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Three-term amplitude-versus-offset (avo) inversion revisited by curvelet and wavelet transforms Hennenfent, Gilles; Herrmann, Felix J. 2004

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Three term Amplitude-Versus-Offset (AVO) inversion revisited by curvelet and wavelet transformsGilles Hennenfent and Felix Herrmann, EOS, University of British ColumbiaSummaryWe present a new method to stabilize the three term AVO in-version using curvelet and wavelet transforms. Curvelets arebasis functions that effectively represent otherwise smoothobjects having discontinuities along smooth curves. The appliedformalism explores them to make the most of the continuityalong reflectors in seismic images. Combined with wavelets,curvelets are used to denoise the data by penalizing highfrequencies and small contributions. This approach is based onthe idea that rapid amplitude changes along the ray-parameteraxis are most likely due to noise. The AVO inverse problem islinearized, formulated and solved for all (x,z) at once. Usingdensities and velocities of the Marmousi model to define thefluctuations in the elastic properties, the performance of theproposed method is studied and compared with a conventionalmethod. We show that our method better approximates thetrue data after the denoising step, especially when noise levelincreases.IntroductionIn oil exploration industry, reflection seismology is widely usedto image the subsurface structure. Pressure waves are emittedby a source at the surface into the Earth. Due to the propertycontrast between two consecutive layers, one part of the energyis transmitted and the other is reflected back up to be detectedby receivers at the surface. Signals are recorded over a range ofsource and receiver offsets.The reflectivity varies significantly along the ray-parameter axisand it can be used to make inferences about the properties ofthe layers. This technique is called Amplitude Versus Offset(AVO) inversion. Due to the ill-conditioned nature of the inverseproblem, it is difficult to obtain accurate estimates for theseproperties. In other words, a small amount of noise may lead tolarge errors in the estimates.In the first part of this paper, we present how curvelets andwavelets can be used to stabilize the three term AVO inversion.We explore the fact that seismic images can be efficiently repre-sented by curvelets since discontinuities (i.e. interfaces betweenlayers) occur along curves. By working on the coefficients of thecurvelet transform instead of the data, we consider the point aspart of an environment in opposition to an isolated point. Con-sequently, information regarding the near-neighborhood of eachpoint can be used to better denoised the data. Intuitively, thecurvelet transform can be seen as a local extraction of majorevents combined with a local averaging of the noise. Based onthe idea that rapid amplitude changes along the ray-parameteraxis result from noise (Kuehl and Sacchi, 2003), we denoise thedata by thresholding high frequencies and small contributions.The denoised data are inverted for all (x,z) at once.In a second part, we illustrate our method using densities and ve-locities of the Marmousi model to define the fluctuations in theelastic properties. The denoised data and the recover model arecompared with results obtained by using a conventional methodwhere only wavelets could have been used to smooth AVO re-sponses and the inverse problem were carried out point-by-point.Wavelet and curvelet transformsMulti-resolution transforms have proven to be successful insignal processing applications (Mallat, 1999). Among these,wavelets are probably the most famous and widely used. Be-cause they are localized and multi-scale, their ability to preserveand characterize point singularities in a noisy signal is provento be better than discrete Fourier transform. However their poororientation selectivity prevents to represent higher-dimensionalsingularities effectively.The curvelet transform is a relatively new multi-scale transformwith strong directional character in which elements are stronglyanisotropic at fine scales, with effective support shaped accord-ing to the parabolic scaling principle length2 similar width (Can-des and Donoho, 1999). Curvelets provide stable, efficient, andnear-optimal representation for seismic data with reflectors onpiece-wise smooth curves.Figure 1: Some curvelets at different scales (Courtesy Em-manuel Cand`AVO inversion of PP dataAt position (xi,zi), the linearized Zoeppritz equation for PPreflection coefficients (RCs) can be written for small angles andcontrasts as (Aki and