Seismic analysis of semi-sine shaped alluvial hills above subsurface circular cavity

In this study, a seismic analysis of semi-sine shaped alluvial hills above a circular underground cavity subjected to propagating oblique SH-waves using the half-plane time domain boundary element method (BEM) was carried out. By dividing the problem into a pitted half-plane and an upper closed domain as an alluvial hill and applying continuity/boundary conditions at the interface, coupled equations were constructed and ultimately, the problem was solved step-by-step in the time domain to obtain the boundary values. After solving some verification examples, a semi-sine shaped alluvial hill located on an underground circular cavity was successfully analyzed to determine the amplification ratio of the hill surface. For sensitivity analysis, the effects of the impedance factor and shape ratio of the hill were also considered. The ground surface responses are illustrated as three-dimensional graphs in the time and frequency domains. The results show that the material properties of the hill and their heterogeneity with the underlying half-space had a significant effect on the surface response.

     

Uniform half-plane elastodynamic problems by an approximate boundary element method

In this paper the applicability of an approximate Boundary Element Method to uniform half-plane elastodynamic problems is investigated. This method employs the concept of images to construct approximate fundamental solutions for the half-plane and does not require any half-plane surface discretization. The method is formulated in the frequency domain for the case of harmonic disturbances or the Laplace transform domain for the case of transient disturbances. In the latter case a numerical inversion of the transformed solution is necessary to obtain the time domain response. The proposed method can be used as an alternative to boundary element methods that either utilize the infinite plane fundamental solution and thus require a half-plane surface discretization, or employ the exact half-plane fundamental solution, which even though leads to no surface discretization, is of a very lengthy and complicated form. Two characteristics numerical examples are used to illustrate the proposed method and study its advantages and disadvantages.     

Surface motion of multiple alluvial valleys for incident plane SH-waves by using a semi-analytical approach

In this paper, the degenerate kernels and Fourier series expansions are adopted in the null-field integral equation to solve the exterior Helmholtz problems with alluvial valleys. The main gain of using degenerate kernels in integral equations is free of calculating the principal values for singular integrals by locating the null-field point exactly on the real boundary. An adaptive observer system is addressed to fully employ the property of degenerate kernels for circular boundaries in the polar coordinate. Image concept and technique of decomposition are utilized for half-plane problems. After moving the null-field point to the real boundary and matching the boundary conditions, a linear algebraic system is obtained without boundary discretization. The unknown coefficients in the algebraic system can be easily determined. The present method is treated as a “semi-analytical” solution since error only attributes to the truncation of Fourier series. Earthquake analysis for the site response of alluvial valley or canyon subject to the incident SH-wave is the main concern. Numerical examples including single and successive alluvial valleys are given to test our program. Limiting cases of a single canyon and two successive canyons are also addressed. Amplification of soft basin is also observed in this study. The validity of the semi-analytical method is verified. Our advantages, well-posed model, principal value free, elimination of boundary-layer effect and exponential convergence and mesh-free, by using the present method are achieved.     

Focusing phenomenon and near-trapped modes of SH waves

In this study, the null-field boundary integral equation method (BIEM) and the image method are used to solve the SH wave scattering problem containing semi-circular canyons and circular tunnels. To fully utilize the analytical property of circular geometry, the polar coordinates are used to expand the closed-form fundamental solution to the degenerate kernel, and the Fourier series is also introduced to represent the boundary density. By collocating boundary points to match boundary condition on the boundary, a linear algebraic system is constructed. The unknown coefficients in the algebraic system can be easily determined. In this way, a semi-analytical approach is developed. Following the experience of near-trapped modes in water wave problems of the full plane, the focusing phenomenon and near-trapped modes for the SH wave problem of the half-plane are solved, since the two problems obey the same mathematical model. In this study, it is found that the SH wave problem containing two semi-circular canyons and a circular tunnel has the near-trapped mode and the focusing phenomenon for a special incident angle and wavenumber. In this situation, the amplification factor for the amplitude of displacement is over 300.     

