# Computation of Optimized Electrode Arrays for 3-D Electrical Resistivity Tomography Surveys
## 1 Introduction  
Over the last twenty years, electrical resistivity tomography (ERT) has been transformed into a widely used geophysical method for approaching several different near-surface applications. This was boosted by the improvement of the hardware's capabilities, the development of multichannel resistivity systems and the compilation of automated inversion and reconstruction algorithms to image and monitor the subsurface geoelectrical properties [1]. Two dimensional (2-D) ERT method is nowadays routinely applied in approaching diverse problems including the study of saline water intrusion in coastal aquifers [2], the tracing of water content dynamics in unsaturated zones [3], in geotechnical studies [4] and the monitoring of subsurface conductive pollutants [5]. However, 2-D approaches underperform in cases of complicated stratigraphy and intense inhomogeneity of the subsurface resistivity. Thus, significant research efforts have been placed on proposing and investigating the resolving capabilities of different measuring and processing strategies to map the subsurface resistivity structure within a three-dimensional (3-D) context [6; 7; 8; 9; 10]. Every surface or cross-hole ERT experiment involves the arrangement of a number of electrodes along a line for 2-D or on the nodesof a regular or irregular grid for 3-D resistivity surveying, respectively. In either case, the linearly independent apparent resistivity measurements are a function of the maximum number of electrodes that are simultaneously connected to any multiplexed/multichannel resistivity instrument. The number of measurements for an array using four-electrode configurations increases exponentially with the respective number of electrodes [11; 12]. This renders the field collection of all the possible measurements practically impossible, even for an array composed of a relatively small number (\(\sim\)50) of electrodes. Thus, the routine practice relies on the collection of tomographic data using standard electrode arrays [13] and the integration of dense parallel or orthogonal 2-D lines to characterize the 3-D resistivity structure [14]. A step forward to exploit the full capability spectrum of the modern acquisition instruments was firstly introduced by the pioneering work of [PERSON] et al. [15], who presented an experimental optimization method to select electrode arrays that maximize the resolving capabilities of the inversion images. Their approach was based on the compilation of the comprehensive data set including all the standard and non-standard arrays, the associated resolution matrix and a goodness function that ranks the sensitivity of every possible measurement to changes in the subsurface parameters. Extensive comparisons among the reconstructed models resulting from the inversion of the comprehensive, standard, and optimized data sets revealed the significant potential of the proposed technique. [PERSON] et al. [16] developed and tested some improved strategies for the automatic selection of the optimized ERT data sets using a linearized estimate of the model resolution matrix that quantifies the resolving capabilities of the observed data. One of the methods (Compare R) directly compared the model resolution matrices to enhance the spatial resolution while another one used a modified version of the goodness function (Modified GF) to ensure a high degree of orthogonality to the base data set. The effectiveness of the algorithms was validated through synthetic and real data, outlining their relative superiority in the optimization and resolving capabilities performance in relation to [PERSON]'s method, though being more time consuming than [PERSON]'s approach. [PERSON] et al. [17] presented an adaptive optimization method based on Compare R to compile optimum arrays for time-lapse monitoring surveys. [PERSON] et al. [18] addressed the time-consuming issues related to the automatic selection of the optimized data set using a fast computation strategy to reduce the traffic between the computer main memory and the CPU registers.