An efficient source wavefield reconstruction scheme using single boundary layer values for the spectral element method
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摘要
In the adjoint-state method, the forward-propagated source wavefield and the backward-propagated receiver wavefield must be available simultaneously either for seismic imaging in migration or for gradient calculation in inversion. A feasible way to avoid the excessive storage demand is to reconstruct the source wavefield backward in time by storing the entire history of the wavefield in perfectly matched layers. In this paper, we make full use of the elementwise global property of the Laplace operator of the spectral element method(SEM) and propose an efficient source wavefield reconstruction method at the cost of storing the wavefield history only at single boundary layer nodes. Numerical experiments indicate that the accuracy of the proposed method is identical to that of the conventional method and is independent of the order of the Lagrange polynomials, the element type, and the temporal discretization method. In contrast, the memory-saving ratios of the conventional method versus our method is at least N when using either quadrilateral or hexahedron elements, respectively, where N is the order of the Lagrange polynomials used in the SEM. A higher memorysaving ratio is achieved with triangular elements versus quadrilaterals. The new method is applied to reverse time migration by considering the Marmousi model as a benchmark. Numerical results demonstrate that the method is able to provide the same result as the conventional method but with about 1/25 times lower storage demand. With the proposed wavefield reconstruction method, the storage demand is dramatically reduced; therefore, in-core memory storage is feasible even for large-scale three-dimensional adjoint inversion problems.
In the adjoint-state method, the forward-propagated source wavefield and the backward-propagated receiver wavefield must be available simultaneously either for seismic imaging in migration or for gradient calculation in inversion. A feasible way to avoid the excessive storage demand is to reconstruct the source wavefield backward in time by storing the entire history of the wavefield in perfectly matched layers. In this paper, we make full use of the elementwise global property of the Laplace operator of the spectral element method(SEM) and propose an efficient source wavefield reconstruction method at the cost of storing the wavefield history only at single boundary layer nodes. Numerical experiments indicate that the accuracy of the proposed method is identical to that of the conventional method and is independent of the order of the Lagrange polynomials, the element type, and the temporal discretization method. In contrast, the memory-saving ratios of the conventional method versus our method is at least N when using either quadrilateral or hexahedron elements, respectively, where N is the order of the Lagrange polynomials used in the SEM. A higher memorysaving ratio is achieved with triangular elements versus quadrilaterals. The new method is applied to reverse time migration by considering the Marmousi model as a benchmark. Numerical results demonstrate that the method is able to provide the same result as the conventional method but with about 1/25 times lower storage demand. With the proposed wavefield reconstruction method, the storage demand is dramatically reduced; therefore, in-core memory storage is feasible even for large-scale three-dimensional adjoint inversion problems.
In the adjoint-state method, the forward-propagated source wavefield and the backward-propagated receiver wavefield must be available simultaneously either for seismic imaging in migration or for gradient calculation in inversion. A feasible way to avoid the excessive storage demand is to reconstruct the source wavefield backward in time by storing the entire history of the wavefield in perfectly matched layers. In this paper, we make full use of the elementwise global property of the Laplace operator of the spectral element method(SEM) and propose an efficient source wavefield reconstruction method at the cost of storing the wavefield history only at single boundary layer nodes. Numerical experiments indicate that the accuracy of the proposed method is identical to that of the conventional method and is independent of the order of the Lagrange polynomials, the element type, and the temporal discretization method. In contrast, the memory-saving ratios of the conventional method versus our method is at least N when using either quadrilateral or hexahedron elements, respectively, where N is the order of the Lagrange polynomials used in the SEM. A higher memorysaving ratio is achieved with triangular elements versus quadrilaterals. The new method is applied to reverse time migration by considering the Marmousi model as a benchmark. Numerical results demonstrate that the method is able to provide the same result as the conventional method but with about 1/25 times lower storage demand. With the proposed wavefield reconstruction method, the storage demand is dramatically reduced; therefore, in-core memory storage is feasible even for large-scale three-dimensional adjoint inversion problems.
引文
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Capdeville,Y.,and Marigo,J.J.(2007).Second order homogenization of the elastic wave equation for non-periodic layered media.Geophys.J.Int.,170(2),823-838.https://doi.org/10.1111/j.1365-246X.2007.03462.x
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De Basabe,J.D.,and Sen,M.K.(2007).Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equation.Geophysics,72(6),T81-T95.https://doi.org/10.1190/1.2785046
De Basabe,J.D.,and Sen,M.K.(2010).Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping.Geophys.J.Int.,181(1),577-590.https://doi.org/10.1111/j.1365-246X.2010.04536.x
Dussaud,E.,Symes,W.W.,Williamson,P.,Lemaistre,L.,Singer,P.,Denel,B.,and Cherrett,A.(2008).Computational strategies for reverse-time migration.In Proceedings of the 78th Annual International Meeting,SEG,Expanded Abstracts(pp.2267-2271).Las Vegas:Society of Exploration Geophysicists.
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Bozda?,E.,Trampert,J.,and Tromp,J.(2011).Misfit functions for full waveform inversion based on instantaneous phase and envelope measurements.Geophys.J.Int.,185(2),845-870.https://doi.org/10.1111/j.1365-246X.2011.04970.x
Bozda?,E.,Peter,D.,Lefebvre,M.,Komatitsch,D.,Tromp,J.,Hill,J.,Podhorszki,N.,and Pugmire,D.(2016).Global adjoint tomography:first-generation model.Geophys.J.Int.,207(3),1739-1766.https://doi.org/10.1093/gji/ggw356
Brossier,R.,Gholami,Y.,Virieux,J.,and Operto,S.(2010).2D frequency-domain seismic wave modeling in VTI media based on a Hp-adaptive discontinuous Galerkin method.In Proceedings of the 72nd EAGE Conference&Exhibition incorporating SPE EUROPEC.Barcelona,Spain:SEG.
