# Geo-Electrical Detection of Impermeable Membranes in the Subsurface
## 2 Principles of ERT  
Electrical resistivity tomography (ERT) is a geophysical technique that has traditionally been used for imaging sub-surface structures. It can be applied using electrodes from the surface or using electrodes in one or more boreholes. The electrical currents in the subsurface are influenced by the composition of the matrix (e.g., sand, gravel, clay) and the composition of the water in the pores. In principle, the resistivity is measured using electrodes at different positions. A tomographic image of the resistivity distribution of the subsurface is obtained through inversion. A typical acquisition system for Direct Current resistivity measurements is shown in Figure 1a,b. The basic setup comprises a four-electrode array with a voltmeter and a current meter. For each measurement, two current electrodes are used: one to inject the current into the subsurface or waterbody and the other to retrieve the same amount of current from it. By convention, these electrodes are named A and B, respectively. The electric field is measured (at least) with two other electrodes (M and N), called the potential electrodes. The way in which the current and potential electrodes are arranged is called an array. The choice of a particular resistivity array for a survey is based upon considerations regarding theoretical advantages and drawbacks of the array and its signal-to-noise ratio [2]. Some of the arrays utilize remote electrodes, i.e., electrodes that are installed relatively far away and can be considered as infinitely far. In general, as the distance between the electrodes increases, the depth of investigation, or in the volume through which the current flows, also increases. The main characteristic of an array is its geometrical factor, which is uniquely related to the respective distances between the different current and potential electrodes. The geometrical factor \(g\) describes the relation between the so-called apparent resistivity \(\rho_{a}\) and the measured resistance \(R\) which represents a weighted average of the true resistivity of the subsurface:  
\[\rho_{a}=R\cdot g \tag{1}\]  
The apparent resistivity is equal to the true resistivity in the case of homogeneous material. The equation of the geometrical factor, in case of half space, depends only on the position of the electrodes. For special cases, e.g., a laboratory experiment in a tank, the geometrical factor needs to be numerically calculated. The true resistivity can be obtained after inversion of the apparent resistivity data. The inversion procedure aims to find a resistivity model that explains (or \"predicts\") the given field measurements best according to a misfit definition. More detailed information about the ERT method can be found in [1]. The ERT method is used in a variety of applications such as in hydrogeology and for the assessment of geohazards. Examples are imaging of sinkholes [3], cavities [4], landslides [5]. It is also one of the most widely used near-surface geophysical survey for civil engineering applications [6]. Other more recent applications can be found in [1]. In a typical application of the ERT method, the electrodes are placed on the surface and a tomographic image of the subsurface can be obtained. The depth of investigation relates to the electrode spacing, total number of electrodes used and the geoelectrical properties of the subsurface. The spatial resolution of the recovered model is decreasing with the  
Figure 1: (**a**) A typical ERT system with the two current electrodes (A and B) and two potential electrodes (M and N). (**b**) Schematic representation of an array of electrodes. (**c**) Cross-section showing the two modes used in the simulation in a T-shaped tool. (**d**) Prototype of ERT tool with ring electrodes on the pole and pin electrodes on the cross. increasing depth, making it harder to determining the exact positioning of deeper targets. Electrodes can also be placed inside boreholes to increase the resolution at depth, typically using dipole-dipole and pole-dipole arrays [7].