Solution of a Green’s function for a saturated porous medium in a half-space subjected to a torsional force

Based on the solutions of the Green’s function for a saturated porous medium obtained by the authors,and using transformation of axisymmetric coordinates,Sommerfeld integrals and superposition of the influence field on a free surface,the authors have obtained displacement solutions of a saturated porous medium subjected to a torsional force in a half-space.The relationship curves of the displacement solutions and various parameters(permeability,frequency,etc.)under action of a unit of torque are also given in this paper.The results are consistent with previous Reissner’s solutions,where a two-phase medium decays to a single-phase medium.The solution is useful in solving relevant dynamic problems of a twophase saturated medium in engineering.     

Dynamic stiffness matrix of a poroelastic multi-layered site and its Green＇s functions

Few studies of wave propagation in layered saturated soils have been reported in the literature. In this paper, a general solution of the equation of wave motion in saturated soils, based on one kind of practical Biot‘s equation,was deduced by introducing wave potentials. Then exact dynamic-stiffness matrices for a poroelastic soil layer and halfspace were derived, which extended Wolf‘s theory for an elastic layered site to the case of poroelasticity, thus resolving a fundamental problem in the field of wave propagation and soil-structure interaction in a poroelastic layered soil site. By using the integral transform method, Green‘s functions of horizontal and vertical uniformly distributed loads in a poroelastic layered soil site were given. Finally, the theory was verified by numerical examples and dynamic responses by comparing three different soil sites. This study has the following advantages: all parameters in the dynamic-stiffness matrices have explicitly physical meanings and the thickness of the sub-layers does not affect the precision of the calculation which is very convenient for engineering applications. The present theory can degenerate into Wolf‘s theory and yields numerical results approaching those for an ideal elastic layered site when porosity tends to zero.     

In-plane dynamic Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic half-space

The dynamic stiffness method combined with the Fourier transform is utilized to derive the in-plane Green’s functions for inclined and uniformly distributed loads in a multi-layered transversely isotropic(TI)half-space.The loaded layer is fixed to obtain solutions restricted in it and the corresponding reactions forces,which are then applied to the total system with the opposite sign.By adding solutions restricted in the loaded layer to solutions from the reaction forces,the global solutions in the wavenumber domain are obtained,and the dynamic Green’s functions in the space domain are recovered by the inverse Fourier transform.The presented formulations can be reduced to the isotropic case developed by Wolf(1985),and are further verified by comparisons with existing solutions in a uniform isotropic as well as a layered TI halfspace subjected to horizontally distributed loads which are special cases of the more general problem addressed.The deduced Green’s functions,in conjunction with boundary element methods,will lead to significant advances in the investigation of a variety of wave scattering,wave radiation and soil-structure interaction problems in a layered TI site.Selected numerical results are given to investigate the influence of material anisotropy,frequency of excitation,inclination angle and layered on the responses of displacement and stress,and some conclusions are drawn.     

Numerical Green’s function method:Application to quantifying ground motion variations of M7 earthquakes

The concept of "numerical Green’s functions" (NGF or Green’s function database) is developed. The basic idea is: a large seismic fault is divided into subfaults of appropriate size, for which synthetic Green’s functions at the surface (NGF) are calculated and stored. Consequently, ground motions from arbitrary kinematic sources can be simulated, rapidly, for the whole fault or parts of it by superposition. The target fault is a simplified, vertical model of the Newport-Inglewood fault in the Los Angeles basin. This approach and its functionality are illustrated by investigating the variations of ground motions (e.g. peak ground velocity and synthetic seismograms) due to the source complexity. The source complexities are considered with two respects: hypocenter location and slip history. The results show a complex behavior, with dependence of absolute peak ground velocity and their variation on source process directionality, hypocenter location, local structure, and static slip asperity location. We concluded that combining effect due to 3-D structure and finite-source is necessary to quan- tify ground motion characteristics and their variations. Our results will facilitate the earthquake hazard assessment projects.     

