## 3 Results
### Test Case 1: Predicting the Parameter Field
For both prior models, the spatial correlation \((n\times n)\)-matrix \(\mathbf{\Sigma}_{r}^{\rho}\) is defined by the second order exponential spatial correlation function \(\rho_{\cdot}(\tau)=\exp{(-\tau^{2}/\delta^{2})};\delta=0.15\), with interdistance \(\tau\). Figure 5 contains realizations from the prior model of the log-diffusivity field and their associated spatial histograms. The prior model is specified to be spatially stationary except for boundary effects with bi-modal spatial histograms. The selection set \(A\subset\mathbb{R}^{n}\) for the prior model is chosen to obtain bi-modal marginal distributions with a very dominant mode centered slightly above the value for \(\lambda_{1}\) and a very small mode centered slightly below the value for \(\lambda_{2}\). The prior is therefore not centered at the true values. Note that the joint random field \([\log(\lambda),\mathbf{r}_{0}]\) will appear as a bi-variate selection-Gaussian random field, see [21]. The SEnKF operates on the \(3n\)-vector \([\log(\mathbf{\bar{\lambda}}),\mathbf{r},\mathbf{r}_{0}]\), and therefore we generate an initial ensemble with \(n_{e}=\) 10,000 ensemble members that are sampled from the Gaussian \(3n\)-vector \([\log(\mathbf{\bar{\lambda}}),\mathbf{v},\mathbf{r}_{0}]\) with pdf,  
\[ \begin{bmatrix}\log(\mathbf{\bar{\lambda}})\\ \mathbf{r}_{0}\end{bmatrix} \sim\varphi_{3n}\left( \begin{bmatrix}\log(\mathbf{\bar{ \lambda}})\\ \mathbf{v}\\ \mathbf{r}_{0}\end{bmatrix} ; \begin{bmatrix}\mathbf{\mu}_{r}^{\lambda}\\ \mathbf{\mu}_{r}^{\lambda}\\ \mathbf{\mu}_{r}\end{bmatrix} , \begin{bmatrix}\sigma_{r}^{\lambda}\mathbf{\Sigma}_{r}^ {\rho}&\gamma^{\lambda}\sigma_{r}^{\lambda}\mathbf{\Sigma}_{r}^{\rho}&0\\ \gamma^{\lambda}\sigma_{r}^{\lambda}\mathbf{\Sigma}_{r}^{\rho}&\gamma^{\lambda} \mathbf{\Sigma}_{r}^{\rho}+(1-\gamma^{\lambda}{}^{2})\mathbf{I}_{n}&0\\ 0&0&\sigma_{r}^{\tau}\mathbf{\Sigma}_{r}^{\rho}\end{bmatrix} \right). \tag{25}\]  
The EnKF operates on the \(2n\)-vector \([\log(\mathbf{\lambda}),\mathbf{r}_{0}]\), and therefore we generate an initial ensemble with \(n_{e}=\) 10,000 ensemble members that are sampled from the selection-Gaussian distribution \(f(\log(\mathbf{\lambda}),\mathbf{r}_{0})\). The variables \(\log(\mathbf{\lambda})\) and \(\mathbf{r}_{0}\) are independent, so we generate them independently: 10,000 samples from the selection-Gaussian \(n\)-vector \(\log(\mathbf{\lambda})\) with parameters \(\mathbf{\Theta}_{\mathbf{\bar{\lambda}}}^{\text{SC}}\) and 10,000 samples from the Gaussian \(n\)-vector \(\mathbf{r}_{0}\) with parameters \(\mathbf{\Theta}_{r}^{G}\). It is important to understand that both ensemble algorithms are initiated with an ensemble from an identical selection-Gaussian random field prior  
Figure 5: Realizations from the initial selection-Gaussian distribution of the log diffusivity \(f(\log(\mathbf{\lambda}))\) at time \(T=0\) (upper panels) and associated spatial histogram (lower panels). Lower panels: the horizontal axes represent the log-diffusivity, the vertical axes represent the relative prevalence of each log-diffusivity value for the realization in the panel right above. model for \([\log(\lambda),r_{0}]\) at \(T=0\), which reflects the bi-modality of the prior model. Due to the size of the ensemble relative to the dimension of the problem, we are using neither localization nor inflation in the algorithms. To illustrate the differences between the SEnKF and the EnKF, we present the following results for both algorithms:  
1. The marginal posterior distributions \(f(\log(\lambda_{i})|\mathbf{d}_{0:T})\) of the log-diffusivity field at four monitoring locations denoted \(1,2,3,4\) on Figure 2, at time \(T=0,50,80,100\). 2. The marginal maximum a posteriori (MMAP) prediction of the log-diffusivity field at time at time \(T=0,50,80,100\). 3. Realizations from the posterior distribution \(f(\log(\lambda)|\mathbf{d}_{0:T})\) at time \(T=100\). 4. The root mean square errors (RMSE) of the MMAP prediction of the log-diffusivity field relative to the true log-diffusivity field at time \(T=100\).