#
Analysis of the Influence of the GPS Errors Occurred While Collecting Electrode Coordinates on the Electrical Resistivity of Tumuli^{ †}

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area

#### 2.2. Geophysical Measurements: Field Survey and First Elaboration

_{AB}) as a constant value (12 V, 25 V, 50 V, 100 V, 200 V, 400 V or 800 V) or to set a constant voltage (Vp) that has to be read at the M and N (20 mV—called Save Energy mode—, 50 mV, 200 mV, and 800 mV). This second set-up lets the instrument modify at each iteration the input voltage that could be at least equal to a maximum V

_{AB}value forced by the operator. Of course, the imposed values are reached, thanks to the auto-ranging, only if the instruments limits (e.g., a maximum current of 2.5 A) are satisfied. For the DD array 5 different input were chosen (V

_{AB}equal to 200 V, 400 V, and 800V and Vp equal to 800 mV and Save Energy with a V

_{AB-max}equal to 800 V), why only two for the PD one (V

_{AB}equal to 800V and Vp equal to Save Energy with a V

_{AB-max}equal to 800 V). The data acquisition sequences were generated to take into account 10 and 5 increments of n and a, respectively, that are the distance between B and M and the fixed distance between two consecutive electrodes [3,27]. Therefore, each DD and PD ERT has 2475 and 2625 acquisitions, respectively. Thanks to the multi-channel system each ERT takes about 10–15 min. The acquisition parameters are summarized in Table 1.

^{TM}was used [4,12,13,34]. It employs a finite element method that divides the subsoil model into triangular cells. Thus, it is more flexible with rough topography or structures [13,28]. The software is able to model a flat surface or take into account the actual electrode position. It also implements a finest data noise management, using the Occam’s regularization [41] as developed by Morelli and LaBrecque [31]. This allows to estimate and assume for the inversion different percentage values of the standard deviation noise, according to the quality of the dataset [12,13]. Finally, starting from a homogeneous resistivity half-space, it calculates the quality of an inversion minimizing the misfit function between the field and modelled data [4,13]. The starting apparent resistivity was chosen equal to 195 Ωm, i.e., equal to the mean values of the whole acquired and simulated dataset (see Section 2.4). Given the good quality of the acquired data, only few measures were removed from the DD and PD dataset (Table 1) on the basis of the standard deviation percentage value fixed lower than 2%. Moreover, the “data noise error” was set equal to 1%. Given the purposes of the present work, all the ERTs were elaborated both with (topo in the following) and without (flat in the following) taking into account the real geographic coordinates of the electrodes.

#### 2.3. Monte Carlo (MC) Simulations to Take into Account the GPS-Error

_{a}) for each quadripoles of the acquired sequence; (6) repetition of the steps 4 and 5 N times; and (7) collection of the final results.

_{flat}and δk

_{topo}, respectively) was calculated, for both the flat and topo models, defined as follows (see also: [29,30]):

_{GPSflat}and k

_{GPStopo}are the geometric factor calculated at step 5 considering, at step 4, relative (flat) or absolute (topo) values of the electrodes coordinates, k

_{theoflat}is the geometric factor obtained considering a flat surface and a fixed electrode distance equal to a = 1.5 m, and k

_{theotopo}is the geometric factor computed taking into account the real electrode location (the collected GPS values).

#### 2.4. MC Simulated ERT Comparison

_{a}), while mean is the average of the whole apparent resistivity values (ρ

_{a}) or of the apparent resistivity at a given level/depth (ρ

_{alevel}), that means for each combination of (a) and (n).

_{simm}) or not (ρ

_{orig}) resistivity inverted models the ERT-error (δERT) was defined as follow (Equation (5)):

_{max}is the maximum value of the simulated dataset for each different voltage input.

## 3. Results

#### 3.1. Survey Results

#### 3.2. MC Simulations Results

_{AB}= 800 V). These two distributions were obtained extracting the positions of A, B, M, and N according to a uniform (in orange in the figure) and a normal (in blue in the figure) distributions (see Section 2.3).

