## 3 Results
### Test Case 2: Reconstructing the Initial Field  
The focus of the study is to evaluate the unknown initial state of the temperature field \(\mathbf{r}_{0}\) based on the observations \(\mathbf{d}\). The posterior model \(f(\mathbf{r}_{0}|\mathbf{d})\) is assessed using the SEnKS, see Appendix A, and the results are compared to those from the traditional EnKS. The true initial temperature field \(\mathbf{r}_{0}\) is set at 20 \({}^{\circ}\)C except for a square shaped region with temperature set at 45 \({}^{\circ}\)C, see Figure 12. The temperature field is formally defined as  
\[\mathbf{r}_{0}(\mathbf{x})=\tau_{1}\mathbf{1}(\mathbf{x}\in D_{1})+\tau_{2}\mathbf{1}(\mathbf{x}\in D_{ 2}), \tag{26}\]  
where \(D_{1}\subset D\) is the low temperature area and \(D_{2}\subset D\) is the high temperature area, and \(\tau_{1}=20\)\({}^{\circ}\)C and \(\tau_{2}=45\)\({}^{\circ}\)C. Figure 12 shows the true log-diffusivity \(n\)-vector \(\log(\mathbf{\lambda})\). The diffusivity \(\lambda\) is always positive. To ensure that ensemble updates do not lead to negative diffusivity values, we work on \(\log(\mathbf{\lambda})\). The heat contained in the high temperature area will diffuse towards the rest of the field according to the diffusion equation in Equation (20), see Figure 13. The temporal observations are collected at five different observation locations according to the likelihood model in Equation (23), see Figure 12. Figure 14 displays the observations \(\mathbf{d}\) where it is clear that the observed temperature increases substantially only at the observation locations close to the high temperature area. The measurements are taken every second from T = 0 to T = 50. The unknown initial temperature field \(\mathbf{r}_{0}\) is assigned a stationary selection-Gaussian random field prior model with parameters \(\mathbf{\Theta}_{r}^{SG}=(\mu_{r},\mu_{v},\sigma_{r},\mathbf{\Sigma}_{r}^{\rho},\gamma,A)\). The parameter values are listed in Table 3. The unknown log-diffusivity field \(\log(\mathbf{\lambda})\) is assigned a stationary Gaussian random field prior model with parameters \(\mathbf{\Theta}_{\lambda}^{G}=(\mu_{\lambda},\sigma_{\lambda},\mathbf{\Sigma}_{\lambda }^{\rho})\) with expectation and variance levels \(\mu_{\lambda}=-8.5\) and \(\sigma_{\lambda}^{2}=2\), respectively. For both prior models, the spatial correlation \((n\times n)\)-matrix \(\mathbf{\Sigma}^{\rho}\) is defined by the second order exponential spatial correlation function \(\rho. (\tau)=\exp{(-\tau^{2}/\delta^{2})};\delta=0.15\), with interdistance \(\tau\). Figure 15 contains four realizations from the prior model of the temperature field and their spatial histograms. The marginal initial distributions of the realizations are bi-modal and spatially stationary except for boundary effects. The selection set \(A\subset\mathbb{R}^{n}\) in the prior model is chosen to obtain a bi-modal marginal distribution with one large mode approximately centered about 20 \({}^{\circ}\)C and a smaller mode centered close to 45 \({}^{\circ}\)C.  
Figure 11: EnKF approach: Realizations of the posterior distribution of the log diffusivity \(f(\log(\mathbf{\lambda})|\mathbf{d}_{0:T})\) at time \(T=100\) (**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. The SenKS operates on the \(3n\)-vector \([\bar{\mathbf{r}}_{0},\mathbf{v},\log(\lambda)]\), and therefore we generate an initial ensemble with \(n_{e}\) = 10,000 ensemble members that are sampled from the Gaussian \(3n\)-vector \([\bar{\mathbf{r}}_{0},\
u,\log(\lambda)]\) with pdf,  
\[ \begin{bmatrix}\bar{\mathbf{r}}_{0}\\ \mathbf{v}\\ \log(\lambda)\end{bmatrix} \sim\varphi_{3n}\left( \begin{bmatrix}\bar{\mathbf{r}}_{ 0}\\ \mathbf{v}\\ \log(\lambda)\end{bmatrix} ; \begin{bmatrix}\mathbf{\mu}_{\eta}\\ \mathbf{\
u}_{\
u}\\ \mathbf{\mu}_{\lambda}\end{bmatrix} ,