Peak surface motion due to scattering of plane harmonic P,SV, or Rayleigh waves by a rough cavity embedded in an elastic half-space

Scattering of plane harmonic P, SV, or Rayleigh waves by a two-dimensional rough cavity completely embedded in an isotropic elastic half-space is investigated by using a direct boundary integral equation method. The cavity’s roughness is assumed to be in the form of periodic or random perturbations of arbitrary amplitude superimposed to a smooth elliptical shape. For the randomly corrugated cavities the normal or the uniform probability distribution functions are assumed. Based on multiple random cavity results, the corresponding average surface response is computed. These are compared with the corresponding periodically corrugated and smooth cavity responses. The surface response is evaluated for different cavity shapes and incident waves and for a range of frequencies. The surface motion results are used to determine the peak surface motion frequencies. They depend strongly upon the basic inclusion shape (the principal axes) and the nature of the incident wave. Strong similarity in the peak surface motion frequencies can be observed for the rough and smooth cavity models for both circular and elliptical shapes. In order to quantify the importance of the cavity corrugation upon the surface motion, a roughness influence factor is defined in terms of the rough and smooth cavity surface responses. This factor strongly depends upon the type of the incident wave, the nature of the cavity corrugation, the basic cavity shape, and the frequency. The factor clearly shows the effect of the cavity roughness upon the surface motion.     

Time-domain two-dimensional site response analysis of non-homogeneous topographic structures by a hybrid BE/FE method

In this paper, an advanced formulation of the time-domain, two-dimensional hybrid finite element–boundary element method (FEM/BEM) is presented, and applied to carry out site response analysis of homogeneous and non-homogeneous topographic structures subjected to incident in-plane motions. Seismic responses of half-plane, horizontally layered site, alluvial valley and ridge sections subjected to incident P and SV waves are analyzed in order to demonstrate the applicability and efficiency of the presented method. The numerical results show that hybrid BE/FE methods require smaller time steps than those needed by BEM schemes. They also show that in case of surface irregularities with height to half-width ratio of up to one, the topography effect could be noticeable, if incident waves have wavelengths of less than approximately eight times the width.     

COMPUTATION OF VLF RESPONSE OVER HALF-PLANE AND WEDGE MODELS*

A theoretical solution is presented to the problem where a VLF anomaly is generated by a conducting half-plane or a perfectly conducting wedge below a stratified overburden. The solution is obtained by the use of a scattering matrix for plane-wave eigenfunctions. VLF anomalies have been computed for different values of the conductance and dip of the half-plane. The phase of the VLF anomaly due to a conducting half-plane depends on the conductance and the distance to the half-plane. Close to the half-plane the tilt angle and ellipticity are of opposite sign for a perfect conductor, but the ellipticity will change sign for a poor conductor. The VLF anomaly for a perfectly conducting wedge is essentially determined by the position of the upper surface of the wedge, i.e. the anomaly will closely resemble the anomaly of a perfectly conducting half-plane in the same position as the upper surface of the wedge.     

Scattering of elastic waves in cracked media using a finite difference method

This paper presents a simple, flexible way of introducing stress-free boundary conditions for including cracks and cavities in 2D elastic media by a finite difference method (FDM). The surfaces of cracks and cavities are discretized in a staircase on a rectangular grid scheme. When zero-stress is applied to free surfaces, the resulting finite difference schemes require a set of adjacent fictitious points. These points are classified based on the geometry of the free surface and their displacement is computed as a prior step to later calculation of motion on the crack surface. The use of this extra line of points does not involve a significant drain on computational resources. However, it does provide explicit finite difference schemes and the construction of displacement on the free surfaces by using the correct physical boundary conditions. An accuracy analysis compares the results to an analytical solution. This quantitative analysis uses envelope and phase misfits. It estimates the minimum number of points per wavelength necessary to achieve suitable results. Finally, the method is employed to compute displacement in various models with cavities in the P-SV formulation. The results show suitable construction of the reflected P and S waves from the free surface as well as diffraction produced by these cavities.     

近地表地震地质复杂性的一种定量分析方法

复杂近地表散射衰减所有的地面观测波场,形成半随机半相干的近地表强散射噪音背景,弥漫整个炮集,淹没深层反射信号,是导致地震资料极低信噪比的主要原因.如何研究和评价近地表散射强度一直是石油勘探未解决的问题,这与起伏地表的粗糙度、近地表速度横向变化和结构倾角分布密切相关.基于前期复杂近地表边界元法波动方程数值模拟研究,本文提出一种复杂近地表散射振幅矩阵方法来分析近地表散射强度.首先对复杂近地表结构进行边界元配置方法离散,根据边界积分方程生成矩阵方程.我们不求解该矩阵方程(涉及海量计算),只是利用矩阵分析技术来解析矩阵方程中的散射振幅系数矩阵,研究复杂近地表结构对不同频率波场的散射强度.该方法利用边界元对近地表结构几何特征的精确表征,研究起伏地表和非规则地质分界面对地震波传播的影响,由基本解及其在边界上的法向导数经过高斯数值积分计算得到的散射振幅系数矩阵,不仅描述了任意两点之间的相互影响,同时还刻画了边界形状特征的影响,为评价不同地质结构的散射强度提供了可能性.作为初步评价手段,我们采用矩阵元素总和与矩阵维数之比作为表征散射振幅系数矩阵散射特征的标量复杂系数,通过理论和实际模型测试,形成了一套行之有效、计算快速的近地表复杂性分析方法.     