_{flat}and δk

_{topo}for both the DD and PD arrays is show in Figure 6. It is possible to observe that, up to a depth of 5 m, δk

_{flat}and δk

_{topo}strongly suffer the electrodes mispositioning effect. These results confirm what known from the literature, i.e., that the DD percentage error is higher (quite double) than that of the PD array [14,25]. It is also important to note that considering a flat topography, at depth higher than 7 m, the PD error is no more influenced by the GPS-error, but it increases with the depth [29].

#### 3.3. Simulated ERT Results

_{norm}, see Equations (5) and (6)) of the mean simulated datasets, obtained for each input voltage, is shown in Figure 8. For the DD array, apart for the input voltage equal to 200 V, the highest errors of the inverted models are located within the first 5 m of depth from the ground surface and at a distance of [35 m–40 m] in correspondence of a high resistivity anomaly. The normalized ERT-errors of each simulated dataset for the 200 V input and the PD δERT

_{norm}show the highest variability.

## 4. Discussion

_{norm}).

_{flat}and δk

_{topo}values against with/respect to the investigation depth for both the DD and PD arrays (Figure 6) it is possible to observe that, within the first 5 m, geometric factor distribution is influenced by the GPS-error. Moreover, the PD measurements are also affected by a greater error at a depth deeper than 7 m (Figure 6). These results are in agreement with observations of Razafindratsima and Lataste [29]. The trend of this error, in fact, is comparable to that of the error induced by considering a finite position of the remote pole instead of an infinite theoretical one. Therefore, this result confirm, as known from literature, that (a) at shallow depth, the DD array is more influenced by the GPS-error than the PD array and (b) introducing the topography in the inversion procedure, it reduces the error in depth generated by the finite position of the remote pole, that seems to have an exponential trend. It is also possible to find in the literature that the geometric error sensitivity could be used to remove measurements more sensitive to the electrode-position error [15,23]. Nevertheless, this function is quite complicate. At the contrary, the proposed parameter δk can be easily used. Considering that archaeologists are often interested in investigating the first few meters of the underground, δk can be employed to evaluate which is the GPS-error influence on the acquired dataset and to decide which quadripoles/measurements are more affected by the GPS-error and have to be removed. Given the purposes of the presented work, no measurements were removed from the datasets, and all the apparent resistivity values were taken into account in the inversion procedure.

_{a}) between the most energetic input (that with an AB voltage of 800 V) and the others. Apart for some quadripoles, as highlighted in the Δρ

_{a}zoom (Figure 10), the Δρ

_{a}is around zero.

_{max}) and the half of the length of the maximum acquired array (D). They considered the maximum quadripole according to their remote pole position, i.e., in the first half of the electrodes line. A graphical description of the involved quantities is presented and values of Q and AÔB are summarized in Figure 11. Given the position of the remote pole, the maximum values of the AO distances were obtained for each values of the increment of (a), considering quadripoles with the minimum length. Because of graphical clarity, each increment of (a), belonging to the same ERT line, has been drawn on a different row. Consistently with the results of Razafindratsima and Lataste [29], if Q is in the range [2,3,4,5] and AÔB is about 100°, the influence of the finite position of the remote pole is not so relevant. Moreover, higher the value of Q, lower the influence. Taking into account the obtained values, it is possible to exclude an influence of the finite remote pole position on the inverted data, and therefore it is possible to assess that the observed ERT-error in the PD acquisitions is caused only by the GPS-error.

_{norm}, Figure 8) of the PD acquisitions is comparable to that of the DD ones, evidencing the absence of the influence of the finite remote pole position on the dataset. Furthermore, the hypothesis that the DD is able to better capture the anomalous resistivity values is confirmed considering that, for each simulated dataset, the highest DD-δERT

_{norm}is located within the first 5 m from the ground surface and in particular at a distance between 35 m and 40 m, in correspondence of the high resistivity anomaly interpreted by the archaeologist as a portion of the dromos (see Section 3.1). It can be assessed with a certain degree of confidence that this δERT

_{norm}is generated by the anomaly itself and not by the GPS-error, because the electrodes over the anomaly are affected by a very low GPS-error (Figure 2 and Figure 6).

