Resolution of 3-D Electrical Resistivity Images from Inversions of 2-D Orthogonal Lines

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1 339 Resolution of 3-D Electrical Resistivity Images from Inversions of 2-D Orthogonal Lines Dr. Mehran Gharibi* 1 and Dr. Laurence R. Bentley 2 1 Researcher, Department of Geology & Geophysics, University of Calgary, Calgary, Alberta T2N 1N4 Canada mgharibi@ucalgary.ca 2 Professor, Department of Geology & Geophysics, University of Calgary, Calgary, Alberta T2N 1N4 Canada lbentley@ucalgary.ca *Corresponding author: Mehran Gharibi; mgharibi@ucalgary.ca ABSTRACT Three-D electrical resistivity imaging (ERI) using sets of orthogonal of 2-D survey lines provides an efficient and cost effective tool for site characterization in environmental and engineering investigations. A 3-D survey design using sparse sets of lines reduces the survey time at the expense of the resolution. The effects of line spacing on the resolution of 3-D electrical resistivity images were investigated using numerical modeling with synthetic and field data for two standard configurations, dipole-dipole and Wenner arrays. Synthetic data studies indicate that dipole-dipole configuration produces a more accurate map of the subsurface than the Wenner configuration. A severely undersampled 3-D survey could result in introducing small-scale shallow spurious artifacts at the surface of the resistivity model caused by the projection of the anomalies located in the deeper parts of the model. Results from inversion of the real and synthetic data showed that lines should be separated by no more than four electrode spacings and, if the shallow subsurface is important, by no more than two electrode spacings. The dipole-dipole array performs better than the Wenner array, but it requires more acquisition effort and is more sensitive to noise. These modeling results provide insight into quantitative survey designs that produce sufficient information to meet survey objective within a given field efforts. Introduction Adequate and cost-effective characterization of the subsurface poses challenging problems. Minimally invasive geophysical methods have been increasingly used to provide detailed images of the subsurface in contaminated and environmentally sensitive areas. Electrical Resistivity Imaging (ERI) is a geophysical technique that has been widely used for site characterization and monitoring of subsurface processes such as remediation progress, fluid migration and, steam injection in environmental studies (e.g., Park, 1998; Ramirez et al., 1993, 1995). Over the past decade, significant progress has been made in the development of field equipment and inversion programs for conducting and processing 3-D ERI surveys (e.g., Loke and Barker, 1996a, b; Zhang et al., 1995; Li and Oldenburg, 1992; Sasaki, 1994). Because of simplicity in field implementation, twodimensional resistivity surveys are still used in most investigations, however, they can lead to distorted and misleading results in heterogeneous areas. Recently, Bentley and Gharibi (2004) demonstrated distortions in 2-D images and described a 3-D ERI survey design in which orthogonal sets of 2-D resistivity survey lines are combined to create a 3-D dataset. The 2-D lines are jointly inverted to produce a 3-D image. They concluded that appropriately designed 3-D arrays can be used efficiently for site characterization. Survey design and layout strategies that produce optimum information using different ERI configurations and set up in different geological settings have been the topic of several studies (e.g., Stummer et al., 2004; Alumbaugh and Newman, 2000; Maurer et al., 2000; Dahlin and Loke, 1998). Numerical modeling or some measure of the Jacobian matrix of partial derivatives is usually used to quantitatively evaluate sensitivity of datasets and to assess the resolution of the resistivity images. The resolution information is not only useful in designing field experiments, but also it is helpful when interpreting resistivity models derived from inversion processes. It can be used to identify artifacts in the model and the level to which the details can be interpreted within the model. The primary goal of this study is to numerically analyze the resolution power of 3-D ERI surveys that are composed of sets of orthogonal 2-D lines. A series of modeling examples systematically demonstrate the trade off JEEG, December 2005, Volume 10, Issue 4, pp