Richards, 1980)Rpp(p) = 12(1 -4?2Sp2 deltarho?rho ) + 12cos2 theta deltacP?cP-4?2Sp2 deltacS?cS(1)where rho is the density, cP (resp. cS) the velocity of P-waves(resp. S-waves), and the ray parameter p ? ?P = sintheta.Three term Amplitude-Versus-Offset (AVO) inversion revisited by curvelet and wavelet transformsUsing the following substitutions (van Wijngaarden, 1998) forthe acoustic impedance ZdeltaZ?Z =deltacP?P +deltarho? , (2)and for the shear modulus ?delta?? =deltarho? + 2deltacS?S , (3)the RCs can be approximated byRpp(p) approxequal 12 deltaZ?Z +bracketleftBig12deltacP?P -2parenleftBig ?cS?PparenrightBig2 delta??bracketrightBig?2P p2. (4)For N ray parameters, the local forward model for the seismicdata becomesdpts =parenlefttpparenleftexparenleftexparenleftexparenleftbt1/2 12parenleftBig?2P p211-?2P p21parenrightBig-2 ?c2S?c2P?2P p21... ... ...1/2 12parenleftBig?2P p2N1-?2P p2NparenrightBig-2 ?c2S?c2P?2P p2Nparenrighttpparenrightexparenrightexparenrightexparenrightbtmpts +npts(5)withdpts =parenleftBigRpp(p1),...,Rpp(pN)parenrightBigT , (6)mpts =parenleftBigdeltaZ?Z ,deltacP?P ,delta??parenrightBigT (7)dpts is the seismic data (i.e. reflectivities) for one point in thespace domain along the ray parameter axis, mpts the model (i.e.elastic properties) for this point, and npts the noise. Withoutloss of generality, we will assume the additive noise to be whiteGaussian with the same mean and variance for all the points.In the conventional approach for the estimation of contrasts inelastic properties, one carry out for each point of the space do-mainminmpts||dpts -Kptsmpts||2 (8)where Kpts stands for the linearized reflection operator. By re-peating the process, the sections for each contrast parameter arebuilt point-by-point. Due to the ill-conditioned nature of the in-verse problem, it is difficult to obtain accurate estimates for thecontrast parameters.We, on the other hand, consider the RC cube as a whole eventhough it is still ill-conditioned. The seismic data becomesd = Km + n (9)where d is the whole RC cube, K the corresponding linearizedreflection operator, m the 3 contrast parameter sections, and nthe white Gaussian noise. In this case, one can introduce in thecost function to be minimized global a priori knowledge in orderto help to converge to the model. Sacchi propose to minimize thefollowing cost function (Kuehl and Sacchi, 2003)F(m) = ||W(d -Km)||2 + lambda2||partialdiffp(Km)||2 (10)where W is a (diagonal) weighting operator and lambda2 a tradeoffparameter depending on the noise level. The second term inthe objective function imposes a relative smoothing constraintin the ray parameter domain but it doesn?t include any a prioriinformation on the lateral continuity along reflectors.Our concern is to account for both smoothness in the ray-parameter domain and along edges in the space domain. Thegeneral idea is to use curvelets in the space domain (2-D) tobenefit from information regarding edges and wavelets in theray-parameter domain (1-D) to denoise AVO responses.Improvement of the data SNRThe denoising of the data does not require the problem to belinearized. Instead of using a smoothing constraint term inthe cost function, we use the multi-scale property of wavelets.The smoothness condition is then equivalent to penalize highfrequency and small contribution coefficients of the wavelettransform in the ray-parameter domain by thresholding. Thesecoefficients are most likely due to noise in the data. But first,to make the most of the continuity along reflectors, we take acurvelet transform in the space domain. This operation can beseen as a local averaging of the noise and a very efficient way tosparsely represent our signal. Chronologically speakingCxzd = CxzKm + Cxzn (11)WpCxzd = WpCxzKm + WpCxzn (12)??d = ThetaG(WpCxzd) approxequal WpCxzKm (13)where Cxz is the curvelet transform in the space domain, Wpthe wavelet transform in the ray-parameter domain, ThetaG(.) a hardthresholding with a threshold level G, and ??the approximate ofthe data in the curvelet-wavelet domain.G is estimated by evaluating the composition of curvelet andwavelet transforms of a few standard white noise signals (Starcket al., 2002)G proportionalbracketleftBig 1QQsummationdisplayi=1parenleftbigWpCxzN(0,1)parenrightbig2bracketrightBig12 (14)Since the amplitudes are important in our case, it is important toonly consider hard thresholding, which preserve the amplitudesunlike soft thresholding. Note also that the threshold level Gdoes not prevent strong events in the high frequencies to remain.Thus, it is possible to apply this method to data containing post-critical angles.From Eq. 13, by taking the inverse wavelet and inverse curvelettransforms, one have an approximation of the data ??d = C-1xz W-1p ThetaG(WpCxzd) approxequal Km. (15)At this point, we assume that the noise was removed from thedata and we carry out the inverse problem using ?to obtain therecovered model mrmr = minm|| ?