## 5. Conclusions

_{norm}), the global (AE) and level (lAE) anomaly effects. The results of the analysis suggest that: (i) the GPS-error influence the apparent resistivity variations within the array; (ii) the PD array is more affected by the GPS-error, especially at higher depth; (iii) within the first 5 m the GPS-error influence is higher, compared with that observed at a depth between 5 m and 25 m; (iv) given the homogeneous conductivity distribution of the soil at the Poggio Pepe site, also a low voltage input (V

_{AB}of 200 V of Vp Save Energy) was sufficient to collect a good quality dataset not strongly affected by the GPS-error, therefore, in this study it was not possible to define if the input voltage can help to compensate the GPS-error and this potential relation must be further investigated, (v) the GPS-errors induce artifacts of a lower amplitude compared with that generated by considering a flat model instead of one with the real topography. Therefore, even if the influence on the inverted model is not so relevant from a qualitative point of view, it is not possible to exclude the influence of the GPS-errors on the apparent resistivity and therefore on the inverted resistivity model. In case of archaeological application, where the area of interest is limited to the first meters underground, errors in collecting the GPS electrodes coordinates can generate anomalies that cannot be ignored, especially in presence of a flat topography.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Busato, L.; Boaga, J.; Peruzzo, L.; Himi, M.; Cola, S.; Bersan, S.; Cassiani, G. Combined geophysical surveys for the characterization of a reconstructed river embankment. Eng. Geol.
**2016**, 211, 74–84. [Google Scholar] [CrossRef] - Jodry, C.; Lopes, S.P.; Fargier, Y.; Côte, P.; Sanchez, M. A cost-effective 3D electrical resistivity imaging approach pplied to dike investigation. Near Surf. Geophys.
**2017**, 15, 27–41. [Google Scholar] [CrossRef] - Loke, M.H.; Chambers, J.E.; Rucker, D.F.; Kuras, O.; Wilkinson, P.B. Recent developments in the direct-current geoelectrical imaging method. J. Appl. Geophys.
**2013**, 95, 135–156. [Google Scholar] [CrossRef] - Pazzi, V.; Ceccatelli, M.; Gracchi, T.; Masi, E.B.; Fanti, R. Assessing subsoil void hazards along a road system using H/V measurements, ERTs and IPTs to support local decision makers. Near Surf. Geophys.
**2018**, 16, 282–297. [Google Scholar] [CrossRef] - Pazzi, V.; Di Filippo, M.; Di Nezza, M.; Carlà, T.; Bardi, F.; Marini, F.; Fontanelli, K.; Intrieri, E.; Fanti, R. Integrated geophysical survey in a sinkhole-prone area: Microgravity, electrical resistivity tomographies, and seismic noise measurements to delimit its extension. Eng. Geol.
**2018**, 243, 282–293. [Google Scholar] [CrossRef] - Pazzi, V.; Morelli, S.; Fanti, R. A review of the advantages and limitations of geophysical investigations in landslide studies. Int. J. Geophys.
**2019**, 2019, 2983087. [Google Scholar] [CrossRef] [Green Version] - Santos, F.A.M.; Afonso, A.R.A. Detection and 2D modelling of cavities using pole-dipole array. Environ. Geol.
**2005**, 48, 108–116. [Google Scholar] [CrossRef] - Uhlemann, S.; Wilkinson, P.B.; Chambers, J.E.; Maurer, H.; Merritt, A.J.; Gunn, D.A.; Meldrum, P.I. Interpolation of landslide movements to improve the accuracy of 4D geoelectrical monitoring. J. Appl. Geophys.
**2015**, 121, 93–105. [Google Scholar] [CrossRef] [Green Version] - Binley, A.; Cassiani, G.; Deiana, R. Hydrogeophysics: Opportunities and challenges. Boll. Geofis. Teor. Appl.
**2010**, 51, 267–284. [Google Scholar] - Binley, A.; Hubbard, S.S.; Huisman, J.A.; Revil, A.; Robinson, D.A.; Singha, K.; Slater, L.D. The emergence of hydrogeophysics for improved understanding of subsurface processes over multiple scales. Water Resour. Res.