Simulation of ground motion in the Moscow region using the empirical Green’s function

Historically, the Moscow region regularly experienced rather weak but quite perceptible seismic vibrations produced by intermediate-depth earthquakes of the Vrancea zone (Romania), located at a distance of 1400 km from Moscow. The coincidence of a number of unique factors such as a slowly varying focal depth, predominant source mechanisms, weak attenuation of seismic radiation in the north-northeast direction provide favorable conditions for application of the empirical Green’s function method. Using the digital seismogram of the Vrancea Mw-5.8 earthquake recorded at the Moscow seismic station, we simulated synthetic seismograms of a scenario (expected maximum) earthquake with Mw = 8.0, by application of the empirical Green’s function method adjusted for the given conditions. The calculation procedure was verified using analog records of strong earthquakes available at the Moscow seismic station. Digital records of the Obninsk seismic station included in the Incorporated Research Institutions for Seismology (IRIS) system were used for additional control. Here, the scenario earthquake was modeled using the data on a much stronger earthquake of 1990 (MW = 6.9). It is shown that, despite a certain scatter (quite adequately assessed in the scope of the method), the ultimate estimates of expected seismic impacts are quite reliable and can be recommended for practical use.     

Broadband strong motion simulation in layered half-space using stochastic Green’s function technique

The stochastic Green’s function method, which simulates one component of the far-field S-waves from an extended fault plane at high frequencies (Kamae et al., J Struct Constr Eng Trans AIJ, 430:1–9, 1991), is extended to simulate the three components of the full waveform in layered half-spaces for broadband frequency range. The method firstly computes ground motions from small earthquakes, which correspond to the ruptures of sub-faults on a fault plane of a large earthquake, and secondly constructs the strong motions of the large earthquake by superposing the small ground motions using the empirical Green’s function technique (e.g., Irikura, Proc 7th Japan Earthq Eng Symp, 151–156, 1986). The broadband stochastic omega-square model is proposed as the moment rate functions of the small earthquakes, in which random and zero phases are used at higher and lower frequencies, respectively. The zero phases are introduced to simulate a smooth ramp function of the moment function with the duration of 1/fc s (fc: the corner frequency) and to reproduce coherent strong motions at low frequencies (i.e., the directivity pulse). As for the radiation coefficients, the theoretical values of double couple sources for lower frequencies and the theoretical isotropic values for the P-, SV-, and SH-waves (Onishi and Horike, J Struct Constr Eng Trans AIJ, 586:37–44, 2004) for high frequencies are used. The proposed method uses the theoretical Green’s functions of layered half-spaces instead of the far-field S-waves, which reproduce the complete waves including the direct and reflected P- and S-waves and surface waves at broadband frequencies. Finally, the proposed method is applied to the 1994 Northridge earthquake, and results show excellent agreement with the observation records at broadband frequencies. At the same time, the method still needs improvements especially because it underestimates the high-frequency vertical components in the near fault range. Nonetheless, the method will be useful for modeling high frequency contributions in the hybrid methods, which use stochastic and deterministic methods for high and low frequencies, respectively (e.g., the stochastic Green’s function method + finite difference methods; Kamae et al., Bull Seism Soc Am, 88:357–367, 1998; Pitarka et al., Bull Seism Soc Am 90:566–586, 2000), because it reproduces the full waveforms in layered media including not only random characteristics at higher frequencies but also theoretical and deterministic coherencies at lower frequencies.     

