Characterization of the bridge pillar foundations using 3d focusing inversion of DC resistivity data

Journal of Applied Geophysics 172 (2020) 103875

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Characterization of the bridge pillar foundations using 3d focusing inversion of DC resistivity data N. Yıldırım Gündoğdu ⁎, İsmail Demirci, Cem Demirel, M. Emin Candansayar Ankara University, Faculty of Engineering, Department of Geophysical Engineering, Geophysical Modelling Group (GMG), 06830 Gölbaşı, Ankara, Turkey

a r t i c l e

i n f o

Article history: Received 4 December 2018 Received in revised form 25 October 2019 Accepted 25 October 2019 Available online 02 November 2019 Keywords: DC resistivity Focusing inversion Minimum gradient support

a b s t r a c t We investigated the effectiveness of a focusing regularization technique for the inversion of direct current (DC) resistivity data for a typical engineering problem. A smoothing stabilizer (Laplacian of model parameters) is generally preferred in the inversion (OCCAM's inversion) of DC resistivity data. Smooth reconstructions may be produced with this stabilizer, but some specific problems might require more focused images for adequate interpretations. For this reason, we investigated the capabilities of the minimum gradient support (MGS) stabilizer for providing shaper results. This stabilizer allows the a sharper reconstruction because its main effect is to minimize the area where strong differences occur between adjacent model parameters. We also analyze the effects of the focusing parameter, which is the parameter in the MGS expression controlling the level of sharpness of the final result. Our strategy for the selection of the optimal focusing parameter allows the resolution of distinct resistivity contrasts. Moreover, some artifacts that may arise in the use of the a very small focusing parameter disappear while using the normalized focusing parameter. We demonstrate these results by using both synthetic and field data examples. In the field data test, the subsurface image reconstructed using the proposed MGS approach matches well with the lithology inferred from borehole drillings. © 2019 Elsevier B.V. All rights reserved.

1. Introduction The direct current (DC) resistivity method is a well-established technique, which is used to tackle geotechnical and engineering problems where a complex subsurface is present (Cardarelli et al., 2007; Crawford et al., 2018; Danielsen and Dahlin, 2009; Kul Yahşi and Ersoy, 2018; Santarato et al., 2011). As with most geophysical methods, one of the main reason why DC resistivity is preferred in such studies is that it is a method that is (almost) non-invasive and non-destructive (Park et al., 2003; Rucker et al., 2013; Sentenac et al., 2018). Although two-dimensional (2D) resistivity imaging routines are probably the most used, the best way to investigate complex 3D geology is via three-dimensional surveys. However, 3D resistivity surveys are timeconsuming and costly when compared to 2D imaging (Rucker et al., 2009). Nevertheless, the popularity of 3D resistivity surveys has been rapidly increasing due to recent developments in field equipment and interpretation software over the last 20 years (Chambers et al., 2006; Jones et al., 2012). Nowadays, thousands of data units can be collected in a few hours with multi-channel and multi-electrode resistivity measurement systems (Loke et al., 2013). Moreover, large data sets can be ⁎ Corresponding author. E-mail addresses: [email protected] (N.Y. Gündoğdu), [email protected] (İ. Demirci), [email protected] (C. Demirel), [email protected] (M.E. Candansayar).

https://doi.org/10.1016/j.jappgeo.2019.103875 0926-9851/© 2019 Elsevier B.V. All rights reserved.

processed by using academic or commercial inversion algorithms. In these algorithms, the widely used regularized optimization is applied since the DC resistivity inversion problem is clearly ill-posed. The usual approach is that second-order derivative regularization is used to maximize the smoothness of in the final model (Binley and Kemna, 2005; Ellis and Oldenburg, 1994; Günther et al., 2006; Loke and Barker, 1996; Marescot et al., 2008; Pain et al., 2002; Papadopoulos et al., 2011; Sasaki, 1994; Yi et al., 2001).

Fig. 1. Synthetic model representing the bridge pillar (1000 Ωm) and the clayey zone (20 Ωm) below it. The homogeneous resistivity is 100 Ωm (the homogeneous medium is transparently presented for emphasizing the anomaly structures).

