Constraining of the zero total surface charge in galvanic modelling

The galvanic problem is frequently solved by a Fredholm integral equation of the second kind based on a single layer source formulation. At higher conductivity contrasts between the model and its surroundings the homogeneous part of the integral equation approaches an eigenvalue equation. With infinite contrast the solution of this limiting integral equation is non-unique, but in the subspace of zero total charge the solution is unique. This mathematical property of the integral equation is reflected in its numerical solution with the result that large numerical errors may appear and convergence of the solution becomes very slow. Errors are, for the most part, related to the computed excess charge generated in the numerical solution. The effect is studied by comparing the results computed from the solution of the integral equation alone with those computed from a particular solution where the requirement of zero total charge is used as a constraint. The model examples clearly show that the use of the constraint condition significantly improves the accuracy of the results.     

BOUNDARY ELEMENT METHOD FOR THE ARBITRARY INHOMOGENEITIES PROBLEM IN ELECTRICAL PROSPECTING*

The generalized integral equation for the electric potential governed by a quasi-harmonic equation can be derived via a variational formulation. For surface current distributions it is not always a Fredholm integral equation of the second kind. Numerical solutions of the general heterogeneous problem can be obtained with the “reciprocal averaging technique”, where the solution is obtained a second time after exchange of source and field points.     

Ground motion at canyons of arbitrary shape under incident sh waves

A method for calculating the two-dimensional scattering of incident SH waves by canyons of arbitrary shape is presented. The problem is formulated in terms of a Fredholm integral equation of the first kind with the integration path outside the boundary. Point-source discretization and a least-squares scheme are used. Numerical results are compared with the known analytic solution for a semi-cylindrical canyon. Spatial variations of surface amplitudes are computed for triangular and half-cycle sinusoidal canyons as well.     

Does Poisson’s downward continuation give physically meaningful results?

The downward continuation (DWC) of the gravity anomalies from the Earth’s surface to the geoid is still probably the most problematic step in the precise geoid determination. It is this step that motivates the quasi-geoid users to opt for Molodenskij’s rather than Stokes’s theory. The reason for this is that the DWC is perceived as suffering from two major flaws: first, a physically meaningful DWC technique requires the knowledge of the irregular topographical density; second, the Poisson DWC, which is the only physically meaningful technique we know, presents itself mathematically in the form of Fredholm integral equation of the 1st kind. As Fredholm integral equations are often numerically ill-conditioned, this makes some people believe that the DWC problem is physically ill-posed. According to a revered French mathematician Hadamard, the DWC problem is physically well-posed and as such gives always a finite and unique solution. The necessity of knowing the topographical density is, of course, a real problem but one that is being solved with an ever increasing accuracy; so sooner or later it will allow us to determine the geoid with the centimetre accuracy.     

Vertical and horizontal vibrations of a rigid disc on a multilayered transversely isotropic half-space

A half-space containing horizontally multilayered regions of different transversely isotropic elastic materials as well as a homogeneous half-space as the lowest layer is considered such that the axes of material symmetries of different layers and the lowest half-space to be as depth-wise. A rigid circular disc rested on the free surface of the whole half-space is considered to be under a forced either vertical or horizontal vibration of constant amplitudes. Because of the involved integral transforms, the mixed boundary value problems due to mixed condition at the surface of the half-space are changed to some dual integral equations, which are reduced to Fredholm integral equations of second kind. With the help of contour integration, the governing Fredholm integral equations are numerically solved. Some numerical evaluations are given for different combinations of transversely isotropic layers to show the effect of degree of anisotropy of different layers on the response of the inhomogeneous half-space.     

位场曲化平积分方程的迭代解

提出了位场曲化平的新方法. 给定观测曲面S上的位场、S对下方水平面P的相对高程,确定P上的位场. 利用由P向上延拓到S的积分式,建立这两个面上位场及相对高程三者所满足的方程,它是第一类Fredholm积分方程. 用Fourier逆变换式把这一空间域积分式化为波数域积分式,再由指数函数的Taylor展开进一步化为级数式. 积分方程的解采用逐次逼近法迭代计算,即用S上的位场观测值作为P上位场的初始迭代值,用导出的级数式求得S上的位场计算值、由S上的位场观测值与计算值之差校正P上的位场,多次迭代,直到满足迭代终止准则. 我们还给出该积分方程的波数域迭代计算方法. 模型算例表明,重力异常曲化平的均方差和磁异常曲化平的均方差分别为0.0008 mGal和0.0019 nT,在主频为2.26 GHz的笔记本电脑运行,2048×2048数据量,计算时间是975 s. 野外磁场实际资料处理也证实这种方法的有效性.     

