We simulate an intermediate-scale quasi two-dimensional sandbox of dimension Lx × Ly × Lz. The domain is discretized by nx × nz elements of size Δx × Δz. The geometric parameters are listed in Table 1. The domain is assumed to be electrically insulated, so that no-current boundary conditions can be applied at all boundaries. The true hydraulic conductivity distribution in the sandbox is synthetically generated by using the spectral method of Dietrich and Newsam  with an exponential covariance function for the log conductivities. The resulting field is illustrated in Figure 1a. The top and bottom of the sandbox are no-flow boundaries, while we assume fixed hydraulic head conditions along the right and left boundaries. We simulate the injection of a tracer solution with concentration cin through the left boundary during a specified period of time tinj. The simulated sandbox is equipped with electrodes with a horizontal and vertical spacing of 10 cm. In our test case we use 48 electrodes (one row of 28 on the top and four columns with another five electrodes each in the vertical direction). All the electrodes extend through the whole width of the sandbox, so that the electrical field arising from current injection is two dimensional. The electrodes situated near the top of the sandbox approximately represent surface electrodes, while the others simulate boreholes. Our laboratory type of setup is of course a simplification, as the electrical field in common field applications would be three dimensional. However, we try to reproduce the common situation in which only a few boreholes at distances larger than the aquifer thickness are available, whereas the surface is accessible for a comparably large number of electrodes. We simulate injection and extraction of a current using pairs of electrodes, while the electrical potential is measured at other pairs of electrodes. Various electrode arrays were tested, including surface, cross-hole, surface-to-borehole and single-hole measurements. The chosen example includes a total of 448 surface (336 configurations, including a mixture of Wenner and Schlumberger arrays) and surface-to-borehole (dipole-dipole) measurements. We also considered only the surface measurements, to see whether the absence of boreholes would still lead to satisfactory results. Eight head measurements, at the top and bottom of each borehole, were also included in the inversion. In a field campaign this would require packing off various sections of the borehole to prevent hydraulic shortcuts in the boreholes.
Figure 1. (a) Synthetically generated true log10 hydraulic conductivity field, (b and d) estimated log10 hydraulic conductivity field resulting from the inversion, and (c and e) standard deviation of estimation. Open circles, position of the electrodes used in the inversion; crosses, position of the head measurements.
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Table 1. Parameters Applied in the Test Case
|Geometric Parameters and Discretization|
|Lx × Ly × Lz||length × width × height of domain||2.8 × 0.05 × 0.6 m|
|nx × nz||number of cells in x and z directions||280 × 60|
|Δx × Δz||grid spacings in x and z directions||0.01 × 0.01 m|
|Hydraulic and Transport Parameters|
|cin||concentration in inflow||0.4 g/ℓ|
|αl||longitudinal dispersivity||0.01 m|
|αt||transverse dispersivity||0.001 m|
|Dm||pore diffusion coefficient||10−9 m2/s|
|σ0||base electrical conductivity||0.03 S/m|
|κ||linear dependence of electrical conductivity on concentration||0.06 Sm2/kg|
|tinj||duration of injection||2 h|
|Δt||time discretization in transient calculation||5 min|
|Geostatistical Values of Log Hydraulic Conductivity Field|
|λx × λy||Correlation lengths||0.4 × 0.05 m|
|exp(β*)||prior mean||10−7 m/s|
|Rββ||uncertainty of prior mean||10|
|σh||hydraulic head||10−3 m|
 Generally, a higher number of independent measurements leads to more accurate inversion results, but in real experiments the number of possible measurements is limited. This particularly holds for time-lapse monitoring considered here. Each time-lapse survey requires a certain amount of acquisition time. During this time, the plume keeps on moving through the domain. One advantage of using temporal moments is that we do not need to take into account the moving of the plume during the acquisition cycle, as each single measurement is registered together with the exact time of acquisition, and each temporal moment is calculated from the time series of a single configuration. However, if an acquisition cycle takes too much time, the plume moves too far between two cycles to be captured in the monitoring. For the evaluation of temporal moments it is of importance that key features of the electrical potential time curves, such as the time and magnitude of maximum perturbation, are resolved, implying that a large time step between the measurements could lead to an insufficient amount of data for accurately determining the temporal moments. Mimicking a laboratory experiment using a multichannel ERT device, it is realistic to assume that a single acquisition cycle including a few hundred measurements takes several minutes. We have accounted for that in our simulations by using an appropriate time step.
