## 4 Results
### Application of VAR and RNN Methods to Field Dataset  
This first example shows the predictive results when the Vector Autoregressive (VAR) and Recurrent Neural Network (RNN) methods are applied to multi temporal CH-ERT field dataset described in Section 3.1. The objective was to predict the displacement of the saline tracer if the pump had not broken. As described in Section 2.2.1, VAR is a statistical multivariate forecasting algorithm that is used when two or more time series influence each other. Typical examples of multivariate time series are time-lapse electric resistivity measurements. These are used for retrieving models of the resistivity distribution that change over time for some geological reason. \begin{table}
\begin{tabular}{c c c} \hline
**Electrical Resistivity Dataset** & **Number of Surveys in Training Data** & **Number of Surveys in Test Data** \\ \hline Field data & 2–20 CH-ERT & 5 CH-ERT \\  & (1836 data each) & (1836 data each) \\ Laboratory data & 2–20 CH-ERT & 5 CH-ERT \\  & (504 data each) & (504 data each) \\ \hline \end{tabular} Cross-hole electrical resistivity tomography (CH-ERT).  
\end{table} 
Table 1: Description of the two datasets used in the present manuscript for testing Vector Autoregressive (VAR) and Recurrent Neural Network algorithms (RNN). Figure 5: Subset of data (in terms of apparent resistivity) taken at several measurement points, in the set-up of Figure 4, over 20 time steps. Q is quadrupole. To retrieve a defined number of \"predicted\" pseudo sections of apparent resistivity values, firstly the VAR algorithm was applied to the data recorded over a sequence of 18 daily CH-ERT surveys (33,048 data). Then, the data belonging to five sets of measurements (9180 data) were used as \"test data\" for checking the predictive capabilities of the VAR algorithm. In detail, the procedure for testing the VAR method is done as follows: First, the correlated historical data were used for creating a model of future trends of electric properties (such as apparent resistivity as well as resistances) over time, for each measurement point by applying Equations (4) and (5). Next, such predicted measurements were inverted for retrieving predicted resistivity models. The test data and models (effectively observed) were used for comparing predicted and true resistivity distributions. Table 2 shows an example of a subset of experimental data (resistances measured at 10 quadrupoles) and the corresponding prediction for five future steps. Furthermore, RNN was tested to the same dataset. Like in the VAR method, the crucial step consists of creating a predictive model using historical data. In this case, a Recurrent Neural Network was used with an architecture similar to the one shown in Figure 2. As explained in the Section 2.2.2, a Long-Short Term Memory (LSTM) algorithm is used for retrieving a multivariate predictive function from various sequences of correlated historical data. For that purpose, different numbers of monitoring surveys were tested (ranging from 2 to 20, 1 per day), in order to compare various predictive models. Since the electrical measurements taken at adjacent electrodes are strongly correlated, it is possible to assume that a number between 10 and 20 monitor surveys should be sufficient for building an effective historical dataset for Recurrent Neural Network application. In general, the performance of the prediction can be assessed by estimating the average forecast error on known data. What is done is to verify how far the forecast of the observations is wrong for known observations, assuming that they are not known to the predictive algorithm. As an example, Figure 6 shows a chart of predictions (predicted future) vs. observations (true future) for a given quadrupole (Q1). The red and blue dots represent the normalized resistance values (ordinate axis) for a single measurement quadrupole, for various time steps (abscissa axis). In detail, 10 historical surveys were proved to be the right compromise between prediction reliability and number of surveys in test data.