**2015**, 51, 3837–3866. [Google Scholar] [CrossRef] [Green Version] - Maillet, G.M.; Rizzo, E.; Revil, A.; Vella, C. High resolution electrical resistivity tomography (ERT) in a transition zone environment: Application for detailed internal architecture and infilling processes study of a Rhône River paleo-channel. Mar. Geophys. Res.
**2005**, 26, 317–328. [Google Scholar] [CrossRef] - Fischanger, F.; Morelli, G.; Ranieri, G.; Santarato, G.; Occhi, M. 4D cross-borehole electrical resistivity tomography to control resin injection for ground stabilization: A case history in Venice (Italy). Near Surf. Geophys.
**2013**, 11, 41–50. [Google Scholar] [CrossRef] - Santarato, G.; Ranieri, G.; Occhi, M.; Morelli, G.; Fischanger, F.; Gualerzi, D. Three-dimensional electrical resistivity tomography to control the injection of expanding resins for the treatment and stabilization of foundation soils. Eng. Geol.
**2011**, 119, 18–30. [Google Scholar] [CrossRef] - Clement, R.; Moreau, S. How should an electrical resistivity laboratory test cell be designed? Numerical investigation of error on electrical resistivity measurement. J. Appl. Geophys.
**2016**, 127, 45–55. [Google Scholar] [CrossRef] - Wilkinson, P.B.; Chambers, J.E.; Lelliot, M.; Wealthall, G.P.; Ogilvy, R.D. Extreme sensitivity of crosshole electrical resistivity tomography measurements to geometric errors. Geophys. J. Int.
**2008**, 173, 49–62. [Google Scholar] [CrossRef] [Green Version] - Pazzi, V.; Tapete, D.; Cappuccini, L.; Fanti, R. An electric and electromagnetic geophysical approach for subsurface investigation of anthropogenic mounds in an urban environment. Geomorphology
**2016**, 273, 335–347. [Google Scholar] [CrossRef] [Green Version] - Tejero Andrade, A.; Argote Espino, D.L.; Cifuentes Nava, G.; Hernandez Quintero, E.; Chavez, R.E.; Garcia Serrano, A. ‘Illuminating’ the interior of Kukulkan’s Pyramid, Chichén Itza, Mexico, by means of a non-conventional ERT geophysical survey”. J. Archaeol. Sci.
**2018**, 90, 1–11. [Google Scholar] [CrossRef] - Tsourlos, P.; Papadopoulos, N.; Yi, M.J.; Kim, J.H.; Tsokas, G. Comparison of measuring strategies for the 3-D electrical resistivity imaging of tumuli. J. Appl. Geophys.
**2014**, 101, 77–85. [Google Scholar] [CrossRef] - Tsokas, G.N.; Tsourlos, P.I.; Kim, J.H.; Yi, M.J.; Vargemezis, G.; Lefantzis, M.; Fikos, E.; Peristeri, K. ERT imaging of the interior of the huge tumulus of Kastas in Amphipolis (northern Greece). Archaeol. Prospect.
**2018**, 25, 347–361. [Google Scholar] [CrossRef] - Arato, A.; Piro, S.; Sambuelli, L. 3D inversion of ERT data on an archaeological site using GPR reflection and 3D inverted magnetic data as a priori information. Near Surf. Geophys.
**2015**, 13, 545–556. [Google Scholar] [CrossRef] - Papadopoulos, N.G.; Tsourlos, P.; Tsokas, N.G.; Sarris, A. Efficient ERT measuring and inversion startegies for 3D imaging of buried antiquities. Near Surf. Geophys.
**2003**, 5, 349–361. [Google Scholar] [CrossRef] - Papadopoulos, N.G.; Yi, M.J.; Kim, J.H.; Tsourlos, P.; Tsokas, N.G. Geophysical investigation of tumuli by means of surface 3D Electrical Resistivity Tomography. J. Appl. Geophys.
**2010**, 70, 192–205. [Google Scholar] [CrossRef] - Oldenborger, G.A.; Routh, P.S.; Knoll, M.D. Sensitivity of electrical resistivity tomography data to electrode position errors. Geophys. J. Int.
**2005**, 163, 1–9. [Google Scholar] [CrossRef] [Green Version] - Szalai, S.; Koppán, A.; Szarka, L. Effect of positional inaccuracies on multielectrode results. Acta Geod. et Geoph. Hung
**2008**, 43, 33–42. [Google Scholar] [CrossRef] [Green Version] - Zhou, B.; Dahlin, T. Properties and effects of measurement errors on 2D resistivity imaging surveying. Near Surf. Geophys.
**2003**, 1, 105–117. [Google Scholar] [CrossRef] [Green Version] - Aizebeokhai, A.P.; Olayinka, A.I. Anomaly effect of orthogonal paired-arrays for 3D geoelectrical resistivity imaging. Environ. Earth Sci.