Use of Green’s function for determining the disturbing potential of an ellipsoidal Earth

A spherical approximation makes the basis for a majority of formulas in physical geodesy. However, the present-day accuracy in determining the disturbing potential requires an ellipsoidal approximation. The paper deals with constructing Green’s function for an ellipsoidal Earth by an ellipsoidal harmonic expansion and using it for determining the disturbing potential. From the result obtained the part that corresponds to the spherical approximation has been extracted. Green’s function is known to depend just on the geometry of the surface where boundary values are given. Thus, it can be calculated irrespective of the gravity data completeness. No changes of gravity data have an effect on Green’s function and they can be easily taken into account if the function has already been constructed. Such a method, therefore, can be useful in determining the disturbing potential of an ellipsoidal Earth.     

Strong ground motion simulation of the 2003 Bam, Iran, earthquake using the empirical Green’s function method

The 2003 Bam, Iran, earthquake caused catastrophic damage to the city of Bam and neighboring villages. Given its magnitude (M _w ) of 6.5, the damage was remarkably large. Large-amplitude ground motions were recorded at the Bam accelerograph station in the center of Bam city by the Building and Housing Research Center (BHRC) of Iran. We simulated the Bam earthquake acceleration records at three BHRC strong-motion stations—Bam, Abaraq, and Mohammad-Abad—by the empirical Green’s function method. Three aftershocks were used as empirical Green’s functions. The frequency range of the empirical Green’s function simulations was 0.5–10 Hz. The size of the strong motion generation area of the mainshock was estimated to be 11 km in length by 7 km in width. To estimate the parameters of the strong motion generation area, we used 1D and 2D velocity structures across the fault and a combined source model. The empirical Green’s function method using a combination of aftershocks produced a source model that reproduced ground motions with the best fit to the observed waveforms. This may be attributed to the existence of two distinct rupture mechanisms in the strong motion generation area. We found that the rupture starting point for which the simulated waveforms best fit the observed ones was near the center of the strong motion generation area, which reproduced near-source ground motions in a broadband frequency range. The estimated strong motion generation area could explain the observed damaging ground motion at the Bam station. This suggests that estimating the source characteristics of the Bam earthquake is very important in understanding the causes of the earthquake damage.     

Singularity-free Green’s function for EM sources embedded in a stratified medium

We present a method to unify the calculation of Green’s functions for an electromagnetic (EM) transmitting source embedded in a homogeneous stratified medium. A virtual interface parallel to layer interfaces is introduced through the source location. The potentials for Green’s function are derived by decomposing the partial wave solutions to Helmholtz’s equations into upward and downward within boundaries. The amplitudes of the potentials in each stratum are obtained recursively from the initial amplitudes at the source level. The initial amplitudes are derived by coupling with the transmitting sources and following the discontinuity of the tangential electric and magnetic fields at the source interface. Only the initial terms are related to the transmitting sources and thus need to be modified for different transmitters, whereas the kernel connected with the stratified media stays unchanged. Hence, the present method can be easily applied to EM transmitting sources with little modification. The application of the proposed method to the marine controlled-source electromagnetic method (MCSEM) demonstrates its simplicity and flexibility.     

Solution of dynamic Green׳s function for unsaturated soil under internal excitation

The closed form three-dimensional Green׳s function of a semi-infinite unsaturated poroelastic medium subjected to an arbitrary internal harmonic loading is derived, with consideration of capillary pressure and dynamic shear modulus varying with saturation. By applying the Fourier expansion techniques and Hankel integral transforms to the circumferential and radial coordinates, respectively, the general solution for the governing partial differential equations is obtained in the transformed domain. A corresponding boundary value problem is formulated. The integral solutions for the induced displacements, pore pressure and net stress are then determined considering the continuity conditions. The formulas are compared with the degenerated solution of saturated soils and confirmed. Numerical results reveal that the response of the unsaturated half-space depends significantly on the saturation by altering dynamic shear modulus to account for the effects of matric suction on soil stiffness. Slight differences between the results occur if only the saturation is taken into account. Moreover, a large source-depth results in a pronounced contribution to the reduction of surface displacement amplitudes. The analytical solutions concluded in the study offer a broader application to dynamic response associated with axi-symmetric and asymmetric conditions.     