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Fig. 2. True model with embedded bodies (1000 and 10 Ωm) in a homogeneous half-space (100 Ωm) and the associated inversion results obtained with the smoothing (SM) and MGS stabilizers (with βsmall and βnormalized). Each row corresponds to a different depth.

In many engineering problems, targets often consist of anomalies with a sharp resistivity contrast. In such situations, we need to clearly image blocky targets. A prior information set must be used to better detect and reconstruct the relevant sharp interfaces. Many approaches have been suggested to achieve a reliable image of the subsurface (Auken and Christiansen, 2004; de Groot-Hedlin and Constable, 2004; Kim et al., 2009; Olayinka and Yaramanci, 2000; Smith et al., 1999). Portniaguine and Zhdanov (1999) described an alternative approach to obtain sharper reconstruction. This regularization strategy is based on the “minimum gradient support (MGS)” stabilizer. This approach is a modification of the “minimum support (MS)” stabilizer, which was originally proposed by Last and Kubik (1983) to diminish the smoothness effects.

Since then, the MGS/MS stabilizers have been applied to many different kinds of geophysical data. Pagliara and Vignoli (2006) used the MGS and Total Variation (TV) stabilizers for the inversion DC resistivity data on finite bodies. Candansayar (2008) compared the effects of the different stabilizers (including the MGS stabilizer) on 2D inversion of synthetic and real MT data. Blaschek et al. (2008) used the MGS stabilizer for the inversion of spectral induced polarization data. Zhang et al. (2012) compared smoothing and focusing inversion on 3D inversion of MT data. Fiadanca et al. (2015) used the minimum support (MS) stabilizer for the regularization in time of time-lapse electrical tomography data. Hermans et al. (2014) investigated smoothness, MGS and geostatistical constraints for inversion of time-lapse cross-hole electrical resistivity tomography data. Nguyen et al. (2016) discussed

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Fig. 3. The true model and the inversion results for the profile crossing the embedded bodies in a perpendicular direction (y = 20).

the effects of the MGS stabilizer on 2D time-lapse inversion of DC resistivity data. Ye et al. (2015) also used this stabilizer for inversion of the synthetic DC resistivity data. Xiang et al. (2017) used MGS stabilizer on regularized the 1D and 2D inversion of the synthetic MT data. In addition, MGS, and its MS variation, have also been applied to also traveltime tomography data (both have been applied, on radar e.g. Vignoli et al. (2012) and seismic e.g. Zhdanov (2006) data). Qiang et al. (2017) used MGS regularization to carry out AVO inversion. Guo et al. (2017) developed an adaptive sharp boundary inversion scheme for transient electromagnetic data inversion using the MGS constraint, whereas Vignoli et al. (2015), Ley-Cooper et al. (2015) and Vignoli et al. (2017) used the MGS/MS approaches in the framework of the spatially constrained inversion of electromagnetic data. Kazei et al. (2017) compared several regularization approaches, which include the MGS stabilizer for full-waveform inversion. Gündoğdu and Candansayar (2018) examined the effects of seven different stabilizers on the 3D inversion of DC resistivity data. In the present research, we invert 3D electrical tomography data by means of a novel version of the MGS stabilizer and we apply it to investigate the resistivity changes induced under a bridge pillar foundation. In the following text, the standard MGS stabilizer is briefly introduced and a synthetic study is used to demonstrate its effectiveness. Similarly to other previous works (e.g. Ajo-Franklin et al., 2007; Fiadanca et al., 2015; Kim and Cho, 2011; Last and Kubik, 1983; Rosas Carbajal et al., 2012; Vignoli et al., 2012; Vignoli et al., 2015; Zhdanov et al., 2006;

Zhdanov and Tolstaya, 2004) we also investigated the effect of the “focusing parameter” in the MGS stabilizer definition. 2. 3D focusing inversion of resistivity tomography data Since the DC resistivity inverse problem is ill-posed, finding a stable and unique solution is critical. Therefore, some regularization must be used to address the nature of the problem. Tikhonov and Arsenin (1977) proposed to minimize not simply the distance between the calculated and the observed data, but an objective functional, consisting of the linear combination of the data misfit functional Ф(d, f(m)) and stabilizing term S(m): PðmÞ ¼ Ф ðd; f ðmÞÞ þ αSðmÞ:

ð1Þ

In Eq. (1), α is the real, positive, number controlling the balance between the importance of: i) the data (d is the observed data vector, whereas f(m) is the vector of the simulated responses based on the model m) and ii) the prior information available about the investigation system; this prior information is formalized through the stabilizing functional S(m). Clearly, the final solution depends strongly on the stabilizer choice; for example, the final model m will be different if we choose a MGS stabilizer or a more “standard” minimum norm (MN) stabilizer minimizing the difference between m and an a priori model (possibly inferred from a well).

