# Signal Processing and Waveform Re-Tracking for SAR Altimeters on High Mobility Platforms with Vertical Movement and Antenna Mis-Pointing

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## Abstract

**:**

^{−10}and the RMSE of τ obtained by the re-tracking method fitted by the proposed model is less than 0.2 m, which indicates the high applicability of the model and accuracy of the re-tracking method.

## 1. Introduction

## 2. Numerical Mean Echo Power Model for SAR Altimetry

#### 2.1. Altimetric Scenario

#### 2.2. Radar Signal Model

#### 2.3. Pulse Compression

#### 2.4. Along-Track FFT

#### 2.5. Backscattered Waveform Power Model

## 3. Semi-Analytical Echo Model of Airborne SAR Altimeters

#### 3.1. Convolution Model

#### 3.2. Semi-Analytical Expression of $FSIR\left({t}_{r},k\right)$

#### 3.3. Multilooking

Algorithm 1 Delay Compensation for Airborne SAR Altimetry |

Input: range r, Doppler frequency ${f}_{d}$, The reflected power ${P}_{I}\left(r,{f}_{d}\right)$, minimum height ${h}_{0}$- 1:
- resolution of the along-track $dy=vcos\mu N\mathrm{PRI}$;
- 2:
- $\mathrm{ylimit}=\lambda {f}_{d,max}{r}_{max}/2vcos\mu $;
- 3:
- ${y}_{n}=-\mathrm{ylimit}+htan\mu :dy:\mathrm{ylimit}+htan\mu $;
- 4:
**for**each i in r**do**- 5:
- ${f}_{n}$ according to (9);
- 6:
- $P\left(r,{f}_{n}\right)=Interpolation\left({f}_{d},P\left(r,{f}_{d}\right),{f}_{n}\right)$;
- 7:
**end for**- 8:
- ${r}_{n}=\sqrt{r\xb7r+{\left({y}_{n}\xb7{y}_{n}\right)}^{\mathrm{T}}}$
- 9:
**for**each n in ${y}_{n}$**do**- 10:
- $P\left({r}_{n},{f}_{n}\right)=Interpolation\left(r,P\left(r,{f}_{n}\right),{r}_{n}\right)$;
- 11:
**end for**
Output: The reflected power after delay compensation $P\left({r}_{n},{f}_{n}\right)$ |

#### 3.4. Analysis of the Semianalytical Echo Model

## 4. Re-Tracking Algorithm Based on Echo Model Fitting

## 5. Results

#### 5.1. Analysis of the FSIR Approximations

#### 5.2. Comparison with Numerical Waveforms

#### 5.3. Analysis of Re-Tracking Precision

## 6. Conclusions

- (1)
- An analytical expression that considers mis-pointing angles, circular antenna patterns, and vertical velocity is introduced for the echo model. The proposed model shows that the across-track mis-pointing angle affects the shape and the amplitude of the altimetric echo, whereas the along-track mis-pointing angle mainly affects the amplitude of the echo. When we consider $\xi \le {20}^{\circ}$ ($\xi <{\theta}_{3\mathrm{dB}}/2$), the expansion order $m=4$ is adequate to achieve the required level of error, which means that the global NQE is less than ${10}^{-10}$.
- (2)
- A novel delay compensation method based on sinc interpolation is proposed to obtain the multilook echo in order to optimally handle non-integer delays and maintain the signal frequency characteristics. Compared with the generalized model, the proposed semi-analytic model is closer to the fully numerical waveform at different values of mis-pointing angle ${\psi}_{ac}$ and ${\psi}_{al}$, especially when the roll angle ${\psi}_{ac}$ is large.
- (3)
- A five-parameter estimation strategy using the least squares procedure is proposed for SAR altimeters with vertical movement. The performance of the proposed model is evaluated using simulated data for typical airborne scenarios, analyzing the influence of different mis-pointing angles and flight path angle on parameter estimation. When the mis-pointing angles are within 10 degrees, the RMSE of $\tau $ obtained by the re-tracking method fitted by the proposed model is less than 0.2 m, and increases more slowly compared to the one fitted by the generalized model.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Delay–Doppler mapping. Each delay–Doppler bin is associated with two delay–Doppler cells on the surface.

**Figure 3.**Effect of the flight path angle on the DDM: (

**a**) $\mu ={0}^{\mathrm{o}}$; (

**b**) $\mu ={6}^{\mathrm{o}}$; (

**c**) $\mu ={12}^{\mathrm{o}}$; (

**d**) $\mu ={18}^{\mathrm{o}}$ (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, and ${\psi}_{ac}={0}^{\mathrm{o}}$).

**Figure 4.**Effect of flight path angle on (

**a**) the multilook echoes and (

**b**) the normalized multilook echoes (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, and ${\psi}_{al}={\psi}_{ac}={0}^{\mathrm{o}}$).