Rocking vibrations of rigid disk on saturated poroelastic medium

The dynamic response of a rigid disk on a saturated poroelastic half space and subjected to harmonic rocking excitation is studied. The mixed boundary-value problem for the case of relaxed contact condition between the disk and the poroelastic half space is reduced to a Fredholm integral equation of the second kind, which is solved numerically. The dynamic compliance coefficient for the rocking vibration of a rigid disk on a poroelastic half space is presented.     

Lateral dynamic compliance of pile embedded in poroelastic half space

The time-harmonic response of a pile in a poroelastic half space and under lateral loadings is studied. By treating the pile as a one-dimensional structure and the half-space as a three-dimensional poroelastic continuum, the dynamic interaction between a pile and a poroelastic medium is formulated as a Fredholm integral equation of the second kind. Green's functions for a distributed lateral force field acting inside a poroelastic half space is an important ingredient of this paper. Numerical results for lateral dynamic compliance functions are presented to illustrate the dynamic characteristics of a pile in a poroelastic half space.     

Forced vertical vibration of rigid circular disc buried in an arbitrary depth of a transversely isotropic half space

This paper is concerned with the investigation of the vertical vibration of a rigid circular disc buried at an arbitrary depth in a transversely isotropic half space in such a way the axis of material symmetry of the half space is normal to the surface of it and parallel to the vibration direction. By using the Hankel integral transforms, the mixed boundary-value problem is transformed to a pair of integral equations called dual integral equations, which generally can be reduced to a Fredholm integral equation of the second kind. With the aid of complex variable or contour integration, the governing integral equation is numerically solved in the general dynamic case. Two degenerated cases (i) the disc is buried in a transversely isotropic full space, and (ii) rigid circular disc is attached on the surface of the half space are discussed. The reduced static case of the dual integral equations is solved analytically and the vertical displacement, the contact pressure and the static impedance/compliance function are explicitly found. It is shown that the vertical pressure and the compliance function reduced for isotropic half space are identical to the previous solutions reported in the literature. The dynamic contact pressure under the disc and the impedance function are numerically evaluated in general dynamic case and graphically shown that the singularity exists in the contact pressure at the edge of the disc is the same as the static case. In addition, the impedance functions evaluated here for the isotropic domain are collapsed on the solution given by Luco and Mita. To show the effect of different material anisotropy, the numerical evaluations are given for some different transversely isotropic materials and compared.     

Potential field of a stationary electric current in a stratified medium with a three-dimensional perturbing body

Summary The potential of the electric field of a stationary current in a two-layered Earth is calculated by applying Green's formula in the case where a three-dimensional inhomogeneity of different conductivity is located in the basement of the layer. It is proved that the potential outside and inside the perturbing body can be calculated from the potential of an electric double-layer distributed on the surface of this body. An integral equation of the Fredholm type is derived for the surface density of the double-layer, together with some of its integral properties. A similar procedure can be applied to computing the magnetic anomalies of three-dimensional magnetized bodies, geothermal anomalies due to three-dimensional inhomogeneities of different heat conductivity, as well as to potential problems of theoretical electrical engineering.     

Vertical dynamic response of a disk on a saturated poroelastic half-space

This paper considers the vertical dynamic response of a disk on a saturated poroelastic half-space. Firstly the pressure-solid displacement form of the harmonic equations of motion for a poroelastic solid are developed from the form of the equations originally presented by Biot. These equations are solved by a new method. Then the mixed boundary value problem for the vertical harmonic vibration of a disk on a poroelastic half-space is studied. The two types of drainage conditions at the surface of the poroelastic half-space are considered: (a) the surface of the poroelastic half-space is assumed to be completely pervious both within and exterior to the plate; (b) The interface between the plate and the poroelastic half-space is assumed to be impervious and the exterior region is assumed to be pervious. By using the Hankel transform techniques, the paper develops the governing dual integral equations. These governing integral equations are further reduced to systems of standard Fredholm integral equations of the second kind by Abel transform.     