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where i denotes the iteration number, We(i) is the linear multiplication operator and this matrix is updated at the current iteration from the model parameters obtained in the previous iteration (Candansayar, 2008). In this more general case, the objective functional to be minimized becomes:

2 2 PW ðmÞ ¼ kWd ðd− f ðmÞÞk þ α
WmeðiÞ m


ð4Þ

where Wme(i) = We(i)Wm. Our inversion algorithm implementation of 3D DC resistivity problem is based on the code developed by Gündoğdu and Candansayar (2018). A seven-point discretization finite-differences method (Dey and Morrison, 1979) is used to calculate the forward response f(m). Eq. (4) is minimized by using a Gauss-Newton scheme. At each iteration, the model updated vector is given as follows:  −1   Δmi ¼ AT WTd Wd A þ αWTmei Wmei AT WTd Wd Δd−αWTmei Wmei mi ð5Þ

Fig. 4. Relative error (%) between simulated and calculated data for smoothing stabilizer, βsmall and βnormalized across the vertical plane y = 20 m.

The measured geophysical data are always contaminated by some noise. Model parameters can be weighted in different ways, for example, to make the data equally sensitive to the components of the model parameter vector. One way to do so is via data and model weighting matrices; respectively: Wd and Wm. Hence, instead of minimizing the objective functional in Eq. (1) in the usual data and model space, it can be convenient to perform that minimization in modified weighted spaces. In this case, the objective functional becomes: PW ðmÞ ¼ Ф ðWd d; Wd f ðmÞÞ þ αSðWm mÞ:

ð2Þ

Very often, the model weighting matrix is simply the matrix approximation of the first derivative. In this case, we deal with the minimum gradient norm stabilizer and, accordingly, we look for the solution characterized by the smallest spatial variation (hence, a quite smooth solution). In the framework of a reweighting strategy, where the model weighting matrix is actually changing iteration by iteration, the stabilizer expression can be further generalized as follows:

SðmÞ ¼ α
WeðiÞ Wm m


ð3Þ

Table 1 Numerical results of the inversion with SM and MGS stabilizer with βsmall and βnormalized for the synthetic data example. The average resistivity values of the homogeneous medium and the embedded resistive and conductive bodies and their misfit error (%) relative to the true model.

SM MGS (βsmall) MGS (βnormalized)

Hom. medium (100 Ωm)

% Misfit

Resistive bodies (1000 Ωm)

% Misfit

Conductive bodies (10 Ωm)

% Misfit

94.93 89.84 97.31

5.07 10.16 2.69

268.69 314.82 391.58

73.13 68.51 60.84

39.86 19.19 16.57

298.65 91.91 65.74

Bold indicate the results of the proposed method. Also these results are more acceptable compared to the other method.

where A is the sensitivity matrix and it is calculated by using reciprocity theorem (de Lugão and Wannamaker, 1996). A “Cooling approximation” approach (Newman and Alumbaugh, 1997) was used to calculate the regularization parameter α and QR factorization method was used to solve parameter correction vector Δm in every iteration step. The initial value of α is the largest eigenvalue of the (ATA) product in the first iteration step. This value is decreased by 10% in each iteration. The minimization process stops when either the numerical value of the parametric functional was not reduced, or the RMS value between two consecutive iteration steps was not smaller than 2%. The RMS value is calculated as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 kWd ðd−f ðmÞÞk2 RMS ¼ N

ð6Þ

Where N is the number of data points. We used three stabilizing functionals in this study. One is the minimum gradient norm stabilizer (we denoted this stabilizer as SM to emphasize smoothing), which is quite commonly used for the inversion of DC resistivity data. The others are: the standard MGS and one variation of it. The expression for the SM and MGS stabilizers are, respectively:

