Sparse seismic deconvolution by method of orthogonal matching pursuit

Transcription

1 From the SelectedWorks of Michael Broadhead 21 Sparse seismic decovolutio by method of orthogoal matchig pursuit Michael Broadhead Available at:

2 P395 Sparse Seismic Decovolutio by Method of Orthogoal Matchig Pursuit M.K. Broadhead (Saudi Aramco) & T.L. Toellot* (Saudi Aramco) SUMMARY The method of orthogoal matchig pursuit is used to solve the sparse determiistic decovolutio problem for poststack seismic data. A descriptio of the theory is give ad algorithmic implemetatio is discussed. This method is the applied to sythetic ad real data. The orthogoal matchig pursuit method, like other sparse spike techiques, produces high apparet resolutio (spiky appearace) i the output. What value this ad other such sparse spike estimators might brig to stratigraphic iterpretatio, if ay, is still uder assessmet. 72 d EAGE Coferece & Exhibitio icorporatig SPE EUROPEC 21 Barceloa, Spai, Jue 21

3 Itroductio Sparse methods have log bee used to obtai reflectivity estimates from seismic data. This is because real reflectio coefficiets of sedimetary rock layers are leptokurtic (Walde, 1985). The literature o sparseess ad sigal processig spas more tha three decades ad is too extesive to cite here. Curret iterest is still strog. A few refereces iclude: Claerbout ad Muir (1973), O'Brie, et al., (1994), Daubechies, et al. (24) ad Dossal ad Mallat (25). We cocetrate o oe particular algorithm from the wavelet processig commuity, called orthogoal matchig pursuit (OMP) (Pati, et al., 1993), that ca be adapted to the sparse decovolutio problem. A earlier variat of the method, called matchig pursuit (MP), (Mallat ad Zhag), has already bee applied to the blid seismic decovolutio problem by Herrma (23). I his approach, the source sigature was represeted by a parametric form. Our approach assumes the source sigature is available. Also, we do't use a parametric form, but rather use the actual sigature itself. Fially, we use OMP rather tha MP. However sigificat these differeces may or may ot be, our mai obective is to itroduce this type of approach to the oil idustry sice Herrma s paper is published i a oural ot widely read i this busiess. I this paper, we give the backgroud theory for OMP for the geeral sigal decompositio problem, show how to adapt it to the decovolutio problem, ad the apply the method to sythetic ad real data to obtai a reflectivity estimate. Note, the emphasis i this paper is o the algorithm rather tha applicatio due to the recet appearace of commercial packages (uder the ame of spectral iversio ) for which it may be difficult to ascertai the uderlyig priciples or repeat the results of i oe s ow software. The algorithm we preset should be easy to implemet ad test. Theory The followig developmet follows closely to that of Mallat ad Zhag (1993). Let H be the Hilbert space of complex valued fuctios such that () 2 + f = f t dt < +. The ier product of ( f, g) H is defied by f, g = + () () f t g t dt where the bar over g deotes complex cougate. The problem they cosider is the time-frequecy decompositio problem, M f () t = a () gγ t where γ () = 1 g t are elemets of a chose family of fuctios D { gγ() t } =, called a dictioary, ad γ Γis the vector of parameter values (e.g., scale, traslatio, frequecy, etc.). Γ is the fiite set of all such parameter combiatios uder cosideratio. Because D is usually highly redudat, our goal is to pick out a small subset of the fuctios to actually expad over. Equivaletly, we could say that we desire a to be sparse (mostly zeros). Oe algorithm for accomplishig this is called the method of matchig pursuit. Let's choose oe elemet gγ D ad proect f oto this directio, obtaiig f = f, g g + R, γ γ γ Γ 72 d EAGE Coferece & Exhibitio icorporatig SPE EUROPEC 21 Barceloa, Spai, Jue 21

4 where R orthogoal. For is the residual. Note that, assumig g γ = 1, it ca be show that f, g γ γ This implies maximizig f, g, which ca easily be see geometrically i Figure 1. Therefore, we eed to compute f, g for all [1,, M ], ad choose the directio (value of ) for which the ier product is a maximum. The residual ca ow successively be proected oto the remaiig g i the same maer, fially obtaiig M f ( t ) = R, () M 1 gγ g γ t + R +. = γ g to be a best approximatio to γ a g γ ad R are f, we require that R be a miimum. At each stage, the coefficiet is give by the ier product f, g. A modificatio of the γ algorithm, called orthogoal matchig pursuit, adusts all of the coefficiets simultaeously after each 2 iteratio by least squares adustmet, i.e., choose the a such that f R is miimized. γ Figure 1 f, g g grows as R shriks. This suggest maximizig f, g γ γ γ to miimize R. Algorithm for Deco Problem The algorithm was implemeted i Matlab. We foud the scripts by Peyré (29) very helpful i this regard. We assume that we have obtaied a source sigature estimate w which we the use to write the covolutio matrix W (which acts as the dictioary). The colums of W are successively lagged copies of w, meaig that our γ vector has oe elemet, amely, traslatio. Each colum of W is a discretee represetatio of oe of the g fuctios. Therefore, our sigal decompositio problem becomes d = Wr, γ which, whe d ad W are kow, is the determiistic decovolutio problem. Here, the reflectio coefficiets (rc) r are equivalet to the expasio coefficiets a. d is the seismic trace uder cosideratio. The ext step is to proect d otoo W ad compute the maximum. This proectio step

