Francisco J. Sánchez-Sesma

Retrieval of the Green’s Function from Cross Correlation: The Canonical Elastic Problem

In realistic materials, multiple scattering takes place and average field intensities or energy densities follow diffusive processes. Multiple P to S energy conversions by the random inhomogeneities result in equipartition of elastic waves, which means that in the phase space the available elastic energy is distributed among all the possible states of P and S waves, with equal amounts in average. In such diffusive regimes, the P to S energy ratio equilibrates in a universal way independent of the particular details of the scattering. We study the canonical problem of isotropic plane waves in an elastic medium and show that the Fourier transform of azimuthal average of the cross correlation of motion between two points within an elastic medium is proportional to the imaginary part of the exact Green’s tensor function between these points, provided the energy ratio ES / EP is the one predicted by equipartition in two and three dimensions, respectively. These results clearly show that equipartition is a necessary condition to retrieve the exact Green’s function from correlations of the elastic field. However, even if there is not an equipartitioned regime and correlations do not allow to retrieve precisely the exact Green’s function, the correlations may provide valuable results of physical significance by reconstructing specific arrivals.

Site effects on strong ground motion

Abstract A review of some of the available methods to study the effects of site conditions on strong ground motion is presented. The need of unified treatment of source, path and side effects in the assessment of seismic risk is pointed out.

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

A spectral element method for the approximate solution of linear elastodynamic equations, set in a weak form, is shown to provide an efficient 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, defined 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 efficient 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 efficiency 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 diffraction is obtained at a low computational cost. The method is shown to provide an efficient tool to study the diffraction of elastic waves and the large amplification of ground motion caused by three-dimensional surface topographies. Copyright

The Optimal Use of Horizontal-to-Vertical Spectral Ratios of Earthquake Motions for Velocity Inversions Based on Diffuse-Field Theory for Plane Waves

The coda of earthquake motions and microtremors are sometimes referred to as diffuse-wave fields. They are generated by the multiple scattering due to the complexity of the Earth. It is accepted that the average cross correlation between the diffuse-field motions at pairs of receivers, in the frequency domain, is proportional to the imaginary part of the Green’s function between these locations. The average autocorrelation of a single receiver is also proportional to the imaginary part of the Green’s function when both the source and receiver are the same. In this study we explored the application of diffuse-field concepts to analyze earthquake records at a site when its site effect can be described using a 1D model. We derived a corollary of Claerbout’s result for a 1D layered medium. We found that the imaginary part of the Green’s function at the free surface is proportional to the square of the absolute value of the corresponding transfer function for a plane, vertically incident wave. We considered a set of incoming plane waves (of P , SV , and SH types) with varying azimuths and incidence angles. After summing up a few hundred synthetics with inclined incidences we obtained horizontal-to-vertical (H/V) spectral ratios that match the ratios estimated from the simple theory of diffuse field. By using observed records in Japan, we found that the earthquake H/V ratios are quite stable and converge rapidly regardless of what part of the waveform is used, except the P -wave part. We also found that their spectral characteristics can be reproduced well by the velocity structures estimated in previous studies. However, theory and observation were not in perfect agreement, which in turn means that the inversion of a 1D structure could be accomplished by adopting the proposed theory for earthquake H/V spectral ratios.

Nucleation of rupture under slip dependent friction law: Simple models of fault zone

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 slip-weakening 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.

Energy Partitions among Elastic Waves for Dynamic Surface Loads in a Semi-Infinite Solid

We examine the energy partitions among elastic waves due to dynamic normal and tangential surface loads in a semi-infinite elastic solid. While the results for a dynamic normal load on the surface of a half-space with Poisson ratio of 1/4 is a well-known result by Miller and Pursey (1955), the corresponding results for a dynamic tangential load are almost unknown. The partitions for the normal and tangential loads were computed independently by Weaver (1985) versus Poisson ratio (0≤ ν ≤1/2), using diffuse-field concepts within the context of ultrasonic measurements. The connection with the surface load point was not explicit, which partially explains why these results did not reach the seismological and engineering literature. The characteristics of the elastic radiation of these two cases are quite different. For a normal load, about 2/3 of the energy leaves the loaded point as Rayleigh surface waves. On the other hand, the tangential load induces a similar amount in the form of body shear waves. It is established that the energies injected into the elastic half-space by concentrated normal and tangential harmonic surface loads are proportional to the imaginary part of the corresponding components of the Green’s tensor when both source and receiver coincide. The relationship between the Green’s function and average correlations of motions within a diffuse field is clearly established.

Boundary integral equations and boundary elements methods in elastodynamics

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.

Multiple Scattering of Elastic Waves by Subsurface Fractures and Cavities

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 displace- ment fields, where one can observe conspicuous peaks in the frequency domain and important wave interactions in the time domain. Finally, we show how our dimen- sionless graphs can be used to deal with materials like clay, sand, or gravel and compare the response with finite-element analysis of elastic beams.

Ground motion in Mexico City during the April 25, 1989, Guerrero earthquake

Abstract Instrumental observations of ground motion in Mexico City during the April 25, 1989, Guerrero earthquake were analyzed. Our aim was to understand various aspects of the seismic response of the valley that had not been completely resolved. Such understanding of the basic mechanisms that control the seismic behavior of the valley sediments is crucial in any modeling attempt. The study of vertical motion for this event, which was shown to be practically unaffected by site conditions, lead to the identification of a prominent long-period Rayleigh wave. This, together with the availability of absolute time for some stations, allowed the establishment of a common time reference for all recordings. Horizontal motion, in contrast, was significantly amplified, with large increases in duration, at lake bed sites. In order to interpret the observed complexity of ground motion we studied two simplified models of soft alluvial valleys. One of these is two-dimensional and it is excited by plane S waves with variable polarization and incidence angles. This model allows three-dimensional response. The other is a three-dimensional axi-symmetric flat valley with a rigid base. Computations were performed in the frequency domain by means of a boundary integral method for the two-dimensional model and using a collocation least-squares technique for the three-dimensional one. Seismograms were obtained through Fourier synthesis. It was found that the irregular soft layer response produces polarization patterns which are similar to the observations, suggesting that the latter are a consequence of three-dimensional effects.

Two perspectives on equipartition in diffuse elastic fields in three dimensions.

The elastodynamic Green function can be retrieved from the cross correlations of the motions of a diffuse field. To extract the exact Green function, perfect diffuseness of the illuminating field is required. However, the diffuseness of a field relies on the equipartition of energy, which is usually described in terms of the distribution of wave intensity in direction and polarization. In a full three dimensional (3D) elastic space, the transverse and longitudinal waves have energy densities in fixed proportions. On the other hand, there is an alternative point of view that associates equal energies with the independent modes of vibration. These two approaches are equivalent and describe at least two ways in which equipartition occurs. The authors gather theoretical results for diffuse elastic fields in a 3D full-space and extend them to the half-space problem. In that case, the energies undergo conspicuous fluctuations as a function of depth within about one Rayleigh wavelength. The authors derive diffuse energy densities from both approaches and find they are equal. The results derived here are benchmarks, where perfect diffuseness of the illuminating field was assumed. Some practical implications for the normalization of correlations for Green function retrieval arise and they have some bearing for medium imaging.