Elastic wave propagation in an irregularly layered medium

Vai, Rossana; Castillo-Covarrubias, José M.; Sánchez-Sesma, F. J.; Komatitsch, Dimitri; Vilotte, J. P.

doi:10.1016/s0267-7261(98)00027-x

Cited by 44 publications

(21 citation statements)

References 21 publications

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“…The computations are based on two different numerical techniques, namely, the Fourier-Chebyshev pseudospectral method (PSM) [12][13][14] and the spectral finite-element method (SEM) [15][16][17][18][19][20]. The first is based on global differential operators in which the field is expanded in terms of Fourier and Chebyshev polynomials, while the second is an extension of the finite-element method that uses Legendre polynomials as interpolating functions.…”

Section: Time-domain Modeling Methodsmentioning

confidence: 99%

“…Each of these terms, integrated over an element Ω e , is easily computed by substituting the expansion of the fields (14), computing gradients using (17) and the chain rule (18), and using the GLL integration rule (16). After this spatial discretization with spectral elements, imposing that (12) holds for any test vector w N , as in a classical FEM, we have to solve an ordinary differential equation in time.…”

Section: The Spectral-element Methodsmentioning

confidence: 99%

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Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

et al. 2011

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Nathalie Favretto-Cristini. Elastic surface waves in crystals -part 2: cross-check of two full-wave numerical modeling methods. Ultrasonics, Elsevier, 2011, 51 (8) AbstractWe obtain the full-wave solution for the wave propagation at the surface of anisotropic media using two spectral numerical modeling algorithms. The simulations focus on media of cubic and hexagonal symmetries, for which the physics has been reviewed and clarified in a companion paper. Even in the case of homogeneous media, the solution requires the use of numerical methods because the analytical Green's function cannot be obtained in the whole space. The algorithms proposed here allow for a general material variability and the description of arbitrary crystal symmetry at each grid point of the numerical mesh. They are based on high-order spectral approximations of the wave field for computing the spatial derivatives. We test the algorithms by comparison to the analytical solution and obtain the wave field at different faces (stress-free surfaces) of apatite, zinc and copper. Finally, we perform simulations in heterogeneous media, where no analytical solution exists in general, showing that the modeling algorithms can handle large impedance variations at the interface.

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Section: Time-domain Modeling Methodsmentioning

confidence: 99%

Section: The Spectral-element Methodsmentioning

confidence: 99%

Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

et al. 2011

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“…This allows us to make use of the well-known exact Green's function for the full elastic space [38; 57]. The application of the IBEM in the context of irregular layers has been described elsewhere [39]. The spectral element computational grid is depicted in Figure 7, together with the source and receivers locations.…”

Section: An Irregular Layermentioning

confidence: 99%

The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems

Komatitsch¹,

Vilotte²,

Vai³

et al. 1999

Int. J. Numer. Meth. Engng.

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174

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SUMMARYA spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an e cient tool for simulating elastic wave propagation in realistic geological structures in two-and three-dimensional geometries. The computational domain is discretized into quadrangles, or hexahedra, deÿned with respect to a reference unit domain by an invertible local mapping. Inside each reference element, the numerical integration is based on the tensor-product of a Gauss -Lobatto -Legendre 1-D quadrature and the solution is expanded onto a discrete polynomial basis using Lagrange interpolants. As a result, the mass matrix is always diagonal, which drastically reduces the computational cost and allows an e cient parallel implementation. Absorbing boundary conditions are introduced in variational form to simulate unbounded physical domains. The time discretization is based on an energy-momentum conserving scheme that can be put into a classical explicit-implicit predictor=multicorrector format. Long term energy conservation and stability properties are illustrated as well as the e ciency of the absorbing conditions. The accuracy of the method is shown by comparing the spectral element results to numerical solutions of some classical two-dimensional problems obtained by other methods. The potentiality of the method is then illustrated by studying a simple three-dimensional model. Very accurate modelling of Rayleigh wave propagation and surface di raction is obtained at a low computational cost. The method is shown to provide an e cient tool to study the di raction of elastic waves and the large ampliÿcation of ground motion caused by three-dimensional surface topographies.

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“…The amount of computational resources required to process such immense data volume is also vast for which both software and hardware developments are needed to make the modelling efforts more tractable. The current study focuses on the software branch of this effort to particularly reduce the CPU time requirements by facilitating the Multilevel Fast ITURRARÁ N-VIVEROS et al, 2004;CUMMINS et al, 1997), complex-screen method (e.g., XIE and WU, 2001;WU et al, 2000), boundary element method (e.g., TADEU et al, 2006;VAI et al, 1999) and those distinct methods (e.g., GAO and ZHANG, 2006;FU and BOUCHON, 2004;HAINES et al, 2004;FACCIOLI et al, 1997). There are also several other numerical techniques that are not cited herein, but equally capable of solving complex wave propagation problems.…”

Section: Introductionmentioning

confidence: 99%

Simulation of 3-D Teleseismic SV-waves Accelerated by the Multilevel Fast Multipole Method

Çakır

2008

Pure appl. geophys.

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We use first-order perturbation theory to represent the three-dimensional (3-D) seismic structure of the Earth. The background structure is assumed one-dimensional (1-D) with variations in only vertical direction. The perturbations are assumed 3-D volumetric inclusions with certain velocity contrast to the embedding structure. A convolutional type integral equation employing the Green's function of the 1-D medium is utilized to solve the total wavefield at any point in the 3-D medium. The 3-D perturbations acting as secondary sources are allowed to have arbitrary shapes and sizes, and the integral equation is solved using the numerical techniques. A flat Earth structure is assumed in the mathematical formulation and the spherical-to-flat transforms are employed to account for the Earth's sphericity. The 1-D Green's function is extracted using the stable reflectivity method. The integral equation consists of five integrals in the spectral and spatial domains. The Multilevel Fast Multipole Method (MLFMM) employed for the two integrals on the horizontal plane greatly helps reduce the computational time. The speed gain is characterized as logarithmic versus exponential if a traditional technique instead of the MLFMM is applied. The difference between the logarithmic and exponential time complexities becomes evident especially when the data size enlarges.The proposed algorithm is demonstrated on some selected numerical examples where the seismic source creating the incident waves is assumed as planar teleseismic SV waves (i.e., S receiver function). The numerical examples utilizing the Born single scattering approximation are somewhat idealized but still effective to demonstrate the potential use of the algorithm to model the actual S receiver functions.

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Elastic wave propagation in an irregularly layered medium

Cited by 44 publications

References 21 publications

Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

Elastic surface waves in crystals – Part 2: Cross-check of two full-wave numerical modeling methods

The spectral element method for elastic wave equations—application to 2-D and 3-D seismic problems

Simulation of 3-D Teleseismic SV-waves Accelerated by the Multilevel Fast Multipole Method

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