## 3 Synthetic Cases
### Four-Layer Inversion  
In a first synthetic toy case, we simulate FDEM and VES data for a subsurface that consists of four layers with contrast in electrical conductivity. The synthetic true model is shown in Figure 1a. The VES data is simulated for an electrode spacing of 30 cm for the voltage electrodes and 30 cm for the current electrodes (compare Figure 1c). The FDEM data is simulated for one transmitter coil and four distinct receiver coils, two in horizontal co-planar (HCP) configuration in 1 and 2 m distance from the transmitter, and two in perpendicular (PRP) configuration in 1.1 and 2.1 m distance from the transmitter coil (compare Figure 1b). The simulated forward responses for the synthetic true model are listed in Figure 1b,c. Small Gaussian white noise of 0.1% was added to all simulated measurements and, as such, considered for the construction of the measurement ensemble **D**. We consider a uniform Gaussian prior model, i.e., the same Gaussian PDF for all model parameters, to exclude further spurious correlation from the inverse processing. Additionally, we define a simple diagonal covariance matrix for the prior model parameters. For the construction of the prior PDF, mean and standard deviation of the logarithmic electrical conductivity of the synthetic true model are considered in steps of 10 cm. Using this mean and STD, we construct the same Gaussian prior PDF for all four model parameters. In particular, the prior mean is set to 0.0581 S/m, with the prior ranging within one STD to the upper limit of 0.1195 S/m and the lower limit of 0.0282 S/m. The Gaussian prior PDFs are sampled to construct the ensemble matrix **A**, and **A** is used to compute the forward response ensemble matrix **G** (Equation (9)). The results for the separate and joint inversion of VES and FDEM data are given in the covariance matrix-like plot in Figure 2[8]. The posterior estimates of the four model parameters are given on the diagonal of the figure. Additionally, the ensemble mean estimates, i.e., the best fits, together with the synthetic true model, are shown in the lower left part of Figure 2. On the off-diagonals, we show scatter plots of pairs of estimated model parameters. The point clouds in the scatter plots on the off-diagonals in Figure 2 illustrate the range of parameter estimates that are compatible with the measurement data. Some interesting aspects are linked to the shapes of these point clouds. First, the point clouds have different shapes and locations for the three applied inversions. This implies that the two data sets do include mutually different information regarding the subsurface. Second, the point clouds for FDEM inversion are smaller than the ones for VES inversion. This is in accordance with the expectations for low-noise FDEM data,  
Figure 2: Covariance matrix-like arrangement of scatter plots of the 1000 KEG samples of all parameter pairs from the posterior models for separate (blue and green) and joint (red) inversion of FDEM and VES data, including a larger black dot for the synthetic true solution. The four plots on the diagonal show the respective Gaussian posterior probability functions. The true subsurface model is shown in the lower left corner (black solid line) with its layer boundaries (black dashed line) being accompanied by the best fit (posterior mean) estimates derived from the respective separate and joint inversions. as signal generation is limited to distinct flows of the induced eddy currents, making the FDEM data, in theory, less prone to non-uniqueness. Third, the range compatible with both data sets is smaller than the range compatible with both separate data sets. This complies with the expectation that a joint inversion approach reduces parameter uncertainty. The parameter estimates that are shown on the diagonal in Figure 2 allow further conclusions regarding the performance of the joint inversion. First, all posterior solutions enclose the synthetic true solution. Second, for all four model parameter estimates, the joint posterior PDF shows the smallest STD.