Open AccessAn Integral Equation and its Solution for some Two‐and Three‐Dimensional Problems in Resistivity and Induced Polarization
Author(s) -
Lee T.
Publication year - 1975
Publication title -
geophysical journal of the royal astronomical society
Language(s) - English
Resource type - Journals
eISSN - 1365-246X
pISSN - 0016-8009
DOI - 10.1111/j.1365-246x.1975.tb05851.x
Subject(s) - integral equation , electrical resistivity and conductivity , galerkin method , overburden , mathematical analysis , conductivity , summation equation , mathematics , cylinder , electrical conductor , polarization (electrochemistry) , physics , geometry , chemistry , geology , thermodynamics , quantum mechanics , geotechnical engineering , finite element method
Summary. The potential, U , about a point electrode, at the surface of a layered ground in which there is an heterogeneity embedded, satisfies the integral equation:Here, U * and σ* are the corresponding quantities for the potential and conductivity without the heterogeneity. The integral is taken over the surface of the heterogeneity, ∂ U /∂ n is the normal derivative (in the direction of the outward normal) of U , and G is a Green's function. Solutions to this equation can readily be found by using the Galerkin method of solving integral equations. The solutions of this equation when the heterogeneity is a sphere or a cylinder in a uniform ground or beneath a conductive overburden are the most readily found. When the solution of the integral has been found for the potential it is a simple matter to calculate the apparent resistivity or chargeability for any electrode configuration.