2 340 Journal of Environmental and Engineering Geophysics of line spacing as well as the grid orientation on the final images are discussed. The results provide guidelines for survey design, specifically, the line spacing required to meet a given survey objective. Three-Dimensional Numerical Modeling Figure 1. Three-D conductivity model of six high conductivity bodies (1 -m) imbedded in a resistive background (100 -m). The anomalies are of 4 3 4, and, meter lateral extents. The blocks of 4 3 4and are at a depth of m and m, respectively. Two of the m blocks are located at a shallow depth of m and the other two are at a depth of m. between the spacing between parallel lines and the ability of inversions to resolve small-scale electrical conductivity (EC) anomalies. In addition to the synthetic modeling examples, a series of inversions from a sub-sampled field ERI survey are used to study the trade-off between line spacing and resolution. In the following, we will present results from 3-D inversions of synthetic and field ERI data constructed with sets of orthogonal lines with different line spacing. Results from two commonly used electrode arrays, dipole-dipole and Wenner array, are presented and the effect Three-D numerical forward and inverse modeling is used to demonstrate the effect of line spacing in 3-D ERI surveys using sets of orthogonal 2-D ERI lines. The model grid is (nominally m). The conductivity model (Fig. 1) consists of six conductive anomalies of 1 -m embedded in a relatively resistive background of 100 -m. The conductive blocks are 4 3 4, and meters in lateral extent. The blocks of and are at a depth of m and m, respectively. Two of the m blocks are located at a shallow depth of m and the other two are at a depth of m. Size, depth, and relative location of the conductive bodies with respect to the surface ERI line alignments simulate a variety of relatively small targets that may be similar to targets of interest in environmental surveys. The 3-D electrical resistivity responses for simulated surveys were calculated using the finite-difference method (Loke and Barker, 1996a). A series of 45 ERI lines separated by 1 m spacing were simulated in the x-direction and another 45 lines in the y-direction. Each line consists of 45 electrode locations separated by 1 m. The apparent resistivities along a cross-section are calculated using the potential values at the electrode position along each line and the geometrical factors determined by the relative location of the current and potential electrodes. The horizontal location of the measured apparent resistivity is placed at the midpoint of the 4-electrode set up used in the measurement and its vertical position is placed at the corresponding median or nominal depth of penetration. The nominal depth of penetration in a homogeneous half space is defined as the depth at which the area under the sensitivity-depth function is equal to half the total area (Edward, 1977). For example, an analytic expression of sensitivity-depth function for a pole-pole array in a homogeneous half space is defined as (Roy and Apparao, 1971); 2z FðzÞ ¼ ð1þ pða 2 þ 4z 2 Þ 3 2 where a is the current-potential electrode separation and z is depth from the ground surface. Accordingly, for a 4-electrode configuration, the contributions from all four possible current-potential pairs, i.e., pole-pole, are superimposed to construct sensitivity curve for a general array. Analytic expressions of the sensitivity-depth function that are used for plotting apparent resistivities are not functions of the resistivity of the half-space. However, the

3 Gharibi and Bentley: Resolution of 3-D Inversions of 2-D Orthogonal Lines 341 real sensitivity values are dependent on the distribution of resistivity in the subsurface. In the sensitivity plots that follow, sensitivity values are calculated using partial derivatives of the observed potentials with respect to the changes in the subsurface resistivity model. The value of the sensitivity in the plots shows the sum of the squares of the partial derivative of potential measurements with respect to the resistivity of the model at that particular grid location. Large values of the sensitivity means that changes in the model resistivity value have a large effect on the values of the measurements, and, consequently, the model resistivity value is well constrained. Small values of the sensitivity indicate that the model resistivity value at that location is not well constrained. The cross-sections from each line are combined to form a 3-dimensional dataset of the apparent resistivity values. Figure 2 shows apparent resistivity responses for the dipoledipole configuration in x-direction. Responses from all bodies are significant and extend beyond the nominal penetration depth of about 9 m. The two m anomalies, that are located at a greater depth compared to the other anomalous bodies, do not exhibit clear signatures and are overprinted by the responses from the large m anomaly in their vicinity. Responses from the Wenner arrays display similar behavior, however with smaller lateral coverage at depth. A set of inversions, using a 3-D resistivity inversion code (Loke and Barker, 1996a; Loke and Dahlin, 2002), were carried out to investigate the resolution power of different ERI survey configurations and the effects of different line spacing. The inversion