Elastostatic response of a pile embedded in a transversely isotropic half‐space under transverse loading

In the framework of elastostatics, a mathematical treatment is presented for the boundary value problem of the interaction of a flexible cylindrical pile embedded in a transversely isotropic half‐space under transverse loadings. Taking the pile region as a stiffened subdomain of an extended half‐space, the formulation of the interaction problem is reduced to a Fredholm integral equation of the second kind. The necessary set of Green's functions for the transversely isotropic half‐space is obtained by means of a method of potentials. The resulting Green's functions are incorporated into a numerical procedure for the solution of the integral equation. The theoretical response of the pile is presented in terms of bending moment, displacement and slope profiles for a variety of transversely isotropic materials so that the effect of different anisotropy parameters can be meaningfully discussed. Copyright © 2013 John Wiley & Sons, Ltd.     

Interpolation with Splines in Tension: A Green's Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances; however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.     

Displacement ring load Green's functions for saturated porous transversely isotropic tri‐material full‐space

Based on the Biot's poroelastic theory and using scalar potential functions both the ring load and point load displacement Green's functions for a transversely isotropic saturated porous full‐space composed of an upper half‐space, a finite thickness middle layer and a lower half‐space is analytically presented for the first time. It is assumed that each region consists of a different transversely isotropic material. The equations of poroelastodymanics in terms of the solid displacements and the pore fluid pressure are uncoupled with the help of two scalar potential functions, so that the governing equations for the potential functions are either a second order wave equation or a repeated wave‐heat transfer equation of sixth order. With the aid of Fourier expansion with respect to circumferential direction and Hankel integral transforms with respect to the radial direction in cylindrical coordinate system, the response is determined in the form of line integrals in the real space, followed by theorem of inverse Hankel integral transforms. The solutions degenerate to a single phase elastic material, and the results are compared with previous studies, where an excellent agreement may be observed with the results provided in the literature. Some examples of displacement Green's functions are finally given to illustrate the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Reconstructing the Green's function through iteration of correlations

Correlations of ambient seismic noise are now widely used to retrieve the Earth response between two points. In this study, we reconstruct the surface-wave Green's function by iterating the correlation process over the tail of the noise-based correlation function. It has been demonstrated that the so-called C³ function shows the surface-wave part of the Green's function. Using data from 150 continuously recording stations in Europe, the C³ results help in the extraction of the travel-times from noise-based measurements, especially through the suppression of effects caused by non-isotropic source distributions. We present the results of the next iterative step (i.e. C⁵), which show that some coherent signal is still present in the coda of the C³ function, and we investigate the evolution of the reconstruction of the Green's function throughout the iteration process. Finally, we discuss the interest of combining information from the different correlation functions to improve noise-based tomography analysis.     

Wave scattering and diffraction of subsurface cavities in layered half-space for incident SV-P and SH waves

A novel approach is proposed to deal with the problem of wave scattering and diffraction of subsurface cavities embedded in stratified half-space. The subsurface cavity with complex surroundings is treated as a substructure. The continuity condition at the interface between the substructure and the far field of stratified half-space is maintained by applying a free-field approach. As the boundary of the free-field ground is regular, the construction of the dynamic matrices and the evaluation of the wave input on the interface become considerably easier. Based on the previous work with some improvement, a novel approach for evaluation of Green's functions in stratified half-space is presented. The wave equation is decoupled into the one for SV-P wave components and the other one for SH wave component. The precise integration technique ensures high accuracy of the solution of wave equations. The layer merging technique and the dual form equation make it possible to obtain Green's function in closed-form solution of matrix equations. Numerical examples validate accuracy and efficiency of the proposed approach.     

Uniform asymptotic Green's functions for efficient modeling of cracks in elastic layers with relative shear deformation controlled by linear springs

We present a uniform asymptotic solution (UAS) for a displacement discontinuity (DD) that lies within the middle layer of a three‐layer elastic medium in which relative shear deformation between parallel interfaces is controlled by linear springs. The DD is assumed to be normal to the two interfaces between the elastic media. Using the Fourier transform method we construct a leading term in the asymptotic expansion for the spectral coefficient functions for a DD in a three‐layer‐spring medium. Although a closed‐form solution will require a solution in terms of an infinite series, we demonstrate how this UAS can be used to construct highly efficient and accurate solutions even in the case in which the DD actually touches the interface. We compare the results using the Green's function UAS solution for a crack crossing a soft interface with results obtained using a multi‐layer boundary element method. We also present results from an implementation of the UAS Green's function approach in a pseudo‐3D hydraulic fracturing simulator to analyze the effect of interface shear deformation on the fracture propagation process. These results are compared with field measurements. Copyright © 2008 John Wiley & Sons, Ltd.     

