## 2 The TL-ERT Method: Data Processing and Inversion  
### Basic Principles  
The TL-ERT method is based on the analysis of a dataset of 2D or 3D resistivity images obtained at different time intervals: it combines 2 or 3 spatial dimensions with a temporal dimension (4D ERT). The TL-ERT or 4D ERT method adopts multi-electrode arrays and the data acquisition procedures already used for the classical 2D and 3D ERT surveys, but the acquisition is generally controlled by a remote system, and it is repeated over time. The main experimental errors associated with the TL-ERT measurements are connected to the presence of man-made electrical noises, galvanic contact changes, electrode corrosion, and low signal-to-noise ratio in voltage signals (well-known sources of errors in 1D, 2D and  
Figure 2: (**Left**) Geological map of the Montaguto earth-flow (Campania region, southern Italy) with the location of direct and indirect investigations. Red lines: ERT profiles. Green line: water drainage structures. Green dot: borehole. Blue triangle: piezometers. (**Right**) Fence diagram showing the locations of the 2D ERT (ERT 3–ERT 13) within a 3D space (after [44]).  
3D ERT measurements). The stacking techniques and algorithms for spectral analysis are generally used for extracting the weak voltage signals generated by the current injection into the subsurface, also in the condition of low signal-to-noise ratio [1; 42]. However, it is necessary to consider other time-related errors connected to temperature and galvanic changes, such as rain precipitation and variations of electrode spacings due to surface ground movements [45].  
As it concerns the interpretation and the inversion of the 4D ERT experimental data, the simpler and more common approach is based on the analysis of the differences between the resistivity images independently inverted during the acquisition phase (_independent approach_). However, the time-lapse of the resistivity images can easily produce an amplification of the inversion artifacts [46].  
Another largely applied method is the _difference inversion_ of the ERT data [47]. The Occam algorithm was modified to allow the inversion of the differences between the background or reference resistivity images and the other 3D resistivity images obtained at a different time. The resistivity obtained by the inversion of background data serves as a priori model in the difference inversion. The method allows a rapid convergence of the inversion process and reduces the errors due the field array configurations.  
A _ratio inversion_ was proposed by [PERSON] 1992 [48], in which the inversion is performed using the ratio between two successive ERT measurements multiplied with a resistivity value representing a homogenous subsurface model. An alternative approach is the _cascaded inversion_: this method uses the inversion result of the initial dataset as the starting and reference model for the inversion of the subsequent datasets [49].  
Another critical error affecting the interpretation of the TL-ERT data is the assumption of the time-invariant (static) subsurface model during the acquisition phase. This hypothesis is realistic for slow phenomena, but in field experiments we could observe rapid changes of the subsurface model (i.e., the migration of a fluid in a fractured material). To overcome this problem, [PERSON] et al. (2009) [50] introduced a subsurface resistivity model that is continuously changed over the time, the model is completely defined in the space-time domains, and the inversion algorithm considers all data available (_Full 4D TL-ERT data inversion_). To reduce the computing time and to stabilize the inversion procedure, some regularization criteria are generally introduced.