Source time-function as discriminant using empirical Green’s function

Since the sign up of CTBT (Comprehensive Test Ban Treaty), the nuclear monitoring and distinguishing has gained great attention by the treaty countries. Seismology is one of the primary technical means available for monitoring compliance to a CTBT. However, there does not exist a single criterion, so far, that can effectively distinguish explorations from earthquakes. It has been known that source time function (abbreviated to STF) of an underground nuclear explosion contains important p…     

The algorithms for calculating synthetic seismograms from a dipole source using the derivatives of Green’s function

The problem of calculating complete synthetic seismograms from a point dipole with an arbitrary seismic moment tensor in a plane parallel medium composed of homogeneous elastic isotropic layers is considered. It is established that the solutions of the system of ordinary differential equations for the motion–stress vector have a reciprocity property, which allows obtaining a compact formula for the derivative of the motion vector with respect to the source depth. The reciprocity theorem for Green’s functions with respect to the interchange of the source and receiver is obtained for a medium with cylindrical boundary. The differentiation of Green’s functions with respect to the coordinates of the source leads to the same calculation formulas as the algorithm developed in the previous work (Pavlov, 2013). A new algorithm appears when the derivatives with respect to the horizontal coordinates of the source is replaced by the derivatives with respect to the horizontal coordinates of the receiver (with the minus sign). This algorithm is more transparent, compact, and economic than the previous one. It requires calculating the wavenumbers associated with Bessel function’s roots of order 0 and order 1, whereas the previous algorithm additionally requires the second order roots.     

Green’s function of the linearized Saint-Venant equations in laminar and turbulent flows

In the present paper, an analytical expression of the Green’s function of linearized Saint-Venant equations (LSVEs) for shallow water waves is provided and applied to analyse the propagation of a perturbation superposed to a uniform flow. Independently of the kinematic character of the base flow, i.e., subcritical or supercritical uniform flow, the effects of a non-uniform vertical velocity profile and a non-constant resistance coefficient are accounted for. The use of the Darcy-Weisbach friction law allows a unified treatment of both laminar and turbulent conditions. The influence on the wave evolution of the wall roughness and the fluid viscosity are finally discussed, showing that in turbulent regime the assumption of constant friction coefficient may lead to an underestimation of both amplification and damping factors on the wave fronts, especially at low Reynolds numbers. This conclusion has to be accounted for, particularly in describing hyper-concentrated suspensions or other kinds of Newtonian mixtures, for which the high values of the kinematic viscosity may lead to relatively low Reynolds numbers.     

Estimated Green’s function extracted from superconducting gravimeters and seismometers

Estimated Green’s function (EGF) between stations has been extracted from ambient seismic noise, direct surface wave and coda waves. It is also confirmed by laboratory experiments on ultrasonics and theoretical derivations assuming diffusive wave field, equi-partition of modes or random sources on an enclosed surface. This method provides a new approach to study the crust and mantle structure at regional scale, continental scale and global scale. Following the achievements with seismometer records, the reco...     

Inverse gravity problems for models composed of bodies of Sretenskii’s class

An inverse gravity problem is solved for a geological model consisting of bodies of Sretenskii’s class. The position of the middle plane is fixed for each body. It is required to determine the upper and lower boundaries of a body, which are described by analytical functions and are parameterized. The solution of the problem is illustrated by an example.     

Green＇s functions for uniformly distributed loads acting on an inclined line in a poroelastic layered site

Based on one type of practical Biot＇s equation and the dynamic-stiffness matrices ofa poroelastic soil layer and half-space, Green＇s functions were derived for unitformly distributed loads acting on an inclined line in a poroelastie layered site. This analysis overcomes significant problems in wave scattering due to local soil conditions and dynamic soil-structure interaction. The Green＇s functions can be reduced to the case of an elastic layered site developed by Wolf in 1985. Parametric studies are then carried out through two example problems.