 In our geostatistical inversion method, we need to compute the sensitivity matrix H, i.e., the partial derivatives of all measurements with respect to all parameters. The sensitivities show the effect of slight changes in a parameter on the measurements. Figure 2 shows two sensitivity patterns for a particular electrode configuration: the sensitivity of electrical resistance with respect to tracer concentration and the sensitivity of the mean arrival time of resistance perturbation with respect to log hydraulic conductivity. The former may be interpreted as the measurement function of concentration and shows a pattern typical in geoelectrics with high sensitivities at the electrodes and changing signs. The sensitivity of the arrival time of the electrical signal on log hydraulic conductivity shows a narrow negative sensitivity stripe which follows the main plume path in upstream direction. We can understand this pattern as the convolution of the first sensitivity with the sensitivity of local concentration arrival time on log conductivity [e.g., Cirpka and Kitanidis, 2000]. By combining various electrode configurations different parts of the hydraulic conductivity field become sensitive. This is a prerequisite to resolve the conductivity field by inversion. The sensitivity patterns in our particular application are influenced by the fact that we consider a sandbox which is bounded on all sides. In a field application with boundaries situated at a large distance from the zone of interest, the sensitivity would be smaller, resulting in a more difficult recovery of the hydraulic conductivity field.
Figure 2. (a) Sensitivity of electrical resistance with respect to tracer concentration and (b) sensitivity of mean arrival time of electrical resistance perturbation with respect to log hydraulic conductivity. Crosses, position of the current electrodes; circles, position of the potential electrodes.
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 The ratio of the first over the zeroth moment gives us the mean arrival time of the perturbation for each electrode configuration. The moments of the noisy data were taken as our measurements in the inversion procedure. In order to assess the uncertainty of the arrival times of resistance perturbations, we follow a bootstrap procedure, in which additional white noise is added to the Δϕ′(t) time curves. For each breakthrough curve we generated a set of 1000 bootstraps, and calculated the temporal moments for each one of them. The resulting standard deviation of the ratio of the first over the zeroth moment of potential perturbation was taken as the uncertainty of the measurement. To simulate noisy measurements of hydraulic heads, we added a random error with zero mean and standard deviation of 1 mm to the error-free hydraulic heads at the corresponding nodes. The value of 1 mm was also considered as measurement uncertainty. While this accuracy might be optimistic for field applications, it should be achievable in laboratory experiments.
 In our synthetic case study, we assumed for simplicity that the proportionality factor κ linking concentration to electrical conductivity is constant and known throughout the domain. In real applications, additional experiments may be required to obtain κ as a spatial field. This spatial variability can be accounted for in the inversion. In the planned laboratory experiments, the variability of κ may be obtained by performing two sets of ERT measurements, one when the tracer concentration is zero throughout the box, and the other when it has reached a maximum value everywhere within the box. Standard ERT inversion would yield the electrical conductivity distributions for two concentrations, which can be converted to the spatial distributions of the base electrical conductivity σ0 and the proportionality factor κ. In field applications, this approach may be difficult to apply, because it is much more difficult to enforce a uniform concentration distribution. Additional geophysical surveying of the aquifer might be necessary to obtain an estimate of the spatial variability of κ (e.g., Linde et al. , using GPR surveys). Alternatively, the proportionality constant could be treated as a free parameter in the inversion.
 The hydraulic conductivity estimate obtained by the inversion and its uncertainty would be statistically correct if the deviations between the true and estimated parameters had zero mean and covariance matrix RYY∣d. With 16800 parameters, unfortunately, testing the full conditional covariance matrix is computationally very demanding. For this reason we restrict the test to the estimation variance by computing the normalized error ɛnorm,i for each parameter, which is the difference between the true and estimated value normalized by the standard deviation of estimation i of each log hydraulic conductivity value:
 The set of normalized errors ɛnorm,i follows a standard normal distribution, if the computed conditional mean and variance are correct.
 To quantify the overall error, we also compute the normalized root-mean-square error (NRMSE) from the normalized residuals for the different test cases:
 For an unbiased estimate, the NRMSE should be close to unity. We also compute the root-mean-square error (RMSE), which does not account for the standard deviation of estimation and expresses the overall discrepancy between the real and estimated log hydraulic conductivity values. A low RMSE indicates that the estimated hydraulic conductivity field agrees very well with the true field, regardless of the estimation uncertainty. To test the performance of the inversion approach, we also compare the true mean values and variances of the log conductivity field with the estimated ones.
