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"ABSTRACTSeismic reflector characterization by a multiscale detection-estimation methodSeismic transitions of the subsurface are typically considered as zero-order singularities(step functions). According to this model, the conventional deconvolution problem aimsat recovering the seismic reflectivity as a sparse spike train. However, recent multiscaleanalysis on sedimentary records revealed the existence of accumulations of varying or-der singularities in the subsurface, which give rise to fractional-order discontinuities.This observation not only calls for a richer class of seismic reflection waveforms, but italso requires a different methodology to detect and characterize these reflection events.For instance, the assumptions underlying conventional deconvolution no longer hold.Because of the bandwidth limitation of seismic data, multiscale analysis methods basedon the decay rate of wavelet coefficients may yield ambiguous results. We avoid thisproblem by formulating the estimation of the singularity orders by a parametric nonlin-ear inversion method.EAGE 69th Conference &amp; Exhibition ? London, UK, 11 - 14 June 2007IntroductionThe earth?s subsurface consists of layers of different materials separated by interfaces,also called transitions. Transitions are characteristic of regions where the acoustic prop-erties of the earth vary rapidly compared to the length-scale of the seismic sourcewavelet. Extracting information on the locations and nature of the transitions fromseismic data has recently received increasing interest, as it provides quantitative infor-mation that can be used in various applications, ranging from improving the geologicalinterpretation of the subsurface to detecting changes in the lithology.Seismic deconvolution is aimed at finding the locations of seismic reflectors and isbased on certain assumptions on the reflectivity. For instance, deconvolution methodshave been developed for a reflectivity that behaves as a (colored) Gaussian random pro-cess (Saggaf &amp; Robinson, 2000). Other approaches are based on the sparsity assump-tion, where the reflectivity is considered to be given by a sparse spike train. Multiscaleanalysis on well and seismic data have shown that neither assumption is rich enoughto describe the different types of transitions present in sedimentary basins (Herrmann,2001). These observations have led to the introduction of a new typeof parametrization, where the observed reflectivity is written as a superpositionof parametrized waveforms. This parametrization is designed to reflect the presenceof transitions other than strictly zero-order transitions (blocked wells) and includesfractional-order transitions. The aim of this paper is two-fold, namely finding the lo-cations of the reflectors, delineating the stratigraphy, and extracting information on thenature of the transitions. We present a new detection-estimation method, where the re-flection events are first detected, then segmented, followed by an estimation based ona descent method (Boyd &amp; Vandenberghe, 2004). The estimated parameters provideinformation on the transition sharpness that is related to the lithology (Herrmann, 2001;Liner et al., 2004), possibly through a critical point in the elastic moduli (Herrmann &amp;Bernab? 2004).The Earth?s Model: We represent a vertical 1-D profile of the earth as a superpositionof parametrized waveforms of the following types(z) =summationdisplayjcjDalphaj psi(z -zj), (1)where z is depth, cj the amplitude for the jth transition, and Dalpha alpha-order operator forfractional differentiation (alpha &gt; 0) or integration (alpha &lt; 0). The psi is some wavelet takento be the Ricker wavelet. The parameters (attributes) of interest in this case are thelocation zj, the amplitude cj and the order alphaj.The characterization problemGiven the above signal representation, our task is to recover the different attributes froma seismic trace. We divide this task into a detection stage, where the main events in thedata are detected, and an estimation stage, where the parameters of the individual wave-forms are estimated. Since our problem does not fit into the classical deconvolutionframework, we use a multiscale wavelet technique to locate the main events. After seg-mentation, the individual waveforms are submitted to a nonlinear inversion procedureto estimate the attributes. This procedure uses rough estimates for the location and scalefrom the detection stage.Event location by multiscale edge detection: The variety of different orders of tran-sitions in the subsurface calls for a seismic-event-detection technique that does not makeEAGE 69th Conference &amp; Exhibition ? London, UK, 11 - 14 June 2007any assumptions regarding the type of transitions. Edge detection based on the multi-scale continuous (complex) wavelet transform modulus maxima (Mallat, 1999) offersan approach that is robust for different waveforms, reflecting different types of transi-tions. First, the method calculates the forward wavelet transformWs(sigma,t) = parenleftbigs asteriskmath ?sigmaparenrightbig(t), (2)where ?sigma(t) = 1radicalsigmapsiasteriskmath(-tsigma ). The range of scales sigma for the wavelet is adapted to the seis-mic source function. After forming the modulus maxima lines (MML) from the waveletcoefficients (Mallat, 1999), the maximum points along these lines are calculated, yield-ing rough estimates for the scale (= band width) and position of the reflection events.The result of this stage is a set of locations and scales {tau(n),sigma(n)} with n = 1 ? ? ? N, andN is the number of detected maxima, which corresponds to location and scale. Theseapproximated values are subsequently used as initial guesses as part of the nonlinearinversion during the estimation stage.(a) Synthetic seismic signal.