# Uncertainty of the 2D Resistivity Survey on the Subsurface Cavities

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Forward-Inverse Modeling

_{1}, T

_{2}, T

_{3}, T

_{4}, T

_{5}, and T

_{6}, with the depths of 2.2, 4.2, 6.2, 8.2, 10.2, and 12.2 m, respectively (as shown in Figure 1). The target cavity T

_{2}was used for further analysis as its depth is more common in cavernous limestone terrain. We numerically modelled the conductive cavities by using a filling of dry and saturated clay; however, the resistivity of the cavity filling material is a function of the particle size, saturation, and resistivity of the pore fluid [39].

^{TM}2D software [49] for the model simulations. The finite-element modeling calculates the resistivity data using a partial differential equation [50]. Throughout the entire lateral and vertical mesh formulations, 1.0 m thickness and depth increment factor were used. Finer meshes of 0.5 × 0.5 m were applied to increase the forward modeling accuracy. The simulation was perturbed using Gaussian noise (a random value generated with zero mean and 3% standard deviation) in order to integrate the field data-acquisition noise effects.

#### 2.2. Data Analysis

_{ρ}) for the measured apparent resistivities (ρ

_{i}) with the mean value (µ

_{ρ}) was calculated as follows:

_{01}and m

_{02}) using 0.1 and 10 times of the average apparent resistivity. DOI value of the bottom model (R

_{b}) was used to normalize the DOI index, and it is presented as:

_{1}and m

_{2}indicate the inverted resistivity values of the same cell for the two inversions.

_{i}is the inverted resistivity data points, ρ

_{a}is the actual model resistivity data points, µ

_{i}is the mean value for the inverted resistivity data, and µ

_{a}is the mean value for the actual resistivity data.

## 3. Results and Discussion

#### 3.1. Array Detection Ability

_{1}(at 2.2 m depth), resistivity data exhibited the variances of 24,400, 9058, 6341, and 2401 Ω·m for the DD, PD, WS, and PP arrays, respectively. In contrast, for the target cavity T

_{6}(at 12.2 m depth), resistivity data showed a statistical variance of 1100, 1044, 936, and 900 Ω·m for the DD, PD, WS, and PP arrays, respectively. The DD array depicted the maximum variance in resistivity data, whereas the PD and WS arrays produced moderate variances. The resistivity data from the PP array indicated the least variance. The results also show a linear dependence between the variance of the resistivity data and the accuracy of the cavity model recovered. Figure 2 shows that the decrease in resistivity data variance with increasing cavity depth is likely due to decreasing measurement sensitivity with survey depth [22].

_{1}. In contrast, the arrays DD, PD, WS, and PP exhibited the lowest anomaly effects of 0.245, 0.24, 0.19, and 0.18 respectively, for the deepest cavity T

_{6}. The dielectric nature of the cavity causes the arrays to acquire resistivity data with higher anomaly effects, mainly for the shallow cavity depth [11]. The DD array provided maximum anomaly effect, while the PP array exhibited minimum anomaly effect, similar to other studies [55]. In comparison, the PD and WS arrays depicted moderate anomaly effects. The anomaly effect shows a linear relationship with the measurement sensitivity of the array [47]. This could be the reason behind the decline of anomaly effects with an increasing cavity depth (Figure 3). The higher sensitivity near the electrode location may enable the arrays to get a more pronounced anomaly effect. On the other hand, the low sensitive zone of the far electrode position might limit the arrays to recover the higher anomaly effect.

#### 3.2. Inversion Error

_{2}. About 5% of the data points situated on the upper and lower limit of the scatter plots were specified as outliers. The predicted resistivity was significantly overestimated for smaller values of measured resistivity data points that may be related to the background medium (host limestone unit). On the other hand, the predicted resistivity was considerably underestimated for the extreme values of measured resistivity, probably due to the smoothing of the resistivity contrast between the background medium and the cavity by the inversion method [59].

#### 3.3. Model Accuracy, Resolution, and Sensitivity

#### 3.3.1. Model Accuracy

#### Depth of Cavity

_{6}) to 0.82 (T

_{1}). On the other hand, the PD and WS models depicted moderate image correlation, ranging from 0.28 (T

_{6}) to 0.75 (T

_{1}) for the PD models and 0.24 (T

_{6}) to 0.81 (T

_{1}) for the WS models. The lowest correlation values were exhibited by the PP array, varying from 0.19 (T

_{6}) to 0.71 (T

_{1}). The correlation coefficients were directly associated with the accuracy of the obtained models.

_{1}and T

_{2}), intermediate (T

_{3}and T

_{4}), and deeper (T

_{5}and T

_{6}) based on the cavity situated depths.

