# An Efficient Imaging Algorithm for GNSS-R Bi-Static SAR

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## Abstract

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## 1. Introduction

- The complex expression for range history: The range history is due to the motion of both the GNSS satellite and the receiving platform, which makes it extremely difficult to obtain a precise analytical solution to the stationary phase point of the Doppler phase.
- The translational variant Doppler history: Unlike mono-static SAR, the Doppler history of the echo signal of GNSS-R BSAR is translationally variant since the trajectories of the transmitter and the receiver are non-parallel and their velocities are also different. This means that imaging becomes a two-dimensional spatially varying filtering process.

## 2. Materials and Methods

#### 2.1. Modelling and Analysis

_{r}, and the transmitter is mounted on the GNSS satellite moving along its Kepler orbit. The target is positioned on the ground plane (x-y plane). The bi-static angle θ is defined as the angle between the target-to-satellite baseline and the target-to-receiver baseline. The receiver works in the broadside mode, and the beam centre of the receiver antenna falls at the y axis at t = 0. Here, t represents the azimuth slow time. The beam of the transmitter antenna covers most area of the Earth’s surface, and the antenna pattern is considered as constant in the imaging duration. For convenience of expression, in the following part of this paper, the slant range between the target and the satellite is referred to as the transmitter range, and the slant range between the target and the receiver is referred to as the receiver range.

_{0}. Then, the echo data of target k after synchronization and SAR data formation could be expressed as:

_{k}(t) is the total slant range, and R

_{k}(t) = R

_{trk}(t) + R

_{tsk}(t). R

_{trk}(t) is the receiver range for target k, and R

_{tsk}(t) is the transmitter range for target k.

_{0}has the same minimum transmitter range as target k. Then:

_{0}is the equivalent velocity and φ is the equivalent squint angle.

_{r}on the ground plane is:

_{ts}on the ground plane, which is equal to the cosine of the elevation angle α of the satellite. When φ increases, $\frac{{V}_{s}^{2}{cos}^{2}\phi}{\lambda {R}_{ts}^{2}}$ decreases, and cosα increases. When φ increases to φ

_{max}, cosα has the maximum value of one. Therefore:

_{ts0}is the height of the satellite relative to the ground when φ = 0 and V

_{s}is the velocity of the satellite.

_{0}becomes:

_{tsc}(t) is the transmitter range for target c, which is positioned at the centre of the scene.

_{shiftk}is the position shift of the signal in the echo window. The Doppler frequency f

_{d}and its derivative can be expressed as a function of the parameters of the satellite and the target.

_{a}is the Doppler FM rate for target k caused by the motion of the receiver. The phase φ

_{1}is omitted because it has no impact on the derivation.

- The spatially varying Doppler centroid: It is formed by the linear component of the residual transmitter range and represented by the spatially varying delay time t
_{dk}. This means the equivalent squint angle is also spatially varying. - The translational variant Doppler FM rate: It is formed by the constant component of the residual transmitter range and represented by the shifting factor R
_{shiftk}. The shifting factor R_{shiftk}indicates that the echo data for targets with the same minimum receiver range (thus the same Doppler FM rate) will not appear in the same range cell. In other words, the signals that appear in the same range cell have a different Doppler FM rate.

- The scene space in which the echo data are received.
- The echo space in which the echo data are stored and processed.

_{shiftk}, for targets with the same minimum receiver range (thus the same Doppler FM rates since they are caused only by the motion of the receiver after bulk compensation for transmitter range) in the scene space, their echo data will not appear in the same range cell in the echo space. In other words, for the echo data in the same range cell, their Doppler FM rates are different. As an example, target k has the same minimum receiver range as target k

_{0}and thus has the same Doppler FM rate as target k

_{0}. However, the echo data of target k appear in the same range cell as target k

_{1}rather than target k

_{0}in the echo space, since ${R}_{k}\left({t}_{0}\right)+{R}_{\mathrm{shiftk}}={R}_{{k}_{0}}\left(0\right)+{R}_{\mathrm{shiftk}}={R}_{{k}_{1}}\left(0\right)$.

_{shiftk}.

_{shiftk}is caused by the difference in transmitter range for different targets at t = 0. This term is positively related to the bi-static angle θ. Since θ varies widely in different receiving geometry, this term also varies widely and in general is also the dominating term of R

_{shiftk}. This term is a linear function of the azimuth time. The second term is caused by the variation of the Doppler frequency along the direction of the receiver velocity, and this is a quadratic function of the azimuth time. The third term is caused by the variation of the Doppler frequency along the cross direction of the receiver velocity, and it is also a linear function of the azimuth time, but the slope varies with y

_{0}.