Dynamic Green's functions for an anisotropic poroelastic half-space

The dynamic responses of an anisotropic poroelastic half-space under an internal point load and fluid source are investigated in the frequency domain in this paper. By virtue of Fourier transform and Stroh formalism, the three-dimensional (3D) general solutions of the anisotropic Biot's coupling dynamics equations are derived in the frequency domain. Considering the two surface conditions, permeable and impermeable, the analytical solutions for displacement fields and pore pressure in half-space under a point source (point load or a fluid source) are obtained. When the material properties are isotropic, the numerical results of the poroelastic half-space are in excellent agreement with the existing analytical solutions. For anisotropic half-space cases, numerical results show the strong dependence of the dynamic Green's functions on the material properties.     

An indirect boundary integral equation method for poroelasticity

This paper presents an indirect boundary integral equation method for analysis of quasi-static, time-harmonic and transient boundary value problems related to infinite and semi-infinite poroelastic domains. The present analysis is based on Biot's theory for poroelastodynamics with fluid viscous dissipation. The solution to a given boundary value problem is reduced to the determination of intensities of forces and fluid sources applied on an auxiliary surface defined interior to the surface on which the boundary conditions are specified. A coupled set of integral equations is established to determine the intensities of forces and fluid sources applied on the auxiliary surface. The integral equations are solved numerically in the Laplace domain for quasi-static and transient problems, and in the frequency domain for time-harmonic excitations. The kernel functions of the integral equation correspond to appropriate Green's functions for a poroelastic full space or half-space. The convergence and numerical stability of the present scheme are established by considering a number of bench mark problems. The versatility of the present method is demonstrated by studying the quasi-static response of a rigid spheroidal anchor, and time-harmonic and transient response of a rigid semi-circular tunnel.     

Impedances of rigid cylindrical foundations embedded in transversely isotropic soils

A complete formulation and implementation for assessment of the response to dynamic loads of cylindrical rigid structures embedded in transversely isotropic elastic half‐spaces is presented. The analysis is performed in the frequency domain and the steady‐state structure response is obtained. The method is based on a non‐singular version of the indirect boundary element method which uses influence functions, instead of Green's functions, as fundamental solutions. These influence functions are the response of an elastic half‐space to distributed, internally applied loads. The proposed method imposes full bonding contact between the foundation and the surrounding soil. Numerical results for displacement (vertical and horizontal) and rotation (twisting and rocking) impedances, showing the influence of the soil anisotropy, are presented. Results for the soil–structure interface tractions and for the displacement field throughout the half‐space are also shown. Copyright © 2006 John Wiley & Sons, Ltd.     

Three‐dimensional time‐harmonic fundamental solutions for a fluid‐saturated poroelastic half‐space with partially permeable free surface

By virtue of a pair of scalar potentials for the displacement of the solid skeleton and the pore fluid pressure field of a saturated poroelastic medium, an alternative solution method to the Helmholtz decomposition is developed for the wave propagation problems in the framework of Biot's theory. As an application, a comprehensive solution for three‐dimensional response of an isotropic poroelastic half‐space with a partially permeable hydraulic free surface under an arbitrarily distributed time‐harmonic internal force field and fluid sources is developed. The Green's functions for the poroelastic fields, corresponding to point, ring, and disk loads, are reduced to semi‐infinite complex‐valued integrals that can be evaluated numerically by an appropriate quadrature scheme. Analytical and numerical comparisons are made with existing elastic and poroelastic solutions to illustrate the quality and features of the solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic medium

This paper presents a rigorous analysis for the static interaction of a cylindrical thin‐walled pile with an inhomogeneous isotropic elastic half‐space under vertical, horizontal, and torsional forces individually applied at the top of pile. The inhomogeneity is specified with the exponential variation of shear modulus along depth of the embedding medium, and the Poisson's ratio is assumed to be constant. By means of a set of Green's functions for pile and soil medium and satisfying the compatibility conditions between the 2 interacting media, the formulation is reduced to coupled Fredholm integral equations. Using the adaptive‐gradient elements, capable of capturing the singular stress transfer at both ends of the pile, a numerical procedure is developed and utilized for evaluating the relevant integral equations and studying the inhomogeneity effect on the soil‐pile interaction responses. The analysis results have been validated for different soil‐pile modulus ratios under axial load and for a Poisson's ratio of 0.3 under lateral load. The procedure does not consider the nonlinear behavior of the soil medium or plastic yielding in the pile section, and the impact of the unreliable results for the case of high Poisson's ratio is not examined.     