_{1}) and 4.2 m (T

_{2}) depths were accurately reconstructed by the DD, PD, and WS arrays, as depicted in Figure 7, Figure 8 and Figure 9a,b. The obtained images have also shown a strong statistical correlation with actual models (Figure 6). The PP array yielded a relatively lower resolution than the tested arrays (Figure 10a,b). The resistivity imaging displayed relatively wider anomaly sizes regarding the actual cavity models (Figure 7, Figure 8, Figure 9 and Figure 10a,b). The obtained geometry accuracy was considerably varied for the upper and lower anomaly boundaries. For instance, the upper boundary of the T

_{1}anomaly coincided with an actual cavity model, while the bottom boundary extended beyond that of the actual models (Figure 7a). Generally, inversion artefact increases with the survey depth; however, the inversion for high resistivity contrast may also provide a noticeable artefact, especially nearby the anomalous body [62]. Even though we applied the robust inversion algorithm, inversion for the shallow depth cavities has generated considerable artefacts, as depicted in Figure 7, Figure 8, Figure 9 and Figure 10a,b. The significant artefacts in inversions performed in all the air-filled cavity modeling had the resistivity value below 950 Ω·m, that is often misinterpreted as subsurface physical features.

_{3}) and 8.2 m (T

_{4}) depths showed reasonable model resolution in the tested arrays (Figure 7, Figure 8, Figure 9 and Figure 10c,d). The DD array displayed slightly better resolution than the other arrays. Figure 6 shows the moderate correlation between the inverted images and the actual models. The obtained anomaly sizes were noticeably overestimated, whereas the anomaly depths were underestimated by about 0.4–1.3 m with regard to actual cavity models. In Figure 7, Figure 8, Figure 9 and Figure 10c,d, the artefacts were clearly observed on the shallow part of the inverted models, particularly on the DD models. However, the resistivity imaging showed the intermediate cavity anomalies, the size overestimation and depth underestimation were limited the proper inference of the cavity information.

_{5}) and 12.2 m (T

_{6}) depths were not recovered adequately by any of the arrays (Figure 7, Figure 8, Figure 9 and Figure 10e,f). The obtained image correlations to the real model were weak (Figure 6). As shown in Figure 7, Figure 8, Figure 9 and Figure 10e,f, the anomaly sizes were highly overestimated, making it very challenging to interpret the cavity geometries without prior subsurface information. The result also showed about 1.6–3.0 m anomaly depth underestimation relative to the actual cavity locations. As the cavity depth increases, significant resistivity underestimation occurred in anomaly zones. Furthermore, the shallow parts of the final models were contaminated by patched inversion artefacts. These poor model accuracies can lead to inappropriate interpretation in resistivity imaging.

_{2}, which crossed the mid-point of the cavity at 4.2 m depth (as shown in Figure 11). We extracted the resistivity values along the 1D profile for the DD, PD, WS, and PP arrays. The DD array showed the highest (steepest) anomaly gradient over the cavity zone, with the peak resistivity value of 2220 Ω·m. In contrast, the PP array exhibited the lowest (gentlest) anomaly gradient, with the peak resistivity value of 1400 Ω·m. The 1D profile of the PD and WS arrays showed moderate anomaly gradients that have the peak resistivity value of 1850 and 1700 Ω·m, respectively. The arrays that provided the steep anomaly gradient can recover cavity boundaries more accurately, while the arrays that yield the gentle anomaly gradient prevent the inference of the cavity boundaries.

_{1}, T

_{2}, T

_{3}, T

_{4}, T

_{5}, and T

_{6}at depths of 2.2, 4.2, 6.2, 8.2, 10.2, and 12.2 m, respectively. The shallower cavity models showed a more robust resistivity amplitude than the cavity set at the relatively deeper position. For instance, the upper cavity T

_{1}indicates the maximum anomaly amplitude with the peak resistivity value of 6470 Ω·m. In comparison, the bottom cavity T

_{6}exhibits a minimum anomaly amplitude with a peak resistivity value of 1080 Ω·m. The substantial decrease in resistivity amplitude (gradient) as an increase in cavity depth can limit obtaining the anomaly geometries (Figure 12).

#### Size of Cavity

_{2}for the sake of comparison (Figure 14). Scenario 2 was adequately resolved in the DD, PD, and WS arrays; however, the PD and WS arrays’ resolution were slightly lower than the DD array, whereas the PP array poorly recovered the cavity in scenario 2. Moreover, all the tested arrays resolved scenario 3 cavities (Figure 15), but the PP array showed a relatively lower resolution than the other arrays. For increasing cavity size, the model resolution significantly improved, while the inversion artefacts considerably increased. The result showed that the overestimation in anomaly size was less pronounced as cavity size increases. By considering the homogenous host medium, cavity targets larger than the anomaly size in scenario 1 could be recovered effectively by the DD array. The DD and WS arrays could be effective for the cavity targets greater than or equal to the anomaly size in scenario 2. The PP array could be effective for the cavity size having a size greater than or equal to the anomaly in scenario 3.

