- Open Access
Geophysical assessments of renewable gas energy compressed in geologic pore storage reservoirs
© al Hagrey et al.; licensee Springer. 2014
- Received: 11 January 2014
- Accepted: 30 April 2014
- Published: 25 May 2014
Renewable energy resources can indisputably minimize the threat of global warming and climate change. However, they are intermittent and need buffer storage to bridge the time-gap between production (off peak) and demand peaks. Based on geologic and geochemical reasons, the North German Basin has a very large capacity for compressed air/gas energy storage CAES in porous saltwater aquifers and salt cavities. Replacing pore reservoir brine with CAES causes changes in physical properties (elastic moduli, density and electrical properties) and justify applications of integrative geophysical methods for monitoring this energy storage. Here we apply techniques of the elastic full waveform inversion FWI, electric resistivity tomography ERT and gravity to map and quantify a gradually saturated gas plume injected in a thin deep saline aquifer within the North German Basin.
For this subsurface model scenario we generated different synthetic data sets without and with adding random noise in order to robust the applied techniques for the real field applications. Datasets are inverted by posing different constraints on the initial model. Results reveal principally the capability of the applied integrative geophysical approach to resolve the CAES targets (plume, host reservoir, and cap rock). Constrained inversion models of elastic FWI and ERT are even able to recover well the gradual gas desaturation with depth. The spatial parameters accurately recovered from each technique are applied in the adequate petrophysical equations to yield precise quantifications of gas saturations. Resulting models of gas saturations independently determined from elastic FWI and ERT techniques are in accordance with each other and with the input (true) saturation model. Moreover, the gravity technique show high sensitivity to the mass deficit resulting from the gas storage and can resolve saturations and temporal saturation changes down to ±3% after reducing any shallow fluctuation such as that of groundwater table.
- Renewable energy
- Compressed air/gas energy storage (CAES)
- Elastic full waveform inversion (FWI)
- Electric resistivity tomography (ERT)
- Gravity method
- Petrophysical rock parameters
One unprecedented challenge facing the human being is the energy resources, and its coupling with global climate changes and warming from greenhouse gases (GHG). Mitigation of anthropogenic GHG, including CO2 emissions in the terrestrial atmosphere demands developments of viable alternative of renewable energy resources including hydroelectric, biomass, solar, wind, marine (wave/tides) and geothermal sources. Most of these sources produce energy only when suitable weather conditions are prevailing and not when energy directly demanded. These sources are intermittent and need buffer storage to bridge the time-gap (disparity) between off-peak production and demand peaks. The underground geology offers an adequate option for short- and long-term energy storage such as compressed air or gas energy storage, CAES (e.g., Crotogino et al. 2001; Succar and Williams 2008). The North German Basin delivers favourable conditions (geological, geochemical) for underground space utilization and has a huge capacity for CAES in porous brine aquifers and salt caverns (natural and artificial). Advantages of renewable energy storage are (1) balancing power demand and fluctuating renewable energy production, (2) bridging temporal mismatch between renewable energy production (off-peaks) and demand (peaks), i.e., storing off-peak energy supply to use it during peak demand periods, and (3) offering large buffer capacity to meet any disruptions in energy supply.
In a geological gas storage in saline formations, the gas replaces the pore brine causing strong changes in elastic moduli, density and electric resistivity. These physical contrasts justify the application of integrative geophysical techniques for monitoring this geostorage. These include techniques of elastic full waveform inversion (FWI), electric resistivity tomography (ERT), gravity and electromagnetic induction (EMI) of time- (TEM) and frequency-domain (FEM). EMI techniques (ground and air based) are applied usually for monitoring shallow targets, e.g., leakages in groundwater.
Since some years ago Germany practices a turnaround in the energy policy (German “Energiewende”) and is currently leading in the production of solar and wind energy (IEA 2013). Wind energy are produced mainly at the coastal areas (on-shore and off-shore) of North Germany which is characterized by high wind speeds. We started 2012 an interdisciplinary joint research project ANGUS+ dealing with impacts of using geologic subsurface as a thermal, electric or material storage in context with alternative energy resources (Bauer et al. 2013). This includes dimensioning, risk analyses and impact predictions as a base for future space planning of the subsurface. Our main task is to develop a geophysical monitoring strategy using integrative approach of geophysical techniques (FWI, ERT, EMI and gravity) on almost realistic scenarios in the North German Basin.