2

SSM ðmÞ ¼
∇2 m


SMGS ðmÞ ¼

ð7Þ

M   X ð∇mi ∇mi Þ= ∇mi ∇mi þ β2

ð8Þ

i¼1

where ∇2 is the Laplacian operator, and β is a positive constant (Portniaguine and Zhdanov, 1999) the “focusing parameter” (Xiang et al., 2017). The SM stabilizer is in quadratic form, so there is no problem in the implementation of Eq. 7. In order to apply the MGS stabilizer, this can be expressed in pseudo-quadratic form since it is dependent on m. The pseudo-quadratic form of MGS is as follows (Candansayar, 2008): 2 WMGS ei

 1 6 2 2 ¼ diag6 4∇mi = ð∇mi Þ þ β

1   2 mi þ β2 2

 3 2

7 7 5

ð9Þ

In Eq. (9), any gradient value significantly larger than β has a contribution equal to one. Thus, the final model with sharp boundaries can be obtained. On the other hand, gradients much smaller than β have (almost) zero contribution to the stabilizer penalization functional. Therefore, the choice of the focusing parameter is a critical issue and affects the final solution.

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Fig. 5. Location map of the Fatih Bridge in Ankara and the DC resistivity survey area (102.5 m × 59 m) (yellow rectangle). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

There are various approaches for the selection of the optimal β value. Last and Kubik (1983) selected a β value such that the focusing parameter was close to machine precision. Portniaguine and Zhdanov (1999) and Candansayar (2008) used a small and fixed value for gravity and MT data inversion, respectively. Portniaguine and Castagna (2005) set the value of the focusing parameter to 10−8max(mi) in the seismic application. Zhdanov and Tolstaya (2004) used a procedure similar to the L-curve method and they chose the focusing parameter to be the value at the maximum convex curvature point. Ajo-Franklin et al. (2007) fixed the focusing parameter value by experience between 10−4 and 10−7 in their travel time tomography studies. Blaschek et al. (2008) used sensitivity-controlled regularization to obtain the optimum value of β. Their aim was to allow smooth variations within compact structures. Rosas Carbajal et al. (2012) used normalized model covariance to number of parameter as focusing parameter and they have obtained satisfactory solutions for all cases in focused time-lapse inversion of RMT and AMT data.

These studies show that the selection of the β value may vary depending on the geophysical data and the inversion problem type. In this study, firstly, we used very small value (βsmall), around machine epsilon ≈ 10−16. In this situation, we have observed some instabilities (small scale anomalies) in our final model. In order to prevent these, instead of finding β value by trial and error where it is possible to disrupt the focusing in such a situation, we used a normalized focusing parameter value (βnormalized). In this case, we checked the numerical value of ∇mi ∇ mi in every iteration step. If this value is lower than a certain tolerance value, we normalized the β value by using the gradient of the model parameter:

SMGS ðmÞ ¼

M    X ð∇mi ∇mi Þ= ∇mi ∇mi þ β2 =j∇mi j

ð10Þ

i¼1

Thus, too large or too small values that may occur in calculating the stabilizer are normalized depending on the change in the model parameters. The β’s initial value is equal to βsmall. We showed the effects of these two different β value selections and we compared these results with the result obtained with the smoothing stabilizer. These comparisons are performed on both synthetic and field data sets. 3. Synthetic data example

Fig. 6. A view from the profiles located under the bridge (profile E) and on the campus area (profile H).

The model of the synthetic case is designed to mimic simple bridge pillar foundations. The resistivity model consists of a homogeneous half-space of 100 Ωm with some inclusions. The embedded bodies simulate the bridge foundation pillars (1000 Ωm) and a relatively conductive zone (10 Ωm) in which these pillars are placed (Fig. 1). The electrode layout (number of electrodes and spacing), and data acquisition settings are the same used (and discussed) for the field example. A total of 4644 synthetic apparent resistivity data were simulated. Gaussian noise (3%) was added to the (V/I) values. The starting model for the inversion is a homogenous resistivity of 98 Ωm. This value is the average of the synthetic data created. The inverted resistivity models using the smoothing stabilizer and the MGS with the different focusing parameters are shown for all depth sections (xy) and for profile located y = 20 m (xz), which is crossing the embedded bodies in a perpendicular direction in Figs. 2 and 3, respectively. The true model, the parametric functional and

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Fig. 7. (a) Electrode positions on the field study area. The black rectangles show the bridge pillar locations. (b) Plot of the dipole-dipole electrode configuration used in the measurements. (c) The measured data location in the profile. The pseudo-depth values are given as in Edwards (1977).