#### Orientation of Cavity

#### Geometry of Cavity

#### Conductive Cavity Modeling

#### Experimental Cavity Study

_{2}was used. Thus, the plastic box’s center with 0.15 × 0.15 × 0.6 m size was situated at 0.42 m depth. The experimental resistivity datasets were measured using 89 electrodes of a 4-point light 10W (LGM Lippmann) resistivity meter with 0.1 m spacing (Figure 21).

#### 3.3.2. Spatial Resolution

#### 3.3.3. Sensitivity to Resistivity Contrasts

## 4. Conclusions

_{1}) set at 2.2 m depth. In contrast, the PP array showed the lowest anomaly effect (0.18) and variance (900 Ω·m) for the deeper cavity (T

_{6}) set at 12.2 m depth. The anomaly effect and the variance of resistivity data decreased for increasing cavity depths, limiting the recovery of the cavity information. The DOI threshold depths were delineated, and the inverted models were inspected. As cavity depth increases, the statistical image correlation between the inverted and actual models was significantly decreased. The anomaly sizes were considerably overestimated as cavity depth increases, which can highly restrict the recovery of accurate geometries.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Resistivity synthetic models that represent a target cavity set in limestone unit at six different depths: (

**a**) Target cavity T

_{1}at 2.2 m, (

**b**) target cavity T

_{2}at 4.2 m, (

**c**) target cavity T

_{3}at 6.2 m, (

**d**) target cavity T

_{4}at 8.2 m, (

**e**) target cavity T

_{5}at 10.2 m, (

**f**) target cavity T

_{6}at 12.2 m depth. The depth indicates the mid-point of the cavity. For better illustration, we presented only 40 m horizontal section of the models.

**Figure 2.**The variance of the numerically measured resistivity data for air-filled cavity situated at different survey depths.

**Figure 4.**The scatter plot of the model predicted and synthetically measured apparent resistivity data for the arrays: (

**a**) dipole-dipole (DD) array, (

**b**) pole-dipole (PD) array, (

**c**) Wenner–Schlumberger (WS) array, and (

**d**) pole-pole (PP) array. Overestimated and underestimated data outliers are displayed on the upper and lower boundaries.

**Figure 6.**The statistical correlation between the inverted resistivity models and actual air-filled cavity models.

**Figure 7.**The inverted resistivity model using dipole-dipole (DD) array for the cavity targets: (

**a**) T

_{1}, (

**b**) T

_{2}, (

**c**) T

_{3}, (

**d**) T

_{4}, (

**e**) T

_{5}, (

**f**) T

_{6}. The rectangular boxes indicate the actual air-filled cavities. The broken lines represent the depth of investigation (DOI) threshold depths. For better anomaly illustration, we presented only a 40 m lateral model section.

**Figure 8.**The inverted resistivity model using pole-dipole (PD) array for the cavity targets: (

**a**) T

_{1}, (

**b**) T

_{2}, (

**c**) T

_{3}, (

**d**) T

_{4}, (

**e**) T

_{5}, (

**f**) T

_{6}. The rectangular boxes indicate the actual air-filled cavities. The broken lines represent the DOI threshold depths.

**Figure 9.**Inverted resistivity model using Wenner–Schlumberger (WS) array for the cavity targets: (

**a**) T

_{1}, (

**b**) T

_{2}, (

**c**) T

_{3}, (

**d**) T

_{4}, (

**e**) T

_{5}, (

**f**) T

_{6}. The rectangular boxes indicate the actual air-filled cavities. The broken lines represent the DOI threshold depths.

**Figure 10.**Inverted resistivity model using pole-pole (PP) array for the cavity targets: (

**a**) T

_{1}, (

**b**) T

_{2}, (

**c**) T

_{3}, (

**d**) T

_{4}, (

**e**) T

_{5}, (

**f**) T

_{6}. The rectangular boxes indicate the actual air-filled cavities. The broken lines represent the DOI threshold depths.

**Figure 12.**Resistivity anomaly gradient for the air-filled cavity set at different survey depths using the dipole-dipole (DD) array.

**Figure 13.**The inverted resistivity model for scenario 1 with 0.5 × 6 m cavity size using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target.

**Figure 14.**The inverted resistivity model for scenario 2 with 1.5 × 6 m cavity size using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target.

**Figure 15.**The inverted resistivity model for scenario 3 with 3 × 6 m cavity size using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target.

**Figure 16.**The inverted resistivity model for the inclined (45°) cavity using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The box indicates the actual cavity target.

