ELORANTA, E.H. 1986, Potential Field of a Stationary Electric Current Using Fredholm's Integral Equations of the Second Kind, Geophysical Prospecting 34,856-872.An integral equation method is described for solving the potential problem of a stationary electric current in a medium that is linear, isotropic and piecewise homogeneous in terms of electrical conductivity. The integral equations are Fredholm's equations of the 'second kind ' developed for the potential of the electric field. In this method the discontinuitysurfaces of electrical conductivity are divided into 'sub-areas' that are so small that the value of their potential can be regarded as constant.The equations are applied to 3-D galvanic modeling. In the numerical examples the convergence is examined. The results are also compared with solutions derived with other integral equations. Examples are given of anomalies of apparent resistivity and mise-i-lamasse methods, assuming finite conductivity contrast. We show that the numerical solutions converge more rapidly than compared to solutions published earlier for the electric field. This results from the fact that the potential (as a function of the location coordinate) behaves more regularly than the electric field. The equations are applicable to all cases where conductivity contrast is finite.
The image principle developed for static problems involving an anisotropic half‐space and bounded by either a perfect electric or magnetic conductor is extended to problems with an anisotropic surface impedance boundary. Such a boundary can be applied to approximate a thin layer of anisotropic conducting material above the anisotropic half‐space. The problem is limited by requiring similar anisotropy for the surface impedance and the transverse part of the resistivity dyadic of the half‐space. It is seen that, instead of a point image for a point source, the image consists of a combination of a point image and a line image obeying an exponential law. The effect of the impedance surface on the potential field of a point source is considered in terms of a numerical example.
ESKOLA, L., ELORANTA, E. and PURANEN, R. 1984, A Method for Calculating IP Anomalies for Models with Surface Polarization, Geophysical Prospecting 32, 79-87.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.
The image principle for an isotropic half‐space bounded by perfect electric conductor (PEC) or perfect magnetic conductor (PMC) plane is presented in most elementary textbooks on electromagnetics. It is perhaps not so well known that this principle can also be generalized to anisotropic media in the static case, because it is not covered in leading monographs of geoelectromagnetics (Wait, 1982; Negi and Saraf, 1989; Eskola, 1992). The anisotropic image method can be applied to geologic media that exhibit anisotropic electrical conductivity caused by the fractures and fissures in the rock. Such structures are important in the sites for disposal of nuclear waste. The characterization of these structures by electrical geophysical methods is very essential because they form the main paths for groundwater flow. The air‐ground boundary can be treated as a PMC plane representing the nonconducting medium. Otherwise the medium is assumed to be linear (ohmic) and homogeneous in terms of electrical conductivity. The image method presented is also relevant to problems arising in the traditional ore prospecting where a conducting ore body buried in an electrically anisotropic host rock generates secondary electric fields (Asten, 1974; Eloranta, 1988).
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