algorithm uses an implementation of the least-square method based on an incomplete Gauss-Newton optimization technique. All inversion parameters were kept constant throughout this study. A grid similar to that used in forward modeling with elements and 10,571 model parameters was used for the inversion. Figure 3a shows results of the 3-D inversion of the synthetic apparent resistivity responses for the dipoledipole configuration using both sets of orthogonal lines with 1 m electrode spacing and 1 m line spacing. The maximum distance between the current pair and the potential pair of electrodes was limited to 6 a-spacings. A total of 38,880 measurements were used in the inversion and the total RMS misfit, representing the percentage difference between the logarithms of the measured and calculated apparent resistivity values, was 0.08% after 20 iterations. The lateral extents of the conductive bodies are precisely imaged with sharp contrasts to the surrounding background. The depth to the top of the anomalies has been located correctly, however the bottom boundary of the anomalies is not as sharp. The larger the anomaly and the shallower the anomaly, the better reproduced the value of the resistivity. Although difficult to see, the two m anomalies at the depth of m have been correctly located, but the magnitude of the resistivity low is poorly reproduced. Figure 2. Three-D apparent resistivity responses for the dipole-dipole configuration in x-direction of the model shown in Fig. 1. The 3-D inversion of a simulated Wenner array survey using 1 m electrode spacing and 1 m line separation is seen in Fig. 3b. The total number of measurements used in the inversion and the total RMS misfit after 20 iterations are 28,350 and 0.26%, respectively. Because the synthetic data used in this study is noise-free we allow the optimization process to iterate until a minimum total RMS misfit is reached. The floor for RMS misfit, however, is dictated by the discretization error and the smoothness constraint used in inversion as well as the round off error in computation. The results show halos on top of the shallowest conductive bodies, which are also flanked by lateral conductive lobes. The elevated EC near the bottom of the conductive anomalies extend to a greater depth compared to the results from inversion of the dipole-dipole data. The two

4 342 Journal of Environmental and Engineering Geophysics Figure 3. Three-D inversion of a) simulated dipoledipole ERI survey with 1 m line spacing, b) simulated Wenner ERI survey with 1 m line spacing. Data used in the inversion was simulated by forward modeling using the conductivity model shown in Fig m anomalies at the depth of m have not been resolved. Figures 4a and 4b show 3-D sensitivity values for each model cell on horizontal slices at different depths for the simulated dipole-dipole and the simulated Wenner arrays with 1 m line spacing, respectively. As discussed earlier, the magnitude of the sensitivity values represent the relative reliability of the corresponding model resistivity derived from the inversion. Near the surface, zones of consistently high sensitivity are observed in both array types. It means shallow surface anomalies are well constrained by the survey arrays. The sensitivity values decrease faster with depth in the Wenner array compared to the dipoledipole array. The Wenner array is also less sensitive at the horizontal margins of the survey array. To examine the effect of increasing line spacing a set of 3-D inversions of the apparent resistivity responses from Figure 4. Three-D sensitivity plots for a) the simulated dipole-dipole array with 1 m line spacing, b) the simulated Wenner array with 1 m line spacing. sets of orthogonal lines with 1 m electrode spacing and 2 m and 4 m line spacing using dipole-dipole and Wenner arrays are shown in Figs. 5 to 9. Increasing line spacing to 2 m introduced spurious features in the shallow depths of the resistivity models (Fig. 5). The lateral extents of the anomalies have been blurred and depths to the tops of the anomalies are less well resolved. These distortions are stronger in results from the Wenner array (Fig. 5b) where several false conductive bodies appear at the surface of the model. The presence of the false anomalies is due to the decrease in sensitivity values (Figs. 6a and b), especially near the surface of the resistivity model. These effects are more apparent in arrays with 4 m line spacing. The total measurements used for each inversion in 4 m line spacing (Figs. 7a and 7b) are one-fourth of their corresponding configurations in the survey using 1 m line spacing and their total RMS misfits after 20 iterations are 0.16% and 0.24%, respectively. The position and amplitude of the anomalies are well represented, however, the lateral