**2011**, 64, 2141–2149. [Google Scholar] [CrossRef] - Dahlin, T.; Zhou, B. A numerical comparison of 2D resistivity imaging with 10 electrode arrays. Geophys. Prospect.
**2004**, 52, 379–398. [Google Scholar] [CrossRef] [Green Version] - Hennig, T.; Weller, A.; Canh, T. The effect of dike geometry on different resistivity configurations. J. Appl. Geophys.
**2005**, 57, 278–292. [Google Scholar] [CrossRef] - Razafindratsima, S.; Lataste, J.F. Estimation of the error made in Pole-Dipole Electrical Resistivity Tomography depending on the location of the remote electrode: Modelling and field study. J. Appl. Geophys.
**2014**, 100, 44–57. [Google Scholar] [CrossRef] - Robain, H.; Albouy, Y.; Dabas, M.; Descloitres, M.; Camerlynck, C.; Mechler, P.; Tabbagh, A. The location of infinite electrodes in pole–pole electrical surveys: Consequences for 2D imaging. J. Appl. Geophys.
**1999**, 41, 313–333. [Google Scholar] [CrossRef] - Morelli, G.; LaBrecque, D.J. Robust scheme for ERT inverse modeling. In Proceedings of the Symposium on the Application of Geophysics to Engineering and Environmental Problems, Keystone, CO, USA, 28 April–2 May 1996; pp. 629–638. [Google Scholar]
- Fressard, M.; Maquaire, O.; Thiery, Y.; Davidson, R.; Lissak, C. Multi-method characterization of an active landslide: Case study in the Pays d’Auge plateau (Normandy, France). Grromorphology
**2016**, 270, 22–39. [Google Scholar] [CrossRef] - Kaplan, E. Understanding Gps: Principles and Applications, 2nd ed.; Artech House: Norwood, MA, USA, 2005; p. 644. [Google Scholar]
- Viero, A.; Galgaro, A.; Morelli, G.; Breda, A.; Francese, R.G. Investigations on the structural setting of a landslide-prone slope by means of three-dimensional electrical resistivity tomography. Nat. Hazards
**2015**, 78, 1369–1385. [Google Scholar] [CrossRef] - Catelani, M.; Ciani, L.; Venzi, M. Sensitivity analysis with MC simulation for the failure rate evaluation and reliability assessment. Measurement
**2015**, 74, 150–158. [Google Scholar] [CrossRef] - Moschioni, G.; Saggin, B.; Tarabini, M.; Hald, J.; Morkholt, J. Use of design of experiments and Monte Carlo method for instruments optimal design. Measurement
**2013**, 46, 976–984. [Google Scholar] [CrossRef] - Falchi, I. Vetulonia e la sua Necropoli Antichissima; Coi tipi dei successori le Monnier: Firenze, Italy, 1891. [Google Scholar]
- Colombi, C. La Necropolis di Vetulonia nel Periodo Orientalizzante; «Italiká» 5; Reichert Verlag: Wiesbaden, Germany, 2018. [Google Scholar]
- Edwards, L.S. A modified pseudosection for resistivity and induced-polarization. Geophysics
**1977**, 42, 1020–1036. [Google Scholar] [CrossRef] - Pazzi, V.; Ciani, L.; Cappuccini, L.; Ceccatelli, M.; Patrizi, G.; Guidi, G.; Casagli, N.; Catelani, M. ERT investigation of tumuli: Does the errors in locating electrodes influence the resistivity? In Proceedings of the IMECO TC-4 Internationa Conference on Metrology for Arcaeology and Cultural Heritage, Florence, Italy, 4–6 December 2019; pp. 527–532. [Google Scholar]
- Constable, S.C.; Parker, R.L.; Constable, C.G. Occam’s inversion: A practical algorithm for generating smooth models from electromagnetic sounding data. Geophysics
**1987**, 52, 289–300. [Google Scholar] [CrossRef] - Hubbard, D.W. The Failure of Risk Management, “Why It’s Broken and How to Fix It”; John Wiley and Sons: Hoboken, NJ, USA, 2009. [Google Scholar]
- Oliveira, S.P.; Rocha, A.C.; Filho, J.T.; Couto, P.R.G. Uncertainty of measurement by Monte-Carlo simulation and metrological reliability in the evaluation of electric variables of PEMFC and SOFC fuel cells. Measurement
**2009**, 42, 1497–1501. [Google Scholar] [CrossRef] - Spasic Jokic, V.; Zupunski, L.; Zupunski, I. Measurement uncertainty estimation of health risk from exposure to natural radionuclides in soil. Measurement
**2013**, 46, 2376–2383. [Google Scholar] [CrossRef]