Capdeville,Y.,and Marigo,J.J.(2007).Second order homogenization of the elastic wave equation for non-periodic layered media.Geophys.J.Int.,170(2),823-838.https://doi.org/10.1111/j.1365-246X.2007.03462.x
Carcione,J.M.,and Wang,P.J.(1993).A chebyshev collocation method for the wave equation in generalized coordinates.Comput.Fluid Dyn.J.,2(3),269-290.
Chaljub,E.,Capdeville,Y.,and Vilotte,J.P.(2003).Solving elastodynamics in a fluid-solid heterogeneous sphere:a parallel spectral element approximation on non-conforming grids.J.Comput.Phys.,187(2),457-491.https://doi.org/10.1016/S0021-9991(03)00119-0
Chaljub,E.,and Valette,B.(2004).Spectral element modelling of threedimensional wave propagation in a self-gravitating Earth with an arbitrarily stratified outer core.Geophys.J.Int.,158(1),131-141.https://doi.org/10.1111/j.1365-246X.2004.02267.x
Chen,M.,Niu,F.L.,Liu,Q.Y.,Tromp,J.,and Zheng,X.F.(2015).Multiparameter adjoint tomography of the crust and upper mantle beneath East Asia:1.Model construction and comparisons.J.Geophys.Res.,120(3),1762-1786.https://doi.org/10.1002/2014JB011638
Clapp,R.G.(2009).Reverse time migration with random boundaries.In Proceedings of the 79th Annual International Meeting,SEG,Expanded Abstracts(pp.2809-2813).Houston,Texas:SEG.
Dablain,M.A.(1986).The application of high-order differencing to the scalar wave equation.Geophysics,51(1),54-66.https://doi.org/10.1190/1.1442040
Dai,W.,Wang,X.,and Schuster,G.T.(2011).Least-squares migration of multisource data with a deblurring filter.Geophysics,76(5),R135-R146.https://doi.org/10.1190/geo2010-0159.1
De Basabe,J.D.,and Sen,M.K.(2007).Grid dispersion and stability criteria of some common finite-element methods for acoustic and elastic wave equation.Geophysics,72(6),T81-T95.https://doi.org/10.1190/1.2785046
De Basabe,J.D.,and Sen,M.K.(2010).Stability of the high-order finite elements for acoustic or elastic wave propagation with high-order time stepping.Geophys.J.Int.,181(1),577-590.https://doi.org/10.1111/j.1365-246X.2010.04536.x
Dussaud,E.,Symes,W.W.,Williamson,P.,Lemaistre,L.,Singer,P.,Denel,B.,and Cherrett,A.(2008).Computational strategies for reverse-time migration.In Proceedings of the 78th Annual International Meeting,SEG,Expanded Abstracts(pp.2267-2271).Las Vegas:Society of Exploration Geophysicists.
Feng,B.,and Wang,H.Z.(2012).Reverse time migration with source wavefield reconstruction strategy.J.Geophys.Eng.,9(1),69-74.https://doi.org/10.1088/1742-2132/9/1/008
Feng,B.,Zhou,Y.,and Wang,H.Z.(2013).Reply to comment on‘Reverse time migration with source wavefield reconstruction strategy’.J.Eng.Geophys.,10(2),028002.https://doi.org/10.1088/1742-2132/10/2/028002
Fichtner,A.,Bunge,H.P.,and Igel,H.(2006a).The adjoint method in seismology:I.Theory.Earth Planet.Inter.,157(1-2),86-104.https://doi.org/10.1016/j.pepi.2006.03.016
Fichtner,A.,Bunge,H.P.,and Igel,H.(2006b).The adjoint method in seismology-II.Applications:traveltimes and sensitivity functionals.Phys.Earth Planet.Inter.,157(1-2),105-123.https://doi.org/10.1016/j.pepi.2006.03.018
Fichtner,A.,Kennett,B.L.N.,Igel,H.,and Bunge,H.P.(2009).Full seismic waveform tomography for upper-mantle structure in the Australasian region using adjoint methods.Geophys.J.Int.,179(3),1703-1725.https://doi.org/10.1111/j.1365-246X.2009.04368.x
Fletcher,R.P.,and Robertsson,J.O.A.(2011).Time-varying boundary conditions in simulation of seismic wave propagation.Geophysics,76(1),A1-A6.https://doi.org/10.1190/1.3511526
Gauthier,O.,Virieux,J.,and Tarantola,A.(1986).Two-dimensional nonlinear inversion of seismic waveforms;numerical results.Geophysics,51(7),1387-1403.https://doi.org/10.1190/1.1442188
Jund,S.,and Salmon,S.(2007).Arbitrary high-order finite element schemes and high-order mass lumping.Int.J.Appl.Math.Comput.Sci.,17(3),375-393.https://doi.org/10.2478/v10006-007-0031-2
Komatitsch,D.,and Vilotte,J.P.(1998).The spectral element method:an efficient tool to simulate the seismic response of 2D and 3D geological structures.Bull.Seismol.Soc.Am.,88(2),368-392.
Komatitsch,D.,and Tromp,J.(1999).Introduction to the spectral element method for three-dimensional seismic wave propagation.Geophys.J.Int.,139(3),806-822.https://doi.org/10.1046/j.1365-246x.1999.00967.x
Komatitsch,D.,and Tromp,J.(2002a).Spectral-element simulations of global seismic wave propagation-I.Validation.Geophys.J.Int.,149(2),390-412.https://doi.org/10.1046/j.1365-246X.2002.01653.x
Komatitsch,D.,and Tromp,J.(2002b).Spectral-element simulations of global
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