## 3 Results
### Test Case 1: Predicting the Parameter Field  
The focus of this test case is to predict the unknown diffusivity field \(\mathbf{\lambda}\) based on the observations \(\mathbf{d}\). Because diffusivity is constant in time, smoothing and filtering give an identical prediction of the field. However, filtering is preferred because it does not require updating the ensemble at all future times in addition to the previous one, see [26]. The posterior model is evaluated using the SEnKF, see Appendix A and the results are compared to those from the traditional EnKF algorithm. The true diffusivity \(n\)-vector \(\mathbf{\lambda}\) is displayed in Figure 2. The diffusivity \(\mathbf{\lambda}\) is always positive. To ensure that ensemble updates do not lead to negative diffusivity values, we work on \(\log(\mathbf{\lambda})\). The figure shows a channel in which the diffusivity is higher than in the rest of the field. The diffusivity field is formally defined as  
\[\mathbf{\lambda}(\mathbf{x})=\lambda_{1}\mathbf{1}(\mathbf{x}\in D_{1})+\lambda_{2}\mathbf{1}(\bm {x}\in D_{2}), \tag{24}\]  
where \(D_{1}\subset D\) is the low diffusivity area and \(D_{2}\subset D\) is the high diffusivity channel. The parameter values are \(\lambda_{1}=e^{-12}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}\) and \(\lambda_{2}=e^{-5}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}\). The true temperature field is initially at \(20\,\mathrm{\SIUnitSymbolCelsius}\) and the heat source on the lower border of the high diffusivity channel starts pumping in heat at \(T=0\) at a constant volumetric rate \(q=15\,\mathrm{W}\,\mathrm{m}^{-3}\), see Figure 2. The temporal evoluation of the temperature field, shown in Figure 3, is obtained by solving the diffusion equation in Equation (20) for the log-diffusivity field in Figure 2 and the initial temperature field defined above. The temperature observations \(\mathbf{d}\), see Figure 4, are then collected from the five locations shown in Figure 2 using the likelihood model defined in Equation (23). The measurements are taken every second from \(T=0\) to \(T=100\). As the heat from the source diffuses mostly along the high diffusivity channel, the observed temperature increases substantially at observation locations within the channel. The unknown initial field for log-diffusivity \(\log(\mathbf{\lambda})\) is assigned a stationary selection-Gaussian random field prior model with parameters \(\mathbf{\Theta}_{\lambda}^{\mathrm{SG}}=(\mu_{r}^{\lambda},\mu_{v}^{\lambda},\sigma _{r}^{\lambda},\Sigma_{r}^{\rho},\gamma^{\lambda},A)\), see [21] and Equation (2). The parameter values for the prior model are listed in Table 1.  
Figure 2: Initial log-diffusivity field with observation locations \(\cdot\), monitoring locations \(\times\), and heat source \(\triangle\). Figure 4: Data collected over time (\(\cdot\)) and true temperature evolution (line) at the data collection points. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(\mu_{\chi}^{2}\) & \(-8.5\) \\ \(\mu_{\chi}^{3}\) & \(0\) \\ \(\sigma_{\tau}^{2}\) & \(\sqrt{1.6}\) \\ \(\gamma^{\lambda}\) & \(0.9\) \\ \(A\) & \(]-\infty,-0.3]\cup[0.5,+\infty]^{\text{\tiny{H}}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameter values for the selection-Gaussian initial distribution for the initial log-diffusivity field. Figure 3: True temperature (\({}^{\circ}\)C) field evolution over time. The unknown initial temperature field \(\mathbf{r}_{0}\) is assigned a stationary Gaussian random field prior model with parameters \(\mathbf{\Theta}_{r}^{G}=(\mu_{r},\sigma_{r},\mathbf{\Sigma}_{r}^{\rho})\) with expectation and variance levels \(\mu_{r}=20\) and \(\sigma_{r}^{2}=2\), respectively. The variance level is relatively large as we assume little prior knowledge of the initial temperature field.