Seismic response of lined tunnels in the half-plane with surface topography

In this work, we examine the seismic response of multiple tunnels reinforced with liners and buried within the elastic homogeneous half-plane in the presence of surface relief. The seismic waves are upward propagating, time-harmonic, horizontally polarized shear (SH) waves. More specifically, we examine: (a) the scattered wave fields along the free surface and inside the half-plane with the embedded tunnels; (b) the dynamic stress concentration factors that develop at the soil-liner interfaces; (c) the stresses and displacements that develop inside the tunnel liners. We use a sub-structuring technique that is based on the direct boundary element method to model each constituent part of the problem separately. Then, assembly of the full problem is accomplished through the imposition of compatibility and equilibrium conditions at all interfaces. Next, a detailed verification study is carried out based on comparisons against available analytical and/or numerical results for a series of test examples. Subsequently, detailed numerical simulations are conducted and the results of these parametric studies reveal the influence of the following key parameters on the soil-tunnel system response: (a) the shape of the free-surface relief; (b) the depth of placement of the tunnels and their separation distance; (c) the SH-wavelength to tunnel diameter ratio; (d) the elastic properties of the tunnel lining rings and (e) the dynamic interaction effects between the free-surface relief and the tunnels.     

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.     

Research on the method for retrieving soil moisture using thermal inertia model

1 Introduction Thermal inertia is a bulk property that shows the re- sistance of a material to an input or output of heat. This plays a very important role in certain geological and hydrological studies, and climate modeling. In the 1970s, a simple thermal inertia model was proposed by Watson et al.[1―3]. Pratt (1979)[4] improved the thermal inertia model based on application tests where more factors were considered such as solar ra- diance, thermal conductivity effect, average humidity of g…     

Boundary integral equation method for the diffraction of elastic waves using simplified green's functions

An alternate formulation of the ‘substructure deletion method’ suggested by Dasgupta in 1979¹ has been successfully implemented. The idea is to utilize simple Green's functions developed for a surface problem to replace the more complicated Green's functions required for embedded problems while still being able to generate an accurate solution. Since the exterior medium is usually represented by a continuum model, the interior medium in the present approach will also be represented by a continuum model rather than a finite element model as suggested originally, thereby eliminating the incompatibility between the solutions of the interior and exterior media. Detailed studies of the method's accuracy and limitations were performed using two-dimensional examples in wave scattering of canyons and alluvial valleys, problems which are more suitable for this method than the embedded foundation problem. The results obtained indicate that the alternate formulation gives accurate results only when the vertical dimension of the scattering object is not too large; if the aspect ratio (vertical over lateral) exceeds a certain limit, the results will not approach the known results given by boundary integral equation solutions or indirect boundary integral equations no matter what the refinement of the model may be. The greatest advantage of the present method is that the task of calculating Green's functions is reduced significantly; computational time using this new formulation is approximately five times less than for conventional boundary integral equation methods.     

2-D transient soil–surface foundation interaction and wave propagation by time domain BEM

This paper shows an effective implementation of the half-plane Green function for surface strip impulses (Lamb's problem), which was previously developed in a closed form by the authors, into the time-domain boundary element method for the analysis of related initial boundary value problems. The time-stepping algorithm utilizing Heaviside step function makes the solution process free from the Rayleigh wave front singularity. Illustrative analyses performed include that: First, the response due to an impulsive uniform strip loading is dealt with in order to check the accuracy of the present solution and to interpret the associated wave motion in the medium. Second, a rigid massless strip surface foundation is analysed when subjected to various impulsive loadings in vertical, horizontal and rotational directions to observe which wave is most concerned with the respective foundation motion. The field response is also of interest with respect to distance attenuation. Third, the dynamic cross-interaction between active and passive foundations through soil is investigated when multiple strip foundations are placed separately on a half-space with a certain distance.     

Detection of underground cavities with monofrequency electromagnetic tomography between boreholes in the frequency range 100 MHz to 1 GHz1

In order to detect underground cavities, we have designed a geophysical method based on the interaction of a monochromatic electromagnetic wave in the frequency band 100 MHz to 1 GHz with the ground situated between two boreholes. Three different approaches are involved in the design of this EM tomographic method.

