## 3 Numerical Results
### gPCE Surrogate Model  
Since we use uniform distributions for the input parameters, the gPCE basis functions are multivariate Legendre polynomials, which are orthogonal with respect to the uniform measure. The first objective of this analysis is to test the quality of gPCE as a surrogate of the exact outcome of the forward model. We provide results for different degrees of the polynomial truncation, \(N=\)1,2,3. A full tensor approach is used for \(N=1\) and \(N=2\), leading to a total number \(n_{q}\) of forward model runs equal to 16 and 81, respectively. Since with \(N=3\) we would have needed \(n_{q}=256\) full model runs, a sparse grid approach based on [PERSON]'s coarse tensorization [54] is used, thus reducing \(n_{q}\) to 137. It should be noticed that the use of such sparse grid numerical integration might lead to unacceptable errors in the expansion coefficients with higher order polynomials [55]. This issue is not encountered in the present application, however alternative accurate approaches can be also employed if needed, such as those advanced in References [55; 56].  
A quantitative evaluation of the fitting quality of the gPCE surrogate is provided by employing the leave-one-out (LOO) cross-validation [57; 58], where an estimate of the mean-square error of the residuals between the full model and surrogate solution is given by the SSE\({}_{\text{LOO}}\) defined as:  
\[\text{SSE}_{\text{LOO}}=\frac{1}{n_{q}}\sum_{i=1}^{n_{q}}(\text{u}_{z,i}- \widehat{\text{u}}_{z,i}^{\sim i})^{2}, \tag{29}\]  
where \(\text{u}_{z,i}\) is the full model result at a certain time \(t\) and parameter combination \(i\) and \(\widehat{\text{u}}_{z,i}^{\sim i}\) is the surrogate solution at the same \(t\) and \(i\) with the gPCE built without the point denoted by \(i\). In practice,  
Figure 2: **(a)** 2D view of the model top with the green point indicating the location where the measurements are supposed to be recorded. **(b)** Synthetic sequence of vertical displacements \(u_{z}\) used in the assimilation process for two different parameter sets \(\mathbf{p}^{\dagger}\). The continuous and dashed lines are the outcomes of the geomechanical model, while dots and squares indicate the assimilated observations.  
the \(n_{q}\) residuals can be obtained without building \(n_{q}\) different gPCE but using the original gPCE constructed with the whole set of collocation points [57]. The coefficient \(\text{Q}^{2}\), similar to the coefficient of determination \(\text{R}^{2}\), can be defined as:  
\[\text{Q}^{2}=1-\frac{\text{SSE}_{\text{LOO}}}{\text{SST}}, \tag{30}\]  
where SST is the sum of the squared deviations of \(\text{u}_{zj}\) from their mean values \(\mu_{u}\) at the instant \(t\). Values of \(\text{Q}^{2}\) close to 1.0 indicates a good match between the model outcome and the gPCE surrogate approximation. Table 1 reports the mean and variance of vertical displacements at the times \(t=\)1, 5, and 10 years, along with the \(\text{Q}^{2}\) value. \(\text{Q}^{2}\) progressively increases to the upper limit of 1 for increasing \(N\), proving the convergence of both the polynomial expansion and the high-dimensional quadrature formula. This holds true for \(t>5\) years with a gPCE-surrogate solution less accurate at the onset of the simulation.