On the Theoretical Determination of Earth Resistance from Surface Potential Measurements

Abstract

A general solution for the potential in the steady flow of current from a single point electrode when the conductivity is an arbitrary continuous function of position is given in the form of an infinite series, by using the method of successive approximations, and conditions for the validity of the solution discussed. The converse problem of determining the conductivity when the surface potential is known is then shown to lead to an integral equation, which has no unique solution. The problem can be made determinate, however, by restricting the functional form of the conductivity—in particular by supposing that it is a function of depth only—or by supposing that the electrode is movable, and that the surface potential is known for all positions of the electrode on some curve at the surface. It is shown how the integral equations can be formally solved by the method of successive approximations. For the special case where the conductivity is a function of depth only, the first approximation is worked out in detail, and an approximate method is given for solving the resulting integral equation for the conductivity. This method is critically discussed and compared with the more exact method of Slichter and Langer. A numerical example is worked out where the conductivity is known in advance, namely a special case of the three-layer earth. It is shown that the present method gives a rough indication of the true behavior of the conductivity, but that the Slichter-Langer method and the ``apparent resistivity'' method both fail completely for this case. It is suggested that in certain cases the present method might give a better result than that of Slichter and Langer.