Rayleigh waves cause a rolling motion —like ocean waves. Slowest of the seismic waves (travel at around 3 km/s)[r]

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However, most existing engineering simulations are based on Fourier analyses, which is for stationary random processes, while seismic waves are non-stationary random [r]

The aim of this work is to present a comprehensive review on numerical modeling of **seismic** **waves** based on DGSE methods on hybrid hexahedral/tetrahedral grids. These methods combine the flexibility of discon- tinuous Galerkin methods to connect together, through a domain decomposition paradigm, Spectral Element blocks where high-order polynomials are used. DGSE methods are implemented in SPEED (SPectral Elements in Elastodynamics with Discontinuous Galerkin - http://speed.mox.polimi.it), an open-source code aims at simulating large-scale **seismic** events in three-dimensional complex media: from far-field to near-field including soil-structure interaction effects. SPEED is jointly developed at Politecnico di Milano by The Laboratory for Modeling and Scientific Computing (MOX) of the Department of Mathematics and the Department of Civil and Environmental Engineering. More precisely, we will review the main advantages of this approach in terms of stability, accuracy, dispersion and dissipation properties, as well as computational costs.

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If the earth were perfectly elastic, **seismic** **waves**, whatever their origin, would continue to vibrate indefinitely. Because the earth is not perfectly elastic, vibrational energy is progressively dissipated. As the material through which the vibrations pass is strained, energy is lost - in the form of heat - through what is often collectively referred to as solid friction. The loss of elastic energy in this manner is generally known as anelasticity. The quality factor Q is a useful measure of anelastic damping. For most purposes, Q may be conveniently defined in terms of the dissipation factor Q that is, 2 tt Q x is equal to the fraction of the total strain energy dissipated per cycle. In general a better understanding of

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Abstract— In this paper an attempt has been made to study the propagation of G type **seismic** **waves** in homogeneous layer overlying an elastic half space under initial stress. Here we have taken constant rigidity and density in upper layer and variation in elastic modulus in the lower transversely isotropic half space. We have obtained dispersion equations and the displacement of the wave. We have seen that initial stress has dominant effect on the propagation of G type wave. As a particular case dispersion equation coincides with that of Love wave. Dispersion curves are plotted for different variation in inhomogeneity parameters and initial stress parameters. Variation in group velocity against scaled wave number has shown for different values of initial stress parameters. Finally surface plots of group velocity have drawn with respect to wave number and depth parameter different values of initial stress parameter.

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Relationship between displacement and velocity amplitudes of **seismic** **waves** was examined with data at stations within 200 km from 142 local earthquakes in and near Japan. The expected value of the coefficient for the logarithmic velocity amplitude to the logarithmic displacement amplitude is 0.5 when a self-similar scaling model is assumed. Observed value of the coefficient is about 0.8 ∼ 0.9. This value appears to be valid at least in the magnitude range from 3.0 to 6.5. Although a spectral model simulation suggested that apparent large contents of high-frequency components were required to explain the observed coefficient, no distinct deviation from the ω-square model was found in the observed spectral ratios from earthquakes of different sizes, for which the path effects were virtually excluded. By using an empirical Green’s function which would correct the effects of propagation and site amplification, it was shown that the apparent deviation from the self-similar scaling model was due to propagation effects.

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Tunnels are vital underground structures that can withstand earthquakes. Although underground structures, in comparison to surface structures are of high safety regarding **seismic** **waves**, historical evidence and earthquake reports show that these structures are vulnerable to **waves**, which result from earthquake, and outbreak of damage and destruction is possible.

Abstract. The earth’s crust presents two dissimilar rheolog- ical behaviors depending on the in situ stress-temperature conditions. The upper, cooler part is brittle, while deeper zones are ductile. **Seismic** **waves** may reveal the presence of the transition but a proper characterization is required. We first obtain a stress–strain relation, including the effects of shear **seismic** attenuation and ductility due to shear de- formations and plastic flow. The anelastic behavior is based on the Burgers mechanical model to describe the effects of **seismic** attenuation and steady-state creep flow. The shear Lamé constant of the brittle and ductile media depends on the in situ stress and temperature through the shear viscos- ity, which is obtained by the Arrhenius equation and the oc- tahedral stress criterion. The P and S wave velocities de- crease as depth and temperature increase due to the geother- mal gradient, an effect which is more pronounced for shear **waves**. We then obtain the P -S and SH equations of motion recast in the velocity-stress formulation, including memory variables to avoid the computation of time convolutions. The equations correspond to isotropic anelastic and inhomoge- neous media and are solved by a direct grid method based on the Runge–Kutta time stepping technique and the Fourier pseudospectral method. The algorithm is tested with success against known analytical solutions for different shear viscosi- ties. A realistic example illustrates the computation of sur- face and reverse-VSP synthetic seismograms in the presence of an abrupt brittle–ductile transition.

