## 3 Results
### Test Case 2: Reconstructing the Initial Field
\begin{bmatrix}\sigma_{\eta}{}^{2}\mathbf{\Sigma} _{\eta}^{\rho}&\gamma\sigma_{\eta}{}^{2}\mathbf{\Sigma}_{\eta}^{\rho}&0\\ \gamma\sigma_{\eta}{}^{2}\mathbf{\Sigma}_{\eta}^{\rho}&\gamma^{2}\mathbf{\Sigma}_{\eta} ^{\rho}+(1-\gamma^{2})\mathbf{I}_{n}&0\\ 0&0&\sigma_{\lambda}{}^{2}\mathbf{\Sigma}_{\lambda}^{\rho}\end{bmatrix} \right). \tag{27}\]  
The EnKS operates on the \(2n\)-vector \([r_{0},\log(\lambda)]\), and therefore we generate an initial ensemble with \(n_{e}\) = 10,000 ensemble members that are sampled from selection-Gaussian distribution \(f(\mathbf{r}_{0},\log(\lambda))\). The variables \(\mathbf{r}_{0}\) and \(\log(\lambda)\) are independent, so we generate them independently: 10,000 samples from the selection-Gaussian \(n\)-vector \(\mathbf{r}_{0}\) with parameters \(\mathbf{\Theta}_{r}^{SG}\) and 10,000 samples from the Gaussian \(n\)-vector \(\log(\lambda)\) with parameters \(\mathbf{\Theta}_{\lambda}^{G}\). Due to the size of the ensemble relative to the dimension of the problem, we used neither localization nor inflation in the algorithms. To illustrate the differences between the SenKS and the EnKS we present the following results for both algorithms: \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline \(\mu_{\eta}\) & 28.75 \\ \(\mu_{\
u}\) & 0 \\ \(\sigma_{\eta}\) & \(\sqrt{10}\) \\ \(\gamma\) & 0.8 \\ \(A\) & \(]-\infty,-0.2]\cup[0.5,+\infty[^{n}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter values for the selection-Gaussian initial distribution for the initial temperature prior model. Figure 12: Initial temperature (\({}^{\circ}\)C) field (**left**) with data collection points \(\cdot\) and monitoring locations \(\times\) and reference log-diffusivity field (**right**). Figure 13: True temperature (\({}^{\circ}\)C) field evolution over time. 1. The marginal posterior distributions \(f(r_{0,i}|\mathbf{d}_{0:T})\) of the initial temperature field at four monitoring locations denoted \(1,2,3,4\) on Figure 12, at time \(T=0,20,30,50\). 2. The marginal maximum a posteriori (MMAP) prediction of the initial temperature field at time at time \(T=0,20,30,50\). 3. Realizations from the posterior distribution \(f(\mathbf{r}_{0}|\mathbf{d}_{0:T})\) of the initial temperature field at time \(T=50\). 4. The root mean square errors (RMSE) of the MMAP prediction of the initial temperature field relative to the true initial temperature field at time \(T=50\). The marginal posterior pdfs \(f(\mathbf{r}_{0,i}|\mathbf{d}_{0:T})\) at the four monitoring locations at time \(T=\) 0,20,30,50 are displayed in Figures 16 and 17 for the SEnKS and EnKS algorithms, respectively. At \(T=0\), the prior models for both algorithms are identical and so are the marginal pdfs. Monitoring location 1 is placed  
Figure 14: Data collected over time (points) and true temperature (\({}^{\circ}\)C) evolution at the data collection points (line). inside the high temperature area. As observations are assimilated, the marginal pdf from the SEnKS remain bi-modal, but the high value mode increases steadily. For the other monitoring locations, all placed outside the high temperature area, the bi-modality is reproduced but with a dominant low value mode. The relative size of the modes reflects the distance to the high temperature area and the observation locations. The marginal pdfs from the EnKS lose their bi-modality after a few assimilation steps and from then on the Gaussian-like marginal pdfs are only slightly shifted by the assimilation of observations. Figure 16: SEnKS approach: Marginal posterior distributions of the initial temperature \(f(\tau_{0j}|d_{0:T})\) at time \(T=0,20,30,50\) at monitoring locations (\(i=1,2,3,4\)) denoted \((p_{1},p_{2},p_{3},p_{4})\). The horizontal axes representing temperature are expressed in \({}^{\circ}\)C.  
Figure 15: Realizations from the selection-Gaussian initial distribution of the initial temperature field \(f(\tau_{0})\) at time \(T=0\) (upper panels) and associated spatial histogram (lower panels).