RMS values are also given for inverted models in the corresponding figures. As expected, the boundaries of the targets are sharper when the MGS stabilizers are used and smoother with the SM stabilizer. The

small focusing parameter application produced some artifacts in the homogenous medium and near the boundaries of the model mesh. It can be seen that these artifacts are reduced when the normalized focusing parameter is used and the boundaries of the bodies are shown even

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Fig. 8. The inversion results of the field data using smoothing and MGS stabilizers with βsmall and βnormalized for the selected profiles.

more sharply. The MGS stabilizer with the normalized focusing parameter better approximates the sharp resistivity distribution of the true model. As can be seen in Fig. 4, a small relative error (%) between the simulated and calculated data is obtained for MGS stabilizer with βnormalized. The parametric functional and RMS values show better convergence using the normalized focusing parameter. The average resistivity of the

homogeneous medium and the embedded bodies in the inversion results are given in Table 1 to compare with the true model. The obtained results are important for the interpretation of DC resistivity data observed in typical bridge foundation problem. In the following section, the information obtained by the synthetic data example was applied to a DC resistivity field data collected at the Fatih Bridge located in Ankara, Turkey.

Fig. 9. The ratio of the MGS solutions with different focusing parameters βsmall and βnormalized to SM stabilizer. The black arrow shows test drilling location and white dashed circle shows the conductive target zone.

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Fig. 10. (a) The inversion results of the field data using smoothing and MGS stabilizers with βsmall and βnormalized for the profile at y = 20 m together with the lithology obtained from test drilling location (black arrow). The upper and lower boundaries of the target zone (clayey agglomerate) are shown by using the white lines. (b) The resistivity values which are extracted from inverted models versus depth at the position of the test drilling location. The boundaries of the target zone are shown by using the black lines.

4. Field data example The study area is located in the northern part of the Ankara city center. The main purpose of this study is to detect the boundaries of the weak zones formed by the foundation of the bridge pillar before the construction of pedestrian underpass between the campus areas on either side of the bridge (Fig. 5). In the survey area, Upper Miocene volcanic units, agglomerate and tuff are observed. The younger Pliocene units consist of conglomeratesandstone-mudstone alternations and Quaternary alluvium unconformably overlies all other units. The DC data were collected using sounding-profile measurement techniques at nine parallel profiles, which were named from A to I. Profiles A, B, H, and I were located on the campus areas. The other profiles were on the asphalt road, open to vehicle and pedestrian traffic under the bridge. A view from profiles E and H are given in Fig. 6. on the asphalt and on the campus area, respectively. The measuring points were prepared by drilling on the asphalt. The distances between each of the consecutive profiles are 10, 10, 5, 5, 5, 4.3, 10 and 10 m. Due to the bridge wall, 39 electrodes were used in profile G, whereas 42 electrodes were placed on the remainder of the profiles. In total, 375 electrodes were used. The electrode spacing was 2.5 m. A total of 3834 apparent resistivity data were collected in the dipoledipole electrode array configuration. The measurements were taken at eight n levels, which associated with 2.5, 5 and 7.5 m dipole separation. All electrode locations, data points for one profile and the schematic view of the dipole-dipole configuration are shown in Fig. 7. Approximately, an area of 6078 m2 was scanned using the DC resistivity method. The model mesh consisted of 5160 blocks (43 × 12 × 10). Homogenous earth of 32 Ωm was used as an initial model. This value is the average of the measured apparent resistivity.

