## 1 Introduction
There are numerous types of data analysis approaches available for time series that are suitable for understanding and forecasting patterns of one or more crucial variables that characterize dynamic systems. Among the others, univariate and multivariate analysis approaches and Deep Neural Networks have proven to be very useful for the scope [4; 7; 26; 27]. The various methods show different effectiveness depending on their application to univariate or multivariate time series. A univariate time series is a data series with a single time-dependent variable. On the contrary, in multivariate time series each variable depends both on its past values and, to some extent, also on other variables. That dependency can be used for improving the forecasting of future values and trends [28]. In detail, the Recurrent Neural Networks (RNN), like the other artificial neural networks, utilize training data to learn, with the final objective to find the \"optimal\" weight matrixes that allow for minimizing the misfit between the actual and predicted output. However, RNN's distinctive characteristic consists in their \"memory\" capability: this consists of their ability to take information from prior inputs to influence both current input and output. In fact, in traditional neural networks, all inputs are assumed independent from the outputs; instead, in RNNs, it is assumed that there is some significant correlation between input and output. In detail, the RNNs structure includes a \"time delay unit\" and a \"feedback connection\" to use information from a previous state for inferring a model in the subsequent state. This allows for a wide range of applications to the predictive analysis of many types of time series, including natural language analysis, image captioning,speech and handwriting recognition, text summarization, text generation, word prediction, stock market prediction, call center data analysis, sentiment analysis, face detection, image recognition, and so forth [29]. For instance, RNN's outperform traditional models in some hydrogeological applications, such as groundwater level and quality prediction [30; 31]. In general, statistical and artificial neural networks have been successful applied to data series in near surface geophysics over multiple monitor surveys [32; 33; 34] and have produced consistent results when applied to time-lapse resistivity survey addressed to optimizing oil production and CO\({}_{2}\) geological storage and predicting the future behavior of soils in slopes [35; 36; 37]. However, only one example of RNN application to near surface geophysical data forecasting is reported in literature by [PERSON] et al. [38], who used RNN to produce synthetic time-lapse seismic data. In this study, statistical and deep learning algorithms were tested to predict saline plume migration in two porous aquifers using cross-hole geoelectrical datasets recorded through experiments at different spatial and temporal scales (a real aquifer and a laboratory experimental set-up). In detail, the Vector Auto Regressive (VAR) method and Recurrent Neural Networks (RNN) were selected. As the other types of time series, the multi temporal CH-ERT datasets acquired in the field and laboratory experiments produced a large volume of data that are very suitable for statistical-based approaches and RNN's application. Because of resistivity changes over time in the subsoil due to a fluid movement, high spatial correlation between electrical resistivity measurements taken along the same layout can be reasonably expected. For this reason, these time-lapse datasets represent a multivariate time series. The following sections describe how to apply both VAR and RNN methods with the final goal to retrieve a multivariate forecast model of the aquifers and of fluids displacements over time.