Comparison of different sets of array configurations for ...

Journal of Applied Geophysics 137 (2017) 34–48

Contents lists available at ScienceDirect

Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo

Comparison of different sets of array configurations for multichannel 2D ERT acquisition R. Martorana a,⁎, P. Capizzi a, A. D’Alessandro b, D. Luzio a a b

Dip. Scienze della Terra e del Mare, Università di Palermo, Italy Istituto Nazionale di Geofisica e Vulcanologia, Centro Nazionale Terremoti, Italy

a r t i c l e

i n f o

Article history: Received 1 March 2016 Received in revised form 24 November 2016 Accepted 9 December 2016 Available online 12 December 2016 Keywords: 2D Resistivity Arrays Multichannel Multiple gradient Resolution

a b s t r a c t Traditional electrode arrays such Wenner-Schlumberger or dipole-dipole are still widely used thanks to their well-known properties but the array configurations are generally not optimized for multi-channel resistivity measures. Synthetic datasets relating to four different arrays, dipole-dipole (DD), pole-dipole (PD), Wenner-Schlumberger (WS) and a modified version of multiple gradient (MG), have been made for a systematic comparison between 2D resistivity models and their inverted images. Different sets of array configurations generated from simple combinations of geometric parameters (potential dipole lengths and dipole separation factors) were tested with synthetic and field data sets, even considering the influence of errors and the acquisition velocity. The purpose is to establish array configurations capable to provide reliable results but, at the same time, not involving excessive survey costs, even linked to the acquiring time and therefore to the number of current dipoles used. For DD, PD and WS arrays a progression of different datasets were considered increasing the number of current dipoles trying to get about the same amount of measures. A multi-coverage MG array configuration is proposed by increasing the lateral coverage and so the number of current dipoles. Noise simulating errors both on the electrode positions and on the electric potential was added. The array configurations have been tested on field data acquired in the landfill site of Bellolampo (Palermo, Italy), to detect and locate the leachate plumes and to identify the HDPE bottom of the landfill. The inversion results were compared using a quantitative analysis of data misfit, relative model resolution and model misfit. The results show that the trends of the first two parameters are linked on the array configuration and that a cumulative analysis of these parameters can help to choose the best array configuration in order to obtain a good resolution and reliability of a survey, according to generally short acquisition times. © 2016 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the electrical resistivity tomography (ERT) has become a popular choice for the investigation of the shallow subsurface. The increasing use of ERT for different applications is made possible by the use of new instruments capable to simultaneously perform multiple voltage measurements for each current dipole, significantly reducing the measuring time and, together, allowing the acquisition of big datasets. These multichannel systems consequently have allowed the use of new multi-electrode arrays and sequences. Examples of these types of arrays are the moving gradient and the midpoint-potentialreferred. Dahlin and Zhou (2004) have carried out several numerical simulations to compare the resolution and efficiency of pole-pole, pole-dipole, Wenner, Schlumberger, dipole-dipole, moving gradient and midpoint-potential-referred array. They recommend moving ⁎ Corresponding author. E-mail address: [email protected] (R. Martorana).

http://dx.doi.org/10.1016/j.jappgeo.2016.12.012 0926-9851/© 2016 Elsevier B.V. All rights reserved.

gradient, pole-dipole, dipole-dipole and Schlumberger array, rather than the other, although the type of geology expected, the purpose of the survey and other logistical considerations should determine the final choice. Various 2D simulations using near surface models, by varying the array type in order to assess the best resolution were compared (Martorana et al., 2009). Despite the flexible nature of the multi-electrode arrays, there still is the tendency to acquire apparent resistivity measures using traditional electrode arrays such Wenner, Schlumberger or dipole-dipole. These latter often prove to be a good choice, as their properties are well known, in terms of depth of investigation (Barker, 1989), lateral and vertical resolution (Barker, 1979) and signal-to-noise ratio (Dahlin and Zhou, 2004). However, the choice of these arrays could not be the most efficient if the time or the number of steps permitted for the survey is limited, or if the target is spatially localized. In ERT the choice of the particular set of array configurations can greatly influence the imaging efficiency, because mainly it depends on the resolution of the inverse model. Recently, there has been a growing

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

interest in the study of the design of sets of array configurations to optimize the resolution of ERT imaging. One of the benefits of the optimization of the set of array sequences would be a better reconstruction of the subsurface that aims to reduce the number of data acquired and, consequently, the cost of the survey, without compromising the quality of the reconstruction itself. Many authors have dealt the optimization of acquisition sequences of data, which can be formed by different array types, dipolar lengths and distances, in order to seek the optimal sequences that ensure a realistic imaging with high resilience of the subsoil, without the need for an excessive number of measurements that would compromise the economic viability of the survey. Several optimization methods were introduced (Maurer and Boerner, 1998; Furman et al., 2004; Maurer et al., 2000; Van den Berg et al., 2003; Curtis, 2004a,b; Hennig and Weller, 2005), most of which are based on the evaluation of the sensitivity of certain electrode arrays for discrete and localized variations in resistivity. However, the above studies considered for the design only fourelectrode arrays. Recently the rise of multi-channel systems has led to consider experimental design for multi-electrode arrays (Xu and Noel, 1993; Stummer et al., 2002, 2004; Wilkinson et al., 2006; Coles and Morgan, 2009). Recent works (Loke et al., 2010, 2014, 2015; Wilkinson et al., 2012, 2015) have greatly improved the efficiency on the methods presented by Wilkinson et al. (2006), based on methods that optimize the choice of the array configuration sets by maximizing the model resolution for a homogeneous earth model. Nevertheless, the imaging efficiency of a specific data set depends not only on the resolution of the inverse model but also on the level of data noise. This is linked to multiple factors, including not only the signal/noise ratio in the voltage measurements, but also the electrode positioning error. Voltage data may be subjected to different types of errors, mainly caused not only by malfunctioning of the resistivitymeter, high contact resistances or electrode polarization effects, but also from many other causes, more difficult to determine (Dahlin, 2000; Wilkinson et al., 2012). Generally, in ERT the problem due to errors on data caused by improper electrode positioning is not adequately considered. This error may be higher in steep or heavily vegetated areas and it generates incorrect estimates of the geometric factors. The relationship that exists between the geometric factor K of the array and the measured voltage shows that arrays characterized by higher values of K results in lower voltage values and therefore potentially noisier measures. To reduce the effect of this type of error on the resulting electrical tomography it is possible to select data sets that include array configurations with relatively low geometric factors and therefore less sensitive to position errors (Wilkinson et al., 2008). In practice, therefore, the methods of optimization of the array configurations to maximize the resolution of tomographic image should also take into account the effects of data errors on the resolution of the inverse model and especially on its ability to retrieve correct information of the subsoil. Nevertheless, most of the optimization techniques used so far did not sufficiently consider this problem. A comprehensive comparison of the imaging capabilities of different array data sets is necessary in order to assess the suitability of their behavior in fieldworks, in order to predict what data sets can resolve more in detail specific characteristics of the subsurface in the electrical resistivity images. In addition to design the fieldwork and the subsequent data interpretation, noise sensitivity of the array data set must be considered. In fact, the ideal data set should provide electrode anomalies with high gradients, high resolution throughout the section and a high signal-to-noise ratio. Once the minimum electrode distance is chosen different array data sets can be designed by varying the potential dipole length a and the dipole separation factor n. Theoretically, a complete data set could be carried out, including all possible electrode arrangements, with a very high number of data (Stummer et al., 2004), but this would involve a fieldwork over time even using a multichannel resistivity meter. Moreover, increasing the number of measures usually will enhance the information content but will also increase the

