# Computation of Optimized Electrode Arrays for 3-D Electrical Resistivity Tomography Surveys
## 3 Synthetic Model  
The efficiency of the optimization algorithm was firstly tested with synthetic data in order to evaluate the resolving capabilities of the generated optimized protocols under constrained conditions, where the exact targets' position and resistivity values are predefined and well-known. The resistivity model used in the work of [PERSON] et al. (2015) was extended in the 3-D space, composed of nine layers (D1-D9) with a constant thickness of 0.5 m (Figure 6). A homogeneous half-space of 1000 ohm.m lies below a less resistive superficial layer of 100 ohm.m one meter thick. A highly resistive target 'A' (3 \(\times\) 2 \(\times\) 1 m) with a resistivity of 10,000 ohm.m is buried at 0.5 m from the ground surface lying within the two horizontal layers D2-D3. A target 'B' with larger dimensions (5 \(\times\) 4 \(\times\) 1.5 m) is  
Figure 4: Participation of protocols used at data. Figure 5: Normalized cumulative sensitivity results along a 2-D section for the comprehensive and optimum 3-D data sets resulted by the “optim-2D” and “optim-3D” strategy. Different scaling is used for presentation purposes since the sensitivity values differ. buried at 2 m depth from the surface within the layers D5 to D7. The resistivity of this block gradually decreases inwards from 100 to 10 ohm.m. The 3-D resistivity survey was composed of total 231 electrodes in total arranged in a rectangular grid of 20 m by 10 m along the X and Y axes, respectively. The inter-probe and inter-line spacing were set to 1 m on both axes. The tomographic data were collected along the 2-D profiles on the X-axis using the combination of the four traditional arrays (dd + pd + pt + gr) and the measuring parameters shown in Table 1. A 3-D finite element algorithm (RES3 DMOD) was used to calculate the forward response of the specific model and extract the distribution of the apparent resistivity values for the comprehensive data in the 3-D subsurface space. The synthetic apparent resistivity data were then corrupted with 5% Gaussian noise on the modeled apparent resistivity values. During the inversion procedure, the L1-norm method was used for data misfit and model roughness to account for the sharp boundaries of the targets [30]. A finite-element modelling subroutine was used to calculate the apparent resistivity values. The selected initial value for the damping factor was \(\lambda\) = 0.1, which was increasing with a depth factor of 1.01. The inversion procedure was terminated after five iterations. The common resistivity colored scale (10-10,000 ohm.m) and depth slices (D1-D9) are used in all inversion images for comparison purposes. Black squared rectangles indicate the expected exact position of targets 'A' and 'B'. The RMS error value varies between 3.7 and 4.1% among all the inversion images of the arrays that were used. The comprehensive FULL protocol (Figure 7) reconstructs the targets 'A' and 'B' with comparable accuracy: the calculated resistivity value for target 'A' is approximately 1000 ohm.m. It is located inside the rectangular lines and the corresponding depth slices (D2 and D3). The target's 'B' resistivity value is close to 500 ohm.m. The target's location at the XY axis is well located but not on the Z-axis, where it is extended to layer depth D4. Figure 6: Synthetic Stummer’s model. The Optim-2D data set (Figure 8) is able to reconstruct the target's 'A' position inside the rectangular box but the resistivity value (500-1000 ohm.m) is lower than the comprehensive data set value. The target 'B' is also located inside the expected rectangular area with resistivity values that vary between 400-600 ohm.m and it is expanded at depth D4. Figure 8: Inversion results from optimum array produced by 2D algorithm. Figure 7: Inversion results from FULL comprehensive array. The higher and closer to true value resistivity value of 5000 ohm.m is presented with the Optim-3D data set regarding the reconstruction of target 'A' (Figure 9). Additionally, the position of target 'A' is inside the expected position.