空间域位场延拓新方法研究

在空间域进行位场延拓,需要数值求解第一类Fredholm积分方程,由于所得方程组系数矩阵不是稀疏矩阵,求解该方程组需要的计算机内存大,计算量大,导致延拓算法在一般计算机上难以实现,阻碍了对空间域位场延拓方法的研究.在分析系数矩阵结构特征的基础上,本文证明了方程组系数矩阵是对称的分块Toeplitz型矩阵.利用系数矩阵的对称性和分块Toeplitz型矩阵与向量相乘的快速算法,解决了系数矩阵的存储和计算问题,使得空间域位场延拓成为可能,为研究新的位场延拓方法和分析延拓误差提供了一条新的途径.利用模型数据和实测资料,对空间域位场向上延拓、空间域积分迭代法向下延拓进行了检验,结果证实了空间域位场延拓的可行性和正确性.     

RESISTIVITY AND INDUCED POLARIZATION RESPONSES OF ARBITRARILY SHAPED 3-D BODIES IN A TWO-LAYERED EARTH*

Numerical computations using the integral equation method are presented for resistivity and IP responses due to arbitrarily shaped 3-dimensional bodies in a layered earth. The unknown surface charge density distribution is expressed as the solution of Fredholm's integral equation of the second kind. Use of moment method (with pulse basis function and point-collocation) yields the matrix equations for the unknowns. The contributions to Green's function are solved (a) analytically for the primary and (b) by convolution for the secondary contributions resulting in a fast algorithm. The further step of computing potential, apparent resistivity, chargeability etc., for any electrode system, is straightforward. Our results show a good agreement with those from finite difference methods and physical tank experiments. The CPU time is only 138 s on a super-minicomputer for an apparent resistivity pseudo-section, even with 96 elementary cells as used for discretization. A large number of models for different geological situations were studied; some are presented here.     

Torsional response of a rigid circular foundation on a saturated half-space to SH waves

An analytical approach is used to study the torsional vibrations of a rigid circular foundation resting on saturated soil to obliquely incident SH waves. Biot’s poroelastic dynamic theory is considered to characterize the saturated soil below the foundation, which is solved by Hankel transform later. In order to consider the scattering phenomena caused by the existence of the foundation, the total wave field in soil is classified into free-field, rigid-body scattering field and radiation scattering field. According to the classification of wave field and the mixed boundary-value conditions between the soil and the foundation, torsional vibrations of the foundation are formulated in two sets of dual integral equations. Then, the dual integral equations are reduced to Fredholm integral equation of the second kind to be solved. Combining with the dynamic equilibrium equations of the foundation, the expressions for the torsional vibrations of the foundation are obtained. Numerical results are presented to demonstrate the influence of excitation frequency, incident angle, the torsional inertia moment of the foundation and permeability of the saturated half-space on the torsional vibrations of the foundation.     

Mise-à-la-masse interpretation using a perfect conductor in a piecewise uniform earth

A useful analysis of the mise-à-la-masse problem can be made by considering a perfectly conducting orebody in a piecewise uniform conducting earth. While the use of a perfect conductor is clearly an idealization of the true geological conditions it provides several advantages for the present purpose.
The electric field associated with the above model can be expressed in terms of a surface integral of the normal potential gradient over the boundary of the conductor, where the normal gradient satisfies a well-posed Fredholm integral equation of the first kind. This integral equation formulation remains unchanged when the conductor is arbitrarily located in the conducting earth, including the important case when it crosses surfaces of conductivity discontinuity. Moreover, it is readily specialized to the important case of a thin, perfectly conductive lamina.
Consideration of the boundary value problem relevant to a conductive body fed by a stationary current source suggests that under certain circumstances, equivalent mise-à-la-masse responses will result from any perfect conductor confined by the equipotential surfaces of the original problem. This type of equivalence can only be reduced by extending the potential measurements into or on to the conductor itself.
This ambiguity in the interpretation of mise-à-la-masse surveys suggests a simple if approximate integral solution to the mise-à-la-masse problem. The solution is suitable for modelling the responses of perfect conductors and could possibly be used as the basis of a direct inversion scheme for mise-à-la-masse data.     

