## 2 Mathematical Framework
### gPCE Surrogate Model  
Running the forward geomechanical model multiple times for large and complex systems can be a very demanding task, both in terms of CPU and memory requirements. A gPCE approach [22; 44] is therefore proposed to approximate the outcome of the deterministic simulator and then compute the propagation of the input uncertainty through the forward model. The main idea of gPCE surrogate models is based on using orthogonal polynomial approximations of the random input to project the stochastic model output. In the following, we provide the basic mathematical framework as derived in Reference [44]. Let us consider the random vector \(\mathbf{U}=(U_{1},\ldots,U_{\ell})\in\mathbb{R}^{\ell}\) written as a function of the random vector \(\mathbf{Z}\) of mutually independent random variables \(\mathbf{Z}=(Z_{1},\ldots,Z_{n})\) and distribution function \(F_{Z}(z_{1},\ldots,z_{n})=\mathcal{P}(Z_{1}\leq z_{1},\ldots,Z_{n}\leq z_{n})\), where we are considering a stochastic process in the probability space (\(\Omega\),\(\mathcal{F}\),\(\mathcal{P}\)) with space of events \(\Omega\), \(\sigma\)-algebra \(\mathcal{F}\) and probability measure \(\mathcal{P}\) on \(\mathcal{F}\), see for example Reference [27]. As usual, the independence assumption implies \(F_{Z}(\mathbf{z})=\prod_{i=1}^{n}F_{Z_{i}}(z_{i})\), where \(F_{Z_{i}}(z_{i})=\mathcal{P}(Z_{i}\leq z_{i})\) is the marginal distribution function with \(i=1,\ldots,n\). Since any random variable may be represented as a series of polynomials in uncorrelated and independent Gaussian variables [22] and, in its generalized extension, in non-Gaussian measures, gPCE basis functions of a univariate random variable \(Z_{i}\) are defined as the polynomials \(\{\phi_{k}(Z_{i})\}_{k=0}^{N}\) of \(N\)th-degree satisfying the orthogonality conditions  
\[\mathbb{E}[\phi_{s}(Z_{i})\phi_{r}(Z_{i})]=\int_{\Sigma_{i}}\phi_{s}(z_{i}) \phi_{r}(z_{i})dF_{Z_{i}}(z_{i})=\gamma_{s}\delta_{s,r}\quad 0\leq s,\,r\leq N, \tag{9}\]  
with \(\gamma_{s}=\mathbb{E}[\phi_{s}^{2}(Z_{i})]\) the normalization factors, \(\delta_{s,r}\) the Kronecker delta function and \(\Sigma_{i}\) is the support of \(Z_{i}\). In the multivariate case, the gPCE basis functions \(\Phi_{\mathbf{\alpha}}(\mathbf{Z})\) of degree up to \(N\) are products of the univariate orthogonal polynomials:  
\[\Phi_{\mathbf{\alpha}}(\mathbf{Z})=\phi_{\mathbf{\alpha}_{1}}(Z_{1})\ldots\phi_{\mathbf{ \alpha}_{n}}(Z_{n})\quad\text{with}\quad 0\leq|\mathbf{\alpha}|\leq N, \tag{10}\]  
where \(\mathbf{\alpha}=(\alpha_{1},...,\alpha_{n})\in\mathbb{N}_{0}^{n}\) is a multi-index with \(\ |\mathbf{\alpha}|=\alpha_{1}+\cdots+\alpha_{n}\). The multivariate basis functions are orthogonal polynomials in \(L^{2}_{dF_{Z}}\), that is, the space of all mean-square integrable functions of \(\mathbf{Z}\) with respect to the inner product based on the measure \(dF_{Z}\):  
\[\mathbb{E}[\Phi_{\mathbf{\alpha}}(\mathbf{Z})\Phi_{\mathbf{\zeta}}(\mathbf{Z})]=\int_{\Sigma} \Phi_{\mathbf{\alpha}}(\mathbf{z})\Phi_{\mathbf{\zeta}}(\mathbf{z})dF_{Z}(\mathbf{z})=\gamma_\mathbf{{\alpha}}\delta_{\mathbf{\alpha}\mathbf{\zeta}}, \tag{11}\]  
where \(\Sigma\) is defined by \(\Sigma=\Sigma_{1}\times\Sigma_{2}\cdots\times\Sigma_{n}\). As a consequence, the class of orthogonal polynomials is selected according to the measure \(F_{Z_{i}}\). The polynomials orthogonal for the standard normal distribution are the Hermite polynomials, which form an ideal basis for the output stochastic domain [27]. In the gPCE context, we aim at finding an approximation \(\tilde{\mathbf{U}}_{N}(\mathbf{Z})\) of the random function \(\mathbf{U}(\mathbf{Z})\in\mathbb{R}^{\ell}\) in the \(N\)-th degree polynomial space generated by the basis functions \(\Phi_{\mathbf{\alpha}}(\mathbf{Z})\):  
\[\mathbf{U}(\mathbf{Z})\approx\tilde{\mathbf{U}}_{N}(\mathbf{Z})=\sum_{|\mathbf{\alpha}|\leq N} \mathbf{c_{\mathbf{\alpha}}}\Phi_{\mathbf{\alpha}}(\mathbf{Z}), \tag{12}\]  
where \(\mathbf{c_{\mathbf{\alpha}}}\in\mathbb{R}^{\ell}\) are the coefficients of the expansion.