On the determinability of all of the self‐ and mutual resistances in a grounded electrode array

Abstract

ABSTRACTIn the geophysical electrical resistivity method, arrays of electrodes are commonly deployed on the Earth's surface. In electrical power engineering applications, arrays of electrodes are often connected in parallel in order to reduce the total resistance to ground. The electrical characteristics of an array of N electrodes can be fully described by N self‐resistances and mutual resistances for a total of independent parameters which represent the coefficients of an N by N symmetric matrix. Typically, certain linear combinations of the mutual resistances are measured during a geophysical electrical resistivity survey while protocols of specific measurements are used to determine self‐resistances. In this contribution, I investigate whether it is possible to determine all of the self‐ and mutual resistances for an array and hence capture all possible information. I assume that measurements of potential and current can be made at each electrode and that any combination of series and parallel current injections between the electrodes can be made. I show that all resistances can only be determined uniquely if current and voltage reference electrodes that are external to the array are used. If only an external current electrode or only an external voltage reference is used then only independent measurements can be made and the system is underdetermined. If only electrodes within the array itself are available only, independent measurements can be made and the system is strongly underdetermined. The procedure outlined here also gives a general prescription for calculating the completeness and redundancy of any given array.