35

minimum misfit obtained by the inversion. Furthermore, the most probable presence of data with a high noise level will lead to the production of artifacts (Zhou and Dahlin, 2003). Another parameter that is not considered by these optimization methods is acquiring velocity in field for a given data set. This, for parallel acquisitions, strongly depends on how many measurements can be made simultaneously with a same current dipole. According to the array used and considering a given total number of data of the set, less current dipoles are used, the lower the acquisition time. The ratio of the number of current dipoles and the total number of measurements is then indicative of the acquiring velocity. Ultimately all of the above discussed factors should be taken into account for the choice of an optimal data set that ensures a good imaging resolution, at the same time providing a low signal to noise ratio and a not excessive acquisition time. Some recent studies (Blome et al., 2011; Nenna et al., 2011) have addressed some aspects of this problem. Wilkinson et al. (2012) proposed improvements to the optimization methods to make efficient use of parallel measurement channels and incorporate data noise estimates in the optimization process. However, the optimization methods are still poorly used for application purposes. Many authors in fact continue to use the standard arrays with simple combination of geometrical parameters for practical applications, both for their ease of design, using the most common acquiring software, both for their well-known characteristics in the literature. Therefore it is still useful to study their behavior when changing the combination of the geometric parameters in order to identify common trends and correlations. For this purpose, in this work a comparison of some array data sets has been done, designed considering standard arrays (dipole-dipole, pole-dipole, Wenner-Schlumberger) and combining different potential dipole length values a e dipole separation factors n or more recent multi-channel configurations (multiple gradient) by varying the coverage factor. The aim is to vary the number of current dipoles involved, but also the median of the geometric factor, maintaining approximately comparable measurements amounts among data sets. It was so data sets characterized by different acquiring velocity but also potentially by changes in the signal/noise ratio expected. The goal is to study how the reliability of inverse models depends on the above discussed factors, not neglecting how noise affects inversion. With this in mind, the use of inclusive parameters to estimate the reliability and resolution of inverse models is crucial to have a reasonable and quantifiable comparison of the results. 2. Design of the sets of array configurations Several simulations on 2D model have been made, using the software RES2DMOD (Loke, 2014), to study the changes in resolution and reliability when the array configuration changes. Simulations have provided profiles of 72 electrodes. Data sets of apparent resistivity values have been calculated using a modified version of multiple gradient (MG) array (Dahlin and Zhou, 2004, 2006) and the three most common arrays for multichannel measurements: dipole-dipole (DD), pole-dipole (PD) and Wenner-Schlumberger (WS). Array data sets were chosen considering the capacity to carry out measures by parallel acquisition systems, with multi-channel resistivity meters capable of acquiring different potential measures related to the same current dipole. Actually WS is the only array that takes a poor advantage of parallel acquisition. However, it has been considered in the simulation as it allows K factors generally lower (and therefore less noisy measurements) and a higher resolution. So it is still interesting to compare results of the other data sets with the WS ones. The design of the sets of array configurations is based on the variation of the ratio between the maximum potential dipole spacing amax and the maximum dipole separation factor nmax of the dipole order n, defined as the smallest spacing between a current electrode and a potential electrode, related to the dipole length a, trying to keep a similar

36

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

amount of number of measures. The DD and PD arrays allow simultaneous acquiring of several potential measurements of different order n, using the same current dipole, while WS practically forces to take only a measure at a time. From this point of view, the most suitable array for multi-channel instruments is MG, because it allows, with a single current dipole, to acquire simultaneously all potential measures related to the dipoles comprised between electrodes A and B. 2.1. The multi-coverage multiple gradient (MG) configuration In the MG array (Dahlin and Zhou, 2006) the current dipoles are provided by dividing the maximum length of array L into equal segments by a fixed divisor e and placing the current electrodes at the ends of each segment (Fig. 1). The length of each current dipole is given by l = L/e. In this way if M is the greatest value of the divisor e, the smallest current dipole used has a length lmin = L/M. The number of current dipoles NAB used is then given by the following formula:

NAB ¼

M X

e:

ð1Þ

the forwarding step of the current dipole is the same of the current electrode separation. Consequently, this sequence does not ensure a uniform lateral coverage comparable to the classical sequences in which all the electrodes are used in turn as the current electrodes. To overcome this drawback, we decided to consider a modified configuration of the multiple gradient array, named “multi-coverage multiple gradient (MG) configuration”, obtained by increasing the current dipoles by dividing the forwarding step of the current dipole by a coverage factor c, as it is shown in Fig. 2. In this way the total number of current dipoles used is given by:

NAB ¼

M X ½ðe−1Þc þ 1:

ð2Þ

e¼1

For example, considering e = 4 (Fig. 2), the single coverage (c = 1) would give four current dipoles, the double one (c = 2) seven dipoles, the triple one (c = 3) ten, and so on. Obviously, when coverage c = 1, the configuration coincides with the original multiple gradient array by Dahlin and Zhou (2006).

e¼1

2.2. Choice of the sets of array configurations This particular sequence allows to minimize the number of current dipoles NAB (and therefore time), but maintaining a resolution comparable to those of the other arrays (Fiandaca et al., 2005; Martorana et al., 2009). In this way, for a simulation with 72 electrodes, and M = 8, only 36 different current dipoles are chosen. In this basic sequence,

In Table 1 the sets of array configurations considered are shown. For DD, PD and WS a progression of eight simulated datasets is chosen, increasing the maximum value amax of the length of the potential dipole a and decreasing the maximum value nmax. So for the

Fig. 1. Multiple gradient array: sketch of the position of the current dipoles when the maximum length of the profile L = 72 m and the divisor e varies from 1 to 8.

