# Repositioning Error Compensation in Discontinuous Ground-Based SAR Monitoring

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## Abstract

**:**

## 1. Introduction

## 2. Repositioning Error in D-GBSAR

#### 2.1. Repositioning Error Modeling

_{1}and R

_{2}are the ranges between the target and the radar during two campaigns, respectively. However, since $\mathsf{\Delta}\phi $ represents the principal value of the interferometric phase, the correct deformation estimation requires the reconstruction of the full phase value by calculating the phase ambiguity. In discontinuous mode, due to the instability of the monitoring station or lack of accurate positioning devices, the position offset of the instrument could occur during repositioning. An additional range variation is caused by the repositioning error, and $\mathsf{\Delta}R$ cannot accurately correspond to the target deformation [12].

_{p}, y

_{p}, z

_{p}) and (x, y, z) respectively, R can be expressed as:

#### 2.2. Simulation Comparisons

## 3. Methodology

_{1}is the linear coefficient, B

_{2}is an offset that can appear in the interferometric phase between two different campaigns. Together with Equation (6), for a PS with no deformation, its unwrapped phase can be modeled as:

_{1}, A

_{2}, A

_{3}, B

_{1}, B

_{2}] are the estimated parameters, $({x}_{p},{y}_{p},{z}_{p})$ is the 3D coordinate of the PS, R

_{p}represents the slant range, and e is the unmodeled error phase. By estimating the polynomial parameters based on the LS method, the error phase can be compensated precisely under discontinuous mode. The equation set can be expressed as:

## 4. Experimental Results

#### 4.1. First Case: Urban Area

#### 4.1.1. Experiment Information

_{1}and CR

_{2}. The horizontal axis represents the moments when radar images were acquired. The red lines mark the moments when the radar was moved, and the interferometric phases of both CRs vary significantly. In contrast, during the period of stable monitoring, their interferometric phases are close to zero. Every time the MIMO radar was moved, multiple radar images were acquired and they were coherently added to obtain a composite image. Seven composite images were acquired in this experiment, denoted as M

_{1}~M

_{7}.

#### 4.1.2. Model Validation

_{1}~IM

_{6}. Since the experimental scenario is flat and stable, the interferometric phases mainly contain repositioning error and atmospheric components.

_{1}~IM

_{6}. Due to the millimeter-level positioning error of the total station, there are slight differences between these curves, but it could still prove that the proposed method can model the repositioning error precisely.

#### 4.1.3. Error Compensation

_{i}as the master image and M

_{i}

_{+1}as the slave image (i = 1, 2, …, 6), 6 coherence maps are acquired, the average coherence map is shown in Figure 10a. As mentioned in Section 3, the coherence method is used to select PSs in discontinuous mode. Set the coherence threshold as 0.98 and the SNR threshold as 30 dB, 51,387 PSs are obtained, which are shown in Figure 10b. The pixels corresponding to both CRs and most building structures are selected as PSs.

_{1}as an example for phase analysis, the phase unwrapping is performed based on the Minimum Cost Flow method [23]. The unwrapped phase map of PSs is shown in Figure 11a. The interferometric phases of most PSs deviate from 0 rad due to the repositioning error. The topography of experimental scenario is mainly flat, so the linear range model of the atmospheric phase component in Equation (13) is utilized. Since that the radar was not moved in the height direction, the phase model in Equation (14) can be simplified as:

#### 4.2. Second Case: Mountainous Area

#### 4.2.1. Experiment Information

#### 4.2.2. Experimental Results

_{A}, SD

_{B}, SD

_{C}, as shown in Figure 15a,c,e. Obviously, the phase scatter could be better simulated with Model C. The compensation results are CIM

_{A}, CIM

_{B}, CIM

_{C}, as shown in Figure 15b,d,f. Figure 15g shows the phase distribution curves of the residual phases under different models. Compared with the improved method, there are a large number of PSs deviate from 0 rad with Model A and B. According to the on-site investigation, there were no obvious deformations for the high and steep rocky slope during the experiment. We could conclude that the best compensation performance could be achieved with the improved multi-parameter model. However, the red circle in Figure 15f marks a deformation area, where the negative deformations denote the direction away from the radar.

## 5. Discussion

- Case 1: An equivalent experiment of discontinuous mode is made by slightly moving the radar. The error compensation results are shown in Figure 11. Two corner reflectors are set to validate the effectiveness of the proposed model, as shown in Figure 12. The standard deviation of residual phases for both CRs is about 0.1 rad, which corresponds to 0.14 mm in Ku band, and can satisfy the requirement of sub-millimeter measurement.
- Case 2: The improved method is applied to process the measured data of a mountainous area. Comparisons of the first-order, second-order and proposed model are shown in Figure 15. There are a large number of PSs deviate from 0 rad with conventional models, and the best compensation performance could be achieved with the improved multi-parameter model. Figure 15f shows the compensated curves, which proves that the error phase components can be better compensated with the improved method.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Simulation of repositioning errors: (

**a**) Topography; (

**b**) ${\epsilon}_{x}=1\mathrm{mm},{\epsilon}_{y}=0,{\epsilon}_{z}=0$; (

**c**) ${\epsilon}_{x}=0,{\epsilon}_{y}=1\mathrm{mm},{\epsilon}_{z}=0$; (

**d**) ${\epsilon}_{x}=0,{\epsilon}_{y}=0,{\epsilon}_{z}=1\mathrm{mm}$.

**Figure 4.**Simulation of repositioning errors with three different kinds of topographies (${\epsilon}_{x}={\epsilon}_{y}={\epsilon}_{z}=1\mathrm{mm}$): (

