## 3 Numerical Results
### Bayesian Update
#### 3.4.1 ES and gPCE-ES  
We assimilate a time series of vertical displacements collected at a point on the top surface located approximately at the center of the reservoir (Figure 2a). The synthetic observations are given by the geomechanical model run with what we assume to be the \"true\" parameter set \((\lambda^{*}=0.011,R=1.25)\). The purpose is to constrain the forecast response of both state variables, for example, surface displacements and material parameters. Figure 4 shows the prior and updated ensembles of the uncertain parameters, \(\lambda^{*}\) (left sub-panels) and \(R\) (right sub-panels), obtained by progressively increasing the number of MC realizations. The assimilation is carried out by both the standard ES with the full forward model (red dots) and the proposed gPCE-ES approach (blue dots). As expected, the mean of the updated ensembles tends to approach the \"true\" value for both parameters, though with possible oscillations far from convergence, for example at \(n_{MC}=50\). We provide the results for \(n_{MC}=(16,25,41,50,75,100)\) in order to analyze the cases when the number of realizations is either smaller than, equal to, or larger than, the number of observations \(m=41\). Since the total cost of an ES application can be regarded as roughly equal to the number of forward model runs, we can measure the computational cost, \(\eta\), just by counting the number of MC simulations. Therefore, the computational cost \(\eta_{g}\) associated to gPCE-ES is constant and equal to 16, that is, the number \(n_{q}\) of forward model runs required to compute the surrogate gPCE model. By distinction, with the standard ES the computational cost, \(\eta_{f}\), increases with \(n_{MC}\). The uncertainty reduction is significantly improved if \(n_{MC}\geq m\). In case \(n_{MC}<m\) we retain 99% of the non-zero singular values computed to invert the matrix in Equation (26). Generally speaking, the parameter \(R\) better approaches the \"true\" value as compared to \(\lambda^{*}\). Figure 3: Sobol’ indices versus simulated time. Profiles with markers are used for the single parameters, while continuous profiles are used for the combinations of parameters. (**a**) Case with \(n=4\) parameters and gPCE degree \(N=2\). (**b**) Case with \(n=4\) parameters and gPCE degree \(N=3\) with a sparse grid. (**c**) The same as (**a**) considering the total effect. (**d**) The same as (**b**) considering the total effect. Figures 5 and 6 show the update of the state variable \(\mathbf{u}_{z}\), that is, the vertical displacement in time at the monitoring point, by increasing \(n_{MC}\) with ES and gPCE-ES, respectively. The observations are generally well-captured by both approaches, with no significant differences between the methods. Hence, the proposed surrogate model appears to capture very well the overall behaviour of the forward model. To measure the quality of the approximation provided by gPCE-ES, we define the metrics AE (average error) and AES (average ensemble spread) defined as:  
\[AE=\frac{1}{n_{MC}}\sum_{k=1}^{n_{MC}}|\psi_{k}-\psi^{true}| \tag{31}\]  
\[AES=\frac{1}{n_{MC}}\sum_{k=1}^{n_{MC}}|\psi_{k}-\mu_{\psi}|, \tag{32}\]  
where \(\psi\) generally stands for state variables or parameters, and \(\mu_{\psi}\) its mean. The metrics are computed both for prior and updated ensembles of states and parameters, providing a quantitative measure of the assimilation quality. For the state \(\mathbf{u}_{z}\), time-averaged values are used to compute the metrics. AE and AES reductions (Table 2) larger than 90% are obtained for both ES and gPCE-ES, thus confirming that the surrogate model is able to reproduce almost exactly the full forward model behaviour. Figure 4: Parameter ensembles for increasing number of Monte Carlo realizations \(n_{MC}\). Panels (**a**) and (**b**) use the standard Ensemble Smoother (ES) with the full forward model for \(\lambda^{*}\) and \(R\), respectively. Panels (**c**,**d**) provide the same outcome as (**a**,**b**) by using the gPCE-ES approach.