Analytical formulas for geometrical factor and sensitivity for long electrodes

In the electrical resistivity method, electrodes are usually modelled as point current sources and point voltage measurements. If the burial depth of the electrode is significant compared with the spacing between electrodes, this point approximation may not be accurate. Common situations employing long electrodes include the use of metal-cased boreholes as electrodes and small-scale, high-resolution environmental, engineering and archaeological surveys where electrode spacings may be very small. In this contribution, I present analytical expressions for the mutual resistance between long electrodes modelled as line current sources. Mutual resistances are then used to calculate geometrical factors. Additionally, I present an expression for the current density and use it to derive an analytical expression for the sensitivity of electrode arrays with long electrodes. The sensitivity is, in turn, used to calculate the mean depth and position which can be used as estimates of depth and position of investigation and as pseudosection plot points. Example calculations using the geometrical factor, sensitivity and mean depth are shown, and comparisons are made with simulations and lab-scale experiments.