# Backscattering Analysis at ATR on Rough Surfaces by Ground-Based Polarimetric Radar Using Coherent Decomposition

## Abstract

**:**

## 1. Introduction

#### 1.1. Problem and Tasks

- Select an object decomposition theorem for feature space design and test an acceptable scattering mechanism after calibration
- Conduct data calibration considering the effect of rough ground surface on object recognition
- Explore supervised learning as an ATR tool for polarimetric radar.

#### 1.2. Structure of the Article

## 2. Theoretical Part

#### 2.1. Assumptions

- Kennaugh matrix dichotomy (Kennaugh–Huynen scattering matrix)
- Decomposition of the covariance matrix (Freeman and Durden)
- Coherent scattering matrix decomposition (Pauli, Krogager, Cameron).

- The transmitted electromagnetic wave is a plane monochromatic wave with constant frequency, amplitude, and initial phase in time.
- In the propagation of a polarized wave, there are additive and multiplicative interferences.
- Polarimetric radar generates a signal with strict linear (vertical and horizontal) and strict circular (right and left) polarization.
- The classification procedure involves obtaining labeled data on detected objects; hence it is a supervised learning task.

#### 2.2. Polarized Wave Model

- The Jones calculus. This is a mathematical description of a fully polarized wave in which the Jones vectors and linear elements of the Jones matrix (Equation (1)) determine the polarization:$${E}_{R}=J{E}_{T}:\left|\begin{array}{c}{E}_{x}^{R}{e}^{j{\varphi}_{x}^{R}}\\ {E}_{y}^{R}{e}^{j{\varphi}_{y}^{R}}\end{array}\right|=\left(\begin{array}{cc}{J}_{xx}{e}^{j{\varphi}_{xx}}& {J}_{yx}{e}^{j{\varphi}_{yx}}\\ {J}_{xy}{e}^{j{\varphi}_{xy}}& {J}_{yy}{e}^{j{\varphi}_{yy}}\end{array}\right)\times \left|\begin{array}{c}{E}_{x}^{T}{e}^{j{\varphi}_{x}^{T}}\\ {E}_{y}^{T}{e}^{j{\varphi}_{y}^{T}}\end{array}\right|$$
- Mueller calculus. A mathematical description of arbitrarily polarized scattering is given by the Stokes vector, which is expressed as follows:$${S}^{R}=M{S}^{T}:\left[\begin{array}{c}{s}_{0}^{R}\\ {s}_{1}^{R}\\ {s}_{2}^{R}\\ {s}_{3}^{R}\end{array}\right]=\left(\begin{array}{cccc}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}& {m}_{11}& {m}_{12}& {m}_{13}\\ {m}_{20}& {m}_{21}& {m}_{22}& {m}_{23}\\ {m}_{30}& {m}_{31}& {m}_{32}& {m}_{33}\end{array}\right)\times \left[\begin{array}{c}{s}_{0}^{T}\\ {s}_{1}^{T}\\ {s}_{2}^{T}\\ {s}_{3}^{T}\end{array}\right]$$

- The Mueller calculus has only a phenomenological interpretation and is not related to the electromagnetic theory, whereas the Jones calculus derives directly from this theory.
- The Jones calculus allows for the absolute phase, while the Mueller calculus does not consider the phase at all.
- The elements of the Jones matrix correspond to the radiation amplitude, while the elements of the Muller matrix are related to the scattering intensity.

## 3. Polarimetric Decomposition of Feature Space

#### 3.1. Pauli Decomposition

#### 3.2. Krogager Decomposition

#### 3.3. Cameron Decomposition

#### 3.4. Assessment of Feature Space for the Learning Algorithm

_{1}score is the harmonic mean value of precision and recall.

## 4. Data and Calibration

- Linear polarization (vertical and horizontal plane)
- Circular polarization (right and left rotation).

#### 4.1. Linear Polarization Data

#### 4.2. Circular Polarization Data

_{LP}= 6.8 for linear polarization and SNR

_{EP}= 17.4 for circular polarization, respectively. Thus, the signal attenuation of the wave with linear polarization significantly exceeds the circular polarization attenuation (more than 10 dB).

## 5. Supervised Learning for Polarimetric Recognition

#### 5.1. Modeling Polarimetric Recognition

#### 5.2. Comparison with Similar Methods

## 6. Results and Discussion

#### 6.1. Influence of Different Target Profiles and Weather Conditions

#### 6.2. Estimation of Binary Classifiers

## 7. Conclusions

## Funding

## Conflicts of Interest

## Appendix A. The Coefficients Chart Depending on the Profile (Truck)

**Figure A1.**Linear coefficient data $S={\left[{s}_{0},\hspace{0.33em}{s}_{1},\hspace{0.33em}{s}_{2},\hspace{0.33em}{s}_{3}\right]}^{T}$ for truck: profile angle changes from 0° to 165° (periodicity 15°).