We show here results of numerical simulations of elastic FWI, ERT and gravity techniques in mapping CAES reservoirs with a continuous gradual desaturation with depth. These simulations are applied for synthetic data (before and after adding 3% random noise error) and inverted using constrains on the initial models.
Gas storages in the North German Basin
The seismic forward problem, FWI and model parameterization
where d denotes the density, v i the particle velocity, σ ij the stress tensor, ϵ ij the strain tensor, λ and μ the Lamé parameters, δ ij the Kronecker Delta, f i , T ij the source terms for body and surface forces, respectively.
Equation (1) is a general expression for the conservation of momentum in a continuum. It is independent of the medium state - such as gas, fluid or solid. To describe the behavior of the material correctly a relationship between the forces (stresses σ ij ) acting on the medium and the resulting deformation (strain ϵ ij ) is required. For small forces/deformations and an isotropic medium this relationship is linear (generalized Hooke’s law) depending only on the distibution of two material parameters λ and μ (Lamé parameters). Assuming that these parameters are time-independent the stress–strain relationship can be replaced by the stress–strain-rate equation (2). For a given isotropic elastic medium equations (1–3) can be solved numerically and therefore synthetic seismograms for any acquisition geometry calculated. Based on the solution of the seismic forward problem a high-resolution imaging concept called full waveform inversion (FWI) has been developed in the 1980s by Tarantola (1986). Since then the FWI is significantly improved and applied to a wide range of field applications (Virieux and Operto 2009).
and H m the second derivative of the objective function (Hessian). An explicit calculation of the Hessian in the time-domain is computational very expensive. Therefore, we use the quasi-Newton L-BFGS (Limited-memory Broyden-Fletcher-Goldfarb-Shanno) technique (Nocedal and Wright 2006; Brossier 2011), where the product of the inverse Hessian H m −1 with the gradient G m is iteratively approximated by finite-differences.
The effective calculation of the time-domain gradient directions with the adjoint method for different model parameterizations are described in Tarantola (1986), Mora (1987), Shipp and Singh (2002) and Köhn et al. (2012). The step length τn is estimated by a line-search satisfying the Wolfe-conditions (Nocedal and Wright 2006) to assure a fast and accurate convergence of the L-BFGS algorithm. While Asnaashari et al. (2013a) introduce another regularization term to assure model smoothness, we apply a weak wavenumber domain filter to the estimated search directions at every iteration step for the same purpose.
which denote the difference between the modelled and the field data at time steps t0 (baseline model) and t1 (Denli and Huang 2009). This redefinition of the data residuals leads to a much stronger focusing of the model updates at reservoir level.
Based on the distribution of the air within the storage formation with a maximum gas saturation of 80% (Figure 2), an elastic model of the underground before and after the CAES injection is built. The elastic properties of the rock matrix (P-wave velocity Vp,m, S-wave velocity Vs,m, density d m and porosity Φ) are linked with the physical parameters of the fluid and gas phases based on realistic matrix parameters to derive effective medium parameters.
To test the robustness of the elastic FWI approach for real field data applications we also investigate the influence of noise and added Gaussian noise within the frequency range of the source signal (0–40 Hz) with a signal-to-noise ratio S/N = 100 (1% noise) using the SUADDNOISE program of Seismic Unix (Cohen and Stockwell 2008). The resulting shot gathers are shown in Figure 4b,d for the baseline and time-lapse data, respectively.
ERT modelling and parameterization
At first we introduce briefly the approach for optimized electrode arrays in boreholes applied here. Like surface surveys, ERT data acquisition between two borehole electrode arrays can be conducted in the tripotential quadrupole configurations α (CPPC, C = current electrode, P = potential electrode), β (CCPP) and γ (CPCP). For an N collinear multi-electrode array, a whole comprehensive data set consists of [N(N − 1)(N − 2)(N − 3)/8] independent non-reciprocal quadrupole configurations (Noel and Xu 1991). The effective comprehensive data set results from excluding the redundant configurations of less stable inversions from the whole set, i.e., γ configurations and those of very large geometric factors (Loke et al. 2010). Resulting comprehensive data set is still huge, e.g., a pair of 32 borehole electrodes yield >106 data points. It can map subsurface targets with the highest possible resolution but at very long acquisition times (i.e., poor temporal resolution) and at high costs. Therefore, an optimization approach is based on the model resolution matrix and searches for electrode configurations that maximize the resolution of survey results (e.g., Stummer et al. 2004; Wilkinson et al. 2006). An optimized borehole data sets of practical sizes (15,000 data points) of only 1.5% of the comprehensive data set but with almost the same spatial resolution is generated in this study (e.g., al Hagrey 2012a). Comparative applications of diverse configurations (standard and non-standard) show the superiority of the optimized array results (al Hagrey 2012b). Moreover, assessing our optimized array by the technique of region of investigation index (ROI) showed that its inverted tomograms are best constrained by the data coverage in comparison to that of the other configurations (Oldenburg and Li 1999). All these confirm the effectiveness of our optimization approach applied here to generate a practical optimized data set of high resolution.