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arrival of a later phase after the ﬁrst arriving shear-wave, at stations along and across the fault zone. We found that the time delay between the shear-wave and the later shear- wave phase is proportional to the propagation distance in the anisotropic fault zone and to the degree of anisotropy. If we deploy **seismic** stations around and along the main fault, calculating these time delay could provide a clue to detect the fault zone. We also found that, in the case of a pure strike slip source, the **seismic** **waves** at stations near the fault zone are more affected by the velocity structure than by the anisotropy of the fault zone. In this case, from usual analysis such as the cross-correlation method we may not be able to detect the anisotropic fault zone and not es- timate the correct shear-wave splitting parameters. How- ever, the anisotropic fault zone can be detected, when the fault zone has large crack density, for normal-fault and dip- slip sources located outside the fault zone with the strike of 45 ◦ or parallel to the fault zone, and for normal-fault and dip-slip sources inside the fault zone with the strike of

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possible the pres e nt knowledge regardin g the velocity-density structure in the Earth, the attenuation of seismic waves , the radiation pattern of the source and [r]

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We follow an approach by Stixrude and Lithgow-Bertelloni (2005) (SLB2005). They use an expression of the mineral’s free energy consisting of two parts: a “cold” part which con- tains an expansion of free energy to fourth order in the fi- nite strain tensor and does not depend on temperature, and a “thermal” part which specifies the free energy of the vibra- tions of the crystal lattice in the quasi-harmonic approxima- tion. Besides its explicit temperature dependency the thermal part also varies with strain via the lattice vibrational frequen- cies. The formulation is completely anisotropic and allows for the calculation of the fully anisotropic tensor of elastic constants if corresponding mineral data are available. The first derivative of free energy with respect to a small change of finite strain due to the passage of **seismic** **waves** yields the stress tensor and a further derivative gives the tensor of elastic constants. By evaluating the expressions for the elas- tic constants at conditions where measurements are avail- able (zero finite strain), the expansion coefficients of the free energy can be determined. For the current application, we assume isotropic finite strains and calculate the equivalent isotropic elastic constants of the medium expressed as the bulk (K) and shear modulus (G). To quantify the thermal part of free energy, SLB2005 assume a parabolic Debye density of vibrational states with a maximum frequency that deter- mines the Debye temperature 2 D . This permits an explicit

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Two conditions were examined. In the first condition, in seven separate mole-rat territories (at least 50·m apart), identified above ground through the mounds of excavated soil forming a straight line, we dug a small rectangular ditch (50·cm 60·cm) across a tunnel, bisecting it into two disconnected parts, as previously described (Kimchi and Terkel 2003a,b). For each ditch, six vertical geophones (Geo space GSC-20D, Houston, TX, USA; vibration detection above 20·Hz) were inserted in the ground at 30–40·cm intervals, at a distance of 15–20·cm from the boundaries of the ditch (Fig.·1A), and connected to a multi-channel tape-recorder (6 channels analog tape, TASCAM, Montebello, CA, USA) (Fig.·1A). Using this geophone array we recorded the **seismic** **waves** generated by each mole-rat in its respective territory throughout the entire process of burrowing the bypass tunnel to detour the obstacle and reconnect the two disconnected tunnel parts.

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More specifically, the aim of the modeling performed here is to “back analyze” a landslide event for which there is no completely known destabilizing action, since, due to the near-field conditions, there is uncertainty related to the inci- dence angle of the **seismic** **waves**. For this reason, the pro- posed approach needs a site-specific and well-constrained engineering–geological model of the landslide slope, i.e. a model supported by in situ surveys and investigations. To test the dependence of the rockslide triggering on the inci- dence angle of the **seismic** **waves**, a “sensitivity analysis” was performed by assuming different values of the incident angles. The “validation process” consisted in comparing the obtained results with the location and size of the unstable ar- eas that resulted from the modeling and those observed in the field. Moreover, a “parametric analysis” was performed to provide more robust results, making it possible to evaluate the weight of the errors related to the attributed values of the mechanical parameters.