5 Gharibi and Bentley: Resolution of 3-D Inversions of 2-D Orthogonal Lines 343 Figure 5. Three-D inversion of a) simulated dipoledipole ERI survey with 2 m line spacing, b) simulated Wenner ERI survey with 2 m line spacing. Data used in the inversion was simulated by forward modeling using the conductivity model shown in Fig. 1. resolution is poorer than the simulated survey with 1 m line separation. The boundaries of anomalous bodies have been distorted particularly in the shallower parts of the model where the survey suffers from under-sampling due to the wider line separation. Spurious elevated EC features projected from the anomalous bodies at the depth appear at the surface of the model in both surveys, however, it is more significant in the Wenner configuration. The large areas of little control at the surface cause the under-sampling effect that is creating spurious features at shallow depths. As the line separation increases the lateral and vertical resolution decrease, however, this effect diminishes with depth where a larger volume is being excited by the induced current from electrodes at large separation distances. The sensitivity values for the simulated dipole-dipole and Wenner arrays with 4 m line spacing are displayed as horizontal slices in Figs. 8a and 8b, respectively. In the Figure 6. Three-D sensitivity plots for a) the simulated dipole-dipole array with 2 m line spacing, b) the simulated Wenner array with 2 m line spacing. shallow subsurface, the locations of the simulated ERI lines have high values of sensitivity. However, large line separation results in square-like zones between the crossing lines that have reduced sensitivity. These zones of reduced sensitivity are not well constrained by the data. As a result, the smoothness constraint used in the inversion causes the deeper conductive anomalies to smear out and appear at the surface of the inverse resistivity model as small-scale isolated zones of artificially elevated conductivity in the low sensitivity areas of the upper layers in Figs. 7a and 7b. These inversion artifacts are more significant in Wenner array due to the lower sensitivity compared to the dipole-dipole array. Grid orientation effects can distort the resolution of the ERI images even further (Dahlin et al., 2002). Figure 9a depicts results from the 3-D inversion of dipole-dipole data with 1 m electrode spacing and 4 m line spacing in the x- direction only. The deep anomalous bodies not only have been projected to the near surface of the resistivity model but also stretched out along the y-direction. The sensitivity

6 344 Journal of Environmental and Engineering Geophysics Figure 7. Three-D inversion of a) the simulated dipoledipole ERI survey with 4 m line spacing, b) the simulated Wenner ERI survey with 4 m line spacing. Data used in the inversion was simulated by forward modeling using the conductivity model shown in Fig. 1. plot shown in Fig. 9b can explain the elongated artifacts. Strips of poor sensitivity (up to two orders of magnitude lower) between the survey lines (Fig. 9b) allow the smoothness constraint to dominate the optimization process in these areas and extend the anomalies across the lines. This example demonstrates that, especially in the near surface, a 3-D inversion cannot properly image the subsurface if the 3-D array is poorly designed and areas of low data density cause areas of low sensitivity. Field Example A 3-D survey was conducted at the site of a decommissioned sour gas processing plant with soils that contained glycol and amine. The amine and the degradation products ammonium and acetic acid elevate the electrical conductivity of the soil and groundwater (Ndegwa et al., 2004; Mrklas Figure 8. Three-D sensitivity plots for a) the simulated dipole-dipole array with 4 m line spacing, b) the simulated Wenner array with 4 m line spacing. et al., 2004; Wong et al., 2004). The site is underlain by 6 to 9 m of Quaternary glacial till which overlies Tertiary sandstone and siltstone rocks (Bentley and Gharibi, 2004). The water table varies between 2 and 4 m below ground level. Due to the fine grained nature of the surficial sediments, soil moisture was high except in the upper 0.5 m. A set of 21 dipole-dipole lines with 28 electrodes, 1 m electrode spacing and, 2 m line spacing and an orthogonal set of 8 Wenner array lines with 42 electrodes and 1 m electrode spacing and 4 m line spacing were collected. A total of 5,846 data points were measured and results of the joint 3-D inversion of the data set from both directions is shown in Fig. 10a. The total RMS misfit after 14 iterations is 4.01%. The dry surface is highly resistive and is seen as dark red. Dark blue zones are elevated EC attributed to the presence of acetic acid and ammonium. An impermeable geomembrane liner was previously installed to hydraulically isolate a part of the site. It is an electrical insulator and was installed at an angle of 458 from vertical. The liner appears as an elongated resistive feature that extends to 2 m in depth.