_{0}and y

_{0}, which are negligible when the imaging scene is not too large. Ignoring the second term, we have:

#### 2.2. Imaging Methods

#### 2.2.1. Bulk Compensation and Range Compression

#### 2.2.2. Residual RCMC

_{d}.

- The scaling operation is not applied to the range frequency axis, but to the range time axis, and thus in some sense, it is more appropriate to name the modified algorithm the range scaling algorithm rather than the frequency scaling algorithm.
- The scaling of the range time needs four multiplications rather than three as in the frequency scaling algorithm.
- The scaling factors should all be adapted to the new application.

_{0}(note that the reference point is not always located at the y axis, and target k

_{0}has a non-zero equivalent squint angle), the echo data of the reference point after bulk compensation for transmitter range have the following form:

_{trc}. The four scaling filter functions corresponding to the first step are:

_{fs}is not fixed, but it should make sure that the spectrum shift does not surpass the sampling rate. The purpose of the next three filter functions is similar to their counterparts in the frequency scaling algorithm. For a detailed description of their functions, please refer to [34]. The final bulk RCMC filter function is given by:

#### 2.2.3. Azimuth Phase Perturbation

- The constant component πat
_{0}^{3}: It changes the phase of the signal according to the position of the target, which can be ignored when only the amplitude of the image is concerned. Here:$$a=\frac{{V}_{r}^{2}\left(\frac{\partial {R}_{ts}}{\partial x}{V}_{r}-\lambda \frac{\partial {f}_{d}}{\partial y}{y}_{0}\right)\frac{\partial {R}_{tr}}{\partial R}}{3\lambda {R}_{trk}{}^{2}},$$$${t}_{0}=\frac{{x}_{0}}{{V}_{r}}$$ - The first order component 3πat
_{0}^{2}(t − t_{0}): It adds a small spatially varying Doppler shift to the signal. This shift is a quadratic function of the azimuth time t, which can be incorporated into the Doppler shift caused by the residual transmitter range, i.e.,$$\mathsf{\Delta}{f}_{dk}=\frac{\partial {f}_{d}}{\partial x}{x}_{0}+\frac{\partial {f}_{d}}{\partial y}{y}_{0}-\frac{\left(\frac{\partial {R}_{ts}}{\partial x}{V}_{r}-\lambda \frac{\partial {f}_{d}}{\partial y}{y}_{0}\right)\frac{\partial {R}_{tr}}{\partial R}}{2\lambda {R}_{trk}{}^{2}}{x}_{0}^{2}$$ - The second order compound 3πat
_{0}(t − t_{0})^{2}: It equalizes the Doppler FM rate along the range cell. - The third order compound πa(t − t
_{0})^{3}: It is a cubic phase modulation, which is the same for all targets. It is far smaller than the phase modulation caused by the receiver motion, which can be ignored during the derivation of the azimuth stationary phase point of the signal.

#### 2.2.4. Azimuth Compensation

_{0}, the expression of the echo data of target k

_{0}after phase perturbation is:

_{0}in the range-Doppler domain. Ignoring the cubic phase modulation caused by the phase perturbation during the derivation of the stationary phase point, we have:

## 3. Results

#### 3.1. Simulation Results

#### 3.2. Experimental Results

## 4. Discussion

_{a}and N

_{r}are the number of range and azimuth samples, respectively. The BPA entails two range FTs and one complex multiplication in the range compression step, and its computational load is ${N}_{a}{N}_{r}{\mathrm{log}}_{2}{N}_{r}+{N}_{a}{N}_{r}$. The computational load in the back-projection step can be expressed as ${N}_{kel}{N}_{x}{N}_{y}{N}_{a}$, where N

_{x}and N

_{y}are the numbers of samples of the imaging scene, and N

_{kel}is the length of the interpolation kernel. Thus, the total computational load for the BPA can be represented by ${N}_{a}{N}_{r}{\mathrm{log}}_{2}{N}_{r}+{N}_{a}{N}_{r}+{N}_{kel}{N}_{x}{N}_{y}{N}_{a}$. Clearly, when the size of the imaging scene increases, the computational load will increase rapidly for the BPA.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Illustration of the range cell shift effect. (

**a**) is the scene space, and (

**b**) is the signal space. Five targets and their echoes are shown in the figure; the curvature of the echo is proportional to the Doppler Frequency Modulation (FM) rate of the target.

