# Motion Error Estimation and Compensation of Airborne Array Flexible SAR Based on Multi-Channel Interferometric Phase

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## Abstract

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## 1. Introduction

## 2. Theory

#### 2.1. 3D Imaging Model and Motion Compensation Theory of Airborne Array Flexible SAR

#### 2.2. Central Incidence Angle for Airborne Array Flexible SAR

#### 2.3. Motion Error Calculation of Airborne Array Flexible SAR

## 3. Method

- The rigid channel is imaged using traditional methods and calibrated by the method introduced in Section 2.2 in combination with the information from calibration points to obtain the central incidence angle under the scene;
- The flexible channel combines the distributed POS data for 2D imaging to obtain the tracking data before operation and compensation;
- Following the principle of flexible motion error estimation introduced in Section 2.3, the flexible motion error is estimated by combining the interferogram;
- According to the motion compensation principle introduced in Section 2.1, the flexible motion error is added to the original data before motion compensation;
- Two-dimensional imaging and interference processing are reperformed;
- Steps 2 to 6 are repeated until the coherence coefficient of the interferogram is greater than 0.9;
- Two-dimensional images with good a focusing effect, and multi-channel interferograms with improved interference performance, are obtained;
- The 3D result map of multichannel reconstruction is obtained through multi-channel registration, amplitude and phase correction, sparse reconstruction, and other steps.

Algorithm 1 Iterative solution to array flexible SAR motion error. |

Input: Flexible channel data ${s}_{1}$ and rigid channel data ${s}_{2}$.Initialization: Interference results ${s}_{3}={s}_{1}.*{s}_{2}$, ${s}_{3}$Step 1: Set 3 × 3 sliding window. Use the sliding window to obtain the interferogram result ${s}_{3}$;Step 2: Calculate according to the coherence coefficient calculation formula $r=\frac{\left|dat{a}_{3}\right|}{\sqrt{dat{a}_{1}^{2}\xb7*dat{a}_{2}^{2}}}$while the coherence coefficient r > 0.9;Step 3: Calculate the interference phase $\delta \varphi $ and the correlation coefficient ${r}_{1}$ of data3;Step 4: The maximum row m of the average value of the recorded azimuth correlation coefficient is the starting point;Step 5: Calculate the maximum deviation of the correlation coefficient $\delta \varphi $ between each azimuth and row m;Step 6: Calculate the distance to $\varphi $ points corresponding to one wavelength;Step 7: Calculate the offset according to Section 2.3;Step 8: Calculate the deviation in distance and elevation directions in combination with the lower view angle calculated in Section 2.2;Step 9: Add the deviation back to the data before operation and compensation, and re-image;Step 10: Calculate the correlation coefficient r.endOutput: Output the current compensation data ${s}_{e}$. |

Algorithm 2 Global optimization combined with least squares for the flexible baseline. |

Input: Flexible channel data ${s}_{1}$ and rigid channel data ${s}_{2}$, POS information ${y}_{ref}$ and ${z}_{ref}$, point target information height h, and geodetic coordinate y. Initialization: Interference results ${s}_{3}={s}_{1}.*{s}_{2}$, ${s}_{3}$ and phase unwrapping results $\mathsf{\Delta}\varphi $.Construct initial solutionStep 1: Calculate $B=\sqrt{{({y}_{ref}(1,n)-{y}_{ref}\left(1\right))}^{2}+{({z}_{ref}(1,n)-{z}_{ref}\left(1\right))}^{2}}$ Calculate $\theta =atan(y/(H-h\left)\right)$ Let $A=zeros\left(size\right(y,2),3)$ $A(:,1)=sin\theta $ $A(:,2)=-cos\theta $ $A(:,3)=\frac{\lambda}{4*\pi *ones\left(size\left(y,2\right),1\right)}$ Step 2: Find the least squares solution $B=\frac{{A}^{\prime}*A}{\left({A}^{\prime}*\left(\frac{\lambda *\mathsf{\Delta}{\phi}^{\prime}}{4\pi}\right)\right)}$Calculate global optimal solutionStep 3: Calculate $\alpha =atan\left(\frac{{z}_{ref}(1,n)-{z}_{ref}\left(1\right)}{{y}_{ref}\left(1,n\right)-{y}_{ref}\left(1\right)}\right)$Step 4: Obtain the central incidence angle from the Equation (10);Step 5: According to result, construct the minimization function from the Equation (9);Step 6: Perform global search until function F = 0.Output: Output $\alpha $, B and $\varphi $. |

## 4. System

## 5. Experimental Results and Analysis

#### 5.1. Simulation Experiments

#### 5.2. Real Data Experiments

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SAR | Synthetic Aperture Radar |

3D | Three-dimensional |

DEM | Digital Elevation Model |

ARTINO | Airborne Radar for Three-dimensional Imaging and Nadir Observation |

SRTM | Shuttle Radar Topography Mission |

AIRCAS | Aerospace Information Research Institute, Chinese Academy of Sciences |

POS | Position and orientation system |

LFM | Linear frequency modulation |

2D | Two-dimensional |

ISLR | integral sidelobe ratio |

PSLR | peak sidelobe ratio |

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**Figure 6.**The change in the POS trajectory. (

