Coupling of Elastic Isotropic Medium Parameters in Iterative Linearized Inversion D.V. Anikiev¹, B.M. Kashtan¹, W.A. Mulder² and V.N. Troyan¹
¹Saint Petersburg State University, 1, Ulyanovskaya str., Peterhof, Saint Petersburg, 198504, Russia, e-mail: danikiev@earth.phys.spbu.ru
²Shell Global Solutions International BV & Delft University of Technology, Kessler Park 1, GS Rijswijk, 2288, Netherlands, e-mail: Wim.Mulder@shell.com
An elastic isotropic medium is described with three parameters. In seismic migration the perturbation of one elastic parameter affects the images of all the three, which means that these parameters are coupled. For an effective quantitative reconstruction of the true elastic medium reflectivity one can apply an iterative linearized migration/inversion, where minimization of the misfit functional is done by the conjugate gradient method. The final result of the iterative approach can be obtained directly by Newton’s method, using the pseudo-inverse Hessian matrix.
Calculation of this matrix for a realistic model is an extremely resource-intensive problem, but for a model of a scatterer in a homogeneous elastic medium it is quite feasible. This paper presents the numerical results of elastic linearized inversion for this simple model, calculated both with iterative approach and Newton’s method. Experiments show that in the both cases the elastic parameters have coupled weaker than in the case of migration. The iterative approach allows achieving acceptable quality of the inversion, but requires a large number of iterations. For faster convergence it is necessary to use the preconditioned conjugate gradient method. The optimal preconditioning will improve the convergence of the method as well as the quality of inversion.
Key words: Linearized inversion, seismic migration, elastic isotropic medium, coupling of elastic parameters.
Introduction
A classic seismic migration (Claerbout 1971) allows for qualitative estimation of reflection capability in an elastic medium, but the true values of the medium in this case remain unknown (Zhu et al. 2009). Moreover, the migration result gives information about
heterogeneities of each of the three parameters used to describe an elastic isotropic medium, so the true perturbation of one of them in a migration image can look like as perturbation in each of the three parameters. There have been a number of optimization methods to estimate the elastic parameters both qualitatively and quantitatively (Ампилов et al. 2009; Virieux and Operto 2009). Their application gives more accurate estimations of true characteristics of the medium and reduces the coupling effect of the parameters. Uncoupling the elastic parameters mathematically is an extremely resource-intensive problem due to the optimized character of the solution (Gelis et al. 2007; Virieux and Operto 2009). That is the reason inversion methods are applied first-hand, so the elastic characteristics could be properly accessed with account for the required computation time and operation memory. For instance, one can apply an iterative linearized migration/inversion (Beydoun and Mendes 1989; Jin et al. 1992; Tura et al. 1993), which advantages are described in Оstmo et al. (2002), where the method was applied to an acoustic wave equation with constant density in the frequency domain. If computations are considered, this approach has proved more effective than a full waveform inversion (Tarantola 1986; Mora 1987; Fichtner 2010).
The objective of this research has been studying the mutual effect of elastic parameters when the iterative linearized inversion is applied. This paper presents the method’s theoretical basis and a number of numerical experiments, using a point scatterer in a homogeneous isotropic elastic medium as the model. For such a simple case the solution can be obtained by Newton’s method (Fichtner 2010), which allows for direct estimation of the elastic parameters’ maximum inversion quality, available for the iterative inversion, so one can make a conclusion about applicability of this method for a particular case.
Effectiveness of the iterative approach mainly depends on initial problem’s conditionality. Its optimal preconditioning has significantly improved the iterative approach’s convergence [Axelsson, 1996], so one of the objectives of this study has also been finding an efficient preconditioning method for the iterative linearized inversion.
Method
A system of motion equations for an elastic isotropic medium in the frequency domain can be expressed as Lu = f, where the wave operator L affects the displacement vector u in the following way:
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ning has res he second, w lem conditi wn. nd row capt and 14 – can give complex ease the nversion e number sulted in while the ions, the tures see
8 iterati 1 iterati 56 itera Figure of the G Analogo diagona number other ha impedan (for the continue accomp be mad approxi ions ion ations 6 Inversion Gesse matrix ous situatio al approxim r of iteration and a signif nce perturb e third colum ed until t panied by ex de that the mation of th n results by x. For colum on takes pla mation of the ns has redu ficant numb bance due to mn in Fig. the commo xplicit mutu e both prec he Gesse m CG method mn and row ace when on e Gesse matr uced to six ber of iterat o the desire 7 the soluti on converg ual effect of conditioners matrix have n d preconditi w captures se ne deals wi trix. Figure in the first tions have ed solution ion quality gence crite f the s that are not been opt
ioned by inv ee Fig. 4. th CG prec 7 shows the t case and t been requir quality Q ≥ has only re erion was and pa based on timal. verted diago onditioning e results of to one –in red in the c ≥ 95 % has eached 88% reached. arameters. S diagonal a onal approx g of inverte such invers the second case of true s not been %), so the it This proce So a conclu and block-ximation d block-sion. The . On the e S-wave obtained terations ess was usion can diagonal
Figure of the G Conclu In this medium proved estimate paramet the exa sources Newton has allo three - p approac appropr 7 Results of Gesse matrix sions paper a nu m while mig to be quali e not only ters. The nu ample of a and a vertic n’s inversion owed one to parameter s ch one can riate quality of CG inver x. For colum umerical stu gration and itatively dif the spatia umerical re point scat cal compon n with calcu o estimate th system have obtain an y a significa rsion, preco mn and row udy of inter inversion h fferent from l location esults of ela tterer in a nent of displ ulation of th he maximum e interrelati appropriate ant amount onditioned b w captures se rrelation of has been pre m the classic of reflectio astic parame homogenou lacement ve he full Gess m possible ions that ar e result of t of iteratio by inverted ee Fig. 4. f the three p esented. Th c migration on borders, eters’ restor us isotropic ector as the se matrix fo quality of t re insignific inversion. ns of conju block-diago parameters e iterative l n, since the but also m ration have c medium data for an ollowed by the final res cant. So, ap But for the ugate eleme onal approx of elastic i linear inver first allow medium’s e been prese with vertic inverse pro its pseudoin sult and sho pplying the e solution t ents is requ ximation isotropic rsion has ws one to physical ented by cal-force oblem. nversion owed the iteration to be of uired. To
improve the convergence rate one has to use an appropriate preconditioning. In the paper it has been shown that the obvious solutions including the inverted diagonal and block-diagonal parts of the Gesse matrix are far from being optimal, since they are strongly dependant on the problem conditions. So, finding optimal preconditioning has been an important scientific problem that determined effectiveness of both linearized and nonlinear inversion. The further study should be aimed at finding an optimal CG preconditioner in the context of a linear iterative inversion, in such a way it could be applied as an effective and accurate tool for estimation of the parameters of elastic isotropic medium.
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About the authors
DENIS V. ANIKEEV, Ph. D. student, Research Engineer of Saint-Petersburg State University. E-mail: danikiev@earth.phys.spbu.ru
BORIS M. KASHTAN, Doctor of Science, Professor, Head of Laboratory of Elastic Media Dynamics of Chair of Earth Physics of Saint-Petersburg State University.
WIM MULDER, Ph.D., Professor, Head of Geophysical Imaging Section of Delft University of Technology , Researcher of Shell Global Solution International, the Netherlands.
E-mail: Wim.Mulder@shell.com
VALDIMIR N. TROYAN, Doctor of Science, Professor, Head of Chair of Earth Physics of Saint-Petersburg State University.
