## 2 Background Theory
### Predictive Methods  
#### 2.2.1 Statistical Approaches  
A relatively simple statistical approach for time series analysis is known as Exponential Smoothing. This is based on a weighted sum of past observations and applies an exponentially decreasing weight for past observations [42]. Several variants of this approach exist. The simplest one is called Single Exponential Smoothing (SES, for short). This is effective for univariate data that do not show neither significant trends nor seasonality. A more complex approach is the Double Exponential Smoothing method that includes the information about trends in the data. Triple Exponential Smoothing is a further extension of the previous approaches. It considers the seasonality eventually presents in the time series.  
In case datasets are characterized by variables that show a significant reciprocal correlation, methods based on multivariate analysis (MVA) are used [43; 44; 45]. This correlation can be exploited for forecasting future values of one or more variables. Such a mutual dependency is often a crucial feature of multiple geophysical measurements used for characterizing dynamic geological models.  
Between the various approaches for multivariate time series forecasting, it is possible to applied Vector Auto Regression (VAR) method where each variable is considered a linear function of the past values of itself and the past values of all the other variables [46].  
Figure 1: (**a**) Pseudo section setting for ERT acquisition with Dipole Dipole (DD) array, (**b**) cross borehole, and (**c**) cross borehole with surface. \(A\), \(B\), \(M\), and \(N\) are electrodes. D is borehole depth; S is borehole separation. D/S > 1.5. Adapted from [14].  
Considering the simplest case in which there are only two correlated variables, \(y_{1}\) and \(y_{2}\), it is possible to forecast the values of these two variables at the future time \(t\), using the data recorded for past n values. Furthermore, assuming the correlation between the two variables, the past value of both \(y_{1}\) and \(y_{2}\) will be used.  
Formally, in case just one set of past measurements (n = 1) is available, the following relationships can be written:  
\[y_{1,t}=c_{1}+\phi_{11,1}\,y_{1,t-1}+\phi_{12,1}\,y_{2,t-1}+e_{1,t} \tag{4}\]  
\[y_{2,t}=c_{2}+\phi_{21,1}\,y_{1,t-1}+\phi_{22,1}\,y_{2,t-1}+e_{2,t}, \tag{5}\]  
where \(e_{1,t}\) and \(e_{2,t}\) are white noise processes that may be contemporaneously correlated. The symbols \(c_{1}\) and \(c_{2}\) are constants serving as the intercepts of the model.  
The coefficient \(\phi_{ii,t}\) represents the influence of the \(\ell_{th}\) lag of variable \(y_{i}\) on itself. Instead, the coefficient \(\phi_{ij,t}\) represents the influence of the \(\ell_{th}\) lag of variable \(y_{j}\) on \(y_{i}\), where \"lag\" represents a fixed amount of passing time [47].  
Equations (4) and (5) just state that each individual variable is using the past values of every variable to make the predictions, based on the assumption of significant correlation between these variables. This approach is useful for describing the dynamic behavior of the data and allows for improving the forecast model.