(b) Continuous wavelet transform of signal.Figure 1:A typical example for the detection of a seismic trace (a) with seven reflection events. (b)The modulus of the continuous wavelet transform with warm colors corresponding to large magnitudes.The local maxima for the wavelet coefficients are used as preliminary estimates for the scale and locationof the transitions. The plus signs show modulus maxima.Partitioning: Given the estimates for the location and scale of the detected events,the trace is segmented into separate events (see Fig. 2). During segmentation, eachindividual waveform is calculated withs(n)(t) = W[tau(n),sigma(n)]s(t) with n = 1 ? ? ? N, (3)where tau(n) and sigma(n) are the locations and scales of the nth detected waveform, W[.] is thewindowing operator centered at tau(n), and has a support proportional to sigma(n). The outputof this procedure are N vectors with ?isolated? events. Even though this segmentationprocedure is somewhat arbitrary, e.g. it depends on a width parameter, we found thismethod to perform reasonably well for most cases. Sub-wavelength details are notextracted this way and are left to the ensuing estimation stage.EAGE 69th Conference &amp; Exhibition ? London, UK, 11 - 14 June 2007Figure 2: Segmentation of detected events. The solid lines show events and the dashed lines correspondto the action of windowing function.Estimation: During the second stage of our method, the segmented events are sub-jected to a nonlinear descent-driven inversion procedure. To setup this inversion pro-cedure, we first need to refine our mathematical model for the parametrized family ofwaveforms. Given our choice presented earlier, we define an individual waveform as afractional derivative of a shifted and scaled Gaussianftheta(t) = Dalpha( 12pisigma2 e(t-tau)2/2sigma2), (4)where the scale is denoted by sigma, the location by tau, and the fractional order by alpha. Moti-vated by the work of Wakin et al. (2005), we can parametrize the family of waveformsas M[theta] = {ftheta : theta element Theta} with theta = [sigma,tau,alpha]. The nonlinear estimation procedure foreach segmented waveform consists of minimizinge(n)(theta) = vextenddoublevextenddoubles(n) -M[theta]vextenddoublevextenddouble22 , (5)or ?theta(n) = arg minthetae(n)(theta), (6)for theta element Thetawith Thetaa feasible parameter range. To solve the above minimization problem,a descent method is employed that requires differentiability of the forward model withrespect to its parameters (theta). Under that assumption, analytical expressions for thesepartial derivatives can be derived. With these partial derivatives w.r.t. thetai, i = 1 ? ? ? 3,the descent update can be calculated and is given byJ(n)i = partialdiffe(n)partialdiffthetai = 2angbracketleftbigs(n) -M[theta],gammathetaiangbracketrightbig , (7)with gammathetai = partialdifffthetapartialdiffthetai , and J(n)i is the projected estimation error for each parameter. Duringeach iteration of the method, e(n)(t) is formed, and J(n) = {J(n)i : i = 1 ? ? ? 3} iscalculated, followed by the following update?theta(n),k+1 arrowleft- ?theta(n),k - 12J(n) (8)where k is the number of iterations of the descent method. Our experience using theabove scheme have shown that this iterative optimization method provides acceptablesolutions to the estimation problem (see Fig. 3).EAGE 69th Conference &amp; Exhibition ? London, UK, 11 - 14 June 2007DiscussionIn this paper, we have presented a new characterization method which allows for theestimation of fractional-order discontinuities. These scale attributes may lead in im-provement of geological interpretation from the seismic trace. The examples we havepresented, indicate that proposed characterization method is leads to accurate results.In addition, our method is well suited for the estimation of the scale exponent attributesfrom bandwidth limited data. As opposed to wavelet coefficient decay based methods,such as SPICE (Liner et al.,2004), our method does not lead to possibly ambiguousestimates since we do not rely on ?infinite? bandwidth.Figure 3: Example of characterization with 10 reflectivity events.(top) Initial and final iteration ofparameter estimation for one isolated event, where the actual values, initial guess and estimation aretheta = (12.2, 667, 2.93), thetainit = (7.81, 668, 0.7), and ? = (12.72, 667, 3.01) respectively. The dashedline shows actual component and the solid line the estimation. (bottom left) Estimated seismic signalis formed by superposition of all characterized events and compared with the original seismic trace.(bottom right) The estimated attributes of events (tau, alpha) are compared to their actual values. The Bluediamonds show actual parameters whereas red the circles estimated values.Acknowledgments: This work was in part financially supported by the Natural Sciences and En-gineering Research Council of Canada Discovery Grant (22R81254) and the Collaborative Research andDevelopment Grant DNOISE (334810-05) of Felix J. Herrmann and was carried out as part of the ChaRMproject supported by Chevron.References[1] Boyd S., and Vandenberghe L. [2004] Convex Optimization, Cambridge University Press[2] Herrmann, F.J., Lyons W.J., and Stark, C. [2001] Seismic facies characterization by monoscale anal-ysis, Geophysical Research Letter, 28(19), 3781-3784.[3] Herrmann, F.J., and Bernab? Y. [2004] Seismic singularities at upper-mantle phase transitions: a sitepercolation model. Geophysical Journal International, 159, 949-960[4] Liner, C., Li, C.-F., Gersztenkorn, A., and Smythe, J. [2004] SPICE: A new general seismic attribute,72nd Annual International Meeting, SEG, 433436 (Expanded Abstracts).[5] Mallat, S. [1999] A wavelet tour of signal processing. Academic Press.[6] Saggaf, M.M., and Robinson, E. A. [2000] A unified framework for the deconvolution of traces ofnonwhite reflectivity. Geophysics, 65(5), 16601676.[7] Wakin, M. B., Donoho, D. L., Choi, H., and Baraniuk R. G. [2005] The Multiscale Structure of Non-Differentiable Image Manifolds. SPIE Wavelets XI, 5914, 413-429, San Diego, (Expanded abstract).EAGE 69th Conference &amp; Exhibition ? London, UK, 11 - 14 June 2007"@en
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