Structure of young East Pacific Rise lithosphere from ambient noise correlation analysis of fundamental- and higher-mode Scholte-Rayleigh waves

Inter-station Green's functions estimated from ambient noise studies have been widely used to investigate crustal structure. However, most studies are restricted to continental areas and use fundamental-mode surface waves only. In this study, we recover inter-station surface (Scholte-Rayleigh) wave empirical Green's function (EGFs) of both the fundamental- and the first-higher mode using one year of continuous seismic noise records on the vertical component from 28 ocean bottom seismographs deployed in the Quebrada/Discovery/Gofar transform faults region on the East Pacific Rise. The average phase-velocity dispersion of the fundamental mode (period band 2–30 s) and the first-higher mode (period band 3–7 s) from all EGFs are used to invert for the 1-D average, shear-velocity structure in the crust and uppermost mantle using a model-space search algorithm. The preferred shear-velocity models reveal low velocities (4.29 km/s) between Moho and 25 km depth below sea-surface, suggesting the absence of a fast uppermost mantle lid in this young (0–2 Myr) oceanic region. An even more pronounced low-velocity zone, with shear velocities ～3.85 km/s, appears at a depth between 25–40 km below sea-surface. Along with previous results, our study indicates that the shear velocity in the uppermost oceanic mantle increases with increasing seafloor age, consistent with age-related lithospheric cooling.     

Ocean acoustic noise and passive coherent array processing

Broadband correlation processing for extracting time-domain Green's functions and coherent wavefronts from random ocean noise has been demonstrated recently using experiments and numerical simulations that are consistent with theoretical predictions. Ocean acoustic noise processing presents additional challenges over its seismological counterpart. Mainly, the ocean environment is temporally non-stationary and it is spatially heterogeneous. Further, in the lower underwater acoustic frequency regime of about 20 to 500 Hz, space-time episodic shipping is the dominant noise source. The data from different publications and research groups are gathered here, with the goal being to review recent underwater acoustic research that demonstrates the viability and potential applications of passive coherent array processing in the ocean.     

The March–April 2014 sequence of earthquakes in North Chile

During March–April 2014 a series of earthquakes occurred around the Iquique city located in the northern Chile region. The two largest events of this sequence are the Mw8.2, April 1, 2014 and Mw7.7, April 4, 2014 quakes. Here we computed the nodal planes of eight of the large and well teleseismically recorded events of this series based on grid search, teleseismic moment tensors inversion, empirical Green's function deconvolution and its stack to average the deconvolutions for the Mw = 8.2, April 1, 2014, synthetic Green's function deconvolution and its stack to average the deconvolutions for the same event and 3D static deformation analysis of the above mentioned events based on the AK135 model. Grid search nodal planes and moment tensors suggest the dominance of reverse faulting. Almost all of the calculated teleseismic moment tensors represent a considerable amount of DC (usually more than 90%) and lower amount of CLVD for this sequence of events. Empirical and synthetic Green's function deconvolution showing down dip rupture propagation and 3D static deformation representing higher amount of vertical deformation in comparison with horizontal deformation components plus the existence of uplift and subsidence. According to the aftershocks distribution there is a bilateral distribution of the aftershocks around the first large event of this sequence that occurred March 16, 2014 (Mw6.7) so that they are approximately limited between the Mw8.2 (at north) and Mw7.7 (at south) quakes. Moreover there exist two bands of regional seismicity during early-mid 2014: a shallow off-shore band between the trench and coast and a deeper inland band under the active volcanic chain (both nearly parallel to the trench).     