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

37

Fig. 2. Multi-coverage multiple gradient array: sketch of the position of the current dipoles when the maximum length of the profile L = 72 m, divisor e = 4 and coverage factor c from 1 to 6.

configuration #1 a = 1 and n = 1 … 35, for the configuration #2, a = 1 … 2, n = 1 … 17, and so on up to the configuration #8 with a = 1 … 8 and n = 1 … 3. Table 1 also summarizes the total number of measures Nmeas and of current dipoles NAB used for each dataset considered. In view of the parallel acquisition, the ratio Nmeas/NAB between number of current dipoles and number of measures is in some way related to the data recording velocity in field, although not directly proportional,

because the latter of course depends on many other factors (time duration of the current input, number of available instrumental channels for parallel acquisition, number of maximum potential measures possible for a given current dipole, in a specific data set). Yet surely the same number of measurements to be performed, the data set with higher ratio Nmeas/NAB will be acquired in less time than those with lower ratio. Finally, in Table 1 the median and median absolute deviation

Table 1 Geometric parameters, number of measures Nmeas, number of current dipoles NAB, ratio Nmeas/NAB, median geometry factor K and its median absolute deviation are shown for each array type and configuration designed. Array type Dipole–dipole

DD

Pole-dipole

PD

Wenner-Schlumberger

WS

Multipole gradient

MG

Configuration

Geometric parameters

#1 #2 #3 #4 #5 #6 #7 #8 #1 #2 #3 #4 #5 #6 #7 #8 #1 #2 #3 #4 #5 #6 #7 #8 #1 #2 #3 #4 #5 #6

amax = 1 amax = 2 amax = 3 amax = 4 amax = 5 amax = 6 amax = 7 amax = 8 amax = 1 amax = 2 amax = 3 amax = 4 amax = 5 amax = 6 amax = 7 amax = 8 amax = 1 amax = 2 amax = 3 amax = 4 amax = 5 amax = 6 amax = 7 amax = 8 emax = 8 emax = 8 emax = 8 emax = 8 emax = 8 emax = 8

nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = nmax = c=1 c=2 c=3 c=4 c=5 c=6

35 17 11 8 6 5 4 3 35 17 11 8 6 5 4 3 35 17 11 8 6 5 4 3

N. of measures Nmeas

N. of current dipoles NAB

Ratio Nmeas/NAB

Median geometry factor K

Median absolute deviation of K

1820 1887 1848 1784 1665 1635 1512 1296 1855 1938 1914 1864 1755 1740 1624 1404 1225 1479 1518 1504 1440 1425 1344 1188 533 904 1227 1596 1921 2292

69 135 198 258 315 369 420 468 69 69 69 69 69 69 69 69 1225 1479 1347 1384 1320 1251 1082 1043 36 65 93 122 150 179

26.4 14.0 9.3 6.9 5.3 4.4 3.6 2.8 26.9 28.1 27.7 27.0 25.4 25.2 23.5 20.3 1.0 1.0 1.1 1.1 1.1 1.1 1.2 1.1 14.8 13.9 13.2 13.1 12.8 12.8

15381.2 3110.1 1319.4 753.9 565.4 376.9 376.9 188.5 1507.9 565.4 376.9 263.8 188.5 188.50 150.80 100.53 414.69 226.19 131.95 113.10 75.40 75.40 62.83 43.98 109.96 94.25 85.13 85.13 75.40 75.40

12440.7 2959.4 1244.1 678.6 490.1 301.6 301.6 141.4 1319.5 490.1 301.6 188.5 138.2 138.2 100.5 62.8 351.9 188.5 113.1 75.4 56.5 50.3 44.0 31.4 90.2 79.0 65.7 65.7 62.8 62.9

38

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

Fig. 3. Median of the geometry factor K for each array configurations.

(MAD) of the geometry factor K are also shown. In this approach the number of measurements and the depth of investigation of each dataset does not differ too much. Different arrays considered do not exploit all alike the ability to acquire measures in a parallel mode. In fact, MG and PD arrays are those that derive greater benefit because the ratio Nmeas/NAB between the number of measurements and the number of current dipoles is high (from 20 to 26 in PD data sets and from 12 to 14 in MG). However, even acquisition of DD data sets is accelerated by a multi-channel instrument, provided that the ratio of maximum dipole length amax to maximum dipole separation factor nmax is not too high. In fact the ratio Nmeas/NAB varies from 26.4 (DD1) to 2.8 (DD8). The aim of this work will be to study which data sets gave the best performance, considering parallel acquisitions, in terms of good compromise between data recording velocity and effectiveness obtained. Some data sets albeit of fast acquisition does not allow a high average resolution or a low average model misfit, conversely others get better results but with excessive acquisition times. As the configuration number increases, for the DD datasets, measurements decrease while the number of current dipoles increases: consequently, the data recording velocity severely decreases. In PD datasets measurements decrease while the number of dipoles remains unchanged, in the WS instead the number of current dipoles is clearly higher to the other datasets and the data recording velocity is very low (Nmeas/NAB ~ 1). For MG we chose a progression of six datasets by

progressively increasing the coverage factor c from 1 to 6: in this case the number of measures increases but also the number of current dipoles in proportion and, consequently, the data recording velocity practically does not vary. The MG1 array data set, in which coverage c = 1, actually coincides with the original multiple gradient array by Dahlin and Zhou (2006). In all array types the median of geometry factor K decreases as the configuration number increases. For each array type this trend is shown in Fig. 3. DD configurations are characterized by very high values of median K, while WS and mainly MG show very low K values. Considering that high K values lead to low signal to noise ratios it follows that MG array configuration should provide the datasets of less noisy measures than the other. The modified version of MG allows to increase the number of measures, without substantially changing the ratio Nmeas/NAB nor the median K factor (indeed the latter decreases). Fig. 4 shows for each dataset the correlation between the amount of measures Nmeas and the number of current dipoles NAB used (related to the data acquisition time if a multichannel resistivity-meter is used). Acquisition times of WS arrays, much greater than the others, practically discourages the use of this array configurations with multichannel resistivity-meters. In the DD datasets NAB increases as the configuration number but Nmeas decreases. The PD datasets would seem to suggest very short recording time, but this assumption does not consider the additional long time necessary to place the remote electrode. In the MG datasets when increasing the configuration number (hence coverage c) both Nmeas and NAB increase (practically the information content and the survey times increase). Starting from the data set MG3 and higher, the number of measurements becomes comparable with the other data sets, while the ratio Nmeas/NAB remains generally lower than DD and WS data sets. 3. Forward modelling In applied geophysics forward modeling is a great tool for planning of investigations and, above all, to reduce the time required for data processing. The first phase of a good modeling concerns the design of models that define at best the real conditions. Different resistivity models were adopted for this study, some of which are similar to those used by Szalai et al. (2013). Three of them will be presented and discussed below. Model #1 (Fig. 5, a) shows ten resistive prisms (100 Ωm) of the same square section (2m*2m), equally spaced and a background of 10 Ωm. The depth of the center of the first prism to the left is 2m and the following prisms are gradually deeper 25 cm until a

Fig. 4. Correlation between the amount of measures Nmeas and the number of current dipoles NAB for each array configuration.

