# An Extension of the Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI)

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## Abstract

**:**

## 1. Introduction

## 2. Outline of the PERTI Method

## 3. Extension of PERTI (E-PERTI)

_{N}, made of arbitrary points of the space U. Furthermore, since every nth geoelectric measurement is an individual process that, in ideal conditions where residual currents and polarization effect can be ignored, is not influenced by and does not affect any of the other measurements, we are allowed to consider the N apparent resistivity tests as automatically satisfying the condition of mutual independence. Hence, we can make recourse to the theorem that states that if N given tests are independent, then any Q tests arbitrarily extracted from the given ones are still independent [31]. This conceptual scheme was first considered by James Bernoulli and is known as the Bernoulli scheme [31].

_{q}≤ N for q = 1,2,…,Q and Q arbitrarily large, we extract a set of N

_{q}data (tests) within the N available data. Applying the PERTI approach, for each N

_{q}set Equation (8) becomes

## 4. Synthetic Test

_{n}is the intensity of the primary current injected into the ground through the electrodes A and B in the nth position (n = 1,2,…,N), say A

_{n}and B

_{n}, V

_{n}is the potential difference across the electrodes M and N in the nth position, say M

_{n}and N

_{n}, and K = πak (k + 1)(k + 2) is the DD geometrical factor, where a is the amplitude of the dipoles and k is the sampling integer denoting the increasing distance between the dipoles along the profile. A pseudosection indicates how the apparent resistivity varies with location and depth. For the DD electrode configuration, the data are plotted beneath the midpoint between the dipoles at a depth of half the distance between the dipole centres and then contoured using colour palettes.

#### 4.1. Random N_{q} Tests from N-Dimensional Pseudosection Space

_{q}data (tests) from the whole set of N data of the pseudosection in Figure 2b. In Figure 3 there are reported from top to bottom the resistivity sections resulting from averaging point by point the resistivity values obtained by the PERTI inversion scheme applied to 20 randomly extracted sets of N

_{q}= 50 (Figure 3a), N

_{q}= 100 (Figure 3b), N

_{q}= 150 (Figure 3c), N

_{q}= 200 (Figure 3d) and N

_{q}= 250 (Figure 3e) tests from the N = 295 starting data set.

_{q}increases. The presence of the three targets is better delineated with increasing N

_{q}, mostly that of the central body. Worth noting is the relative stable position of the three main nuclei in the sequence of sections, unlike other rather wandering nuclei, mainly in the topmost depth level. This allows giving a statistical relevance to the final choice of the most probable targets with respect to artifacts.

#### 4.2. Progressive N_{q} Tests by Vertical Scanning

_{q}subsets on a criterion linked to the depth, i.e., we consider the possibility of a vertical scanning of the pseudosection by selecting subsets with gradually increasing pseudodepths.

_{q}subsets are taken at increasing values of the depth factor k that appears in the definition of the geometrical factor K in Equation (16) (from top to bottom k = 5, k = 7, k = 9, k = 10). As in the original PERTI, the depth of each E-PERTI section is related to the pseudodepth of the corresponding subset.

#### 4.3. Sequential N_{q} Tests by Horizontal Scanning

_{q}tests for example by isolating a 10 m wide data window, moved progressively with a step of 1 m along the profile.

## 5. Field Example

_{q}tests depth scanning approach, while the right-hand column shows the sequence of sections, obtained by averaging point by point 10 PERTI sections (for each N

_{q}, 10 data sets were used) elaborated using randomly extracted N

_{q}data (tests) from the N = 91 starting dataset (from top to bottom N

_{q}= 15, N

_{q}= 30, N

_{q}= 45, N

_{q}= 60, N

_{q}= 75 and N

_{q}= 90).

