## 2 Mathematical Framework
### Sobol' Indices in the gPCE Framework  
The objective of a preliminary SA is to quantify the influence of the uncertain input parameters and their combinations on the model output, so as to retain only the most significant contributions. This is important to (i) focus the design of laboratory/field experiments on the collection of measurements of the most important parameters, and (ii) reduce the number of calibration parameters for an easier solution of the inverse problem, especially in real-world applications. In this work, we propose the use of the variance-based SA according to the framework developed in Reference [45]. A variance decomposition of the model output into partial variances of the model input (and combination of more than one parameter), allows to estimate the so-called Sobol' indices, which provide a measure of the relative importance of each uncertain parameter to the model outcome. In the sequel, we provide a brief description of the ideas lying behind Sobol' indices. For completeness, the reader is referred to the work of Reference [45] or to the application developed in Reference [19]. For the sake of simplicity, let us consider the model outcome \(U\), that is, the quantity of interest, as a one-dimensional variable function \(f\) of the \(n\)-variate random vector \(\mathbf{Z}=(Z_{1},Z_{2},\ldots,Z_{n})\) of mutually independent components defined over an \(n\)-dimensional hypercube \(\Sigma^{n}\). Assuming \(f\) to be a square-integrable function, the functional decomposition of \(U=f(\mathbf{Z})\) reads  
\[f=f_{0}+\sum_{i=1}^{n}f_{i}+\sum_{i=1}^{n}\sum_{j>i}^{n}f_{i,j}+\cdots+f_{1,2, \ldots,n}, \tag{16}\]  
where \(f_{0}\) is a constant representing the mean value of \(f\), and \(f_{i}\), \(f_{i,j}\), \(\ldots\), \(f_{1,2,\ldots,n}\) are the uncorrelated random effects associated to the factors in their indices, for example, \(f_{i}=f_{i}(Z_{i})\) are the main effects due to the factors \(Z_{i}\) and \(f_{i,j}=f_{i,j}(Z_{i},Z_{j})\) are the effects related to interactions between the factors \(Z_{i}\) and \(Z_{j}\) with \(j>i\). It has been proved in Reference [45] that a unique expansion of Equation (16) exists for any function \(f(\mathbf{Z})\) integrable in \(\Sigma^{n}\) under the hypothesis of zero mean of all expansion terms with respect to each variable. The expansion (16) for the quantity of interest \(U\) is used to derive the associated variance \(V(U)\). In fact, squaring Equation (16) and using the orthogonality condition holding for the expansion terms [45], it can be demonstrated that \(V(U)\) reads:  
\[V(U)=\sum_{i=1}^{n}V_{i}+\sum_{i=1}^{n}\sum_{j>i}^{n}V_{i,j}+\dots+V_{1,2,\dots,n}, \tag{17}\]  
where \(V_{i},V_{j},\dots,V_{1,2,\dots,n}\) are the partial variances of \(f_{i},f_{j},\dots,f_{1,2,\dots,n}\), respectively. Generally, the partial variance \(V_{i_{1},\dots,i_{s}}\) in (17) reads:  
\[V_{i_{1},\dots,i_{s}}=\int_{\Sigma^{s}}f_{i_{1},\dots,i_{s}}^{2}(Z_{i_{1}}, \dots,Z_{i_{s}})d(Z_{i_{1}},\dots,Z_{i_{s}}) \tag{18}\]  
which can be written in terms of conditional expectations as:  
\[V_{i} = V(\mathbb{E}(U|Z_{i}))\] \[V_{i,j} = V(\mathbb{E}(U|Z_{i},Z_{j}))-V_{i}-V_{j}\] \[V_{i,j,k} = V(\mathbb{E}(U|Z_{i},Z_{j},Z_{k}))-V_{i}-V_{j}-V_{k}-V_{i,j}-V_{ j,k}-V_{j,k}\] \[\vdots\] \[V_{1,\dots,n} = V(U)-\sum_{i=1}^{n}V_{i}-\sum_{i=1}^{n}\sum_{j>i}^{n}V_{i,j}- \dots-\sum_{1\leq i_{1}<\dots<i_{n-1}\leq n}^{n}V_{i_{1},\dots,i_{n-1}}.\]  
Equations (19) provide practical relationships for the computation of partial variances for any indices combination. The Sobol' first and higher-order indices are defined as:  
\[S_{i}=\frac{V_{i}(U)}{V(U)} \tag{20}\]  
\[S_{1,\dots,s}=\frac{V_{1,\dots,s}(U)}{V(U)}, \tag{21}\]  
where \(S_{i}\) measures the relative importance of a single factor \(Z_{i}\) on the total model variance and the higher-order indices \(S_{1,\dots,s}\) represent a measure of the combined model sensitivity to the group of factors \(Z_{1},\dots,Z_{s}\).