-Km||2 (16)Three term Amplitude-Versus-Offset (AVO) inversion revisited by curvelet and wavelet transformsExampleTo illustrate our method, densities and velocities of the Mar-mousi model are used to define the fluctuations in the elasticproperties. Two noisy data sets d1 and d2 are built. d1 (resp.d2) has a SNR = 0 dB (resp. SNR = 6 dB). Without lossof generality, we will assume the additive noise to be whiteGaussian. As a first approximation, small angle and smallcontrast assumptions are made. In other words, the dip is notcorrected.Both data sets are processed using our method and a conven-tional one. The conventional method considers each point sep-arately as formulated in Eq. 5 and imposes a smoothing con-dition on their AVO response. In this case, we apply directlyon the data a wavelet transform in the ray-parameter domain(Eq. 17) and hard threshold (Eq. 18) with a threshold level? = sigmaradicalbig2loge N where sigma is the standard deviation of the noiseand N the number of data samples (?). We finally get the ap-proximated data using the conventional method by taking theinverse wavelet transform in the ray-parameter domain (Eq. 19).Wpdpts = WpKptsmpts + Wpnpts (17)?dprimepts = Theta?(Wpdpts) approxequal WpKptsmpts (18)?dpts = W-1p Theta?(Wpdpts) approxequal Kptsmpts (19)In Fig. 2, we can see that our method to denoise the data out-performs the conventional method in the sense that our approx-imated data is closer to the true data. This is especially true forthe d1 where the noise level is higher.Figure 2: Normalized misfit ||d - ?|| function of the ray-parameter #. +- lines represent misfits using the conventionalmethods, straight lines misfits using our method. In the upperpart, the misfits are related to d1, in the bottom part to d2.For a better understanding, four sample reflectors were chosento compare the methods (Figs 3 & 4). For high level of noise,the conventional method is not able to make the difference be-tween the signal and the noise whereas our method does dueto the neighborhood-effect introduced by curvelets. An illustra-tive example is reflector 3 in Fig. 3 around ray parameter #20.The conventional method tends to follow the noise whereas ourmethod stays close to the true data.DiscussionWe developed and demonstrated in this paper a new method thatuses curvelet and wavelet transforms to stabilize the three-termAVO inversion. Our method was successfully compared with aconventional method on synthetic data for the denoising and theneighborhood-effect of curvelets was highlighted.By using the curvelet transform, we can determine and correctfor the dip.Our method can be applied to data with post-criticals angleswithout any problem.AcknowledgementThe authors would like to thank Emmanuel Cand` and DavidDonoho for making their Digital Curvelet Transform viaUnequally-spaced Fourier Transforms available for evaluation.This work was in part financially supported by a NSERCDiscovery Grant.ReferencesAki, K., and Richards, P. G., 1980, Quantitative seismology:Theory and methods: W. H. Freeman and Co.Candes, E., and Donoho, D., Curvelets: A surprisingly effectivenonadaptive representation of objects with edges:, Technicalreport, Caltech, 1999.Candes, E., and Donoho, D., 2000, Recovering Edges in Ill-Posed Inverse Problems: Optimality of Curvelet Frames.Kuehl, H., and Sacchi, M. D., January-February 2003, Least-squares wave-equation migration for avp/ava inversion: Geo-physics, 68, no. 1, 262?273.Mallat, S., 1999, A wavelet tour of signal processing: AcademicPress.Starck, J., Candes, E. J., and Donoho, D. L., June 2002, Thecurvelet transform for image denoising: IEEE Trans. ImageProc., 670?684.van Wijngaarden, A.-J., 1998, Imaging and characterization ofangle-dependent seismic reflection data: Ph.D. thesis, DelfUniversity of Technology.Three term Amplitude-Versus-Offset (AVO) inversion revisited by curvelet and wavelet transformsFigure 3: Denoised data using either our method (dotted line)and a conventional one (dashed line) compared against eachother with respect to the true data and the noisy data d1 (SNR =0) on the four sample reflectors.Figure 4: Denoised data using either our method (dotted line)and a conventional one (dashed line) compared against eachother with respect to the true data and the noisy data d2 (SNR =6) on the four sample reflectors.

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