# Computation of Optimized Electrode Arrays for 3-D Electrical Resistivity Tomography Surveys
## 2 Methodology  
The experimental selection of the optimum measurements from an initial array based on the Jacobian matrix (see Appendix A) method was firstly introduced by [PERSON] et al. [25] for the case of the 2-D resistivity tomographic problem. The same technique expanded and incorporated within a 3-D resistivity modelling and inversion algorithm [14] to select the optimum tomographic data in the 3-D space. In both algorithms, the adjoint equation technique [29] is used to calculate the sensitivity matrix in a homogeneous 2-D or 3-D resistivity model, which associates variations in the resistivity of the parameters \(\rho_{\rm j}\) with variations in the apparent resistivity data \(\rm d_{\rm i}\). This was carried out to ensure the generality of the optimized protocols regardless of the target's or the background's property values. Figure 1a illustrates a simplified model with 24 surface electrodes (indicated with black dots), on a 3-D rectangular grid with the associated parameterization of the subsurface composed of 60 blocks to describe the earth's resistivity. It is known that the parameters closer to the surface electrodes are more sensitive to any change since they have higher absolute Jacobian values (Figure 1a, green-colored blocks). Gradually, the parameters that are further away from the surface (Figure 1a, yellow-colored blocks) and to the edges (Figure 1a, red-colored blocks) are less sensitive (lower absolute Jacobian values) to any subsurface resistivity change. As an example, let us assume an experiment with 200,000 measurements collected from a network of 24 surface electrodes and a subsurface model discretized using these 60 parameters. The calculated Jacobian matrix will have 200,000 rows representing the measurements and 60 columns associated with the parameters (Figure 1b). The experimental workflow follows the next procedure after the calculation of the Jacobian matrix. Firstly, the normalized cumulative Jacobian value is calculated for each parameter by summation of the absolute sensitivity values of all measurements for each parameter and the resultant value is divided by the number of measurements associated to a specific parameter. Then, all parameters are sorted in ascending order from the least to the most sensitive having the higher absolute sensitivity value. The specific sorting of the parameters from the weakest to the strongest sensitivity is performed to enforce the optimization algorithm to correlate the strongest array configuration to the weakest parameter and enhance the resolving capabilities of the deepest subsurface parameters. For each parameter, the algorithm selects the electrode configuration with the maximum sensitivity value, which is stored as the optimum measurement for the specific parameter (Figure 1b, black-colored text 'MAX'). During the experimental procedure, if one measurement is already selected for a previous parameter (Figure 1b, red-colored text 'MAX'), the algorithm chooses the next best measurement for the specific parameter. In the end, the'strongest' measurement (with the highest sensitivity value) is correlated with the weakest parameter. After the termination of the experimental procedure, the number of optimized measurements will be equal to the number of the parameters forming the model parameter space. The specific optimization technique is repeated through iterative steps to add extra optimum measurements according to the operator's demand for increasing the subsurface resolution. At each iterative step, the extra added optimum measurements will be equal to the number of original model's parameters. In practice, in the above-mentioned example, after two steps, the protocol will have 120 optimum measurements (60 + 60), after three-step 180 optimum measurements (120 + 60), etc. Figure 1: (**a**) Simplified 3-D model with 60 parameters, where the sensitivity value of each parameter is decreasing in larger distances from the surface electrodes (black dots).