where ρ br is the brine resistivity, Φ the porosity, S g the gas saturation and a,m and n are Archie constants. The separate phases (matrix, brine and gas) are assumed without any interaction.
The electrical conductivity of the storage formation is caused mainly by the electrolytes of its pore brine. In the North German Basin, temperature and pressure, and particularly the salinity or total dissolved solids (TDS) increase with depth. Increasing both TDS and temperature causes a dramatic decrease in the resistivity (e.g., Arps 1953; Schlumberger 1985). The TDS rise increases the number of ions carrying electrical currents. The temperature rise increases the salt solubility and decreases the brine viscosity which in turn enhances the ion mobility. The pressure increase with depth, on the other hand, causes a slight increase in the resistivity due to the closure of cracks that are often filled with conductive fluids. However, this effect decreases with increasing depth and is negligible at pressure >0.3 GPa (e.g., Brace et al. 1965). In formations of the North German Basin, the average vertical gradient (with depth) of brine salinities, temperature and pressure approach 100 mg/L/m, 0.03°C/m and 22.6 kPa/m, respectively (e.g., Magri et al. 2009).
The target layers host borehole electrode arrays at 620 m offsets within a depth range of 0.9–1.8 km. Each array consists of 32 electrodes at 20 m spacing. The electric resistivity of the 2D models are parameterized by the bulk rock resistivity values calculated from Archie equation (17). Here we applied the values of 0.2 for Φ, 0.08 Ωm for ρ br (corresponding to TDS ≈ 100 g/l at 1 km depth of the North German Basin) and 1, 2 and 2 for constants a,m and n of, respectively, as typical values for the sandstone aquifer. Values of gas saturation are calculated by a potential function simulating their gradual damping with depth.
A 2.5D forward and inverse ERT modelling is carried out using modern codes (RES2DMOD, RES3DMOD × 64 and RES2DINV × 64) based on algorithms by e.g., Loke et al. (2003). The forward modelling code is applied to generate synthetic data sets between each adjacent pair of borehole electrode arrays (including inhole and crosshole) using optimized electrode configurations. These synthetic data sets are generated after gas injection in the brine reservoir of Rhaet formation. The data quality (0.6% average simulation error) is confirmed by results of tests on a homogeneous model with a constant ρ value. The technique robustness in the field is realized by adding a random error of commonly 3% to data sets in addition to their forward simulation error of 0.6%.
In the ERT inversions, diverse setup constraints (mainly regularizations) are applied. These include the minimization methods of least squares (L2) or robust blocky normalization (L1), and initial models of a constant homogeneous resistivity or an approximate inverse model (e.g., Claerbout and Muir 1973). Two synthetic data sets (generated before and after adding 3% random noise) are inverted with incorporating mapping data of subsurface stratigraphy from prior (seismic) surveys (see next sections). Each of these two data sets was inverted twice, once by incorporating layer interfaces and once by fixing resistivity regions, both are outside the reservoir layer.
Gravity modelling and parameterization
We used here the software IGMAS+ (Interactive Gravity and Magnetic Application System) designed for 3D gravity, gravity gradient and magnetic modelling (e.g., Götze and Lahmeyer 1988; Schmidt et al. 2011). The model is extrapolated outside the volume of interest in all directions (about two times the model length) to avoid any edge effects. We calculated the gravity field components (g z , g y and g z ) and gradients (g zx , g zy and g zz ) before and after CAES injection, respectively, as well as their difference (residual) anomalies (∆g x , ∆g y and ∆g z ). Here we show vertical component ∆g z maps only which reflect the strongest anomalies with respect to CAES reservoirs. We will discuss these gravity anomalies resulting from the two saturations (Sg 1 and Sg 2) and the lower sensitivity boundary determined for the technique at all saturations, i.e., the least measurable gravity anomaly determined by the modern micro-gravimeter accuracy (3–5 μGal).