3.2. Results and Discussion
 After 15–20 iterations, the inversion procedure usually converged or else showed minimal further improvement. The computational time for the inversion and the final transient calculation using the estimated hydraulic conductivity field ranged between 10 and 15 h on an Intel Quadcore Q9300 (2.5 GHz, 8 GB RAM, running in 64 bit MATLAB, on 64 bit Linux). The convergence criterion was set as a maximal absolute difference between the log hydraulic conductivity of two consecutive steps of 0.05, and a maximal change in the likelihood value of the objective function, weighted by the number of measurements, of 0.02. Figures 1b and 1c show the best estimate of the log hydraulic conductivity distribution and its corresponding uncertainty for the described test case. A comparison with the true log hydraulic conductivity field shows that most features are reproduced, but the simulated field is, as expected, smoother than the original one and small-scale variability is lost: the variance of the estimated log hydraulic conductivity field is reduced by about 40% in comparison to the true field. Table 2 shows the root-mean-square errors and the comparison of mean values and variances. Figure 3 shows the distribution of the normalized errors ɛnorm,i for our test case. The assumption that they follow a standard normal distribution must be rejected. The distribution is slightly asymmetric leading to a minimal shift of the mean value toward negative values, and the variance reaches a value of approximately 1.05. This is in agreement with equation (32) giving only a lower limit of the estimation variance. The slight overestimation is caused by the linearizations applied in the inversion procedure.
Table 2. Performance for Different Test Casesa
|True log conductivity field|| || ||−7||1|
|Test case according to Table 1||0.72||1.06||−6.95||0.56|
|False correlation lengths λx = λy = 0.1||0.83||1.02||−6.86||0.60|
|False prior variance σY2 = 2||0.98||1.09||−6.93||0.72|
|False prior variance and correlation lengths||0.93||0.84||−6.86||0.60|
|False prior variance, correlation lengths, and mean value βY = −5||0.90||0.81||−6.88||0.55|
|Higher measurement error σR = 2%||0.80||1.15||−6.89||0.46|
|Higher measurement error σR = 5%||1.04||1.39||−7.00||0.27|
|Pure hydrological inversion||0.97||0.29||−7.09||0.09|
 In order to assess the influence of the statistical prior knowledge on the solution, the inversion was performed for the same hydraulic conductivity field but assuming false variance and correlation length, respectively. The numerical results are shown in Table 2. Assuming a too high variance σY2 of the log hydraulic conductivity fluctuations leads to an increase in the standard deviation of estimation, but the estimated log hydraulic conductivity field is very similar. On the contrary, an underestimation of the prior variance leads to a large reduction of the variability of the field because the latter is penalized. Wrong correlation lengths logically lead to differently shaped features in the log hydraulic conductivity field and in its standard deviation, for example, by choosing the same correlation length in both directions, the features are less stretched. However, the overall pattern remains the same, with the zones of high and low conductivity being detected quite accurately. In fact, in some areas, particularly in the top part of the sandbox, the result is better than in the case with true correlation lengths. Combining erroneous values about prior statistics produces a slight deterioration of the estimate (although the main areas of low and high conductivity are still detected), but the results nonetheless show consistency, as the uncertainty of the estimation also increases significantly. Because the prior variance of β is very high, an erroneous mean value does not cause any additional deterioration of the results.
 Variations of the measurement error also have a considerable impact on the inversion result. While a doubling of the error does not influence the hydraulic conductivity field too much, except for a slight decrease in variability, a 5% error not only causes an even larger reduction in the variability, but also the pattern is poorly reconstructed. The main reason for this is the difficulty of calculating temporal moments from very noisy data. This issue might therefore become more important in field applications, where the measurement errors could be significantly higher.
 The analysis of the RMSE logically shows that the smallest error is obtained by the test case using the true statistical parameters. The test case with an increased error (2%) and the one with false prior correlation lengths are comparatively good too. In particular, the latter shows a NRMSE very close to one, indicating that the estimate is practically unbiased.
 The comparison between the true and estimated mean values and variances indicates that, while the mean values show little variations and no significant bias, the variance, indicating how well the variability of the field is reproduced, is always reduced, usually by about 40 to 50%, meaning that small-scale features are lost. Even more variability, in particular in the lower part of the sandbox, is lost when using only surface measurements. The results of the inversion for this case are shown in Figure 1d (showing the estimated field) and Figure 1e (showing the corresponding uncertainty). While the top part of the domain is reproduced quite well, the pattern in the bottom part is only poorly recovered. Also, the uncertainty of the estimation in the lower half of the sandbox is significantly higher than in the upper half. For field applications, this implies that electrodes within the aquifer are most likely required in high-quality monitoring of salt tracer tests by ERT. The quality would further decrease if conductive layers of fines were overlaying the aquifer under investigation.
