# Full-Wave Modeling and Inversion of UWB Radar Data for Wave Propagation in Cylindrical Objects

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## Abstract

**:**

## 1. Introduction

## 2. Methods and Experimental Set-Up

#### 2.1. Computational Methods

#### 2.1.1. Scattering Green’s Function for Cylindrically-Layered Media with Circular Cross Section

#### 2.1.2. Optimal Integration Path

#### 2.1.3. Far-Field Radar-Antenna Model

#### 2.1.4. Full-Wave Inversion Method

#### 2.2. Numerical Experiments

#### Laboratory Experimental Set-Up

- The Lightweight Radar System

- Measurements on the Cylindrical Models

## 3. Results

^{2}copper sheet, assumed as a perfect electrical conductor (PEC), at different distances ranging unevenly from 0.31 m to 0.57 m. In total, nine measurements were performed. The corresponding Green’s functions for a PEC were then calculated. Together with the collected $S\left(\omega \right)$ data set, Equation (10) was solved for retrieving the antenna characteristic functions.

#### 3.1. The Effects of Conductivity, Radius, and Relative Permittivity on Time Domain GPR Signals

#### 3.2. Full-Wave Inversion Analysis

#### 3.2.1. Topography of the Objective Functions

#### 3.2.2. The Local Optimization Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**Detecting the cylindrical model with the lightweight radar system with vertical polarization.

**Figure 4.**Amplitude and phase of the (

**a**) return loss transfer function (

**b**) global transmission and reflection coefficient function and (

**c**) feedback loss transfer function of the BBHA 9120D horn antenna as a function of frequency.

**Figure 5.**The effects of (

**a**) radius and (

**b**) relative permittivity on nonconductive cylindrical models.

**Figure 7.**The objective function distribution of (

**a**) PVC-1 with d equals 0.15 m, (

**b**) PVC-2 with d equals 0.15 m, (

**c**) PVC-3 with d equals 0.15 m, (

**d**) PVC-1 with d equals 0.4 m, (

**e**) PVC-2 with d equals 0.4 m, (

**f**) PVC-3 with d equals 0.4 m.

**Figure 8.**The objective function distributions of metallic tubes with distances of (

**a**) 0.15 m and (

**b**) 0.4 m.

**Figure 9.**The comparison between the measured time domain signal and the modeled time domain signal of (

**a**) PVC-1, (

**b**) PVC-2, (

**c**) PVC-3, (

**d**) Metal-1, (

**e**) Metal-2, (

**f**) Metal-3 at position 0.15 m.

**Figure 10.**The comparison between the measured time domain signal and the computed time domain signal of (

**a**) PVC-1, (

**b**) PVC-2, (

**c**) PVC-3, (

**d**) Metal-1, (

**e**) Metal-2, (

**f**) Metal-3 at position 0.4 m.

Model | Radius (cm) | Relative Permittivity | Conductivity (S/m) |
---|---|---|---|

1 | 1, 5, 10 | 5 | 0 |

2 | 5 | 2, 6, 12 | 0 |

3 | 1, 5, 10 | 1 | 100 |

4 | 5 | 1 | 1, 10, 100 |

5 | 1, 2, 3, … 10 | 2, 2.5, 3, … 12 | 0 |

6 | 1, 2, 3, … 10 | 1 | 100 |

Model | Radius (m) | Relative permittivity | |||
---|---|---|---|---|---|

Estimated | Measured | Error (%) | Estimated | Measured | |

PVC-1 | 0.0236 | 0.0201 | 17.41 | 1.8428 | 2.35–2.82 |

PVC-2 | 0.0432 | 0.0406 | 6.40 | 2.0641 | 2.35–2.82 |

PVC-3 | 0.0801 | 0.0805 | 0.50 | 2.4316 | 2.35–2.82 |

Metal-1 | 0.0051 | 0.0170 | 70.00 | / | / |

Metal-2 | 0.0082 | 0.0248 | 66.93 | / | / |

Metal-3 | 0.0207 | 0.0640 | 67.66 | / | / |

Model | Radius (m) | Relative permittivity | |||
---|---|---|---|---|---|

Estimated | Real | Error (%) | Estimated | Measured | |

PVC-1 | 0.0210 | 0.0201 | 4.48 | 2.3283 | 2.35–2.82 |

PVC-2 | 0.0419 | 0.0406 | 3.20 | 2.2918 | 2.35–2.82 |

PVC-3 | 0.0815 | 0.0805 | 1.24 | 2.4057 | 2.35–2.82 |

Metal-1 | 0.0124 | 0.0170 | 27.06 | / | / |

Metal-2 | 0.0203 | 0.0248 | 18.15 | / | / |

Metal-3 | 0.0528 | 0.0640 | 17.50 | / | / |

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**MDPI and ACS Style**

Gao, L.; Dachena, C.; Wu, K.; Fedeli, A.; Pastorino, M.; Randazzo, A.; Wu, X.; Lambot, S.
Full-Wave Modeling and Inversion of UWB Radar Data for Wave Propagation in Cylindrical Objects. *Remote Sens.* **2021**, *13*, 2370.
https://doi.org/10.3390/rs13122370

**AMA Style**

Gao L, Dachena C, Wu K, Fedeli A, Pastorino M, Randazzo A, Wu X, Lambot S.
Full-Wave Modeling and Inversion of UWB Radar Data for Wave Propagation in Cylindrical Objects. *Remote Sensing*. 2021; 13(12):2370.
https://doi.org/10.3390/rs13122370

**Chicago/Turabian Style**

Gao, Lan, Chiara Dachena, Kaijun Wu, Alessandro Fedeli, Matteo Pastorino, Andrea Randazzo, Xiaoping Wu, and Sébastien Lambot.
2021. "Full-Wave Modeling and Inversion of UWB Radar Data for Wave Propagation in Cylindrical Objects" *Remote Sensing* 13, no. 12: 2370.
https://doi.org/10.3390/rs13122370