**Figure 17.**The inverted resistivity model for the vertical (90°) cavity size using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target.

**Figure 18.**The inverted resistivity models for the cavity geometries using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The solid line indicates the actual cavity targets.

**Figure 19.**The inverted resistivity model for a cavity filled with dry clay using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target.

**Figure 20.**The inverted resistivity model for the saturated clay-filled cavity using: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the actual cavity target. The broken line represents the actual groundwater level.

**Figure 22.**The inverted resistivity model of the experimental data for the: (

**a**) DD array, (

**b**) PD array, (

**c**) WS array, (

**d**) PP array. The rectangular box indicates the buried empty plastic box.

**Figure 23.**Scenario 1 spatial resolution test for the 1 m (vertical) × 2 m (horizontal) checkerboard cell sizes. (

**a**) Checkerboard model, (

**b**) DD array, (

**c**) PD array, (

**d**) WS array, (

**e**) PP array.

**Figure 24.**Scenario 2 spatial resolution test for the 2 m (vertical) × 4 m (horizontal) checkerboard cell sizes. (

**a**) Checkerboard model, (

**b**) DD array, (

**c**) PD array, (

**d**) WS array, (

**e**) PP array.

**Figure 25.**Scenario 3 spatial resolution test for the 4 m (vertical) × 8 m (horizontal) checkerboard cell sizes. (

**a**) Checkerboard model, (

**b**) DD array, (

**c**) PD array, (

**d**) WS array, (

**e**) PP array.

**Figure 26.**Dipole-dipole (DD) array sensitivity to resistivity perturbations. The inverted model for: (

**a**) ±20% perturbation, (

**b**) ±40% perturbation, (

**c**) ±60% perturbation, (

**d**) ±80% perturbation.

**Figure 27.**Pole-dipole (PD) array sensitivity to resistivity perturbations. The inverted model for: (

**a**) ±20% perturbation, (

**b**) ±40% perturbation, (

**c**) ±60% perturbation, (

**d**) ±80% perturbation.

**Figure 28.**Wenner–Schlumberger (WS) array sensitivity to resistivity perturbations. The inverted model for: (

**a**) ±20% perturbation, (

**b**) ±40% perturbation, (

**c**) ±60% perturbation, (

**d**) ±80% perturbation.

**Figure 29.**Pole-pole (PP) array sensitivity to resistivity perturbations. The inverted model for: (

**a**) ±20% perturbation, (

**b**) ±40% perturbation, (

**c**) ±60% perturbation, (

**d**) ±80% perturbation.

**Table 1.**Adequacy of the resistivity imaging types for cavity study: highly effective = xxx, moderately effective = xx, and less effective = x.

Adequacy of Arrays for Cavity | DD | PD | WS | PP |
---|---|---|---|---|

Shallow depth (≲4 m) ^{1} | xxx | xxx | xxx | xx |

Intermediate depth (≲8 m and >4 m) ^{1} | xx | x | x | x |

Deeper depth (>8 m) ^{1} | x | x | x | - |

Small size (≲0.5 m) ^{2} | x | - | - | - |

Intermediate size (≲3 m and >0.5 m) ^{2} | xxx | xx | xx | x |

Large size (>3 m) ^{2} | xxx | xxx | xxx | xx |

Horizontal cavity ^{1,2} | xxx | xx | xx | x |

Vertical cavity ^{1} | xx | xx | x | x |

Inclined cavity ^{1} | xxx | xxx | x | x |

Dry clay-filled cavity ^{1,2} | xxx | x | xx | x |

Saturated clay-filled cavity ^{1,2} | xxx | xxx | xxx | xx |

^{1}Considering a 1.5 × 6 m cavity size.

^{2}Considering 4.2 m depth. Except for dry and saturated clay, all the scenarios considered an air-filled cavity. The host medium is considered as homogenous limestone. For the heterogeneous background, the checkerboard cell size scenarios could be used.

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**MDPI and ACS Style**

Doyoro, Y.G.; Chang, P.-Y.; Puntu, J.M. Uncertainty of the 2D Resistivity Survey on the Subsurface Cavities. *Appl. Sci.* **2021**, *11*, 3143.
https://doi.org/10.3390/app11073143

**AMA Style**

Doyoro YG, Chang P-Y, Puntu JM. Uncertainty of the 2D Resistivity Survey on the Subsurface Cavities. *Applied Sciences*. 2021; 11(7):3143.
https://doi.org/10.3390/app11073143

**Chicago/Turabian Style**

Doyoro, Yonatan Garkebo, Ping-Yu Chang, and Jordi Mahardika Puntu. 2021. "Uncertainty of the 2D Resistivity Survey on the Subsurface Cavities" *Applied Sciences* 11, no. 7: 3143.
https://doi.org/10.3390/app11073143