**Figure 1.**Maps of the study area. (

**a**) Ancient map of the Etrurian region around Vetulonia in the 6th century BC, with the location of Poggio Pepe Tumulus (red dot). (

**b**) Locations of the 2D-ERT (T1-T4, red lines) and 3D-ERT (C1, C-shaped red line) over the geological map of the study area. Light green dot (RP) represents the remote pole employed in the T1 pole-dipole acquisitions.

**Figure 2.**The GPS-errors, that affect the coordinates measures of each electrode, shown as blue bar (in the insert in the upper right corner) and as red circle around the “real” electrode position in the main panel (x and y axes are the longitude and latitude coordinates, respectively, in the Gauss Boaga-Roma40 projected system). In the insert (upper right corner) the red dashed line at 0.05 m is the imposed GPS threshold employed in the field acquisition, i.e., the error value imposed as acceptable (see text), while the red pointed line at 0.03 m is another possible threshold commonly used in field acquisition.

**Figure 3.**Flow chart of the whole procedure. Four main steps can be identified: 1. The field data acquisition of both electrical soil parameters (“original” data) and geographical coordinates of the electrodes, 2. The MC simulation to take into account the GPS-errors, 3. The “original” and simulated electrical resistivity data inversion with (topo) and without (flat) topography, 4. The models comparison to evaluate the GPS-error influence.

**Figure 4.**T1-ERT resistivity profiles obtained inverting at the same time DD and PD acquired data. (

**a**) ERT profile obtained without take into account the topography (flat) and (

**b**) taking it into account. Anomaly A and B can be associate to the dromos rocky blocks and the shadow of the funeral chamber, respectively.

**Figure 5.**Histograms, for a randomly selected quadripole (the number 759 with A = 15, B = 17, M = 21, and N = 23, acquired with an input voltage of V

_{AB}= 800 V), of the apparent resistivity values obtained by the MC simulation employing 100,000 different positions of the electrodes selected using both uniform (orange) and normal (blue) distributions. For the normal distribution, values of 1st percentile (green dashed line), 25th percentile (pink line), mean (black line), median (red line), 75th percentile (blue line), and 99th percentile (light blue line) are reported. The 759 quadripole apparent resistivity is 926.625 Ωm.