7 Gharibi and Bentley: Resolution of 3-D Inversions of 2-D Orthogonal Lines 345 The effect of increasing line separation was investigated by creating a sub-sampled 3-D data set with orthogonal lines and 4 m line spacing by removing every second dipole-dipole array line from the original data set. Figure 10b shows results from 3-D inversion of this data set with 4,098 data points. A total RMS misfit of 3.19% was achieved after 15 iterations. The resulting image is similar to the image with 2 m spacing, but the anomalies are not as precisely imaged and the magnitudes of the extreme values are too low. In addition, the geomembrane liner is less well resolved. Results of the 3-D inversion of the even further sub-sampled data set with 8 m line spacing are shown in Fig. 10c. Much less detail in the shallow subsurface is visible in this inversion when compared to the results from inversion of the data sets with 2 and 4 m line spacing (Figs. 10a and 10b). The geomembrane liner was detected but its lateral and vertical extents are not correctly located. The inverse model shows a more homogenous shallow and subsurface structure due to under-sampling of the smallscale anomalies. The acquisition footprint is clearly visible as linear features in Fig. 10c. The effects of ERI line orientation on the results from inversion of field are shown in Fig. 11. Figure 11a depicts results from the 3-D inversion of twenty-one dipole-dipole field survey lines with 1 m electrode spacing and 2 m line spacing in the y-direction. It nearly resembles the results from joint inversion of the dipole-dipole and Wenner data in both directions (Fig. 10a). Geometries and magnitudes of most of the small-scale shallow anomalies have been successfully reproduced and no clear indication of line orientation is observed. The good results indicate that the 2 m line spacing is a good design with adequate data density and resolution power for this particular site. The results from inversion of eight Wenner field survey lines with 1 m electrode spacing and 4 m line spacing in x-direction (Fig. 11b), however, exhibit a strong acquisition footprint that is not evident when the cross lines are present (Fig. 10b). Shallow anomalies have been magnified along the lines and smeared out across the lines direction. The inverse model gives rise to a false impression of the subsurface anomaly distribution due to under-sampling. Discussion Three-D electrical resistivity imaging using sets of orthogonal ERI lines provides a practical method for implementing field operations and data interpretation in electrical resistivity surveys (Bentley and Gharibi, 2004). Although this design makes the 3-D surveys cost-efficient, survey time can still be a concern in a relatively large area. A 3-D survey design using sparse sets of orthogonal ERI lines can significantly reduce the data acquisition time at the expense of the resolution. There is a trade off between the Figure 9. a) Three-D inversion of a simulated dipoledipole ERI survey in x-direction only with 4 m line spacing. Data used in the inversion was simulated by forward modeling using the conductivity model shown in Fig. 1. b) sensitivity plots. resolution and the density of the data, i.e., line separation, in a survey. The question is for a given structure, what is the maximum line spacing in a survey that produces an image that can be regarded as a true 3-D map of the subsurface? It is an optimization problem and depends on the survey objectives and specifications of the targets. The factors which contribute to the final results are the resistivity distribution of the subsurface in terms of the size, depth, their relative locations with respect to the ERI lines, resistivity contrast between background and anomalous bodies and, the array type and configuration that is used in the survey. Different array types give rise to different sensitivity patterns and coverage in lateral and vertical extents. In the particular examples presented in this paper, the results from inversions of synthetic apparent resistivity data from the dipole-dipole configuration with 1 m electrode spacing and 1 m (Fig. 3a), 2 m (Fig. 5a), and 4 m (Fig. 7a) line spacing showed that the surveys were successful in

8 346 Journal of Environmental and Engineering Geophysics Figure 10. Joint 3-D inversion of field ERI data. The corresponding ERI survey arrays and electrode positions (solid circles) are shown at the top of each inverse model. a) eight lines using Wenner array with 4 m line spacing in x-axis and twenty-one lines using the dipole-dipole array with 2 m line spacing in y-axis, b) eight lines using Wenner array with 4 m line spacing in x-axis and eleven lines using the dipole-dipole array with 4 m line spacing in y-axis, c) four lines using Wenner array with 8 m line spacing in x-axis and six lines using the dipole-dipole array with 8 m line spacing in y-axis. resolving all the anomalies. However, their geometries are not as well resolved in 2 m and 4 m line spacing when compared to the results from 1 m line spacing survey. Discretization error in forward modeling and also the smoothness constraint used in optimization process may affect the inverse models and may account for a part of the degraded results. In the synthetic example, the dipole-dipole configuration generally created a more accurate geometry of the anomalous bodies with less distortion and false features than the results from Wenner array (Figs. 3b, 5b, and 7b). Of course, this may not be the case when dealing with field data, because the Wenner configuration generally has a higher signal to noise ratio than the dipole-dipole array. The higher signal to noise ratio of the Wenner array is due to the location of the potential electrodes between the current electrodes which causes larger potential drops than the dipole array.