 We also performed inversions using other configurations and a different number of boreholes, the results of which are not shown here. The addition of single-hole and cross-hole measurements did not yield any improvement. The inclusion of more boreholes logically led to a higher resolution and greater variability of the inferred hydraulic conductivity field throughout the sandbox. However, this is obviously associated to higher costs.
 In general, the estimation of the hydraulic conductivity in the part of the sandbox nearest to the inflow was more accurate than in the parts near the outflow. The main reason for this can be found in the sensitivity patterns described previously, with the higher-sensitivity areas situated near the electrodes and upstream of them. This implies that the zone at the beginning of the sandbox is much better resolved, while in the zone near the outflow only the measurements including the electrodes situated in that area yield information on the hydraulic conductivity distribution.
 In order to show the additional value of geoelectrical data compared to the use of hydraulic data only, we performed a pure hydrological inversion using only the eight hydraulic head values as measurements. The result was, as expected, very bad: the hydraulic conductivity field obtained was practically homogeneous. An increase of the number of measurements led to slightly better results, but only the use of a very large number of boreholes, associated with a reduction of the measurement error, permitted to obtain a hydraulic conductivity field comparable to the one resulting from the coupled inversion.
3.3. Comparison of Transient Behavior
 At the heart of our approach is the characterization of the measured electrical signals by their mean arrival time. This is the only information used in the inversion. If these data are informative enough to resolve the underlying hydraulic conductivity field, the full time series of resistances should be recovered when performing transient simulations using the estimated field. We test this by simulating and comparing transient breakthrough curves of electrical resistance perturbations using the true and estimated hydraulic conductivity distribution. Two examples for our main test case are shown in Figure 4.
Figure 4. Comparison of the breakthrough curve of electrical resistance perturbation for two particular electrode configurations, computed using the true (solid line) and estimated (dashed line) hydraulic conductivity distribution: (a) unimodal resistivity time curve and (b) bimodal resistivity time curve.
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 While most of the breakthrough curves (even the ones with secondary peaks) are reproduced quite well, like the one illustrated in Figure 4a, some curves show qualitative differences, such as the shift of the secondary peak in Figure 4b. Also, the root-mean-square difference between the true and estimated breakthrough curves is generally higher than the measurement error. We conclude that the full time series of time-lapse ERT data contain more information than covered by the mean arrival time. To retrieve this information in fully coupled hydrogeophysical inversion would be computationally quite demanding. Luckily, major features of the hydraulic conductivity field are already resolved by the inversion of lower-order temporal moments presented here.
 We also simulated the propagation of the tracer plume through the sandbox using the true and estimated hydraulic conductivity fields and compared the concentrations obtained and the mean arrival time of the plume throughout the entire sandbox. Figure 5 shows the mean arrival time of the plume in the sandbox, i.e., the ratio of the first over the zeroth moment of concentration. The propagation of the plume is reproduced fairly accurately and there are no significant differences between the true solution and the estimation. Even the loss of smaller-scale heterogeneities, caused by the smoothing of the hydraulic conductivity field in the inversion, does not lead to a great error in the prediction of solute transport. By construction, the mass balance of the tracer is met, which is in contrast to two-step geoelectrical inversion [e.g., Singha and Gorelick, 2005].
Figure 5. Comparison of the mean arrival time (in h) of the tracer plume for (a) the true and (b) the estimated hydraulic conductivity field.
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 In most tested cases the results were satisfactory. The main problems were encountered when increasing the measurement error too much, which led to significant differences in the plume arrival time in some parts of the sandbox. Also, when all prior statistical parameters (variance, correlation lengths and mean value) differed significantly from the true values, the results obviously deteriorated. If only one of these values was erroneous, plume propagation was still predicted well. This holds even by choosing a wrong correlation length both in horizontal and vertical direction. While this false assumption was clearly visible in the comparison of the true and estimated log hydraulic conductivity field, in which the features were well reproduced but had a slightly wrong shape, practically no effect could be detected in the prediction of solute transport. In the simulation using only the top row of electrodes representing surface geoelectrical surveying, the low resolution in hydraulic conductivity in the bottom part of the domain also led to an almost complete loss of spatial variability in the pattern of solute arrival time.