A METHOD FOR CALCULATING IP ANOMALIES FOR MODELS WITH SURFACE POLARIZATION*

A numerical method is given for calculating resistivity and induced polarization anomalies produced by a surface polarization model. Surface polarization is generated when a purely electronic conductor is located in an electrolyte environment. The system that develops on the boundary between the conductor and the electrolyte is described macroscopically by a net surface charge distribution and an electric double layer. An integral equation is derived for the potential by assuming that the electronic conductor forms an equipotential system and that the polarization impedance across the boundary is linear. The integral equation is solved by means of the method of subsections. As an application some numerical modeling results are presented. The surface impedance values used in calculations are based on laboratory measurements that are briefly described. Implications of the results for scale modeling are discussed.     

Horizontal vibration of a rigid disk buried in a poroelastic half-space in contact with a fluid half-space

This paper addresses the horizontal vibration of a rigid disk embedded in a poroelastic half-space in contact with a fluid half-space using the poroelastic theory of potentials. The solution of this problem is expressed in terms of dual integral equations that are converted into Fredholm integral equations of the second kind and solved numerically. Selected numerical results for the horizontal dynamic impedance coefficient are examined based on different poroelastic materials, embedment depths, and excitation frequencies; furthermore, the results are analyzed for the cases in which there is and is no fluid overlying the poroelastic medium to examine the effect of fluid. The results of this study are helpful for designing a foundation embedded in the seabed due to dynamic horizontal forces.     

ASPECTS OF CHARGE ACCUMULATION IN d. c. RESISTIVITY EXPERIMENTS1

When an electric current is introduced to the earth, it sets up a distribution of charges both on and beneath the earth's surface. These charges give rise to the anomalous potential measured in the d. c. resistivity experiment. We investigate different aspects of charge accumulation and its fundamental role in d. c. experiments. The basic equations and boundary conditions for the d. c. problem are first presented with emphasis on the terms involving accumulated charges which occur wherever there is a non-zero component of electric field parallel to the gradient of conductivity. In the case of a polarizable medium, the polarization charges are also present due to the applied electric field, yet they do not change the final field distribution. We investigate the precise role of the permittivity of the medium. The charge buildup alters the electric fields and causes the refraction of current lines; this results in the well-known phenomenon of current channelling. We demonstrate this by using charge density to derive the refraction formula at a boundary. An integral equation for charge density is presented for whole-space models where the upper half-space is treated as an in-homogeneity with zero conductivity. The integral equation provides a tool with which the charge accumulation can be examined quantitatively and employed in the practical forward modelling. With the aid of this equation, we investigate the effect of accumulated charges on the earth's surface and show the equivalence between the half-space and whole-space formulations of the problem. Two analytic examples are presented to illustrate the charge accumulation and its role in the d. c. problem. We investigate the relationship between the solution for the potential via the image method and via the charge density. We show that the essence of the image method solution to the potential problem is to derive a set of fictitious sources which produce the same potential as does the true charge distribution. It follows that the image method is viable only when the conductivity structure is such that the effect of the accumulated charge can be represented by a set of point images. As numerical examples, we evaluate quantitatively the charge density on the earth's surface that arises because of topography and the charge density on a buried conductive prism. By these means, we demonstrate the use of the boundary element technique and charge density in d. c. forward modelling problems.     

Horizontal vibrations of a disk on a poroelastic half-space

This paper analytically examines the horizontal vibration of a rigid disk on a saturated poroelastic half-space. The pressure-solid displacement form of the harmonic equations of motion for asymmetric dynamic problem are developed from the form of the equations originally presented by Biot. Making use of a new method the solution of the above equations is obtained. According to the mixed boundary -value conditions, the dual integral equations of the horizontal vibration of a rigid disk on a saturated poroelastic half-space are established. By appropriate transforms, it is shown that the dual integral equations can be reduced to a pair of Fredholm integral equations of the second kind, whose solutions are then computed. Numerical results for the horizontal dynamic compliance coefficient are given at the end of this paper.