Results of elastic time-lapse FWI
FWI workflow: corner frequencies (ƒ c ) of Butterworth-lowpass filter for the sequential frequency inversion and time-damping coefficients
Time-damping coefficients [1/s]
100, 50, 5, 1
100, 50, 5, 1
Gas quantification by elastic time-lapse FWI
Impact of errors in the baseline model on the elastic time-lapse FWI results
ERT inversion results
As mentioned before the two synthetic data sets (generated before and after adding 3% random noise) are inverted with incorporating mapping data of subsurface stratigraphy from prior surveys (e.g. seismic, borehole logs). These two apparent resistivity data sets were inverted by incorporating layer interfaces and fixing resistivity regions, respectively, both are outside the reservoir layer. ERT data inversions reconstruct directly the true subsurface resistivity tomograms including the study gas plume of downward gradual desaturation. This implies no model differencing, unlike seismic results showing the model difference before and after the gas injection. Of all differently independent inversions, every best-fitting tomogram shows least root mean square (rms)-errors of <0.5% and iteration number almost of 5, and is optimized with the L1 norm for sharp interfaces. Also this low rms-error value is explained by the good convergence of the synthetic data sets toward the final solution. This L1 norm yields significantly more accurate results than L2 norm, where the actual subsurface resistivity changes abruptly at sharp target boundaries (cf. Loke et al. 2013). It is more likely to suppose that considerable subsurface information is already available during monitoring from the detailed baseline survey and any other subsequent survey. The a priori incorporation of this mapping data in ERT inversions minimizes the ambiguity of the solution and enhances the resolution of results. Here each data set inversion is constrained by incorporating resistivity regions and interfaces outside the reservoir, respectively.
Obviously, the ERT inversion results of reconstructed tomograms visualize well the storage targets, particularly the gas plume of a downward gradual desaturation within the brine reservoir for all studied four cases (Figure 11). It is clear that constrained inversion models incorporating resistivity regions are better resolved than these incorporating boundaries. Also the addition of 3% random error to the synthetic data sets increases the misfit of rms-error values (between input and output response) by a factor of 5–8 but slightly decreases the mapping resolution as reflected by low residual ∆ρ rise from 4.5 and 7.8% to 4.9 and 7.9%, respectively. Loke et al. (2013) found that the model inverted with the L1-norm is less sensitive to random noise compared with the L2-norm.
Resulting residual Δρ r tomograms as a measure for resolution confirm generally the good mapping capability of the applied constrained ERT technique, where all average Δρ r have low values of 4.5–7.9% with minor deviations (Figure 11). This Δρ r distribution shows that the resulting model reliability (inverse of Δρ r ) is least for the noisy data set inverted by incorporating boundaries and best for the data set without adding random noise inverted with fixing ρ regions. An accurate investigation of Δρ r tomograms show that the resolution suppression due to the addition of 3% random errors to the generated data (average Δρ r increases from 4.5 to 4.9 for fixing ρ region, and from 7.7 to 7.9 for incorporating boundaries) is lower than that resulting from incorporating boundaries instead of fixing ρ regions (average declines from 4.5 to 7.7 for data without random noise, and from 4.9 to 7.9% for noisy data).
Constrained inversion tomograms (using any available subsurface data as an a priori information in the ERT inversion) show better resolution (inverse of ∆ρ r , Figure 11) than their corresponding unconstrained or even partly constrained inversion tomograms (not shown here, see al Hagrey et al. 2013). However, these residual ∆ρ maps still reflect the common smearing effects and artifacts of varying degrees of the ERT technique. This negative effect is particularly visible within the thin storage formation (∆ρ r up to ±20%) with spatially varying gas saturation (i.e., resistivity amplitude). On the other hand the non-varying (homogeneous) formations above (cap rock) and below (aquitard) the storage formation show a ∆ρ r of almost ±0. Obviously almost all inversion uncertainties are related to the monitored gas phase within the host reservoir. Thus they deliver an error estimate of the injected gas quantities monitored by this constrained inversion technique.