## 6. Discussion

_{q}tests from the original N-dimensional top pseudosection, after damping the influence of the electrode effects. A 20% reduction factor has been applied to all those apparent resistivity values that, after extraction of the N

_{q}tests, resulted in at least 3 points to the corrupted strips. In the final E-PERTI section we now observe that there are not resistive zones on the right part anymore, what seems to exclude the presence of a tomb, which was expected to be a highly resistive target inside a less resistive background. This exercise was done to show the possibility of using E-PERTI to distribute different weights always in a probabilistic key.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic depiction of the characteristics of a dipole-dipole electrode device used to perform a 2D geoelectrical profile and the rule used to construct an apparent resistivity pseudosection.

**Figure 2.**A synthetic test: (

**a**) assigned 2D three-prism model, (

**b**) simulated pseudosection, (

**c**) model reconstruction by the original PERTI method.

**Figure 3.**Average sections, obtained by averaging point by point 20 PERTI sections elaborated using randomly extracted N

_{q}data (tests) from the N = 295 starting dataset. N

_{q}= 50 (

**a**), N

_{q}= 100 (

**b**), N

_{q}= 150 (

**c**), N

_{q}= 200 (

**d**) and N

_{q}= 250 (

**e**).

**Figure 4.**Section resulting from the estimation of ${\rho}_{m}$ with a linear best-fit procedure applied point by point to the resistivity values belonging to the PERTI sections in Figure 2.

**Figure 5.**Extractions of the N

_{q}tests in depth (from top to bottom k = 5, k = 7, k = 9, k = 10) and estimation of ${\rho}_{m}$ with linear best-fit procedure (last section).

**Figure 6.**Extraction of the N

_{q}tests by a stepwise displacement of 2 m of a 10 m wide data window along the array axis. The reduced sections are classical PERTI results over the whole set of tests within each window.

**Figure 7.**A perspective view of the same synthetic experiment as in Figure 6, but with PERTI sections displayed every 5 m along the profile axis, and estimation of ${\rho}_{m}$ with linear best-fit procedure (frontmost section).

**Figure 8.**Example of dispersion of resistivity values at 2 m depth from all overlapping segments of Figure 6, against offset distance x along the entire profile length.

**Figure 9.**Pseudosection corresponding to the three-prism model in Figure 2a, corrupted by a 30% random noise in the left side portion (

**a**); classical PERTI section (

**b**); section resulting from the estimation of ${\rho}_{m}$ with a weighted best-fit procedure applied point by point to the resistivity values obtained by a sequential horizontal scanning (

**c**).

**Figure 10.**Chapultepec castle on the Chapulín hill in Mexico City (

**a**), the Baths of Moctezuma (

**b**) with indication of the survey line, the dipole-dipole (DD) pseudosection (

**c**) and the PERTI section (

**d**).

**Figure 11.**Extension of PERTI for the field example of Figure 10 showing progressive N

_{q}tests by vertical scanning (

**left column**) and random N

_{q}tests from the N-dimensional pseudosection space (

**right column**).

**Figure 12.**E-PERTI section resulting from the estimation of ${\rho}_{m}$ with a linear best-fit procedure applied point by point to the resistivity values belonging to the PERTI sections in Figure 11 (

**a**) and discrepancy section between the E-PERTI section in (

**a**) and the classical PERTI section in Figure 10b (

**b**).

**Figure 13.**Fading procedure of a systematic noise effect from a field dataset: the DD field pseudosection (upper section), the classical PERTI section (intermediate section), the filtered best-fit E-PERTI section (lower section).

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**MDPI and ACS Style**

Cozzolino, M.; Mauriello, P.; Patella, D. An Extension of the Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI). *Geosciences* **2020**, *10*, 380.
https://doi.org/10.3390/geosciences10100380

**AMA Style**

Cozzolino M, Mauriello P, Patella D. An Extension of the Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI). *Geosciences*. 2020; 10(10):380.
https://doi.org/10.3390/geosciences10100380

**Chicago/Turabian Style**

Cozzolino, Marilena, Paolo Mauriello, and Domenico Patella. 2020. "An Extension of the Data-Adaptive Probability-Based Electrical Resistivity Tomography Inversion Method (E-PERTI)" *Geosciences* 10, no. 10: 380.
https://doi.org/10.3390/geosciences10100380