The inversion results of the field data using the smoothing stabilizer and the MGS with the different focusing parameters are given for three selected profiles located near the bridge pillar in Fig. 8. The RMS value is 1.53 for the smoothing stabilizer after nine iterations, 1.31 for the MGS stabilizer with βsmall focusing parameter after 11 iterations and 1.07 for the MGS stabilizer with βnormalized focusing parameter after ten iterations. These RMS values are acceptable for 3D inversion when considering the number of data. The locations of the bridge pillar on each profile and the inferred relatively conductive zones below these pillars are marked with the black arrows and the white dashed circles, respectively. These conductive zones are thought to be weakened clay-rich zones that are formed by the bridge pillar. It is important to determine the boundaries of any weak zones more clearly, as the potential underpass is planned to be constructed into a strong geological unit that can be represented by a relatively more resistive unit from the bottom of this zone. When all the inversion results are compared, the inverse models corresponding to the conductive target zones are observed to vary more smoothly when using the smoothing stabilizer. These anomalies seem more focused and more sharply defined when using the MGS stabilizers. However, these results are not as clear as in the synthetic data example, since the noise content of the field data is quite high. The high noise content reduces the effect of the focusing inversion method. On the other hand, when comparing different focusing parameters, the target zones show more contrast using the βnormalized focusing parameter and these zones are more distinguishable from their surroundings. The ratio of the different MGS solutions to SM stabilizer are given in Fig. 9 for a better understanding of this situation. Here, we used SM results to obtain the ratio since there is not a true model for the field study example. It is clear that the amplitude and boundaries of the conductive target zones are found more clearly using the MGS stabilizer

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with the βnormalized focusing parameter. The resistivity ratio values of the conductive target zones are smaller with the βnormalized focusing parameter. This indicates that the low resistivity zone is more distinguishable with this focusing parameter. After the inversion studies, a drilling test location (x = 50 m) was used as tentative ground truthing (Fig. 10a). An agglomerate layer was seen at a depth of approximately 0.63 m from the surface (after the asphalt cover), followed by a deeper clay-rich unit. This low resistive unit is our target zone as it is associated with poor geotechnical properties. Below the clay layer, a relatively high resistivity conglomerate unit was found. This latter lithology is compatible with the geology of the study area. The formation boundaries are clearly compatible in resistivity for the three inversion schemes with the borehole. The conductive target zones are also observed in all solutions. The variation of resistivity values versus depth is also given in Fig. 10b. According this figure, the amplitude of the conductive target zone is recovered using the MGS stabilizer with the βnormalized focusing parameter lower than other stabilizers. 5. Conclusions In this study, we compared the effects of the smoothing and the focusing inversion techniques on the inversion of DC resistivity data, which was compiled in order to solve civil engineering problems. We investigate the capability of the focusing inversion implemented via an MGS stabilizer with the normalized focusing parameter in order to obtain a sharp subsurface image. Our studies with synthetic data show that the resistive or conductive structures with sharp boundaries are determined to be close to the true model when using the focusing inversion technique with MGS stabilizer. Moreover, the use of the focusing parameter normalized with the gradient of the model parameters in the MGS definition help us to recover the resistivity and boundaries of these structures better. Furthermore, no significant time difference during the inversion is observed with different stabilizers and different focusing parameters. The inversion results of the field data measured to determine bridge pillar foundations are also lead to similar conclusions. The inversion results are in agreement with the lithology of the study area. The resistivity of conductive target zones is observed as being lower using the MGS stabilizer with normalized focusing parameter. The clayey weak zones below the bridge pillar foundations are more clearly distinguished from its surrounding background with this focusing inversion. This improved knowledge will definitely affect the choices for the construction of the underpass. Declaration of Competing Interests The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Acknowledgement The authors thank three anonymous reviewers for their constructive comments. References Ajo-Franklin, J.B., Minsley, J.B., Daley, T.M., 2007. Applying compactness constraints to differential traveltime tomography. Geophysics 72, R67–R75. https://doi.org/10.1190/ 1.2742496. Auken, E., Christiansen, A.V., 2004. Layered and laterally constrained 2D inversion of resistivity data. Geophysics 69, 752–761. https://doi.org/10.1190/1.1759461. Binley, A., Kemna, A., 2005. Electrical methods. In: Rubbin, Y., Hubbard, S.S. (Eds.), Hydrogeophysics. Springer, Netherlands, pp. 129–156.

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