A network of nine broad-band seismographs was operated from March to May 1994 to study the propagation of seismic waves across the Mexican Volcanic Belt (MVB) in the region of the Valley of Mexico. Analysis of the data from the network reveals an amplification of seismic waves in a wide period band at the stations situated in the southern part of the MVB.The group velocities of the fundamental mode of the Rayleigh wave in the period range 2-13s are found to be lower in the southern part of the MVB than in its northern part and in the region south of the MVB. The inversion of dispersion curves shows that the difference in group velocities is due to the presence of a superficial lowvelocity layer (with an average S-wave velocity of 1.7 km s-' and an average thickness of 2 km) beneath the southern part of the MVB. This low-velocity zone is associated with the region of active volcanism.Numerical simulations show that this superficial low-velocity layer causes a regional amplification of 8-10 s period signals, which is of the same order as the amplification measured from the data. This layer also increases the signal duration significantly because of the dispersion of the surface waves. These results confirm the hypothesis of Singh et al. (1995), who suggested that the regional amplification observed in the Valley of Mexico is due to the anomalously low shear-wave velocity of the shallow volcanic rocks in the southern MVB

[1] The initiation of frictional instability is investigated for simple models of fault zone using a linearized perturbation analysis. The fault interface is assumed to obey a linear slipweakening law. The fault is initially prestressed uniformly at the sliding threshold. In the case of antiplane shear between two homogeneous linearly elastic media, space-time and spectral solutions are obtained and shown to be consistent. The nucleation is characterized by (1) a long-wavelength unstable spectrum bounded by a critical wave number; (2) an exponential growth of the unstable modes; and (3) an induced off-fault deformation that remains trapped within a bounded zone in the vicinity of the fault. These phenomena are characterized in terms of the elastic parameters of the surrounding medium and a nucleation length that results from the coupling between the frictional interface and the bulk elasticity. These results are extended to other geometries within the same formalism and implications for three-dimensional rupture are discussed. Finally, internal fault structures are investigated in terms of a fault-parallel damaged zone. Spectral solutions are obtained for both a smooth and a layered distribution of damage. For natural faults the nucleation is shown to depend strongly on the existence of a internal damaged layer. This nucleation can be described in terms of an effective homogeneous model. In all cases, frictional trapping of the deformation out of the fault can lead to the property that arbitrarily long wavelengths remain sensitive to the existence of a fault zone.INDEX TERMS: 7209 Seismology: Earthquake dynamics and mechanics; 3220 Mathematical Geophysics: Nonlinear dynamics; KEYWORDS: earthquake nucleation, slip instability, fault zone, frictional instability, seismic rupture Citation: Ampuero, J.-P., J.-P. Vilotte, and F. J. Sánchez-Sesma, Nucleation of rupture under slip dependent friction law: Simple models of fault zone,

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S U M M A R YA Triangular Spectral Element Method (TSEM) is presented to simulate elastic wave propagation using an unstructured triangulation of the physical domain. TSEM makes use of a variational formulation of elastodynamics based on unstructured straight-sided triangles that allow enhanced flexibility when dealing with complex geometries and velocity structures. The displacement field is expanded into a high-order polynomial spectral approximation over each triangular subdomain. Continuity between the subdomains of the triangulation is enforced using a multidimensional Lagrangian interpolation built on a set of Fekete points of the triangle. High-order accuracy is achieved by resorting to an analytical computation of the associated internal product and bilinear forms leading to a non-diagonal mass matrix formulation. Therefore, the time stepping involves the solution of a sparse linear algebraic system even in the explicit case. In this paper the accuracy and the geometrical flexibility of the TSEM is explored. Comparison with classical spectral elements on quadrangular grids shows similar results in terms of accuracy and stability even for long simulations. Surface and interface waves are shown to be accurately modelled even in the case of complex topography with the TSEM. Numerical results are presented for 2-D canonical examples as well as more specific problems, such as 2-D elastic wave scattering by a cylinder embedded in an homogeneous half-space. They all illustrate the enhanced geometrical flexibility of the TSEM. This clearly suggests the need of further investigations in computational seismology specifically targeted towards efficient implementations of the TSEM both in the time and the frequency domain.Many problems in geophysics need to infer the physical and chemical parameter distributions of the Earth's interior from information provided by seismic wave propagation through complex media. Moreover, numerical simulations of earthquake-induced wave propagation within heterogeneous geological structures have important implications in terms of earthquake risk assessment and of strong motion predictions. Continuous improvements during the last decades, both in terms of acquisition techniques and seismic network density, lead today the possibility of exploiting new information provided by high-resolution multicomponent seismograms, over a wide range of frequencies. The development of new seismic interpretation methods, which take advantage of these high-resolution * Presently at: Laego-INERIS, Laboratoire Environnement Géomécanique et Ouvrages, Ecole des Mines de Nancy, France. observations, requires accurate numerical modelling tools for the simulation of the complete waveform field in heterogeneous geological media with complex geometries.Recent developments towards high-order numerical modelling of seismic wave propagation (Seriani et al

SUMMARYA boundary integral formulation is presented and applied to model the ground motion on alluvial valleys under incident P, S and Rayleigh waves. It is based on integral representations for the diffracted and the refracted elastic waves using single-layer boundary sources. This approach is called indirect BEM in the literature as the sources' strengths should be obtained as an intermediate step. Boundary conditions lead to a system of integral equations for boundary sources. A discretization scheme based on the numerical and analytical integration of exact Green's functions for displacements and tractions is used. Various examples are given for two-dimensional problems of diffraction of elastic waves by soft elastic inclusion models of alluvial deposits in an elastic half-space. Results are displayed in both frequency and time domains. These results show the significant influence of locally generated surface waves in seismic response and suggest approximations of practical interest. For shallow alluvial valleys the response and its resonant frequencies are controlled by a coupling mechanism that involves both the simple one-dimensional shear beam model and the propagation of surface waves.

In order to explain damage and observed ground motions in Mexico City during the 1985 Michoacán earthquake, simultaneous consideration must be given to source, path, and site conditions. This is clear from teleseismic records and local vertical displacements. Incident waves had an important part of energy in the frequency band of 0.3-1 Hz. Damage distribution and observed motion in the lake bed zone cannot be satisfactorily explained using one-dimensional theory. The effects of lateral irregularities are required. To assess its effects we describe the stratigraphic setting of the valley and discuss some features of damage distribution with results for one- and two-dimensional wave propagation models. These are useful to establish on quantitative basis the importance of lateral heterogeneity.

Comprehensive studies in geophysics and seismology have dealt with scattering phenomena in unbounded elastic domains containing fractures or cavities. Other studies have been carried out to investigate scattering by discontinuities located near a free surface. In this last case, the presence of fractures and cavities significantly affects wave motion and, in some cases, large resonant peaks may appear. To study these resonant peaks and describe how they can be affected by the presence of other near-free-surface fractures or cavities we propose the use of the indirect boundary element method to simulate 2D scattering of elastic P and SV waves. The geometries considered are planar and elliptic cracks and cavities. This method establishes a system of integral equations that allows us to compute the diffracted displacement and traction fields. We present our results in both frequency and time domains. In the planar cracks located near the free surface, we validate the method by comparing results with those of a previously published study. We develop several examples of various fractures and cavities to show resonance effects and total scattered displacement fields, where one can observe conspicuous peaks in the frequency domain and important wave interactions in the time domain. Finally, we show how our dimensionless graphs can be used to deal with materials like clay, sand, or gravel and compare the response with finite-element analysis of elastic beams.

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.