## 1 Introduction  
The estimation of the soil resistivity as a function of the depth is a key point for modelling any grounding system [1]. There are several methods to determine the soil resistivity. Most of them are indirect methods that use an intermediate magnitude as the apparent resistivity. In one of the most used methods, the resistivity is estimated from a vertical electrical sounding (VES) performed with a four-pin device as in the Wenner and Schlumberger arrays, from which the apparent resistivity is obtained [2]. From the apparent resistivity, it is possible to find out the soil resistivity profile, provided that a constant multi-layered model has been adopted [3, 4]. Even though the VES is a non-invasive procedure, small errors in the measuring procedure could lead to a soil model far from the real one. The ill-conditioned nature of the method to obtain the resistivity profile from the soil's apparent resistivity is at the bottom of the issue [5, 6]. On the other hand, in some sense a VES could be seen as a non-local test since the soil conductivity between the active electrodes of the test array is being checked. Note for instance that in a Wenner array the active electrodes are separated three times the distance between adjacent probes. The soil properties could change horizontally leading to a soil resistivity profile that would not match with the real one. This is especially important when a grounding system makes use of vertical rods as grounding electrodes, such as in the case of lightning protection electrodes. Another indirect procedure sometimes used is the driven-rod method. In this method, the grounding resistance of a rod driven into the ground is measured by the fall-of-potentialprocedure [7]. If the rod is modeled as a cylinder of length L and radius r, where L >> r and with a hemisphere at the end, the grounding resistance in a homogeneous medium of resistivity, rho, is \(R=\frac{\rho}{2\pi L}\ln(\frac{2L}{d})\). On the other hand, if the rod is modeled as a half-ellipsoid of revolution, whose major axis is L and its minor axis is the diameter d, the expression \(R=\frac{\rho}{2\pi L}\ln(\frac{4L}{d})\) must be used. Another expression widely used to theoretically estimate grounding resistance is \(R=\frac{\rho}{2\pi L}[\ln(\frac{8L}{d})-1]\)[8] in which the supplementary assumption is made that the ground leakage current has a constant distribution along the rod. By varying the length of the rod, an apparent resistivity profile \(\rho_{app}(L)\) representing the resistivity seen by the rod of length L can be obtained. Figure 1 outlines the data that are necessary to implement the method. An advantage of the driven-rod method is that it allows obtaining valuable information on the feasibility of a grounding system. Knowing the necessary depth to obtain a given grounding resistance could greatly facilitate the final touch-up of the grounding system design. An important observation is that the above mentioned VES method is not very sensitive to the state or type of electrodes used except for a shallow depth where the electrodes are very close together and their size and shape influence somewhat the potential [9]. In general, it does not matter if the electrodes are corroded or if they have a great internal resistance since what is measured is a potential difference on the ground. In contrast, the driven-rod method is greatly affected by oxidation and the internal resistance of the rod. The previous expressions for the grounding resistance \(R\) as a function of the rod length \(L\), were found under the condition that the rods are ideal conductors without internal resistance. Therefore, it is to be expected that this fact will need to be taken into account when looking for a model for soil resistivity. There are more reasons, in addition to those already exposed, to incorporate the internal resistance of the conductors into the calculation scheme of the electrical properties of a grounding electrode.