## 3 Results
### Test Case 1: Predicting the Parameter Field
Figures 6 and 7 show the marginal posterior pdfs \(f(\log(\lambda_{i})|\mathbf{d}_{0:T})\) at the four monitoring locations at time \(T=0,50,80,100\) for the SEnKF and EnKF algorithms, respectively. Monitoring locations \(1\) and \(2\) are placed within the high diffusivity area while the two other locations are placed far into the low diffusivity area. At \(T=0\), all pdfs are identical, in all locations due to the stationary prior model and for both algorithms due to identical prior models. The SEnKF results appear to preserve bi-modality as observations are assimilated. As more data are made available, the high value mode increases at monitoring locations \(1\) and \(2\) that are inside the high diffusivity area. The low value mode remains dominant at monitoring locations within the low diffusivity areas. These results reflect expected behaviors. The traditional EnKF results are significantly different since the bi-modality of the marginal pdfs disappears already at \(T=50\). The marginal pdfs are Gaussian-like and are gently moved toward high and low values depending on which diffusivity area the monitoring locations are in. This regression toward the mean effect of the EnKF is generally recognized as it gives the best prediction in the squared error sense [27]. Figure 6: SEnKF approach: Marginal posterior distribution of the log diffusivity \(f(\log(\lambda_{i})|\mathbf{d}_{0:T})\) at time \(T=0,50,80,100\) at the monitoring locations \((1,2,3,4)\) denoted \((p_{1},p_{2},p_{3},p_{4})\). Figure 8 displays the MMAP predictions based on the SEnKF and the traditional EnKF at time \(T=0,50,80,100\). At \(T=0\), the predictions from the two algorithms are identical since they use identical prior models. As observations are assimilated, the SEnKF predictions reproduce the high diffusivity area relatively well, with clear contrast. The traditional EnKF predictions also indicate the diffusivity areas, but with less contrast. Figure 9 displays the MMAP prediction at \(T=100\) along the section B-B' shown in Figure 2. The high contrast reliable reconstruction of the high diffusivity channel by the SEnKF algorithm is confirmed. The traditional EnKF predictions appear less reliable. The 80% highest density interval (HDI) [28] covers the true diffusivity values for the SEnKF while these values are far outside the interval for the traditional EnKF results. The results are consistent with the observations made regarding the marginal posterior pdfs in Figures 6 and 7. Figures 10 and 11 show realizations and spatial histograms from the posterior distribution of the log-diffusivity at time \(T=100\) for the SEnKF and traditional EnKF algorithms, respectively. The realizations from the SEnKF largely reproduce the channel with clear contrast while the realizations from the EnKF also reproduce the channel, but with much less contrast. The spatial histograms also underline the difference in contrast in that they are clearly bi-modal for the SEnKS and much more Gaussian-like for the EnKS. Table 2 shows that the RMSE of the MMAP prediction relative to the true diffusivity field for the SEnKF is approximately 30% lower than for the EnKF. Figure 8: MMAP predictions of the log diffusivity field \((\log(\Lambda)|\mathbf{d}_{0,T})\) at time \(T=0,50,80,100\) (upper panels—SEnKF approach, lower panels—EnKF approach). Figure 10: SEnKF approach: Realizations of the posterior distribution of the log diffusivity \(f(\log(\Lambda)|\mathbf{d}_{0,T})\) 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. Figure 9: MMAP predictions of the log diffusivity field with 80% HDI in cross section B-B\({}^{\prime}\) at time \(T=100\) with SEnKF (**left**) and with EnKF (**right**).