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It has been established that the degree of damage to the pipeline during an earthquake depends on a whole range of factors: the strength of the **seismic** impact and the directions in which the **seismic** **waves** travel, the geological and hydrological conditions, the operational loads and impacts, the construction of the pipeline and joints, the characteristics of the pipe steel, and the material of the supports, as well as the degree of ‘wear and tear’ in the pipeline. It is not uncommon to see old pipelines fail even after very weak earthquakes, which are barely felt otherwise.

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One of the main difficulties arising in the problems of geophysics, **seismic** exploration, and problems of calculating oscillations in an infinite medium is integration into infinity. For the transition from an infinite region to a finite one, the introduction of external friction is used. For example, in [22] it was shown that for longitudinal oscillations of a composite semi-infinite elastic rod, the spectral problem is equivalent to the problem of the natural vibrations of a finite rod with external damping of damper type. This method - the introduction of external friction - has also been used to determine the periods of natural oscillations of the chimney [7], consideration of forced oscillations of the axial symmetric viscoelastic cylindrical shell [39,52,58], the intrinsic and forced oscillations of the model of the construction-ground system [7, 8], spatial forms of oscillations of the axis of symmetric structures [31]. In [29, 30], the natural oscillations of a cylindrical layer in an elastic, infinite medium are investigated. The problem is solved numerically and analytically in a flat formulation. The solution of the problem of natural oscillations of rectangular bodies in an elastic medium is considered in [31]. The problem is solved in a flat formulation by the method of separated variables. In this case, the natural frequencies are complex. The imaginary parts of the eigen frequencies denote the attenuation coefficient. The real and imaginary parts of the complex natural frequency have not been studied sufficiently. The intrinsic oscillations of pipelines in an elastic medium with specific parameters

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for the reflected and incident P waves we may calculate the actual value of the initial stress near the core-mantle boundary... FORMULATION AND SOLUTION OF THE PROBLEM..[r]

K for an unsupported cavity, but the presence of a lining reduces the concentration of stresses in the massif by 20- 40%. It is also found that increasing the rigidity of the lining leads to an increase in stress concentrations. That most soils are known to exhibit dynamic inelastic effects, such as many cohesive soils, behave as an essentially visco elastic medium [42, 43]. Elastic idealization in solving diffraction problems for such media may not be accurate enough. In [42] numerical results are presented for the case of diffraction of stepwise **waves** of tension and shear on a concrete cylinder placed in sandstone, which is characterized by the properties of a standard linear body. When calculating the following: the relative thickness of the lining = 0 , 05

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Arrival times, amplitudes, and periods of the seismic phases SKB and SEKS have been investigated for normal, intermediate, and deep earthquakes recorded at Pasaden[r]

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It is thus seen that it would be quite impossible to assign a period if there is only one complete pure wave in the interval; but if n is large , the period corresponding to the central [r]

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Subduction zones are often simplified to a 2D system for modelling purposes. While this reduces the computational costs, and therefore allows greater complexity to be considered, it also means that along strike variations are generalised and 3D effects are ignored. The waveform fitting and coda decay resolution tests require a large amount of models to be run. These models must also be run at relatively high resolution in order to be able to accurately simulate **seismic** **waves** up to at least 5 Hz. Therefore these models are run in 2D as the computational cost of running all these models in 3D would be far too high. To give an indication of the relative costs of modelling a subduction zone system in 2D and 3D figure 4.8 shows the number of calculations that are needed for a 90 second simulation, at various frequencies in 2D and 3D. At the relevant frequencies the 3D model is far more expensive. 3D effects may however be important, especially in relation to waveguide structures. For this reason the **seismic** structures that are resolved by the 2D modelling are tested using the more computationally expensive 3D models. With the computational resources available for this project we are able to benchmark 3D models with a resolution of up to 2.5 Hz. Both the 2D and 3D models are run in parallel, with the model domain split over several processors as described in chapter 3. The specific parallel setups of the 2D and 3D models are described below.

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