**Figure A2.**Linear coefficient data $S={\left[{s}_{0},\hspace{0.33em}{s}_{1},\hspace{0.33em}{s}_{2},\hspace{0.33em}{s}_{3}\right]}^{T}$ for truck: profile angle changes from 180° to 345° (periodicity 15°).

## Appendix B. Confusion Matrix of the True Positive Rate of Classification

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**Figure 1.**Principle of obtaining a polarized target signature. E—vector of electric-field intensity.

**Figure 3.**Polarization diagram of the received signal relative to the transmitted signal unit for a rough surface and free meadow propagation (radial axis is a dimensionless variable; linear polarization; number of marks for each observation—64).

**Figure 5.**Unprocessed linear polarization target signature (truck, profile angle 45°): ${\mathrm{I}}_{\mathrm{L}1},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{L}1}$—horizontal transmission and vertical reception, respectively; ${\mathrm{I}}_{\mathrm{L}2},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{L}2}$ —vertical transmission and vertical reception, respectively; ${\mathrm{I}}_{\mathrm{L}3},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{L}3}$ —vertical transmission and horizontal reception, respectively; ${\mathrm{I}}_{\mathrm{L}4},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{L}4}$ —horizontal transmission and horizontal reception, respectively.

**Figure 6.**Target marks of the received signal (the second letter in the designation) relative to the transmitted signal unit (the first letter in the designation) in polar coordinates before and after calibration (linear polarization, car, profile angle 0°, number of marks for each observation—64).

**Figure 7.**Unprocessed circularly polarized target signature of circular polarization (truck, profile angle 45°): ${\mathrm{I}}_{\mathrm{C}1},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{C}1}$—left-hand circular transmission and right-hand circular reception, respectively; ${\mathrm{I}}_{\mathrm{C}2},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{C}2}$ —right-hand circular transmission and right-hand circular reception, respectively; ${\mathrm{I}}_{\mathrm{C}3},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{C}3}$ —right-hand transmission and left-hand circular reception, respectively; ${\mathrm{I}}_{\mathrm{C}4},\hspace{0.33em}{\mathrm{Q}}_{\mathrm{C}4}$ —left-hand circular transmission and left-hand circular reception, respectively.

**Figure 8.**Target marks of the received wave relative to the normalized transmitted signal in polar coordinates before and after calibration (circular polarization, car, profile angle 0°, number of marks for each observation—64): LL—left-hand circular transmission and left-hand circular reception; RR—right-hand circular transmission and right-hand circular reception; RL—right-hand circular transmission and left-hand circular reception; LR is left-circular transmission and right-hand circular reception.

**Figure 9.**Confusion matrix for linear polarization (algorithm—fine tree (68%); split criterion—Gini’s diversity index).

**Figure 10.**Confusion matrix for circular polarization (algorithm—logistic regression (54%); the hyper parameter option is disabled.

**Figure 11.**Scattering chart of a linear polarized wave from a trihedral reflector obtained at different time intervals during one day.

Parameter | Value |
---|---|

Distance to target | 150 m |

Sight angle | flat |

Angles of profile | 360° (periodicity 15°) |

Radar resolution (range) | 0.5 m |

Radar resolution (azimuth) | 1.8° |

Time registration | 10 ms |

Type of Polarization | F1 Score | MCC |
---|---|---|

Linear (Car) | 0.6282 | 0.3775 |

Linear (Truck) | 0.7206 | 0.3775 |

Circular (Car) | 0.2667 | 0.1260 |

Circular (Truck) | 0.6667 | 0.1260 |

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**MDPI and ACS Style**

Kvasnov, A.V.
Backscattering Analysis at ATR on Rough Surfaces by Ground-Based Polarimetric Radar Using Coherent Decomposition. *Sensors* **2023**, *23*, 3614.
https://doi.org/10.3390/s23073614

**AMA Style**

Kvasnov AV.
Backscattering Analysis at ATR on Rough Surfaces by Ground-Based Polarimetric Radar Using Coherent Decomposition. *Sensors*. 2023; 23(7):3614.
https://doi.org/10.3390/s23073614

**Chicago/Turabian Style**

Kvasnov, Anton V.
2023. "Backscattering Analysis at ATR on Rough Surfaces by Ground-Based Polarimetric Radar Using Coherent Decomposition" *Sensors* 23, no. 7: 3614.
https://doi.org/10.3390/s23073614