## 3 Synthetic Cases
### Multi-Layer Inversion  
As a more application-oriented example of a joint inversion using the KEG, in which the number and thickness of subsurface layers do not need to be known a priori, we introduce a parameter model with a large number of discrete model layers. In this example, the synthetic true model consists of three distinct subsurface layers showing vertical contrast in EC (dotted line in Figure 3): an upper layer of 70 cm thickness and an EC of 0.2 S/m, a resistive (0.08 S/m) intermediate layer of 1 m, and a bottom layer extending to infinite depth with an EC of 0.1 S/m. The amount of discrete layers is chosen so large that it is expected that the boundaries of the discrete layers will not strongly influence the Bayesian update. The discrete model layers are 10 cm thick. We use an ensemble of size 10,000 to sample the defined PDFs. Similar to the procedure for the few-layer inversion, we derive logarithmic mean and STD for the EC of all discrete model layers. Mean and STD define the prior Gaussian PDF for all of the model parameters. Again, no correlation between prior model parameters is implemented. This way, we aim to derive a Gaussian prior model uniform in depth, imposing as little spurious correlation as possible on the inversion procedure. This prior selection results in a mean of 0.1054 S/m with the prior ranging within one STD to the upper limit of 0.1388 S/m and the lower limit of 0.0800 S/m. The measurement setups used for simulation of VES and FDEM data equal the ones that were used for the previous few-layer inversion example. In particular, the VES data is simulated for a voltage electrode offset of 30 cm, the current electrodes add another 30 cm for the first measurement. Thereafter, the current electrode offset is extended in 30 cm steps, up to a total extent of 15 m. The FDEM data are simulated for four receiver coils with offsets and geometrical configurations, as shown in Figure 1. Gaussian noise of 0.1% is added to this simulated measurement values and thereafter used for construction of matrix \(\mathbf{D}\). In Figure 3, the results for separate and joint KEG inversions are compared to the synthetic true model. Overall, the joint parameter estimate shows the smallest uncertainty around the best fit models. For almost all posterior model parameters, the true synthetic model is included within two STDs around this ensemble mean. Exceptions can be found around the upper layer boundary. The large EC contrast at the layer boundaries is not always captured within two STDs due to the smoothing effect of the multi-layer inversion. The different best fit models, i.e., the ensemble means, can be compared to the synthetic true model for their ability to characterize layers of different EC. In general, the best fit of the FDEM data inversion give good estimates for the uppermost layer. In contrast, the VES data inversion better characterizes the EC of the lowermost layer. However, while the top layer is distinctly represented in all three inversion results, for the separate VES and FDEM inversions, intermediate and bottom layer cannot clearly be distinguished by analysis of the best fit models. In contrast, in the best fit model derived from joint inversion, there is a clear difference of the average EC estimate for intermediate and bottom layer. The quality of the fits derived from the separate and joint inversions are quantified computing the root-mean-square error (RMSE) between the synthetic true models and the ensemble mean estimates. The results for the RMSEs are given in Table 1. As expected from visual comparison in Figure 3, the best fit for the joint inversion gives a smaller RMSE than the separate inversions of VES and FDEM data. To provide insight on the efficiency of the joint EEG method, we investigate the above example for different ensemble sizes. We compare the KEG results for an ensemble of size 100, 1000, 2500, 10,000, and 100,000.