SUMMARY We present an introduction to the spectral element method, which provides an innovative numerical approach to the calculation of synthetic seismograms in 3-D earth models. The method combines the £exibility of a ¢nite element method with the accuracy of a spectral method. One uses a weak formulation of the equations of motion, which are solved on a mesh of hexahedral elements that is adapted to the free surface and to the main internal discontinuities of the model. The wave¢eld on the elements is discretized using high-degree Lagrange interpolants, and integration over an element is accomplished based upon the Gauss^Lobatto^Legendre integration rule. This combination of discretization and integration results in a diagonal mass matrix, which greatly simpli¢es the algorithm. We illustrate the great potential of the method by comparing it to a discrete wavenumber/re£ectivity method for layer-cake models. Both body and surface waves are accurately represented, and the method can handle point force as well as moment tensor sources. For a model with very steep surface topography we successfully benchmark the method against an approximate boundary technique. For a homogeneous medium with strong attenuation we obtain excellent agreement with the analytical solution for a point force.

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Introduction to the spectral element method for three-dimensional seismic wave propagation

A new numerical method is presented for propagating elastic waves in heterogeneous earth media, based on spectral approximations of the wavefield combined with domain decomposition techniques. The flexibility of finite element techniques in dealing with irregular geologic structures is preserved, together with the high accuracy of spectral methods. High computational efficiency can be achieved especially in 3D calculations, where the commonly used finite-difference approaches are limited both in the frequency range and in handling strongly irregular geometries. The treatment of the seismic source, introduced via a moment tensor distribution, is thoroughly discussed together with the aspects associated with its numerical implementation. The numerical results of the present method are successfully compared with analytical and numerical solutions, both in 2D and 3D.

2D and 3D elastic wave propagation by a pseudo-spectral domain decomposition method

The series solution of wave functions for 2D scattering and diffraction of plane SH (shear horizontal) waves induced by a U‐shaped canyon is proposed herein to account for the topographic effect of such a canyon. The wave function expansion method has been frequently employed to study the topographic effect because it can reveal the physics of the wave scattering and can test the accuracy of other methods. Through a new domain decomposition strategy, the half‐space having a U‐shaped canyon is divided into three subregions. Hence, we defined three cylindrical coordinate systems. In each coordinate system, the wave field satisfying the Helmholtz equation was represented by means of the separation of variables method, in terms of the series of both Bessel functions and Hankel functions with unknown complex coefficients. Then three wave fields are all represented in the same coordinate system using the Graf addition theorem. The unknown coefficients are solved by satisfying the continuity conditions of the auxiliary boundary and the traction‐free boundary conditions on the bottom of the canyon. To show the effects of symmetrical and nonsymmetrical U‐shaped canyons on the surface ground motion, a parametric analysis is carried out in the frequency domain. Surface and subsurface transient responses in the time domain demonstrate the phenomenon of wave propagating and scattering. It is found that a zone of amplification can obviously take place at the bottom of a U‐shaped canyon with nearly vertical walls.

Effects of Topographic Amplification Induced by a U‐Shaped Canyon on Seismic Waves

SUMMARY

Topographies have significant effects on seismic waves. The wavefunction expansion method has been frequently employed to study the topographic effect because this method can reveal the physics of the wave scattering and can testify the accuracy of numerical methods. The 2-D scattering and diffraction of plane SH waves induced by a non-symmetrical V-shaped canyon is examined here by using the wavefunction expansion method. Through a new domain decomposition strategy, the half-space containing a V-shaped canyon is divided into three subregions. Then the wavefield for every subregion is constructed in terms of an infinite series of wavefunctions with unknown coefficients in three coordinate systems, respectively. After that, three wavefields are all represented in the same coordinate system using the Graf addition theorem. The unknown coefficients are obtained by satisfying the continuity conditions of the auxiliary boundary. The proposed series solution of wavefunctions for a non-symmetrical V-shaped canyon can be reduced to a symmetrical case. Results in this study for the symmetrical case agree very well with those in the published literature. To show the effects of the non-symmetrical-geometry topography on the surface ground motion, a parametric study is carried out in the frequency domain. Surface and subsurface transient responses in the time domain demonstrate the phenomenon of wave propagating and scattering. Finally, the proposed model is successfully applied to two idealized cross sections of Pacoima canyon and Feitsui canyon.

Scattering of SH waves induced by a non-symmetrical V-shaped canyon

We review the application of boundary integral equation (BIE) methods to elastic wave propagation problems. BIE methods express the wavefield as an integral equation defined over the boundary of the domain studied. They can be grouped into two families. “Direct” BIE relate the wavefield (generally the displacements and tractions) to the values of the wavefield at the boundary of the domain, while “Indirect” BIE rely on an intermediate unknown, which is usually a distribution of fictitious sources along the boundary. We present the mathematical bases of the methods and discuss and review their various numerical implementations. We illustrate some of the applications for seismic wave propagation.

Boundary Integral Equations and Boundary Elements Methods in Elastodynamics

SUMMARY The diffraction of P, S and Rayleigh waves by 3-D topographies in an elastic halfspace is studied using a simplified indirect boundary element method (IBEM). This technique is based on the integral representation of the diffracted elastic fields in terms of single-layer boundary sources. It can be seen as a numerical realization of Huygens’ principle because diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. A Fredholm integral equation of the second kind for such sources is obtained from the stress-free boundary conditions. A simplified discretization scheme for the numerical and analytical integration of the exact Green’s functions, which employs circles of various sizes to cover most of the boundary surface, is used. The incidence of elastic waves on 3-D topographical profiles is studied. We analyse the displacement amplitudes in the frequency, space and time domains. The results show that the vertical walls of a cylindrical cavity are strong diffractors producing emission of energy in all directions. In the case of a mountain and incident P, SV and SH waves the results show a great variability of the surface ground motion. These spatial variations are due to the interference between locally generated diffracted waves. A polarization analysis of the surface displacement at different locations shows that the diffracted waves are mostly surface and creeping waves.

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Diffraction of P, S and Rayleigh waves by three-dimensional topographies

A method is proposed to approximately compute the seismic response of shallow three-dimensional (3D) alluvial valleys. It is based on the exact solution for a simple configuration. Namely, the antiplane response of a rectangular deposit with a moving rigid base. In such a solution, which is used as a framework for our approximations, the spectral signature of locally generated surface waves is explicit. Thus, an approximate expression is constructed to separately account for one-dimensional (1D) response and lateral effects. Results are presented for two- and three-dimensional cases. Comparisons are provided with rigorous solutions as well. Our formulas allow for fast computations.

Seismic response of shallow alluvial valleys: The use of simplified models

We present the application of high-resolution ground inclination measurements to estimate some elastic properties in shallow stratified terrain. The direct problem for the solution of vertical and radial displacements under static loads is illustrated by means of the well-known Boussinesq problem for the elastic half-space. More realistic (stratified) configurations are taken into account by means of the stiffness matrix method proposed by Kausel and Roesset. In order to test our procedure some inclination surveys were performed in the virgin Texcoco Lake zone in Mexico City. Results are presented for the shear wave velocity and depth of the uppermost strata. A good fit between theoretical and observed inclinations was obtained. This leads to improved estimates of the ground parameters studied. Comparisons with results from independent seismic wave refraction profiles and cone resistance values validate our procedure. The agreement between the results from measurements of the ground surface inclination an...

J. L. Rodríguez-Zúñiga

Papers

Diffraction of P, S and Rayleigh waves by three-dimensional topographies

Seismic response of shallow alluvial valleys: The use of simplified models

Use of in situ measurements of ground inclination for determining shallow elastic parameters