**Figure 6.**Distributions for the DD and PD arrays of the errors in the geometric factor (δk). In (

**a**) and (

**c**) for each quadripole the δk is the radius of a circle, while in (

**b**) and (

**d**) the δk numerical values are plotted versus the depth. In all the panels blue represents the δk values for the flat solution, while red that for the topography one.

**Figure 7.**(

**a**) The DD (marked with dots) and PD (marked with triangles) global anomaly effect (AE) for each original and simulated dataset. (

**b**–

**h**) The DD and PD level anomaly effect (lAE) for the original and simulated (1st percentile, 25th percentile, 75th percentile, 99th percentile, mean, and median) datasets.

**Figure 8.**Normalized ERT-error (δERT

_{norm}), for each input voltage, of the mean simulated dataset.

**Figure 9.**The DD and PD level anomaly effect (lAE) for the original and simulated (25th percentile, 75th percentile, maximum, mean, median, and minimum) datasets obtained with an input voltage of 200 V.

**Figure 10.**(

**a**) The apparent resistivity values collected for each different input voltage and, around zero, the differences between the most energetic input (that with an AB voltage of 800 V) and the others. (

**b**) A zoom on the differences of the measures between 305 and 405 (those highlighted by the red rectangle in panel (

**a**)).

**Figure 11.**(

**a**) Graphical description of the quantities involved in the computation of the infinite length coefficient (Q) and of the angle between the remote pole (A) and the first electrode of the tomography (B), calculated for each increment of the fixed inter electrode distance a according to the definitions provided by Razafindratsima and Lataste [29]. A and B are the current electrodes, M and N the voltage electrodes, O is the center of the maximum quadripole, and D the maximum quadripole half length. In the table, for each increment of a, are indicated the maximum investigation depth (after [39]), the D (the half of BN

_{max}length) and AO lengths, and the values of Q and AÔB. (

**b**) Distribution of the error, induced by the finite remote pole position, with respect to the AO length for each increment of the inter electrode distance (different symbols) and of the n value (different colours).

**Figure 12.**Geometric factor percentage relative error—with respect to both the depth (upper axis) and the number of measures ordered according to the quadripole investigation depth (lower axis)—between the flat and topo dataset (in blue), and between the original and the mean simulated datasets (in orange). (

**a**) Both trends refer to the DD array while in (

**b**) to the PD one.

**Table 1.**Summary of the acquisition carried out along the T1 profile (see Figure 1 for the location) to test the influence of the GPS-error, occurred in collecting the electrode position, on different array (dipole-dipole: DD and pole-dipole: PD) and different input voltage.

Name | Array | Input | a | n | Removed Data | Iterations |
---|---|---|---|---|---|---|

T1-DD-200V | DD | V_{AB} 200 V | 1–5 | 1–10 | 32 | 5 |

T1-DD-400V | DD | V_{AB} 400 V | 1–5 | 1–10 | 4 | 6 |

T1-DD-800V | DD | V_{AB} 800 V | 1–5 | 1–10 | 3 | 6 |

T1-DD-Vp800mV | DD | Vp 800 mV (V_{AB} max 800 V) | 1–5 | 1–10 | 3 | 6 |

T1-DD-Vpsaven | DD | Vp 20 mV—SaveEnergy (V_{AB} max 800 V) | 1–5 | 1–10 | 2 | 6 |

T1-PD-800V | PD | V_{AB} 800V | 1–5 | 1–10 | 2 | 4 |

T1-PD-Vpsaven | PD | Vp 20 mV—SaveEnergy (V_{AB} max 800 V) | 1–5 | 1–10 | 0 | 4 |

_{AB}is the value of the input voltage at the current electrodes set constant and equal to 200 V, 400 V, and 800 V. Vp is the values that was imposed to be read at the potential electrodes (see the text for more details) set equal to 800 mV or 20 mV (Save Energy mode). a is the distance between two adjacent electrodes and n is the distance between B and M. The range values show the number of increments of these distances. The last column indicates the number of iterations necessary to reach the iteration procedure convergence.