In conclusion, the ERT technique with permanently installed borehole electrodes aims at mapping, monitoring and quantifying the gas volume injected into the saline aquifer at any time. Obviously, the resulting tomograms (Figure 11) fulfil well the spatial mapping and monitoring purposes. This permanent electrode installation helps to maximize the reliability of monitoring data. Modelled anomalies minimizes the background effect and thus maximizes the time-varying response caused here by the injected gas quantity. The residuals (Figure 11) assess and prove the high reliability of the results including the quantification capability for the resistivity amplitude. These highly reliable resistivity amplitudes motivated us to derive the gas saturations (see next section).
Gas quantification by ERT
An investigation of resulting models shows that the gradual gas desaturations (S g ERT ) with depth are well mapped within the thin storage formation. The results in Figure 12 show a satisfactory similarity between input (true) and corresponding output (reconstructed) models. The average difference ∆S g relative to S g input (Figure 12) approaches 15–21% for all studied four cases of Figure 11. This small difference confirms again the satisfactory reliability of the results. The gas distribution within the Rhaet reservoir formation shows a good similarity particularly in the region 1–1.4 km (corresponding to the real distribution) and is insignificant outside this region where its amplitude is below the average (∆S g ). Most recovered S g ERT values are lower than their corresponding input ones. This may be related to the smearing effects of the technique. This smearing influences negatively the modelled resistivity amplitude and reduces the resistivity high of the gas plume sandwiched between the two resistivity lows of caprock and aquitard, respectively. High ∆S g values are concentrated mainly at reservoir interfaces and may be related to discretization errors. At the Cranfield site, Mississipi, Carrigan et al. (2013) found that CO2 saturations measured in monitoring wells are higher than the ERT-derived saturations although both show good spatial correlations. They added that ERT provides an integrated response from large volume, whereas gas sensors (dm penetration) provide point measurements and are sensitive to conditions near the well.
3D gravity modelling results
Mitigation of anthropogenic green house gases demand developments of renewable energy resources. However, most renewable energy sources are intermittent and therefore need buffer storage to match public power supply and demand. In geologic storages, replacing pore brine with compressed gas energy within the reservoir formation causes changes in elastic and electric properties and justify applications of integrative geophysical methods for monitoring this energy storage. We apply here elastic FWI, ERT and gravity techniques to map and quantify a thin gas plume of gradual downward desaturation injected in a deep brine aquifer. These aimed tasks are real challenges for any singly applied geophysical monitoring technique.
In this numerical study we simulate a nearly realistic storage scenario by considering: (1) the study site is chosen inside the North German Basin of favourable conditions for energy storages, (2) a storage model scenario is parameterized by real (published) data for this basin, (3) the gas phase plume is simulated with downward gradual desaturation similar to realistic cases, and (4) a common random noise level is added to the synthetic data to robust the technique for real field applications. The aimed resolution is enhanced by applying: (1) an integrative approach of geophysical methods, (2) an optimized approach for data acquisitions with a data coverage constraining well the inversion model and maximizing the resolution, and (3) constrained inversions to minimize interpretation ambiguities (by a priori use of available data, e.g. seismic, logs).
Unlike classical travel-time based tomographic approaches, the elastic FWI is capable to map the extension of the thin gas plume of downward gradual desaturation using only reflection seismic data, if a very accurate background model for the seismic velocities and density can be estimated before the gas injection. Additionally the elastic FWI recovers the changes of isotropic elastic material parameters and density due to the gas injection and subsequent partial drainage of the aquifer. By using an appropriate rock model, changes of the gas saturation can be deduced from the elastic FWI results with an accuracy of 5–30% within the aquifer, depending on the amount of noise (S/N-ratio > 100) present in the data. Due to the finite frequency content of the source signal larger saturation errors up to 60% can occur at the boundaries of the gas plume. Density inversion artefacts outside the aquifer due to noise can lead to fictitious estimates of saturation variations with local maxima of 20%. For a S/N-ratio of 50 the shape of the gas plume is still visible, but estimations of the gas saturation become highly erroneous and a S/N-ratio of 25 seem to be the detection limit for the gas plume. Errors in the elastic baseline model has a very substantial impact on the quality of the elastic time-lapse FWI results. Picking errors in layer interfaces and inaccurate material parameters within the layers lead to results with an approximately correct shape and position of the gas plume, but overestimated wrong elastic material parameters within the gas plume. Therefore, the estimation of accurate elastic baseline models for a successful elastic multiparameter FWI, and a subsequent calculation of the gas saturation distribution, is the greatest challenge for real field applications. Smooth macro-velocity models based on Common-Reflection Surface (CRS) stacking (Mann 2002) and Normal-Incidence Points (NIP)-wave tomography (Duveneck 2004) for P- and SH-wave data combined with the intensive use of prior information from borehole logs seem to be the most promising approach.
