# Circular Regression Applied to GNSS-R Phase Altimetry

^{*}

## Abstract

**:**

## 1. Introduction

## 2. GNSS-R Altimetry Using Phase Measurement

#### 2.1. Height Retrieval Using GNSS Phase Signals

#### 2.2. Phase Observation

## 3. Linear-Circular Regression Applied to GNSS-R Altimetry

#### 3.1. Circular Model

#### 3.2. Log-Likelihood Estimator

#### 3.3. Initialization and Assessment

## 4. Experimentation

#### 4.1. Assessment with Synthetic Data

#### 4.2. Experimental Setup

#### 4.3. Assessment with Real Data

## 5. Discussion

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 4.**RMSE and theoretical standard deviation of the estimated height as a function of the integration time, for various values of $C/{N}_{0}$.

**Figure 5.**Standard deviation of the estimated height as a function of the satellite elevation rate and for various initial elevations.

**Figure 7.**Example of delay-Doppler map obtained with the proposed receiver. The direct and reflected signals are separated in time by a delay ${\tau}^{L}$ = 5.12 $\mathsf{\mu}$s (in addition to the path delay).

**Figure 8.**Experimental vehicle carrying the GNSS-R altimeter. The observed GPS signals are reflected on an artificial basin in Calais, France (50.962689${}^{\circ}$N, 1.857305${}^{\circ}$E).

**Figure 9.**Example of interferometric phase observations obtained for a GPS-L1 reflection on the artificial basin.

**Figure 10.**GPS satellites in view and corresponding footprints on the basin 30 June 2016, 15 h 25 min.

**Figure 11.**(

**Left**) Observed interferometric phase for Satellites 18 and 21 at 15 h 38 min, 30 June 2016. (

**Right**) Observed interferometric phase for satellite 18 and 21 at 15 h 51 min, 30 June 2016.

**Table 1.**Concentration parameter ${\kappa}_{s}$ corresponding to various $C/{N}_{0}$ values of the interferometric signal.

$\mathit{C}/{\mathit{N}}_{0}$ (dB-Hz) | 30 | 35 | 40 | 45 |

${\mathit{\kappa}}_{\mathit{s}}$ | 1.35 | 2.96 | 9.34 | 30.82 |

**Table 2.**Parameters for satellites PRN 18 and 21, 30 June 2016, during 600 s of observation at 15 h 25 min, 15 h 38 min and 15 h 51 min.

15 h 25 min | |||||

Satellite PRN | Interferometric Signal C/${\mathit{N}}_{0}$ (dB-Hz) | Elevation | |||

min. (${}^{\circ}$) | max. (${}^{\circ}$) | mean (${}^{\circ}$) | rate (${}^{\circ}$/s) | ||

18 | 36.5 | 36.44 | 39.21 | 37.85 | 0.0046 |

21 | 34 | 53.73 | 57.56 | 55.65 | −0.0064 |

15 h 38 min | |||||

Satellite PRN | Interferometric Signal $\mathit{C}$/${\mathit{N}}_{\mathbf{0}}$ (dB-Hz) | Elevation | |||

min. (${}^{\circ}$) | max. (${}^{\circ}$) | mean (${}^{\circ}$) | rate (${}^{\circ}$/s) | ||

18 | 37 | 39.99 | 42.23 | 41.15 | 0.0037 |

21 | 34.1 | 48.55 | 52.51 | 50.53 | −0.0066 |

15 h 51 min | |||||

Satellite PRN | Interferometric Signal $\mathit{C}$/${\mathit{N}}_{\mathbf{0}}$ (dB-Hz) | Elevation | |||

min. (${}^{\circ}$) | max. (${}^{\circ}$) | mean (${}^{\circ}$) | rate (${}^{\circ}$/s) | ||

18 | 35 | 42.80 | 44.38 | 43.63 | 0.0026 |

21 | 32.3 | 43.39 | 47.38 | 45.38 | −0.0066 |

Satellite PRN | 15 h 25 min | 15 h 38 min | 15 h 51 min |
---|---|---|---|

${\mathit{h}}_{\mathit{ref}}=12.60$ m | ${\mathit{h}}_{\mathit{ref}}=12.60$ m | ${\mathit{h}}_{\mathit{ref}}=12.60$ m | |

Estimated Height $\widehat{\mathit{h}}$ (m) | Estimated Height $\widehat{\mathit{h}}$ (m) | Estimated Height $\widehat{\mathit{h}}$ (m) | |

18 | 12.68 | 12.87 | 13.02 |

21 | 12.65 | 12.49 | 12.36 |

18 + 21 | 12.61 * | 12.61 * | 12.51 * |

18 + 21 | 12.60 ** |

**Table 4.**Estimated height (m) as a function of the observation duration, obtained by fusion of Satellite 18 and 21 data at 15 h 25 min, 15 h 38 min and 15 h 51 min, 30 June 2016 (reference height: ${h}_{ref}=12.60$ m.)

Sat.: 18 + 21 | 15 s | 30 s | 50 s | 100 s | 150 s | 300 s | 600 s |
---|---|---|---|---|---|---|---|

15 h 25 min | 8.80 | 12.63 | 12.62 | 12.62 | 12.61 | 12.62 | 12.61 |

15 h 38 min | 7.54 | 12.62 | 12.61 | 12.61 | 12.62 | 12.62 | 12.61 |

15 h 51 min | 16.01 | 12.56 | 12.53 | 12.54 | 12.55 | 12.55 | 12.51 |

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**MDPI and ACS Style**

Kucwaj, J.-C.; Reboul, S.; Stienne, G.; Choquel, J.-B.; Benjelloun, M.
Circular Regression Applied to GNSS-R Phase Altimetry. *Remote Sens.* **2017**, *9*, 651.
https://doi.org/10.3390/rs9070651

**AMA Style**

Kucwaj J-C, Reboul S, Stienne G, Choquel J-B, Benjelloun M.
Circular Regression Applied to GNSS-R Phase Altimetry. *Remote Sensing*. 2017; 9(7):651.
https://doi.org/10.3390/rs9070651

**Chicago/Turabian Style**

Kucwaj, Jean-Christophe, Serge Reboul, Georges Stienne, Jean-Bernard Choquel, and Mohammed Benjelloun.
2017. "Circular Regression Applied to GNSS-R Phase Altimetry" *Remote Sensing* 9, no. 7: 651.
https://doi.org/10.3390/rs9070651