**Table 2.**Statistical parameters of the two tested distributions (normal and uniform) for the randomly selected quadripole 759 (A:15; B:17; M:21; N:23) acquired with an input voltage of V

_{AB}= 800 V which apparent resistivity is 926.625 Ωm.

Parameter | Normal Distribution [Ωm] | Uniform Distribution [Ωm] |
---|---|---|

1st percentile | 762.95 | 705.21 |

25th percentile | 874.09 | 820.98 |

Median | 925.53 | 926.78 |

Mean | 930.29 | 940.43 |

75th percentile | 981.32 | 1048.90 |

99th percentile | 1139.00 | 1235.20 |

Original | 1st perc | 25th perc | 75th perc | 99th perc | Mean | Median | ||
---|---|---|---|---|---|---|---|---|

DD-AE | 200 V | 5.524 | 5.401 | 5.427 | 5.572 | 5.904 | 5.495 | 5.507 |

400 V | 5.522 | 5.392 | 5.424 | 5.569 | 5.886 | 5.493 | 5.504 | |

800 V | 5.528 | 5.384 | 5.434 | 5.572 | 5.877 | 5.500 | 5.510 | |

Vp800 mV | 5.517 | 5.371 | 5.424 | 5.561 | 5.854 | 5.490 | 5.500 | |

VpSaven | 5.507 | 5.370 | 5.410 | 5.544 | 5.836 | 5.479 | 5.572 | |

PD-AE | 800 V | 5.051 | 4.438 | 4.815 | 5.148 | 5.629 | 5.004 | 4.977 |

VpSaven | 5.046 | 4.432 | 4.809 | 5.143 | 5.616 | 4.998 | 4.972 |

**Table 4.**Number of iterations, for each original and simulated dataset (1st percentile, 25th percentile, 75th percentile, 99th percentile, mean, and median), to reach the inversion model convergence.

Dataset | T1-DD-200V | T1-DD-400V | T1-DD-400V | T1-DD-Vp800mV | T1-DD-Vpsaven | T1-PD-800V | T1-PD-Vpsaven |
---|---|---|---|---|---|---|---|

original | 5 | 6 | 6 | 6 | 6 | 4 | 4 |

1st perc | 12 | 12 | 12 | 12 | 12 | 9 | 10 |

25th perc | 7 | 7 | 7 | 7 | 7 | 6 | 9 |

75th perc | 8 | 8 | 8 | 8 | 8 | 6 | 9 |

99th perc | 14 | 14 | 14 | 14 | 14 | 10 | 11 |

mean | 6 | 6 | 6 | 6 | 6 | 5 | 8 |

median | 5 | 5 | 5 | 5 | 5 | 4 | 7 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Pazzi, V.; Ceccatelli, M.; Ciani, L.; Patrizi, G.; Guidi, G.; Cappuccini, L.; Casagli, N.; Catelani, M.
Analysis of the Influence of the GPS Errors Occurred While Collecting Electrode Coordinates on the Electrical Resistivity of Tumuli. *Sensors* **2020**, *20*, 2966.
https://doi.org/10.3390/s20102966

**AMA Style**

Pazzi V, Ceccatelli M, Ciani L, Patrizi G, Guidi G, Cappuccini L, Casagli N, Catelani M.
Analysis of the Influence of the GPS Errors Occurred While Collecting Electrode Coordinates on the Electrical Resistivity of Tumuli. *Sensors*. 2020; 20(10):2966.
https://doi.org/10.3390/s20102966

**Chicago/Turabian Style**

Pazzi, Veronica, Mattia Ceccatelli, Lorenzo Ciani, Gabriele Patrizi, Giulia Guidi, Luca Cappuccini, Nicola Casagli, and Marcantonio Catelani.
2020. "Analysis of the Influence of the GPS Errors Occurred While Collecting Electrode Coordinates on the Electrical Resistivity of Tumuli" *Sensors* 20, no. 10: 2966.
https://doi.org/10.3390/s20102966