• 1 The application of an ‘exact’ theory is used to calculate artificial data, based on an integral equation method. These data are inverted using a tomographic algorithm whose basic assumptions are rather coarse. The results show that, however, cavities can very well be detected and their position recovered.
• 2 Data are obtained with a physical reduced-scale model in a water tank, in which all parameters are known in advance. The inversion process confirms the validity of the method.
• 3 Underground cavities are actually detected.

The above approaches are described and discussed and results are shown. The equipment involved and its operation is quite simple: the surface devices are commercially available and only the transmitter and receiver antennae must be specially built. The method is shown to be quite efficient and its cost should be reasonably low.     

Analytical approximations for flow in compressible,saturated, one-dimensional porous media

A nonlinear model for single-phase fluid flow in slightly compressible porous media is presented and solved approximately. The model assumes state equations for density, porosity, viscosity and permeability that are exponential functions of the fluid (either gas or liquid) pressure. The governing equation is transformed into a nonlinear diffusion equation. It is solved for a semi-infinite domain for either constant pressure or constant flux boundary conditions at the surface. The solutions obtained, although approximate, are extremely accurate as demonstrated by comparisons with numerical results. Predictions for the surface pressure resulting from a constant flux into a porous medium are compared with published experimental data.     

Gravity field determination using boundary element methods

The Boundary Element Method (BEM), a numerical technique for solving boundary integral equations, is introduced to determine the earth's gravity field. After a short survey on its main principles, we apply this method to the fixed gravimetric boundary value problem (BVP), i.e. the determination of the earth's gravitational potential from measurements of the intensity of the gravity field in points on the earth's surface. We show how to linearize this nonlinear BVP using an implicit function theorem and how to transform the linearized BVP into a boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation into a linear system of equations. We discuss the major problems of this approach for setting up and solving the linear system. The BVP is numerically solved for a bounded part of the earth's surface using a high resolution reference gravity model, measured gravity values of high density, and a 50 50 m² digital terrain model to describe the earth's surface. We obtain a gravity field resolution of 1 1 km² with an accuracy of the order 10^–3 to 10^–4 in about 1 CPU-hour on a Siemens/Fujitsu SIMD vector pipeline machine using highly sophisticated numerical integration techniques and fast equation solvers. We conclude that BEM is a powerful numerical tool for solving boundary value problems and may be an alternative to classical geodetic techniques.     

3D Diffraction of obliquely incident SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space

This paper studies three-dimensional diffraction of obliquely incident plane SH waves by twin infinitely long cylindrical cavities in layered poroelastic half-space using indirect boundary element method. The approach is validated by comparison with the literature, and the effects of cavity interval, incident frequency, and boundary drainage condition on the diffraction are studied through numerical examples. It is shown that, the interaction between two cavities is significant and surface displacement peaks become large when two cavities are close, and the surface displacement may be significantly amplified by twin cavities, and the influence range with large amplification can be as wide as 40 times of the cavity radius. Surface displacements in dry poroelastic case and saturated poroelastic cases with drained and undrained boundaries are evidently different under certain circumstances, and the differences may be much larger than those in the free-field response.     

Locating near-surface scatterers using non-physical scattered waves resulting from seismic interferometry

We use controlled-source seismic interferometry (SI) and inversion in a unique way to estimate the location of near-surface scatterers and a corner diffractor by using non-physical (ghost) scattered surface and body waves. The ghosts are arrivals obtained by SI due to insufficient destructive interference in the summation process of correlated responses from a boundary of enclosing sources. Only one source at the surface is sufficient to obtain the ghost scattered wavefield. We obtain ghost scattered waves for several virtual-source locations. To determine the location of the scatterer, we invert the obtained ghost traveltimes by solving the inverse problem. We demonstrate the method using scattered surface waves. We perform finite-difference numerical simulations of a near-surface scatterer starting with a very simple model and increase the complexity by including lateral inhomogeneity. Especially for the model with lateral variations, we show the effectiveness of the method and demonstrate the estimation of the subsurface location of a corner diffractor using S-waves. In all models we obtain very good estimations of the location of the scatterer.     

考虑刚度及边界条件的短索索力求解与试验研究

建立了一个非齐次边界拉索力学模型,基于拉索线性振动理论,推导了考虑拉索抗弯刚度及边界条件的拉索频率方程;根据双曲函数的数理特性对频率方程进行了合理简化,得到了实用的频率简化方程,提出了逆过程求解索力的方法和步骤,并探讨了不同支承刚度对索力的影响。设计了室内拉索试验模型并进行了27种工况索力试验,比较了不同工况下的计算索力与实际索力,验证了本文提出的频率简化方程和逆过程求解方法的实用可靠性。