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

39

Fig. 5. Resistivity models used for the simulations. From top to bottom: 1) resistive prisms with square sizes at increasing depth in a conductive medium; 2) a buried resistive dike in a conductive basement covered by a near surface layer; 3) a waste landfill, done in a resistive terrain, filled with heterogeneous materials and a conductive leachate at its bottom.

maximum depth of 3.75 m. Model #2 (Fig. 5, b) shows a resistive buried vertical dyke (300 Ωm) intruded in a conductive rock (50 Ωm) and a near surface layer (200 Ωm). For the above presented models 72 electrodes were considered with a minimum spacing of 1 m. A third model (model #3, Fig. 5, c) simulates a landfill embedded in a subsoil of 200 Ωm. The landfill is filled with waste which forms a very conductive lower layer (2 Ωm) because of the leachate, and a more heterogeneous upper zone. This latter model has been considered in order to compare the results with those of a field test carried out in similar conditions. For model #3 72 electrodes were considered with a minimum spacing of 3 m. Forward problem was solved using RES2DMOD software and finiteelement method. 2D-models used divide the subsurface into a number of cells using a rectangular mesh. For the shallower zone (until depth of five times the electrode distance) a regular distribution of cells of the same size has been considered (quarter of the electrode space). In the deeper zone rectangular cells with size increasing with depth were used. A fundamental role in the forward modelling is the simulation of noise on the predicted data. The study of the influence of errors on the

resolution of the inverse model and especially on its ability to retrieve correct information of the subsurface is important to understand how the performance varies from a few simple parameters such as the total number of measurements of the data set and the distribution of the geometric factor values. For this purpose, rather than consider a random error simulation, noise was added to simulate errors due both to incorrect electrode positioning and to electrical potential, considering errors due to geological reasons. To add noise to geometric factor, errors with standard deviation of 3% were considered for all electrode positions. The potential errors were instead generated by simulating the trend showed by Zhou and Dahlin (2003). We used the formula: noisy data ¼ U ð1 þ R  β=100Þ;

ð3Þ

where U is the potential reading, R is a random number and β = (c1/U)c2 denotes absolute relative errors of the potential observations (Dahlin and Zhou, 2004). From observed data we considered c1 = 104 and c2 = 0.4 (Fig. 6, left) in order to obtain an average error on resistivity data of approximately 5% (Fig. 6, right).

Fig. 6. Simulations of the potential noise for 2D resistivity imaging surveys (left) and corresponding crossplot of noisy vs. noise free resistivity data (right). Data are related to model #1, considering the MG configuration with coverage factor c = 6.

40

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

4. Parameters to estimate the reliability of inversion Inversion of predicted data was performed using the RES2DINV Software (Loke, 2013), considering the same optimized parameter settings, in order to compare the results obtained from noise free data as well as noisy and field data. For all datasets related to a same model, the same initial damping factor and the same minimum damping factor were set to stabilize the inversion process and to highlight the differences in terms of data misfit, model misfit and relative resolution matrix related to each array data set and the same resistivity model. The option to automatically increase the damping factor for layers at different depth was made to take into account the decrease of the resolution. The evaluation of the results was performed by a quantitative analysis of some parameters that could define the ability of the inverse model to approach to the real situation. Generally, the main parameter used to evaluate the reliability of an inversion is the RMS error, which quantifies the misfit between the observed and predicted data vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u obs pred 2 XNmeas u d −d i t i : Фd ¼ i¼1 εi

ð4Þ

However, simulated models give the possibility to define quantitative parameters that describe the discrepancy between the tomographic model and the original one. The inversion program uses an arrangement of cells according to the pseudo-section and to the sensitivity function. This is obviously different from the arrangement of the original model. For this reason, in order to evaluate the resistivity mismatch between inverted and original models, a refined mesh was designed, obtained from the superposition of boundaries of all cells of inverted and original models. For the j-th cell of the refined mesh the model misfit Mj has been defined as: M j ¼ log

ρ j;inv ; ρ j;true

ð5Þ

where ρj, inv and ρj, true are respectively the resistivity of the j-th cell in the inverse model and in the “true” model (Martorana et al., 2009). The use of the logarithm for this parameter takes into account the nonlinear variation of the electrical resistivity. Positive or negative values of Mj indicate, respectively, an overestimation or underestimation of resistivity in the j-th cell. Another useful parameter to evaluate the model resolution can be derived by the model resolution matrix R. This is related to the leastsquares equation (Ellis and Oldenburg, 1994; Loke, 2011) used for the inversion of resistivity data and is calculated by the following equation (Day-Lewis et al., 2005):  −1 JT J; R ¼ JT J þ λF

ð6Þ

where J is the Jacobian matrix of partial derivatives, λ is the damping factor and F is a smoothing matrix defined by deGroot-Hedlin and Constable (1990). The matrix R, considering linear approximations, links the inverted model resistivity vector ρinv to the true resistivity vector ρtrue (Menke, 2015): ρinv ≈ Rρtrue :

ð7Þ

The elements of the main diagonal Rj,j give the ‘degree’ of resolution of the vector ρinv of the inverted parameters and are equal to 1 in the

ideal case of perfect resolution. The off diagonal elements give the degree of cross-correlation with the neighboring model cells. One way to illustrate the resolution is to plot the values of the diagonal elements of the R matrix that shows the degree at which the cell of the inverted model value depends on the corresponding true value. A better choice is plotting the diagonal elements of the resolution matrix for a particular data set divided by the corresponding resolution matrix elements of the data set that in theory assures the best resolution, i.e. a comprehensive data set. The terms of relative model resolution vector S are then given by:

Sj ¼

Rbj; j

ð8Þ

Rcj; j

where Rb is the model resolution of the considered data set and Rc is the model resolution of the comprehensive data set (Loke et al., 2015). The choice of the comprehensive data set depends on the number of electrodes. Considering 72 electrodes the full data set would include more than 3 million of measures (Noel and Xu, 1991; Stummer et al., 2004) and the use of the comprehensive data set defined by Stummer et al. (2004) still require more than 1 million of measures. In view of this we have used the comprehensive data set defined by Loke et al. (2015) for 2D resistivity surveys with a large number of electrodes. This consists of only symmetrical “alpha” and symmetrical “beta” arrays, so drastically reducing the number of measures to 58,380. For this so called “symmetrical array set” the average resolution is about 75 to 80% of the full set. To obtain inclusive parameters that express the overall effectiveness and resolution of each data set, for each model we also calculated the average values of the parameters above described, weighted according to the cell sizes sj of the refined mesh. In this way for each tomography the average model misfit and the average relative model resolution have been calculated. The average model misfit is calculated as follows: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XNcel 2 M j s2j j¼1 ~ ¼ ; M XNcel s j¼1 j