9 347 Gharibi and Bentley: Resolution of 3-D Inversions of 2-D Orthogonal Lines Figure 11. Three-D inversion of field ERI data. The corresponding ERI survey arrays and electrode positions (solid circles) are shown at the top of each inverse model. a) twenty-one lines using the dipole-dipole array with 2 m line spacing in y-axis, b) eight lines using the Wenner array with 4 m line spacing in x-axis. Increasing line spacing in the field survey example showed the same decrease in the lateral and vertical resolution. For instance, the geomembrane liner that had been resolved accurately in joint inversion of dipole-dipole and Wenner arrays (Fig. 10a), and the results were validated with the ground truthing, lost its geometrical precision with increasing the line spacing. In the results from the inversion of ERI data with 8 m line spacing (Fig. 10c) the conductive anomalies in the vicinity of the geomembrane liner at the very shallow surface are reproduced with stronger amplitude and larger lateral extent compared with the results from inversion with smaller line spacing. Lack of sufficient constraint by data at the surface of the resistivity model has caused the strong conductive anomaly situated at depth to project to the surface of the inverse model and expand laterally. Increasing the line spacing has also caused the grid orientation bias in the inverse model where the alignment of the survey lines can be identified as the strips of rapidly varying anomalies. The field example illustrates that the ERI survey carried out in only one direction could nearly reproduce

10 348 Journal of Environmental and Engineering Geophysics results of the orthogonal ERI survey (Fig. 11a). However, when the ERI survey is conducted with inadequate line spacing, the grid orientation effect is stronger in the parallel line configuration than in the orthogonal line configuration. Results from inversion of synthetic and field data (Figs. 9 and 11b) showed that insufficient line spacing in a survey can cause breakdown in continuity of the anomalous bodies or stretching of the anomalies across the line direction, depending on the relative orientation of the line and anomalies. Conclusions The resolution power of 3-D ERI surveys composed of orthogonal 2-D lines is a function of the array type, electrode spacing and line spacing as well as the relative size, depth and location of the anomalous bodies with respect to the ERI lines. Small shallow anomalous features with lateral extents on the order of the electrode spacing used in the survey require closer line spacing in order to be resolved correctly. Small anomalies, on the order of the electrode spacing, located at a depth of several electrode spacing are usually unresolvable or result in incorrect amplitude of the resistivity value. Lateral extent and depth to the top of the larger anomalous features are well resolved with correct amplitude of the resistivity value, however depths to the bottom of the bodies are blurred. This effect is more severe in Wenner array since sensitivity values decrease faster with depth compared to the dipole-dipole array. Also, the dipole-dipole array has better resolution at the sides of the image than the Wenner array. However, the Wenner array reduces acquisition time because it uses fewer measurements to cover the area and is less sensitive to noise. Increase in line spacing between the neighboring ERI lines results in under-sampling of the small-scale anomalies and also loss of resolution in reproducing the geometry and actual resistivity of the large anomalies. Insufficient line spacing in a survey can lead to resistivity artifacts at the surface of the image due to projection of the anomalies located at depth and the image may also fail resolve smaller features at depth. The field and synthetic examples lead to the following set of guidelines for designing 3-D ERI surveys using orthogonal sets of 2-D lines. The dipole-dipole array is generally preferable to the Wenner array unless noise is a problem. The electrode spacing should not be larger than the smallest anomalies to be imaged. If the shallow subsurface, i.e., depths on the order of one electrode spacing, is important, then line spacing should be no more than twice the electrode spacing. If imaging the shallow subsurface is not a critical objective, then line spacing of up to four electrode spacings will generally be adequate. However, the field example indicates that, in cases where grid orientation effects are not anticipated, a preferred design for the same acquisition effort may be a survey with 2 electrode spaces between lines oriented in one direction rather than orthogonal lines separated by 4 electrode spacings. The trade-off is between a more even data coverage giving better control in the near surface and potential problems with grid orientation effects. Acknowledgments We would like to thank Imperial Oil Resources for logistical, technical and financial support. Funding was also provided by the Natural Sciences and Engineering Research Council (NSERC) of Canada. Dr. Meng Heng Loke is thanked for his support and assistance in inversion code related matters. Mehran Gharibi was financially supported in part by the University Technologies International Inc. (UTI). 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