**a**) Topographies; (

**b**) Elevations; (

**c**) Repositioning error phase maps.

**Figure 5.**Compensation results of repositioning error phase with three different models (${\epsilon}_{x}={\epsilon}_{y}={\epsilon}_{z}=1\mathrm{mm}$): (

**a**) $\phi ={a}_{0}+{a}_{1}\mathrm{sin}\theta $; (

**b**) $\phi ={a}_{0}+{a}_{1}\eta +{a}_{2}\xi +{a}_{3}{\xi}^{2}$; (

**c**) $\phi ={a}_{0}+{a}_{1}\frac{{x}_{p}}{{R}_{p}}+{a}_{2}\frac{{y}_{p}}{{R}_{p}}+{a}_{3}\frac{{z}_{p}}{{R}_{p}}$.

**Figure 9.**Comparison of theoretical value and experimental data from both two CRs: (

**a**) CR

_{1}; (

**b**) CR

_{2}.

**Figure 11.**Results of error phase compensation in IM

_{1}: (

**a**) Unwrapped phase map (Before compensation); (

**b**) Corresponding histograms; (

**c**) Compensated phase map; (

**d**) Phase distribution curve.

**Figure 14.**Results of PS selection: (

**a**) Interferometric phase map; (

**b**) Coherence map; (

**c**) Wrapped phases of PSs; (

**d**) Unwrapped phases of PSs.

**Figure 15.**Phase scatter diagrams: (

**a**) SD

_{A}; (

**c**) SD

_{B}; (

**e**) SD

_{C}. Compensation results: (

**b**) CIM

_{A}; (

**d**) CIM

_{B}; (

**f**) CIM

_{C}. Comparison of different models: (

**g**) Phase distribution curves of residual phases.

Type | Residual Phases | |||||
---|---|---|---|---|---|---|

$\mathit{\phi}={\mathit{a}}_{0}+{\mathit{a}}_{1}\mathbf{sin}\mathit{\theta}$ | $\mathit{\phi}={\mathit{a}}_{0}+{\mathit{a}}_{1}\mathit{\eta}+{\mathit{a}}_{2}\mathit{\xi}+{\mathit{a}}_{3}{\mathit{\xi}}^{2}$ | $\mathit{\phi}={\mathit{a}}_{0}+{\mathit{a}}_{1}\frac{{\mathit{x}}_{\mathit{p}}}{{\mathit{R}}_{\mathit{p}}}+{\mathit{a}}_{2}\frac{{\mathit{y}}_{\mathit{p}}}{{\mathit{R}}_{\mathit{p}}}+{\mathit{a}}_{3}\frac{{\mathit{z}}_{\mathit{p}}}{{\mathit{R}}_{\mathit{p}}}$ | ||||

Max/mrad | RMSE/mrad | Max/mrad | RMSE/mrad | Max/mrad | RMSE/mrad | |

Flat | 131.11 | 59.76 | 23.31 | 8.06 | 0.07 | 0.01 |

Slope | 164.39 | 74.94 | 23.91 | 8.08 | 0.07 | 0.01 |

Hillside | 211.09 | 86.12 | 48.49 | 10.79 | 0.06 | 0.01 |

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

Carrier Frequency | 16.2 GHz |

Wavelength | 18.5 mm |

Synthetic Aperture | 1.18 m |

Bandwidth | 1 GHz |

Range Resolution | 0.15 m |

Azimuth Angle Resolution | 7.81 mrad |

Serial | Offsets | |
---|---|---|

${\mathit{\epsilon}}_{\mathit{x}}\left(\mathbf{mm}\right)$ | ${\mathit{\epsilon}}_{\mathit{y}}\left(\mathbf{mm}\right)$ | |

No.1 | 0.11 | 0.32 |

No.2 | −6.28 | 0.32 |

No.3 | −2.62 | −0.15 |

No.4 | 0.17 | −1.31 |

No.5 | −0.46 | 5.16 |

No.6 | −3.48 | 4.75 |

Serial | Residual Phases | |||
---|---|---|---|---|

CR_{1} | CR_{2} | |||

Max/rad | σ/rad | Max/rad | σ/rad | |

Before Compensation | 3.11 | 2.48 | 2.98 | 2.27 |

After Compensation | 0.11 | 0.05 | 0.17 | 0.09 |

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**MDPI and ACS Style**

Hu, C.; Zhu, J.; Deng, Y.; Tian, W.; Yin, P.
Repositioning Error Compensation in Discontinuous Ground-Based SAR Monitoring. *Remote Sens.* **2021**, *13*, 2461.
https://doi.org/10.3390/rs13132461

**AMA Style**

Hu C, Zhu J, Deng Y, Tian W, Yin P.
Repositioning Error Compensation in Discontinuous Ground-Based SAR Monitoring. *Remote Sensing*. 2021; 13(13):2461.
https://doi.org/10.3390/rs13132461

**Chicago/Turabian Style**

Hu, Cheng, Jiaxin Zhu, Yunkai Deng, Weiming Tian, and Peng Yin.
2021. "Repositioning Error Compensation in Discontinuous Ground-Based SAR Monitoring" *Remote Sensing* 13, no. 13: 2461.
https://doi.org/10.3390/rs13132461