ð9Þ

where Mj is the model misfit of the j-th cell of the refined mesh, with size sj, and Ncel is the total number of cells of the refined mesh. The average model misfit gives an overall estimate of how much the tomography is different from the simulated model. Similarly, the average relative model resolution is calculated as follows: XN ~S ¼ Xj¼1 N

S js j

s j¼1 j

;

ð10Þ

where Sj is the relative model resolution of the j-th cell of the refined mesh, with size sj. This second inclusive parameter is commonly used to assess the performance of the array data set. 5. Results and discussions The results of the inversions were compared between themselves and with those obtained from a test on field data carried out in a landfill of the waste site of Bellolampo, near Palermo (Sicily). For the inversion of both simulated and experimental data the same algorithms and parameters were used. Some examples of the results are shown and discussed: these firstly show that the resolution strongly depends on the data set and that the reliability of ERT decreases with the depth of the target investigated. To facilitate the comparison of results we decided to use a single color scale for each parameter (electrical resistivity, model misfit,

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

relative sensitivity model), regardless of the array data set considered, that would include the entire range of parameter variability. A summary of the results of inversions for model #1 are shown as example (Fig. 7). In particular, for DD, PD and WS arrays a comparison

41

is shown between configuration #1 (amax = 1; nmax = 35) and #5 (amax = 5; nmax = 6), and for MG between configuration #1 (e = 1–8; c = 1) and configuration #6 (e = –1–8; c = 6). Finally, the results for the comprehensive data set considered (SDS = symmetrical data

Fig. 7. Model #1. Results of the inversion of different data sets of array configurations: (a) Inverse resistivity models for noisy data sets; (b) inverse resistivity models for noise free data sets; (c) images of the model misfit; (d) images of the relative model resolution. A sketch design of model #1 is above traced in every image.

42

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

set) are also shown for comparison. ERT images obtained from noisy data (Fig. 7a) and noise free data (Fig. 7b), are shown. For noisy data the images of the model misfit M (Fig. 7c) and of the relative model resolution S (Fig. 7d) are also represented. S is calculated by means of Eq. (8), considering as comprehensive data set the symmetrical data set. The original synthetic model is overlaid in schematic form to each section to facilitate the comparison. Results obtained by noisy data (Fig. 7a) are poorer of information than noise free data (Fig. 7b) but the gap between results decreases considering more complex and time requiring data sets. For noisy data the resolution strongly depends on the choice of the data set and on how the information recovery decreases with the target depth. Generally, in all the results M increase with the target depth (Fig. 7c) and, as the configuration number increases, it decreases in correspondence of the targets and overall the shapes are better resolved and artifacts are more limited. This can be explained by the comparison of the correspondent images of relative model resolution S that, as the configuration number increases, show a more uniform distribution and higher values at greater depths (Fig. 7d). This improvement is more pronounced for WS and PD arrays and minor for the MG. For this latter, however, an increasing trend of the lateral homogeneity is also noted. The DD datasets show values of M lower at the center of the targets, but also higher values near the target boundaries. This consequently leads to higher resistivity contrasts but also to more evident artifacts than other arrays. However all arrays show model misfit trends with lateral heterogeneities that tend to fade as the configuration number increases. This phenomenon is more evident in DD and in MG arrays than in PD ones. Similar considerations can be made analyzing the results obtained with the other simulated models. Fig. 8 shows a summary of the trend of the relative model resolution S related to the four arrays analyzed by increasing the configuration number, considering the noisy datasets of model #2. Obviously, in all inverse models S decreases with depth, but significant differences are noted, both between different arrays and between different array configurations. For each array it is observed that an increase of amax/nmax, causes a significant decrease of the vertical gradient in S images. In fact, if for all datasets Sj tends to 1 near surface, in deeper cells it critically

varies: in DD it ranges about from 0.02 (DD1) to 0.4 (DD7); in PD about from 0.1 (PD1) to 0.5 (PD7); in WS from about 0.07 (WS1) to 0.4 (WS7); in MG about from 0.03 (MG1) to 0.2 (MG5). Analyzing more in detail the results obtained with the MG, it is noted that the increasing of the coverage factor c allows to obtain results that are more similar to the original model. In fact biases decrease increasing coverage, while the lateral distribution of the relative model sensitivity S becomes more regular (Fig. 8). 5.1. A field test on a waste disposal site Simulations have shown that it is possible to optimize the acquisition sequences as a function of the ratio amax/nmax between maximum dipole lengths and maximum separation factors used. In order to verify the validity of the simulations in terms of accuracy of the methodological approach and realism of the simulated noise, a test survey was conducted in field by replicating the previously discussed sequences for DD, WS and MG arrays and comparing them with the symmetrical data set. An electrical resistivity tomography has been carried out in the landfill site of Bellolampo (Palermo, Italy), aimed to detect and locate the leachate plumes and to identify the HDPE bottom of the landfill. Results were compared with simulations on model #3, that represent a schematic landfill shielded. The area is characterized by fault slopes, relict river-karst valleys and canyons and karst depressions (Catalano et al., 2013), set up on a succession of calcareous and dolomitic limestones outcropping. These carbonate deposits are fractured and characterized by high permeability, consequently with high risk of polluting fluid infiltration (Contino et al., 1998). Measures has been acquired using a resistivity-meter MAE X612EM+, designed to carry out multi-electrode resistivity profiles using contemporary 96 integrated electrodes, so allowing a very high measurement velocity. Electrical resistivity tomography was carried out using 72 electrodes equally spaced of 3 m, for a total length of the survey equal to 213 m. The same inversion algorithms and parameters used for the simulated data were adopted for the field data.

Fig. 8. Model #2. Images of the relative model resolution for different array configurations obtained from noisy data. A sketch design of model #2 is above traced in every image.

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

43

Fig. 9. Model #3 representing a waste landfill. Results of inversion of different data sets: (a) geophysical model. Inverse resistivity models obtained by: (b) symmetrical data set; (c) dipoledipole array data sets; (d) Wenner-Schlumberger data sets; (e) pole-dipole data sets; (f) multiple gradient data sets.

Fig. 10. Electrical resistivity tomographies carried out in a landfill of the waste disposal site of Bellolampo near the town of Palermo (Sicily). Results of the inversion of different data sets of array configurations are shown.