Simulation of the crosshole method in isotropic and anisotropic media

The crosshole seismic method was simulated using a finite element model with a diagonal mass matrix and a direct integration of the equations of motion in the time domain. The results were compared to those of a more efficient but also more restricted formulation using discrete Green's functions. The effects of the type of excitation, the shape of the applied pulse and the position of the receiver with respect to the source on the shape of the recorded motions were investigated for isotropic and cross anisotropic soil deposits. The computed times of arrival of the different waves were compared to those predicted using curved ray path theory to assess the accuracy of this much simpler procedure as a means to interpret the experimental data and determine the soil properties. Copyright © 1999 John Wiley & Sons, Ltd.     

Crustal structure beneath the Malawi and Luangwa Rift Zones and adjacent areas from ambient noise tomography

Crustal shear wave velocity structure beneath the Malawi and Luangwa Rift Zones (MRZ and LRZ, respectively) and adjacent regions in southern Africa is imaged using fundamental mode Rayleigh waves recorded by 31 SAFARI (Seismic Arrays for African Rift Initiation) stations. Dispersion measurements estimated from empirical Green's functions are used to construct 2-D phase velocity maps for periods between 5 and 28 s. The resulting Rayleigh wave phase velocities demonstrate significant lateral variations and are in general agreement with known geological features and tectonic units within the study area. Subsequently, we invert Rayleigh wave phase velocity dispersion curves to construct a 3-D shear wave velocity model. Beneath the MRZ and LRZ, low velocity anomalies are found in the upper-most crust, probably reflecting the sedimentary cover. The mid-crust of the MRZ is characterized by an ~3.7% low velocity anomaly, which cannot be adequately explained by higher than normal temperatures alone. Instead, other factors such as magmatic intrusion, partial melting, and fluid-filled deep crustal faults might also play a role. Thinning of the crust of a few kilometers beneath the rifts is revealed by the inversion. A compilation of crustal thicknesses and velocities beneath the world's major continental rifts suggests that both the MRZ and LRZ are in the category of rifts beneath which the crust has not been sufficiently thinned to produce widespread syn-rifting volcanisms.     

Elastic lateral dynamic impedance functions for a rigid cylindrical shell type foundation

Elastic lateral dynamic impedance functions are defined as the ratio of the lateral dynamic force/moment to the corresponding lateral displacement/rotation at the top ending of a foundation at very small strains. Elastic lateral dynamic impedance functions have a defining influence on the natural frequencies of offshore wind turbines supported on cylindrical shell type foundations, such as suction caissons, bucket foundations, and monopiles. This paper considers the coupled horizontal and rocking vibration of a cylindrical shell type foundation embedded in a fully saturated poroelastic seabed in contact with a seawater half‐space. The formulation of the coupled seawater–shell–seabed vibration problem is simplified by treating the shell as a rigid one. The rigid shell vibration problem is approached by the integral equation method using ring‐load Green's functions for a layered seawater‐seabed half‐space. By considering the boundary conditions at the shell–soil interface, the shell vibration problem is reduced to Fredholm integral equations. Through an analysis of the corresponding Cauchy singular equations, the intrinsic singular characteristics of the problem are rendered explicit. With the singularities incorporated into the solution representation, an effective numerical method involving Gauss–Chebyshev method is developed for the governing Fredholm equations. Selected numerical results for the dynamic contact load distributions, displacements of the shell, and lateral dynamic impedance functions are examined for different shell length–radius ratio, poroelastic materials, and frequencies of excitation. Copyright © 2016 John Wiley & Sons, Ltd.     

Seismic response of 3D topographical irregularities under incoming elastic waves from point sources

We study some effects that produce 3D topographical irregularities under incoming elastic waves from point sources using the indirect boundary element method. This technique is based on the representation of elastic waves in terms of single-layer boundary sources. In this way reflected and diffracted waves are constructed at the boundaries from where they are radiated by means of boundary sources. The field emited from the point source is computed with the moment tensor for a shear dislocation and analytical expressions of the elastodynamic 3D Green's functions. In this way we can construct a double couple with variable orientation. We compare the technique with that of Bouchon, who used the discrete wave number method for a shear dislocation in a halfspace with a triangular source function. We subsequently apply our method to simulate the seismic response of a mountain of ellipsoidal geometry. The source function that we have considered is a triangular pulse and we show results corresponding to the velocity registered over the surface of the irregularity and that of the halfspace. This motion is presented by means of snapshots showing the evolution of the wavefields that are present in the problem.