## 3 Results
### Test Case 2: Reconstructing the Initial Field
Upper panels: the colorbar gives the temperature in \({}^{\circ}\)C. Lower panels: the horizontal axes represent the temperature (\({}^{\circ}\)C), the vertical axes represent the relative prevalence of each temperature value for the realization right above. Figure 18 displays the MMAP predictions of the initial temperature field based on the SEnKS and the traditional EnKS at time \(T=0,20,30,50\). For the SEnKS, the high temperature area is clearly identifiable with clear contrast from time \(T=30\) while for the EnKS the high temperature area is hardly ever identifiable on the MMAP predictions that show little contrast. Figure 19 displays the MMAP prediction of the initial temperature field at \(T=50\) along the section A-A\({}^{\prime}\), see Figure 12, for the SEnKS and the traditional EnKS. The SEnKS clearly identifies the high temperature area and the 80% HDI covers the truth, while the EnKS clearly fails to identify the high temperature area and the 80% HDI does not even cover it. Figure 17: EnKS approach: Marginal posterior distributions of the initial temperature \(f(r_{0,i}|\mathbf{d}_{0,T})\) 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 18: MMAP predictions of the initial temperature (\({}^{\circ}\)C) field at time \(T=0,20,30,50\) for the SEnKS approach (**upper**) and the EnKS approach (**lower**). Realizations of the posterior model at \(T=50\) based on the SEnKS and the traditional EnKS algorithms are displayed in Figures 20 and 21, respectively. The SEnKS produces realizations that appear bi-modal while the ensemble members from the EnKS display more symmetric spatial histograms. Even though the differences between the realizations are quite subtle, they are consistent with previous results. Table 4 shows that RMSE of the MMAP prediction relative to the true initial temperature field for the SEnKS is approximately 20% lower than for the EnKS. This test case clearly illustrates the ability of the SEnKS to conserve multimodality in the posterior distribution, and it leads to predictions with better contrast and accuracy. We conclude that the SEnKS algorithm provides a more reliable reconstruction of the initial state of the temperature field than the traditional EnKS algorithm. Note that the posterior model for the unknown diffusivity field \(f(\log(\Lambda)|\mathbf{d})\) can also be assessed with the two algorithms. When comparing the MMAP predictions relative to the true diffusivity field, see Figure 12, we observe that none of the algorithms provide reliable predictions. We conclude that the small scale variations in the field are not sufficiently distinct to be identified. Figure 19: MMAP predictions of the initial temperature (\({}^{\circ}\)C) field (\(\tau_{0}|\mathbf{d}_{0:T}\)) with 80% HDI in cross section A-A’ at time \(T=50\) with SEnKS (**left**) and with EnKS (**right**). Figure 20: SEnKS approach: Realizations from the posterior distribution \(f(\mathbf{r}_{0}|\mathbf{d}_{0:T})\) of the initial temperature field at time \(T=50\). Upper panels: the colorbar gives the temperature in \({}^{\circ}\)C. Lower panels: the horizontal axes represent the temperature (\({}^{\circ}\)C), the vertical axes represent the relative prevalence of each temperature value for the realization right above.