44

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

Fig. 11. Comparison between the data misfit (RMS%) obtained by ERT field test of Bellolampo and those obtained by simulations on model #3.

Fig. 9 shows some results by inverting simulated data sets of model # 3 (Fig. 9a), for different arrays, using the same level of noise and the same algorithm parameters of the previously discussed data sets. The symmetrical data set (Fig. 9b) is able to represent with high detail the geometry of the landfill and the inside heterogeneity. However it overestimates the resistivity of the rock below the landfill. Even the DD arrays (Fig. 9c) represent with high accuracy geometries and resistivity values of waste and leachate. Furthermore, the value of the rock resistivity below the landfill is very close to those of the model. Good results are also obtained, though less accurate, by the PD data sets (Fig. 9e). Instead both WS (Fig. 9d) and MG data sets (Fig. 9f) produce much more smoothed inverse models, in which the landfill bottom is undefined and the resistivity of the lower rock underestimated. However, the

efficiency of imaging the waste and the leachate increases at increasing coverage, from MG1 to MG6. The results of the field test on the Bellolampo landfill are shown in Fig. 10. Assuming that the best and more detailed performance is obtained by the symmetrical data set, even if this seems to overestimate the resistivity of the rock bottom, an evident leachate bag is located in the lower central portion of the landfill, while numerous smaller anomalies, resistive or conductive are well detected in the upper zone. The landfill boundary is clearly detected by a high gradient of resistivity that delineates the shape of the landfill. As for simulated data, DD data sets show quite well the near surface heterogeneities of the waste, but for all these data sets the bottom of the landfill is not well-defined, and the resistivity below is much lower than the expected value, however the conductive leachate is much less blurry and the HDPE bottom is more clear in the data set with a high ratio amax/nmax. Deeper zones of the section show more realistic resistivity values in WS and MG data sets, at the expense of a less detailed surface reconstruction. A grater coverage factor e in MG data sets provide an imaging more similar to that of the symmetrical array.

5.2. Average parameters To assess the overall effectiveness of the inversion and the image resolution we averaged the above discussed parameters into the whole section of each inverse model. The data misfits are also compared with those resulted from the field test.

Fig. 12. Data Misfit (RMS %) obtained by inversion of simulated noisy data of (from top to bottom) model #1, #2, #3, as function of the array configuration number (left), the ratio Nmeas/NAB (center), the median of geometry factor K (right).

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

In Fig. 11 the trend of the data misfits (RMS error %) obtained from inversions of the field data is shown and compared with those from the model #3, considering both noise free and noisy data sets. Field and noisy data show sensibly higher values than noise free data. Furthermore, the trends shown by each array are very similar, when increasing configuration number. Data misfit obtained from all simulated noisy data sets are summarized in Fig. 12. Also the comparison between the data misfits resulting from the three simulated models show very similar trends, albeit with different average values between the models, in dependence on different values of electrical resistivity. The trend of the misfit as function of the configuration is therefore almost independent from the resistivity distribution in the subsurface. As the amax/nmax ratio increases the RMS% decreases. For DD it starts from high values and decreases rapidly, for WS and PD it starts from lower values and decreases more slowly. For MG the RMS% slowly increases with the coverage factor c (Fig. 12, left). Another interesting way to analyze the data misfit is to examine its dependence on the ratio between number of measures and number of current dipoles, Nmeas/NAB, related to the velocity of data acquisition (Fig. 2 center). MG survey configurations presenting misfits comparable to WS and DD datasets are generally faster than these ones. Apparently, PD surveys have the highest velocity and a relatively low data misfit, but we must point out that simulations not considered errors and additional time due to the placement of the remote current electrode, theoretically at an infinite distance.

45

Finally, the trend of data misfit versus the median of geometry factor K proves regularly increasing in a similar way for DD, PD and WS (Fig. 12, right). ~ (Fig. 13) is highly depending on both The average model misfit M resistivity values of the subsoil and array configuration. Only for MG ar~ as configuration rays the trends show an almost regular decreasing of M number increases (Fig. 13, left). This is probably related to the increas~ versus Nmeas/NAB ing of number of measures. Observing the trend of M (Fig. 13, center) it is noticed for the MG array that a good compromise ~ and high values of Nmeas/NAB are often shown by inof low values of M termediate configuration values. These datasets have also a low median ~ value of geometry factor K (Fig. 13, right). In the other array data sets M shows irregular behaviors, highly dependent on the model and on the array. The average relative model resolution ~S (Fig. 14) is few influenced by the presence of noise and its trend is very similar between noisy and field data because probably the only difference is due to the different resistivity trend in the subsoil. In a similar way to that observed for data misfit, variations of average relative model resolution are almost independent from the resistivity distribution in the subsurface. Considering the same configuration number, PD data sets show the highest values of ~S and MG ones the lowest. In every array, ~S increases with array configuration number (Fig. 14). For about c ≥ 4 the average relative model resolution of MG datasets is comparable with the lowest configuration number of the other array data sets. Plotting ~S versus Nmeas/NAB (Fig. 14, center) suggests the convenience of MG datasets with high

Fig. 13. Average model misfit obtained by inversion of simulated noisy data of (from top to bottom) model #1, #2, #3, as function of the array configuration number (left), the ratio Nmeas/NAB (center), the median of geometry factor K (right).

46

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

Fig. 14. Average relative model resolution obtained by inversion of simulated noisy data of (from top to bottom) model #1, #2, #3, as function of the array configuration number (left), the ratio Nmeas/NAB (center), the median of geometry factor K (right).

coverage respect to WS and DD arrays, as they show a comparable ~S values but a higher data recording velocity. MG1 data set (Dahlin and Zhou, 2006) shows a high average model misfit and low average model resolution, probably because it lacks a homogeneous lateral coverage of the subsoil. In fact the number of measurements carried out with this array data set is significantly lower than all other data sets and the pattern of model resolution is highly heterogeneous in the horizontal direction. The PD data sets show the best results that however do not consider possible problems and time delays due to the remote electrode. If reasonably we consider the values of Nmeas/NAB N 8 and ~SN0:2 to obtain an acceptable resolution taking advantage of multi-channel resistivity meters (carrying out an average of at least 8 simultaneous measurements), a part PD data sets, only the MG3, MG4, MG5, MG6, DD2 and DD3 data sets satisfy these conditions. Finally, the decrease of ~S with increasing median geometry factor K is regular for all arrays (Fig. 14). For model #1 (resistive square targets) we also estimated the trends of the values of the relative model resolution averaged on windows of the same size of the target, as a function of the depth of the target. Results are shown in Fig. 15. Colored zones show the areas of variation of the parameter for each array, from the lower data set number (dotted line) to the highest one (solid line). Trends show the exponential decrease of the resolution as the target depth increases. The slope of the curves decreases considering data sets with higher value of amax/nmax. This means that not only these latter have a trend for higher model resolution in every area of the section, but also that they resolve better the deeper cells of the models. Considering DD, PD and WS data sets, when the ratio amax/nmax is low, they show different target resolution values

(lower for the DD1, higher for WS1 and PD1) As ratio amax/nmax increases the resolution substantially enhances and tends to conform for DD, PD and WS array data sets. Also for the MG array data sets the target resolution increases with the coverage factor e, but it still remains lower than the others data sets.

