# A Preliminary Numerical Study to Compare the Physical Method and Machine Learning Methods Applied to GPR Data for Underground Utility Network Characterization

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## Abstract

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## 1. Introduction

## 2. Estimation Methods

#### 2.1. Ray-Based Method

#### 2.2. Machine Learning Methods: SVM and SVR

#### Formulation

#### 2.3. SVM Implementation

#### 2.3.1. Feature Selection

#### 2.3.2. Training, Validation and Testing

## 3. Database Generation

## 4. Results

## 5. Discussion

#### 5.1. Ray-Based Method

#### 5.2. SVM

#### 5.3. SVR

^{−5}S m

^{−1}, 1 × 10

^{−3}S m

^{−1}and 1 × 10

^{−1}S m

^{−1}, which is presented in Table 2. Overall mean err and the 95th percentile of err increases with the conductivity in r, d and v estimations. For example, though the mean err is 6.3% in radius estimation, the error has increased from 5.3% to 7.7% when ($\sigma $) is increased from 1 × 10

^{−5}S m

^{−1}to 1 × 10

^{−1}S m

^{−1}. However, the error difference is very small between ($\sigma $) levels 1 × 10

^{−5}S m

^{−1}and 1 × 10

^{−3}S m

^{−1}. The trend is similar for both depth d and velocity v as well. However, The depth d and velocity v estimation mean err are well remained below 1%. Radius estimation mean err are larger at higher conductivity medium due to the fact that the pulse’s frequency components are attenuated by the medium and it cause changes in the pulse shape and shifts the signal peak and causes travel time picking error. Since the radius is highly sensitive to the travel time error, it leads to large error in the radius estimation.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 6.**Representation of travel-time-based feature selection from the hyperbola on a B-scan; ${\epsilon}_{r}=6$, $r=1\mathrm{c}\mathrm{m}$, $d=30\mathrm{c}\mathrm{m}$.

**Figure 8.**Confusion matrix of predicted results for radius estimation based on multi-class SVM classification model. Radius classes: 1 cm, 2 cm, 3 cm, 5 cm, 7 cm and 10 $\mathrm{c}$$\mathrm{m}$, respectively. Blue boxes indicates number of correct predictions and pink boxes represents number of false alarms.

**Figure 9.**Absolute relative error (err) in ray−based estimation of radius at fixed velocity scenario.

**Figure 11.**Absolute relative error (err) variation in SVR−based radius estimation across different depths.

**Figure 12.**Absolute relative error (err) variation in SVR−based radius estimation across different velocities of mediums.

**Figure 15.**Absolute relative error (err) variation in SVR−based depth estimation across different depths.

**Figure 16.**Absolute relative error (err) variation in SVR−based depth estimation across different velocities of mediums.

**Table 1.**Mean relative error and maximum relative error in terms of 95th percentiles (${P}_{95}$) with respect to radius (r), depth (d) and velocity (v) estimation. The last row represents the number of false alarms in SVM.

Method | $\mathit{err}\left(\mathit{r}\right)$% | $\mathit{err}\left(\mathit{d}\right)$% | $\mathit{err}\left(\mathit{v}\right)$% | |||
---|---|---|---|---|---|---|

Mean | ${\mathit{P}}_{95}$ | Mean | ${\mathit{P}}_{95}$ | Mean | ${\mathit{P}}_{95}$ | |

Ray-based concurrent | 260% | 464% | 25.1% | 65% | 11.3% | 22% |

Ray-based fixed velocity | 120% | 353% | - | - | - | - |

Regression (SVR) | 6.3% | 26.5% | 0.39% | 1% | 0.22% | 0.5% |

Classification (SVM) | 2% (10/500) | 0% (0/500) | 1% (5/500) |

**Table 2.**Mean absolute relative error and 95th percentiles (${P}_{95}$) with respect to radius (r), depth (d) and velocity (v) estimation in SVR approach.

Conductivity ($\mathit{\sigma}$) | $\mathit{err}\left(\mathit{r}\right)$% | $\mathit{err}\left(\mathit{d}\right)$% | $\mathit{err}\left(\mathit{v}\right)$% | |||
---|---|---|---|---|---|---|

Mean | ${\mathit{P}}_{95}$ | Mean | ${\mathit{P}}_{95}$ | Mean | ${\mathit{P}}_{95}$ | |

1 × 10^{−5} S m^{−1} | 5.3% | 26.04% | 0.25% | 0.74% | 0.12% | 0.39% |

1 × 10^{−3} S m^{−1} | 5.9% | 25.5% | 0.26% | 0.75% | 0.14% | 0.42% |

1 × 10^{−1} S m^{−1} | 7.7% | 28.4% | 0.52% | 1.1% | 0.32% | 0.79% |

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**MDPI and ACS Style**

Jaufer, R.M.; Ihamouten, A.; Goyat, Y.; Todkar, S.S.; Guilbert, D.; Assaf, A.; Dérobert, X.
A Preliminary Numerical Study to Compare the Physical Method and Machine Learning Methods Applied to GPR Data for Underground Utility Network Characterization. *Remote Sens.* **2022**, *14*, 1047.
https://doi.org/10.3390/rs14041047

**AMA Style**

Jaufer RM, Ihamouten A, Goyat Y, Todkar SS, Guilbert D, Assaf A, Dérobert X.
A Preliminary Numerical Study to Compare the Physical Method and Machine Learning Methods Applied to GPR Data for Underground Utility Network Characterization. *Remote Sensing*. 2022; 14(4):1047.
https://doi.org/10.3390/rs14041047

**Chicago/Turabian Style**

Jaufer, Rakeeb Mohamed, Amine Ihamouten, Yann Goyat, Shreedhar Savant Todkar, David Guilbert, Ali Assaf, and Xavier Dérobert.
2022. "A Preliminary Numerical Study to Compare the Physical Method and Machine Learning Methods Applied to GPR Data for Underground Utility Network Characterization" *Remote Sensing* 14, no. 4: 1047.
https://doi.org/10.3390/rs14041047