## 2 The TL-ERT Method: Data Processing and Inversion
### Novel Algorithms for Data Processing and Inversion
A static algorithm was modified for increasing the resolution within target zones; during the time-dependent period from the image analysis, the zones were interested by the resistivity changes, and the resolution within these zones in the successive ERT acquisition increased. [PERSON] et al., 2016 [59] introduced the concept of the minimum gradient support (MGS) in the TL-ERT data inversion. This parameter was selected using the value of the TL-ERT data root mean square misfit and the first iteration of the inversion procedure. Some recent works are mainly devoted to the development of innovative algorithms for considering and removing the influence of artifacts and errors in the TL-ERT inversion procedure. [PERSON] et al., 2017 [60] approached the problems related to the propagation of the false anomalies in the initial model and their propagation in the successive TL-ERT inversions, and the assumption of the uniform sensitivity to resistivity variations. They introduced a temporal gradient to enhance the vertical and/or horizontal variation of the model parameters, and normalized the observations before inversion. [PERSON] et al., 2017 [61] analyzed the problems related to the quantification of the errors in the TL-ERT data inversion. Their approach was based on the analysis of the differences between normal and reciprocal measurements (i.e., swapping of the energizing and measuring electrodes) acquired at different time. They extended the concept of the static error formed by a linear combination of the systematic and random components [7], the results emphasized the necessity to better evaluate the errors in the time-lapse resistivity monitoring independently by the algorithm inversion. [PERSON] et al., 2017 [62] organized an overview of the main methodologies for removing and minimizing the influence of the experimental errors in the ERT inversion algorithms. They extended the standard error model based on the linearity with the transfer resistance introducing a grouping variable based on the electrode number used for the experimental measurements. This new linear mixed model (LME) was tested using lab and field data, improving the quality of the ERT images. Other works focus their attention on errors due to the presence of an irregular surface topography in the area interested by the field measurements [17], and the acquisition schemes with electrodes installed in the boreholes [63]. Furthermore, there are some studies based on the use of the classical Kalman Filter, Bayesian Markov chain, Monte Carlo simulation and clustering procedures for the TL-ERT data analysis and inversion. Notwithstanding these methods are largely applied in geophysics, there is a limited number of applications on the time-dependent geoelectrical monitoring. [PERSON] et al., 2014 [64], introduced the Kalman filtering approach to analyze the TL-ERT data. This model is useful in practical applications in which data is acquired at a rapid rate, and when changes in subsequent states are small and can be approximated by a random process. [PERSON] et al., 2019 [65] developed a novel method for the TL-ERT data inversion using a Bayesian Markov chain Monte Carlo (McMC) technique. The approach was based on the parametrization of the Bayesian inversion model in terms of the optimal bases, which allows a sparse representation of the desired features leading to dimensionality reduction and Monte Carlo simulations tuned to the physics of the targeted process. The method was tested during a heat-tracer experiment. [PERSON] et al., 2021 [66] proposed the time clustering method as a tool for the post-inversion of the TL-ERT data. The authors tested three different algorithms (K-means, hierarchical agglomerative clustering, and Gaussian Mixture Model) on raw ERT data analyzing the role of the spatial constraints and evaluating the robustness of the method. Recently, there has been growing attention on the development of open sources software and codes, and the use of parallel computing to reduce the time required for geophysical data inversion.