Fig. 15. Model #1. Trend of the average relative model resolution as a function of the depth of the target. Colored zones show the areas of variation of the parameter for each array, from the lower array configuration numbers (dotted lines) to the highest ones (solid lines).

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

6. Conclusions Different sets of array configurations were chosen, varying the combination of the dipole lengths and distance between dipoles, to maintain roughly comparable measure numbers and therefore comparable information content, but by varying the distribution of the geometric factor K, resulting in a decrease in the median value with increasing the configuration number. The sets of array configurations so designed involve different resolution, noise level and acquisition velocity, due to the different combinations of the geometric parameters used. The comparison between 2D synthetic resistivity models and their inversion images, obtained starting from predicted data with opportunely simulated noise, has revealed very useful in order to evaluate the effectiveness of the array configuration chosen. A key role in the simulation process is played by the noise generation on the predicted measures, considering the effects of the propagation of errors on direct estimates of the voltage and the electrode positions. The evaluation of the results was performed using a quantitative analysis of data misfit and parameters that define the capacity to recover the “true” resistivity (model misfit) or the relative model resolution. The results show that the distribution of these parameters is highly dependent on the array configuration as well as on the resistivity of the subsoil. Results of inversions of both synthetic and field data show that data misfit is increasing with the median of the geometric factor K, although with different values in accordance with array considered. Indeed, the correlations between the geometric parameters used and the average model misfit are not so clear. However, it can be stated that the data sets with greater configuration number (lower median factor K) show a higher fit in homogeneous areas, that is, minor artifacts. Distribution of the relative model resolution is highly dependent on the configuration number. In fact although in near surface cells it is always very high, in the deeper ones it increases for those configurations characterized by a high ratio amax/nmax, allowing in theory to resolve deeper targets. Regarding the array type used, the average relative model resolution are overall better for PD and WS array data sets and worse for DD and MG ones. If we want to take advantage of multi-channel resistivity meters, WS array data sets are not suitable because, although they allow a good relative model resolution, they are characterized by a very low acquisition velocity, and relatively high model misfit. The PD array data sets could in theory provide excellent results both in terms of model misfit but above all in terms of acquisition velocity. However, these results do not consider possible problems related to the presence of the remote electrode. So in order to obtain an acceptable resolution and at the same time a high acquisition velocity, DD array data sets with medium amax/nmax ratio (DD2, or DD3) or MG arrays data sets with high coverage are recommended. Tests on field data have allowed a comparison, although limited to the evaluation of the data misfit RMS% and of the model resolution for an evaluation of the reliability of the results obtained. They showed similar trends such as on synthetic data. The comparison of the behaviour of different array configurations, when varying the combination of dipole length and distance between dipoles, is crucial in order to choose array configurations sets that would provide a higher resolution and good reliability, at the same time ensuring an adequate number of data in a short time. This is important both in scientific and especially in professional activities, where it is vital to obtain the best results with the least economic effort. References Barker, R.D., 1979. Signal contribution sections and their use in resistivity studies. Geophys. J. R. Astron. Soc. 59 (1):123–129. http://dx.doi.org/10.1111/j.1365-246X. 1979.tb02555.x. Barker, R.D., 1989. Depth of investigation of collinear symmetrical four-electrode arrays. Geophysics 54 (8):1031–1037. http://dx.doi.org/10.1190/1.1442728.