5 turs out to be equivalet to cross correlatig the wavelet estimate with the seismic trace. The algorithm steps ca be writte as Iitialize: R = r = d Iterate: =1,N T c = W R Compute cross correlatio fuctio (proectio f, g ) over all γ imax c i = max c Fid locatio (idex i max ) of abs max (maximize 1 r i = r i + ci Put a rc there scaled to size of corr. coef. ˆ T ˆ ˆ T ˆ ed ( ε ) f, g ) γ WW+ Ir = Wd Least squares adust all rc s simultaeously (OMP oly) r ˆ ( k) = rk, k = 1,, Restore zeros. (OMP oly) i r + 1 R = d Wr Update residual (oce a portio of d is fitted, stop workig o it) N is the umber of ozero elemets of r. r is the th iterate estimate of r. r ˆ is the cosecutive ozero elemets of r. Ŵ is the matrix formed by removig the colums of W that correspod to zeros i r. i is the vector of elemet idices for r. r Refer to Figure 2 for example results o sythetic seismic data. Reflectio coefficiets (rc s) were computed from a impedace log from a well i Saudi Arabia, where the geology teds to be blocky, ad therefore should be favorable to this kid of aalysis. A wavelet was applied ad the resultig sythetic seismogram was processed with our algorithm. Oly the reflectio coefficiets are show i order to save space. Two cases are show, where the umber of spikes is 133 ad 5, respectively. Results show that the method successfully recovered the rc s i the latter case. Although, ote that sice this parameter is ot kow, ad caot be iferred reliably from the data, the solutio is ouique i that sese. Prior iformatio might be useful i determiig this parameter. Oe would also have to cosider the impact of oise i the data ad ucertaity i the wavelet estimate. Refer to Figure 3 for a example result o real seismic lad data. The wavelet was estimated by spectral smoothig ad zero phase was assumed. Oly whe sufficiet case studies have bee accumulated will we be able to ascertai if this type of sigal processig yields a useful seismic attribute. Coclusios We have discussed the geeral theory of OMP. I additio, we gave a algorithm for applyig it to the determiistic decovolutio problem. This was demostrated o real ad sythetic seismic data. While the method is efficiet, it ca yield sub-optimal solutios. This is due to the fact that, oce a compoet is chose (spike locatio), it does ot chage. Although, the first locatio is ot ecessarily optimal. Refereces Claerbout, J. ad Muir, F., Robust modelig with erratic data, 1973, Geophysics, 38, Daubechies, I., M. Defrise ad C. DeMol, 24, A iterative thresholdig algorithm for liear iverse problems with a sparsity costrait, Comm. Pure Appl. Math, 57, Dossal, C. ad S. Mallat, 25, Sparse spike decovolutio with miimum scale, I Proceedigs of Sigal Processig with Adaptive Sparse Structured Represetatios, d EAGE Coferece & Exhibitio icorporatig SPE EUROPEC 21 Barceloa, Spai, Jue 21

6 Herrma, F. J., 23, Seismic decovolutio by atomic decompositio: A parametric approach with sparseess costraits, Itegr. Computer-Aided Eg., 12, 1, Mallat, S. G. ad Z. Zhag, 1993, Matchig pursuits with time-frequecy dictioaries, IEEE Tras. o Sig. Proc., 41, 12, O'Brie, M., A. Siclair ad S. Kramer, 1994, Recovery of a sparse spike time series by l 1 orm decovolutio, IEEE Tras. o Sigal Process., 42, Pati, Y.C., R. Rezaiifar ad P.S. Krishaprasad, 1993, Orthoormal matchig pursuit recursive fuctio approximatio with applicatios to wavelet decompositio, i Proc. 27th Aual Asilomar Cof. o Sigals, Systems ad Computers. Peyré, G., 29, A Numerical Tour of Sigal Processig, Web site: ~peyre/umerical-tour/. Walde, A. T., 1985, No-Gaussia reflectivity, etropy ad decovolutio, Geophysics, 5, Blue True RC s Red OMP Estimate No. spikes = No. spikes = Figure 2 Compariso of sythetic seismic data: actual rc's overlaid with OMP deco estimates Figure 3 Compariso of lad seismic data (left) ad OMP sparse decovolutio results (right). 72 d EAGE Coferece & Exhibitio icorporatig SPE EUROPEC 21 Barceloa, Spai, Jue 21