## 1 Introduction
### Electrical Resistivity Tomography  
The electrical properties of a natural rock depend mainly on its geometric dimensions, chemical composition, presence of metals, porosity, density, and water content. As stated in [10], the electrical resistivity tomography aims to determine the inner resistivity of a sample, which reveals the intrinsic material characteristics. The electrical resistivity tomography consists of a set of measurements made on the surface of a stone sample using two pairs of electrodes: a pair to inject the current (I), C1\(-\)C2, and another pair to read the potential difference (V) in between, P1\(-\)P2. With these measurements, we can determine the stone apparent resistivity (\(\rho_{\text{A}}\)) using the following Expression (2).  
\[\rho_{\text{A}}=\frac{\text{k\cdot V}}{\text{I}} \tag{2}\]  
where k is the geometrical factor that depends on the used electrode arrangement.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**N.** & **Description** \\ \hline
1 & Draw a grid pattern in both side surfaces of the block with the position where the transducers will align. \\
2 & Apply conductive gel to where the transducers contact the block surface. \\
3 & Unscrew the support brackets and slide the profile holding the transducers to the specified position and tighten the screws. \\
4 & Turn on the UPV testing equipment and insert the distance between transducers. \\
5 & Trigger the reading. \\
6 & Register the values of velocity and time shown in the test equipment and their location. \\
7 & Repeat all the above steps until completing the entire block. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Ultrasonic tomography.  
Figure 2: UPV transducer arrangements.  
This calculated apparent resistivity is not the real resistivity of the sample but rather the mean value assuming the rock as a homogeneous medium, which in reality is not the case. Then, to approach this complex mathematical problem, inversion techniques are used [10].  
The most widely used electrode arrangement is the dipole-dipole because it leads to a low magnetic coupling between the current and the potential circuits (Figure 3). The geometrical factor, k, of this arrangement is presented in Equation (3).  
\[\mathbf{k}=\pi\cdot\mathbf{n}\cdot\mathbf{a}\cdot(\mathbf{n}+1)\cdot(\mathbf{n }+2) \tag{3}\]  
In this case, as shown in Figure 3, the pair of electrodes C1-C2 are placed side by side with a distance \"a\" from each other and, likewise, the pair P1-P2 are also placed side by side, separated by \"a\". The distance between the closer electrodes of each pair, C1 and P1, is \"L\". Finally, a third value to be considered is \"n\", which represents the ratio between \"a\" and \"L\", as shown in Equation (4).  
\[\mathbf{n}=\frac{\mathbf{L}}{\mathbf{a}} \tag{4}\]  
To study the apparent resistivity of deeper stone areas, the \"n\" factor must be incremented, while the \"a\" value should remain the same, although one of the disadvantages of this arrangement is that for high \"n\" values, the required electric current amplitude is very high. To avoid this problem, one of the solutions, if the use of high \"n\" values can not be avoided, the \"a\" value should be significantly increased.  
Usually, the change of distances \"a\" and \"L\" are made manually by moving the electrodes to different locations.  
Electric tomography is done on the upper surface of the block sample (see Figure 1). In the current standard of the marble industry, as this test is an edge technology, the protocol to perform the test is completely manual. To perform an electrical resistivity tomography, an operator must carefully execute the steps mentioned in Table 2 (all tasks are manually operated).