47

Blome, M., Maurer, H., Greenhalgh, S., 2011. Geoelectric experimental design—efficient acquisition and exploitation of complete pole–bipole data sets. Geophysics 76 (1): F15–F26. http://dx.doi.org/10.1190/1.3511350. Catalano R., Avellone G., Basilone L., Contino A. and Agate M., 2013. Note illustrative della Carta Geologica d'Italia alla scala 1:50.000 del foglio 594 “Partinico-Mondello”. Progetto CARG. Coles, D.A., Morgan, F.D., 2009. A method of fast, sequential experimental design for linearized geophysical inverse problems. Geophys. J. Int. 178 (1):145–158. http://dx.doi. org/10.1111/j.1365-246X.2009.04156.x. Contino, A., Cusimano, G., Frias Forcada, A., 1998. Modello Idrogeologico dei Monti di Palermo. Atti del 79° Congresso nazionale della Società Geologica Italiana, Palermo 21 -23 Settembre 1998 vol. A pp. 212–215. Curtis, A., 2004a. Theory of model-based geophysical survey and experimental design: part 1 — linear problems. Lead. Edge 23 (10):997–1004. http://dx.doi.org/10.1190/ 1.1813346. Curtis, A., 2004b. Theory of model-based geophysical survey and experimental design: part 2 — nonlinear problems. Lead. Edge 23 (11):1112–1117. http://dx.doi.org/10. 1190/1.1825931. Dahlin, T., 2000. Short note on electrode charge-up effects in DC resistivity data acquisition using multi-electrode arrays. Geophys. Prospect. 48 (1):181–187. http://dx.doi. org/10.1046/j.1365-2478.2000.00172.x. Dahlin, T., Zhou, B., 2004. A numerical comparison of 2D resistivity imaging with ten electrode arrays. Geophys. Prospect. 52 (5):379–398. http://dx.doi.org/10.1111/j.13652478.2004.00423.x. Dahlin, T., Zhou, B., 2006. Multiple-gradient array measurements for multichannel 2D resistivity imaging. Near Surf. Geophys. 4 (2):113–123. http://dx.doi.org/10.3997/ 1873-0604.2005037. Day-Lewis, F.D., Singha, K., Binley, A.M., 2005. Applying petrophysical models to radar travel time and electrical resistivity tomograms: resolution-dependent limitations. J. Geophys. Res. 110, B08206. http://dx.doi.org/10.1029/2004JB003569 (17 pp.). deGroot-Hedlin, C., Constable, S., 1990. Occam's inversion to generate smooth, twodimensional models form magnetotelluric data. Geophysics 55 (12):1613–1624. http://dx.doi.org/10.1190/1.1442813. Ellis, R.G., Oldenburg, D.W., 1994. Applied geophysical inversion. Geophys. J. Int. 116 (1): 5–11. http://dx.doi.org/10.1111/j.1365-246X.1994.tb02122.x. Fiandaca, G., Martorana, R., Cosentino, P.L., 2005. Use of the linear grid array in 2D resistivity tomography. Near Surface 2005 — 11th European Meeting of Environmental and Engineering Geophysics, A023. Furman, A., Ferré, T.P.A., Warrick, A.W., 2004. Optimization of ERT surveys for monitoring transient hydrological events using perturbation sensitivity and genetic algorithms. Vadose Zone J. 3 (4):1230–1239. http://dx.doi.org/10.2113/3.4.1230. Hennig, T., Weller, A., 2005. Two dimensional object orientated focusing of multielectrode geoelectrical measurements. Near Surface 2005 — 11th European Meeting of Environmental and Engineering Geophysics, A011. Loke, M.H., 2011. Electrical resistivity surveys and data interpretation. In: Gupta, H. (Ed.), Solid Earth Geophysics Encyclopaedia, second ed. Electrical & Electromagnetic. Springer-Verlag, pp. 276–283. Loke, M.H., 2013. Tutorial: 2-D and 3-D Electrical Imaging Surveys. www.geotomosoft. com. Loke, M.H., 2014. RES2DMOD ver. 3.01. Rapid 2D resistivity forward modelling using the finite-difference and finite-element methods. www.geotomosoft.com. Loke, M.H., Wilkinson, P., Chambers, J., 2010. Parallel computation of optimized arrays for 2-D electrical imaging. Geophys. J. Int. 183 (3):1202–1315. http://dx.doi.org/10.1111/ j.1365-246X.2010.04796.x. Loke, M.H., Wilkinson, P.B., Chambers, J.E., Strutt, M., 2014. Optimized arrays for 2D crossborehole electrical tomography surveys. Geophys. Prospect. 62:172–189. http://dx. doi.org/10.1111/1365-2478.12072. Loke, M.H., Wilkinson, P.B., Chambers, J.E., Uhlemann, S.S., Sorensen, J.P.R., 2015. Optimized arrays for 2-D resistivity survey lines with a large number of electrodes. J. Appl. Geophys. 112 (1):136–146. http://dx.doi.org/10.1016/j.jappgeo.2014.11.011. Martorana, R., Fiandaca, G., Casas Ponsati, A., Cosentino, P.L., 2009. Comparative tests on different multi-electrode arrays using models in near-surface geophysics. J. Geophys. Eng. 6 (1):1–20. http://dx.doi.org/10.1088/1742-2132/6/1/001. Maurer, H., Boerner, D.E., 1998. Optimized and robust experimental design: a non-linear application to EM sounding. Geophys. J. Int. 132 (2):458–468. http://dx.doi.org/10. 1046/j.1365-246x.1998.00459.x. Maurer, H., Boerner, D.E., Curtis, A., 2000. Design strategies for electromagnetic geophysical surveys. Inverse Probl. 16 (5):1097–1117. http://dx.doi.org/10.1088/0266-5611/ 16/5/302. Menke, W., 2015. Review of the generalized least squares method. Surv. Geophys. 36 (1): 1–25. http://dx.doi.org/10.1007/s10712-014-9303-1. Nenna, V., Pidlisecky, A., Knight, R., 2011. Informed experimental design for electrical resistivity imaging. Near Surf. Geophys. 9 (5):469–482. http://dx.doi.org/10.3997/ 1873-0604.2011027. Noel, M., Xu, B., 1991. Archaeological investigation by electrical resistivity tomography: a preliminary study. Geophys. J. Int. 107, 95–102. Stummer, P., Maurer, H., Horstmeyer, H., Green, A.G., 2002. Optimization of DC resistivity data acquisition: real-time experimental design and a new multi-electrode system. IEEE Trans. Geosci. Remote Sens. 40 (12):2727–2735. http://dx.doi.org/10.1109/ TGRS.2002.807015. Stummer, P., Maurer, H., Green, A.G., 2004. Experimental design: electrical resistivity data sets that provide optimum subsurface information. Geophysics 69 (1):120–139. http://dx.doi.org/10.1190/1.1649381. Szalai, S., Koppán, A., Szokoli, K., Szarka, L., 2013. Geoelectric imaging properties of traditional arrays and of the optimized Stummer configuration. Near Surf. Geophysics 11 (1):51–62. http://dx.doi.org/10.3997/1873-0604.2012058.

48

R. Martorana et al. / Journal of Applied Geophysics 137 (2017) 34–48

Van den Berg, J., Curtis, A., Trampert, J., 2003. Optimal nonlinear Bayesian experimental design: an application to amplitude versus off-set experiments. Geophys. J. Int. 155 (2):411–421. http://dx.doi.org/10.1046/j.1365-246X.2003.02048.x. Wilkinson, P.B., Meldrum, P.I., Chambers, J.E., Kuras, O., Ogilvy, R.D., 2006. Improved strategies for the automatic selection of optimized sets of electrical resistivity tomography measurement configurations. Geophys. J. Int. 167 (3):1119–1126. http://dx.doi.org/ 10.1111/j.1365-246X.2006.03196.x. Wilkinson, P.B., Chambers, J.E., Lelliott, M., Wealthall, G.P., Ogilvy, R.D., 2008. Extreme sensitivity of crosshole electrical resistivity tomography measurements to geometric errors. Geophys. J. Int. 173 (1):49–62. http://dx.doi.org/10.1111/j.1365-246X.2008.03725.x. Wilkinson, P.B., Loke, M.H., Meldrum, P.I., Chambers, J.E., Kuras, O., Gunn, D.A., Ogilvy, R.D., 2012. Practical aspects of applied optimized survey design for electrical resistivity

tomography. Geophys. J. Int. 189 (1):428–440. http://dx.doi.org/10.1111/j.1365246X.2012.05372.x. Wilkinson, P.B., Uhlemann, S., Meldrum, P.I., Chambers, J.E., Carrière, S., Oxby, L.S., Loke, M.H., 2015. Adaptive time-lapse optimized survey design for electrical resistivity tomography monitoring. Geophys. J. Int. 203 (1):755–766. http://dx.doi.org/10.1093/ gji/ggv329. Xu, B., Noel, M., 1993. On the completeness of data sets with multi-electrode systems for electrical resistivity survey. Geophys. Prospect. 41 (6):791–801. http://dx.doi.org/10. 1111/j.1365-2478.1993.tb00885.x. Zhou, B., Dahlin, T., 2003. Properties and effects of measurement errors on 2D resistivity imaging. Near Surf. Geophysics 1 (3):105–117. http://dx.doi.org/10.3997/18730604.2003001.