of synthetic seismic using Ricker-2 wavelet. Note strong variation around 1.5 s. (B) Two-stack extraction screen

capture: far-stack synthetic and observed traces are appended at the right. (C) ML wavelets vs. wavelet length (mode) with no peak-

constraints. ML wavelet (boldest trace) is shortest mode. (D) Same, with peak-constraints in prior. (E) Recovered wavelet vs. underlying

wavelet used to generate synthetics.


wavelet was used for both stacks, so no waveletstretch-and-scale parameters were used.

Here the XML shows how to set up an extractionon a single seismic line, where the 3D interpolation

is approximated by perpendicular ‘snapping’ ontothe 2D line. The extraction module can be used togenerate AVO gathers of the kind seen in Fig. 5a.The command


% waveletExtractor Bitters.xml --dump-ML-wavelets -v 4 -NLR --make-synthetic-AVO-gather 80 2

was used in this example, which generates 80Hzsynthetics sampled at 2ms. Angles are stored in theSU header word f2. Displays can then be obtainedvia, e.g.,

% cat synth_seis.Bitters.las.AVO-gather.su j[suswapbytes] jsuxwigb x2beg ¼ 0

The synthetic seismic minicubes are generated
using the --make-synthetic freq dt noiseoption, so, e.g.,
% waveletExtractor Bitters.xml -v 4 -NLR--dump-ML-wavelets--make-synthetic 802 2.3

makes a pair of synthetic minicubes using a Ricker 2wavelet of 80Hz band edge, sampled at 2ms, with anoise reflectivity 2.3 times the RMS reflection powermeasured in the logs added onto the log reflectivity.The time-to-depth curve is the prior mean read fromthe checkshot.

The second stack is specified naturally in theXML file, and the extraction run with

% waveletExtractor Bitters.xml --dump-ML-parameters --dump-ML-synthetics --dump-ML-wavelets -v 4 -c -NLR

Significance testing is obtained by the command

% waveletExtractor Bitters.xml --dump-ML-parameters --dump-ML-synthetics --dump-ML-wavelets -v 4 -c -NLR --monte-carlo 10

which specifies a lateral correlation length of 10traces in the Monte Carlo synthetic minicubes. Theposition of the ML noise parameters with respect tothe Monte Carlo noise parameter distribution isfound by examining the file MC_ParameterSum-mary.txt. In this case, they sit about 10% of theway into the Monte Carlo distribution.

With very noisy data sets like this one, use ofdiscontinuous or strongly non-linear terms in thelikelihood such as the phase-constraint or waveletpeak terms is somewhat dangerous. If the extractionis barely significant without these constraints (asuggested first test), it will be even less so when theyare added.

8. Conclusions

We have introduced a new open-source softwaretoolkit for performing well-ties and wavelet extrac-tion. It can perform multi-well, deviated well, andmulti-stack ties based on imaged seismic data,standard log files, and checkshot information.Uncertainties in the time-to-depth conversion andthe imaging and wavelet aesthetic constraints areautomatically included by the use of Bayesiantechniques. The module produces maximum-apriori(‘best’) estimates of all the relevant parameters, plusmeasures of their uncertainty. Stochastic samples ofthe extracted wavelet can be produced to illustratethe uncertainty in the tie, and Monte Carlo testsof the well-tie significance can be performed forhighly suspicious ties. Our experience is that thewavelet and noise estimates produced are of criticalimportance for inversion packages like Delivery andmany other applications.

Inputs for the module are common seismic dataminicubes, well-log files, simple ASCII representa-tions of checkshot information, and a simplywritten XML file to specify all the needed para-meters. Users will be readily able to tackle their ownproblems by starting from the provided examples.

The authors hope that this tool will prove usefulto the geophysical and reservoir modeling commu-nity, and encourage users to help improve thesoftware or submit suggestions for improvements.

Acknowledgements

The first author gratefully acknowledges generousfunding from the BHP Billiton technology program.Helpful comments from the two reviewers are alsoappreciated.

Appendix A. Supplementary data

Supplementary data associated with this articlecan be found in the online version, at 10.1016/j.cageo.2005.10.001.

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http://www1.fpl.fs.fed.us/optimization.html

http://www1.fpl.fs.fed.us/optimization.html

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