## 2 Theoretical Background
According to (4), it can be written as  
\[k_{m}(x)=\frac{R_{G}}{R_{G}-\frac{V_{e}(x)}{I}+\frac{\rho}{2\pi}(\frac{1}{D-x }-\frac{1}{D})}=\frac{k_{\rho}}{k_{\rho}-k_{p}(x)+\frac{1}{2\pi}(\frac{1}{D-x }-\frac{1}{D})} \tag{5}\]where the coefficients \(k_{\rho}=\frac{R_{G}}{\rho}\) and \(k_{p}(x)=\frac{V_{c}(x)}{\rho! }\) have been introduced. These coefficients, \(k_{\rho}\) and \(k_{p}(x)\), turn out to be parameters of the electrodes and can be calculated for single-layered soils of resistivity \(\rho\). Therefore, the proposed method requires precise knowledge of the geometry and placement of the electrode, information that is presumably available, with which to evaluate \(k_{\rho}\) and \(k_{p}(x)\). Once \(k_{m}(x)\) is obtained for a specific position of the potential probe, the electrode resistance \(R_{G}^{meas}(x)\) is measured, and the grounding resistance is calculated as \(R_{G}=k_{m}(x)\cdot R_{G}^{meas}(x)\). Since these expressions have been derived from a drastic simplification, such as considering an extensive electrode as a point source at short distances, it is expected that \(R_{G}\) will depend on \(x\) and actually represents an upper bound to the true grounding resistance value. It should be noted that in the three-pin method, the measured resistance \(R_{G}^{meas}(x)\) is always lower than the real value. As a comment, the denominator of (5) actually contains the touch potential of the electrode at distance x per unit of current and soil resistivity, known as \(k_{c}(x)\) and defined by \(k_{c}(x)=\frac{V_{c}-V_{c}(x)}{\rho! }\), so (5) can also be expressed as  
\[k_{m}(x)=\frac{k_{\rho}}{k_{c}(x)+\frac{1}{2\pi}(\frac{1}{D-x}-\frac{1}{D})} \tag{6}\]  
The magnitude \(k_{c}(x)\) can be evaluated theoretically, just like \(k_{p}(x)\), given the nature, geometry, and positioning of the electrode in the soil. When dealing with extensive electrodes or with auxiliary current electrodes close to the electrode being measured, it becomes necessary to determine the appropriate value of \(D\) to use. It might be useful to rewrite (6) by introducing two distances,  
\[k_{m}(x)=\frac{k_{\rho}}{k_{c}(x)+\frac{1}{2\pi}(\frac{1}{D_{1}-x}-\frac{1}{D_ {2}})} \tag{7}\]  
where \(D_{1}\) represents the distance between the auxiliary current electrode and the coordinate origin used to define the electrode being measured and \(D_{2}\) is the optimal distance from the auxiliary electrode to the electrode being measured to obtain the potential \(V_{A}\) according to the first equation in (3). In critical situations, that is, when \(D_{1}\) is not very large, we propose averaging \(R_{G}\) evaluated with two different values of \(D_{2}\), corresponding to the nearest and farthest distances from the auxiliary electrode to the extensive electrode being measured. It is easy to see that \(D_{1}\approx D_{2}\) if both are large compared to the size of the electrode being measured, so the point current approximation for the grounding electrode becomes quite accurate. In the proposed method, we start with a known \"theoretical\" electrode for which it is possible to calculate \(k_{c}(x)\), \(k_{\rho}\), and therefore \(k_{m}(x)\) according to (6). However, the electrode to be measured may not exactly correspond to the \"theoretical\" electrode. It may be altered for various reasons, so its resistance will not match the theoretical resistance. If we measure \(R_{G}^{meas}(x)\), then applying the correction factor \(k_{m}(x)\) will not result in \(R_{G}\). The main causes include the oxidation or deterioration of the conductors that comprise the electrode. In addition, the measurement can also be altered because the electrode is interconnected with other external grounding grids. This is a common practice that significantly reduces the resistance of the joint ground. With electrode deterioration, the measure typically leads to grounding resistances, \(R_{G}^{meas}(x)\), which are somewhat higher than those of the unaltered electrode, which, when multiplied by the correction factor, results in an overestimation of \(R_{G}\).