**Figure 6.**Layout of the point targets in the simulation and the imaging results. (

**a**) is the layout of the point targets, and (

**b**) is the imaging results.

**Figure 7.**The ranging code sequence of PRN30 and its auto-correlation result. (

**a**) is the ranging code sequence, and (

**b**) is the auto-correlation result.

**Figure 8.**Evaluated results for the chosen targets: each subfigure has four parts, including the azimuth profile (upper left), range profile (left bottom), 3D contour (upper right), and 2D contour (right bottom) of the chosen target. (

**a**–

**e**) are the evaluated results for target 1, 7, 13, 19, 25, respectively.

**Figure 9.**The experimental setups. (

**a**) is the optical image of the experimental area. (

**b**) is the reflect channel antenna and the targets. (

**c**) is the direct channel antenna. (

**d**) is the receiver.

**Figure 12.**Comparison between the optical image and the resultant radar image. (

**a**) is the optical image, and (

**b**) is the radar image.

**Figure 13.**Imaging results using the proposed algorithm and the Back Projection Algorithm (BPA). The size of the imaging area is 100 m × 500 m, and the integration time is 300 s. (

**a**) is the imaging results acquired using the proposed algorithm, and (

**b**) is the imaging results acquired using the BPA.

**Figure 14.**The azimuth profile of the resultant radar image of the gymnasium. The blue line represents the result using the proposed algorithm, while the red line represents the result by the BPA. (

**a**) is the azimuth profile, and (

**b**) is the range profile.

x | y | z | |
---|---|---|---|

Satellite position (km) | −5908 | −12,714 | 16,112 |

Satellite velocity (m/s) | −2475 | −1198 | −1310 |

Receiver position (m) | 0 | 0 | 6000 |

Receiver velocity (m/s) | 60 | 0 | 0 |

Satellite/Signal | GPS L5 |
---|---|

Carrier frequency | 1176.45 MHz |

Signal bandwidth | 10.23 MHz |

Sample rate | 40 MHz |

Integration time | 10 s |

Target | Range | Azimuth | ||||
---|---|---|---|---|---|---|

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

1 | 17.1 | −32.92 | −14.53 | 9.9 | −12.97 | −10.50 |

7 | 16.7 | −34.76 | −13.47 | 9.7 | −13.24 | −10.83 |

13 | 16.5 | −34.68 | −13.45 | 9.6 | −13.27 | −10.90 |

19 | 16.6 | −35.04 | −13.47 | 9.8 | −13.21 | −10.89 |

25 | 16.9 | −33.85 | −13.65 | 10.1 | −12.90 | −10.59 |

Theoretical | 16.5 | −35.00 | −13.45 | 9.6 | −13.26 | −10.90 |

Satellite used | GPS PRN30 |

Signal used | L5 C/A code |

Satellite elevation | 55.7° |

Satellite azimuth | 268.6° |

Carrier frequency | 1176.45 MHz |

Sample rate | 62 MHz |

Signal bandwidth | 10 MHz |

Equivalent PRF | 1000 Hz |

Doppler bandwidth | 151 Hz |

x | y | z | ||
---|---|---|---|---|

Satellite position (km) | start | −11,822 | −300 | 17,341 |

middle | −11,799 | − 735 | 17,341 | |

end | −11,778 | −1172 | 17,332 | |

Satellite velocity (m/s) | start | 173 | −3001 | −2 |

middle | 137 | −2962 | −31 | |

end | 129 | −2998 | −101 |

N_{a} | N_{r} | N_{x} | N_{y} | N_{kel} |
---|---|---|---|---|

3e5 | 1024 | 500 | 500 | 8 |

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## Share and Cite

**MDPI and ACS Style**

Zhou, X.-k.; Chen, J.; Wang, P.-b.; Zeng, H.-c.; Fang, Y.; Men, Z.-r.; Liu, W.
An Efficient Imaging Algorithm for GNSS-R Bi-Static SAR. *Remote Sens.* **2019**, *11*, 2945.
https://doi.org/10.3390/rs11242945

**AMA Style**

Zhou X-k, Chen J, Wang P-b, Zeng H-c, Fang Y, Men Z-r, Liu W.
An Efficient Imaging Algorithm for GNSS-R Bi-Static SAR. *Remote Sensing*. 2019; 11(24):2945.
https://doi.org/10.3390/rs11242945

**Chicago/Turabian Style**

Zhou, Xin-kai, Jie Chen, Peng-bo Wang, Hong-cheng Zeng, Yue Fang, Zhi-rong Men, and Wei Liu.
2019. "An Efficient Imaging Algorithm for GNSS-R Bi-Static SAR" *Remote Sensing* 11, no. 24: 2945.
https://doi.org/10.3390/rs11242945