The applied constrained ERT inversion technique (taking use from previous seismic mapping and well logs) is also able to accurately map storage targets (caprock, reservoir with thin gas and aquitard) in all four applied cases (resulting from inverting noise-free and noisy data by incorporating layer boundaries and resistivity regions, respectively). The technique can even recover the gas plume of downward gradual desaturation with a good resolution. Also inversion models constrained by incorporating resistivity regions are better resolved than these constrained by incorporating boundaries, both are applied outside the reservoir layer. The thin resistive gas plume is sandwiched between the two conductive layers of the overlying caprock and the underlying aquitard. Based on the equivalence principle, the resistance (ρ*h, ρ = resistivity, h = thickness) of this thin resistive plume can hardly be resolved into ρ and h. This normally results in smearing with blurred boundaries and larger volume relative to the input model. These common ERT limitations are minimized here by applying the constrained inversion approach taking use of any available subsurface data. Uncertainties in mapping structures and quantifying the resistivity amplitudes are relatively low reflecting the high reliability of the reconstructed results.
Notably we could quantify reliably the gas saturations indirectly from the density and resistivity models resulting from the inversion by applying common petrophysical equations. The saturation results deduced from ERT technique fit well their corresponding values derived from elastic FWI. Both show reasonable absolute average differences (<20%) relative to the background. However, such results should be cautiously treated, where their validity and uncertainty should be studied in real field data.
The use of synthetic data contaminated with random error may reflect the real world of data. Obviously, adding random noise (typically 3%) to the synthetic ERT data increases the rms-error values by a factor of 5–9 but slightly decreases the mapping resolution. Our results here are in accordance with that obtained by al Hagrey (2012a) for ERT applications in CCS modelling. Using modelling codes (as applied here) and adding a random noise in an ascending order (1, 2, and 5% levels) to the synthetic data sets generally increases the rms-errors by a factor of 2 to 9 but slightly decreases the mapping capability of ERT technique. Ramirez et al. (2005) obtained similar results and concluded that the effect of the random error in ERT is insignificant for anomalies of a large size and magnitude.
Obviously the elastic FWI and ERT modelling using 2.5D codes has been conducted along a 2D section of the geological 3D model applied in the modelling simulation. This 2D model simplification is fully justified by the evidence that this 2D section cuts the main (storage) structure (gentle anticline within almost horizontal layering) along its main strike.
In conclusion results reveal the capability of our applied integrative geophysical approach to resolve the CAES targets and to quantify intrinsic property changes of the injected gas saturation in the reservoir. Constrained inversion models of elastic FWI and ERT are even able to recover well the gradual desaturation with depth. The accurately mapped spatial (seismic and electric) parameters are applied in their respective petrophysical equation to yield precise quantifications of gas saturations from each technique independently. Both resulting saturation models are in accordance with each other and with the input (true) saturation model. A joint elastic FWI and ERT inversion has a high potential to improve the applicability of the approach (e.g. Karaoulis et al. 2012). In a numerical study of a moving gas front within a reservoir, the joint inversion of seismic and electric time-lapse data sets reduces the presence of artifacts, and can retrieve the shape and estimate parameter better than their individual (unconstrained) inversions. For time-lapse data ERT uses permanently installed borehole electrodes, whereas seismic data needs to be repeated such that the source and receiver positions could not be the same and therefore more uncertainties can occur.
Moreover, the applied 3D gravity technique shows high sensitivity to the mass deficit resulting from the storage of the gas phase. The vertical gravity component can resolve saturations and saturation changes down to ±3% assuming that the data is corrected for temporal fluctuation effects of the groundwater table.
We thank D. De Nil for fruitful discussions, D. Wehner, M. Merz, B. Weise for computer work, the Editor and three anonymous reviewers for their constructive comments to improve the paper. Special thanks go to H.-J. Götze and S. Schmidt for supplying the license of the software IGMAS+ used for gravity modelling. This study has been carried out within the framework of research project “ANGUS+” funded by the German Federal Ministry of Education and Research (BMBF).
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