Analytical Expressions for the Depth Investigation Characteristic and Mean Sensitivity Depth for Electrical Resistivity Surveys Using Long Electrodes

ABSTRACT In analyses of the electrical resistivity method, it is usually assumed that electrodes can be represented as point current sources and point potential measurements. However, if electrodes extend into the ground to depths that are similar to their spacing or greater, this approximation may not be valid. The use of metal‐cased boreholes as electrodes and high‐resolution environmental and engineering surveys are situations where electrodes are often long. The analysis of resistivity data has benefited from the existence of a simple analytical formula for the depth variation of the sensitivity termed the depth investigation characteristic. This depth‐dependent function is calculated from the horizontally averaged sensitivity function for an infinite homogeneous half‐space. The peak, median and mean of this function have been used to estimate the depth of investigation and as a depth coordinate for pseudosections. In this contribution, I present an analytical formula for the depth investigation characteristic for long electrodes modelled as line current sources and line potential measurements. The depth investigation characteristic is also used to calculate an analytical expression for the mean depth for pole–dipole and four‐electrode arrays that can be used to estimate the depth of investigation and used as a depth coordinate for pseudosections when one or more long electrodes are used.