# Computation of Optimized Electrode Arrays for 3-D Electrical Resistivity Tomography Surveys
## 2 Methodology
(**b**) Representation of the Jacobian matrix where the higher sensitivity absolute values (rows) are located for the 60 parameters of the model (columns) (detailed description in the text). The comprehensive data set for a single 2-D profile was compiled from the combination of widely used common arrays: dipole-dipole (DD), pole-dipole (PD), pole-tripole (PT) and gradient (GR) (Figure 2). The 2-D and 3-D algorithms were used to produce the 2-D and 3-D optimum protocols, named 'Optim-2D' and 'Optim-3D', respectively. In both cases, the Jacobian matrix is calculated for a homogeneous model to ensure that the generated optimum protocols would apply to general resistivity surveys regardless of the target's or the background's property values. The algorithmic details of the procedure to extract the optimum protocols with the Stummer's method [15] were followed in the specific work and the respective optimized configurations are coded as 'Res-STUM'. A demonstrative way of producing the 3-D optimum measurements from the comprehensive data set is shown in Figure 3. From the initial comprehensive (normal) 2-D tomographic data set (Figure 3, red-colored profile view), the 2-D algorithm is used to select the optimum data for a single 2-D profile (Figure 3, green-colored). Then, this optimum 2-D data set is cloned in multiple parallel profiles equally separated to create the 3-D optimum data set. Figure 3: _Cont._  
Figure 2: Representation of the basic arrays used to generate the comprehensive data set. On the other hand, for the 3-D optimization technique, the initial comprehensive 2-D data are firstly cloned in multiple equally separated parallel profiles (Figure 3, red-colored) and then the 3-D algorithm produces the 3-D optimum data (Figure 3, blue-colored). In the present study, both 2-D and 3-D optimization algorithms were allowed to run for specific steps to maintain equal measurements for the 'Optim-2D' and 'Optim-3D' protocols, respectively, for direct comparison purposes. Table 1 shows the unit electrode spacing 'a' and the separation factor 'n' used to compile the 23,670 measurements of the comprehensive data set, comprised of dipole-dipole (DD), pole-dipole (PD), pole-tripole (PT) and gradient (GR) array, assuming 11 parallel profiles. The final resulted in \"Optim-2D\" and \"Optim-3D\" protocols each being only 14% of the initial comprehensive data set. The \"Res-STUM\" protocol used only 22% of the initial comprehensive data set. Figure 4 shows the percentage participation of each protocol that was used. A comparable quantitative analysis of the resolving capabilities of the different protocols based on the cumulative Jacobian values per parameter is shown in Figure 5. The cumulative values have been normalized according to the total number of the data to gain the flexibility of comparing protocols with uneven measurements. As is expected, higher sensitivity values are the ones closer to the surface. The normalized cumulative Jacobian of the \"optim-3D' data set has higher values than the \"optim-2D\". Furthermore, the sensitivity pattern of the \"optim-3D\" is more intense in the deeper layers, thus indicating a higher capability in resolving deeper resistivity structures. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Protocol** & **a** & **na** & **Comprehensive** & **Optim-2D** & **Optim-3D** & **Stummer’s** \\ \hline FULL & 1–9 & 17 & 23,670 (100\%) & 3390 (14\%) & 3315 (14\%) & 5224 (22\%) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Number of data for FULL protocol (percentage of initial array). Figure 3: Optimization technique using: “Optim-2D” (2-D algorithm) data set (green-colored) and “Optim-3D” (3-D algorithm) data set (blue-colored). Comprehensive data set (red-colored).