**a**) The POS trajectory; (

**b**) the jitter trajectory of the flexible antenna.

**Figure 9.**The comparison of interference fringe results. (

**a**) The comparison method; (

**b**) the proposed method.

**Figure 10.**The comparison results. (

**a**) The comparison method’s coherence coefficient diagram; (

**b**) the proposed method’s coherence coefficient diagram; (

**c**) the comparison method’s coherence coefficient statistical histogram; (

**d**) the proposed method’s coherence coefficient statistical histogram.

**Figure 11.**The comparison of interference fringe results. (

**a**) The comparison method; (

**b**) the proposed method.

**Figure 12.**The comparison results. (

**a**) The comparison method coherence coefficient diagram; (

**b**) the proposed method coherence coefficient diagram; (

**c**) the comparison method coherence coefficient statistical histogram; (

**d**) the proposed method coherence coefficient statistical histogram.

**Figure 13.**The comparison of 2D results of point targets. (

**a**) The comparison method; (

**b**) the proposed method.

**Figure 15.**The comparison of interference fringe results. (

**a**) The main POS imaging; (

**b**) the left POS imaging.

**Figure 16.**The comparison results. (

**a**) The main POS imaging coherence coefficient diagram; (

**b**) the left POS imaging coherence coefficient diagram; (

**c**) the main POS imaging coherence coefficient statistical histogram; (

**d**) the left POS imaging coherence coefficient statistical histogram.

**Figure 17.**The comparison of interference fringe results. (

**a**) The comparison method; (

**b**) the proposed method.

**Figure 18.**The comparison results. (

**a**) The comparison method coherence coefficient diagram; (

**b**) the proposed method coherence coefficient diagram; (

**c**) the comparison method coherence coefficient statistical histogram; (

**d**) the proposed method coherence coefficient statistical histogram.

**Figure 19.**The comparison of interference fringe results. (

**a**) The main POS imaging; (

**b**) The left POS imaging.

**Figure 20.**The comparison results. (

**a**) The comparison method coherence coefficient diagram; (

**b**) the proposed method coherence coefficient diagram; (

**c**) the comparison method coherence coefficient statistical histogram; (

**d**) the proposed method coherence coefficient statistical histogram.

**Figure 21.**The mean values of coherence coefficient. (

**a**) channel 1 and channel 3; (

**b**) channel 1 and channel 2.

**Figure 22.**The comparison of 2D results of point targets. (

**a**) The main POS imaging; (

**b**) the left POS imaging; (

**c**) the comparison method; (

**d**) the proposed method.

**Figure 23.**The comparison of 3D reconstruction point cloud results. (

**a**) The comparison method; (

**b**) the proposed method.

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

Flight height | H | 5 km |

Center Frequency | ${f}_{c}$ | 15 GHz |

Bandwidth | ${B}_{w}$ | 500 MHz |

Rigid baseline length | ${B}_{r}$ | 0.8 m |

Flexible baseline length | ${B}_{f}$ | 1.6 m |

Flight velocity | v | 80 m/s |

Horizontal inclination of baseline | $\alpha $ | ${0}^{\circ}$ |

Parameter | Main POS | Sub POS |
---|---|---|

Height measurement accuracy | 5 cm | 10 cm |

Gyroscopic drift | 0.01°/h | 0.1°/h |

Accelerometer bias | 10 $\mathsf{\mu}$g | 20 $\mathsf{\mu}$g |

Method | Range | Azimuth | ||||
---|---|---|---|---|---|---|

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

method 1 ^{1} | 0.13 | −8.15 | −10.02 | 0.17 | −22.74 | −6.01 |

method 2 ^{2} | 0.13 | −8.26 | −10.67 | 0.17 | −22.49 | −10.23 |

^{1}This is the comparison method [18].

^{2}This is the proposed method.

Method | Range | Azimuth | ||||
---|---|---|---|---|---|---|

Resolution (m) | PSLR (dB) | ISLR (dB) | Resolution (m) | PSLR (dB) | ISLR (dB) | |

method 1 ^{1} | 0.13 | −8.73 | −10.14 | 0.17 | −7.59 | −4.48 |

method 2 ^{2} | 0.13 | −8.98 | −10.17 | 0.17 | −3.61 | −1.89 |

method 3 ^{3} | 0.13 | −8.71 | −10.12 | 0.17 | −10.21 | −6.06 |

method 4 ^{4} | 0.13 | −8.93 | −10.13 | 0.17 | −13.45 | −8.09 |

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## Share and Cite

**MDPI and ACS Style**

Yang, L.; Zhang, F.; Sun, Y.; Chen, L.; Li, Z.; Wang, D.
Motion Error Estimation and Compensation of Airborne Array Flexible SAR Based on Multi-Channel Interferometric Phase. *Remote Sens.* **2023**, *15*, 680.
https://doi.org/10.3390/rs15030680

**AMA Style**

Yang L, Zhang F, Sun Y, Chen L, Li Z, Wang D.
Motion Error Estimation and Compensation of Airborne Array Flexible SAR Based on Multi-Channel Interferometric Phase. *Remote Sensing*. 2023; 15(3):680.
https://doi.org/10.3390/rs15030680

**Chicago/Turabian Style**

Yang, Ling, Fubo Zhang, Yihong Sun, Longyong Chen, Zhenhua Li, and Dawei Wang.
2023. "Motion Error Estimation and Compensation of Airborne Array Flexible SAR Based on Multi-Channel Interferometric Phase" *Remote Sensing* 15, no. 3: 680.
https://doi.org/10.3390/rs15030680