**Figure 5.**Antenna gain with different mis-pointing angles: (

**a**) $\xi ={0}^{\mathrm{o}},\widehat{\varphi}={0}^{\mathrm{o}}$; (

**b**) $\xi ={0}^{\mathrm{o}},\widehat{\varphi}={0}^{\mathrm{o}}$; (

**c**) $\xi ={20}^{\mathrm{o}},\widehat{\varphi}={0}^{\mathrm{o}}$; (

**d**) $\xi ={20}^{\mathrm{o}},\widehat{\varphi}={90}^{\mathrm{o}}$.

**Figure 6.**Effect of ${\psi}_{al}$ on (

**a**) the multilook echoes and (

**b**) the normalized multilook echoes (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, and $\mu ={\psi}_{al}={0}^{\mathrm{o}}$).

**Figure 7.**Effect of ${\psi}_{ac}$ on (

**a**) the multilook echoes and (

**b**) the normalized multilook echoes (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, and $\mu ={\psi}_{ac}={0}^{\mathrm{o}}$).

**Figure 9.**Overallerror versus m for different ${\psi}_{ac}$, showing the global NQE (continuous line) and NQE of echo maximum (crossed line) for ${\psi}_{ac}={6}^{\mathrm{o}}$ (in red), ${\psi}_{ac}={12}^{\mathrm{o}}$ (in green), and ${\psi}_{ac}={18}^{\mathrm{o}}$ (in blue).

**Figure 10.**Overall error versus ${\psi}_{ac}$ and ${\psi}_{al}$ when $m=4$ (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, and $\mu ={0}^{\mathrm{o}}$).

**Figure 11.**Comparison of multilooked power waveforms in typical airborne scenarios: (

**a**) $\left(\mu ,{\psi}_{ac},{\psi}_{al}\right)=\left(0,0,0\right)deg$ and (

**b**) $\left(\mu ,{\psi}_{ac},{\psi}_{al}\right)=\left(18,10,18\right)deg$.

**Figure 12.**Comparison of multilooked power waveforms in typical airborne scenarios: (

**a**) $\left(\mu ,{\psi}_{ac},{\psi}_{al}\right)=\left(10,10,10\right)deg$ and (

**b**) $\left(\mu ,{\psi}_{ac},{\psi}_{al}\right)=\left(0,10,0\right)deg$.

**Figure 13.**RMSE of (

**a**) $\tau $ and (

**b**) SWH versus SWH in the absence of mis-pointing for the G-PRA and PRA algorithms (Pu = 1, $\tau $ = 30 gates, ${\psi}_{al}={0}^{0}$, ${\psi}_{ac}={0}^{0}$).

**Figure 14.**RMSE of (

**a**) $\tau $ and (

**b**) SWH versus ${\psi}_{al}$ for the G-PRA and PRA algorithms (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, ${\psi}_{ac}={0}^{0}$).

**Figure 15.**RMSE of (

**a**) $\tau $ and (

**b**) SWH versus ${\psi}_{ac}$ for G-PRA and PRA algorithms (Pu = 1, $\tau $ = 30 gates, SWH = 2 m, ${\psi}_{al}={0}^{0}$).

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

Frequency | 2.95 GHz |

Wavelength ($\lambda $) | 0.1 cm |

Bandwidth (B) | 100 MHz |

Mean flight altitude (h) | 2000 m |

Pulse repetition frequency (PRF) | 5000 Hz |

3dB Antenna beam width (${\theta}_{3\mathrm{dB}}$) | ${40}^{\circ}$ |

Velocity (${v}_{s}$) | 100 m/s |

Pulse per burst | 100 pulses |

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## Share and Cite

**MDPI and ACS Style**

Wang, Q.; Jing, W.; Liu, X.; Huang, B.; Jiang, G.
Signal Processing and Waveform Re-Tracking for SAR Altimeters on High Mobility Platforms with Vertical Movement and Antenna Mis-Pointing. *Sensors* **2023**, *23*, 9266.
https://doi.org/10.3390/s23229266

**AMA Style**

Wang Q, Jing W, Liu X, Huang B, Jiang G.
Signal Processing and Waveform Re-Tracking for SAR Altimeters on High Mobility Platforms with Vertical Movement and Antenna Mis-Pointing. *Sensors*. 2023; 23(22):9266.
https://doi.org/10.3390/s23229266

**Chicago/Turabian Style**

Wang, Qiankai, Wen Jing, Xiang Liu, Bo Huang, and Ge Jiang.
2023. "Signal Processing and Waveform Re-Tracking for SAR Altimeters on High Mobility Platforms with Vertical Movement and Antenna Mis-Pointing" *Sensors* 23, no. 22: 9266.
https://doi.org/10.3390/s23229266