# Link Budget Analysis for GNSS-R Sea Surface Return in Arbitrary Acquisition Geometries Using BA-PTSM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Bistatic Anisotropic Polarimetric Two-Scale Model

## 3. Link Budget for the GNSS-R Sea Surface Return

- Coherent integration: The signal received through the reflection path is correlated with a properly delayed and frequency-shifted version of either the direct signal or a locally-generated clean replica of the transmitted GNSS signal. The output of this step is a 2-D map of the signal power distribution in the delay-Doppler domain. It is commonly referred to as single-snapshot delay-Doppler map (DDM).
- Incoherent integration: The coherently-integrated DDM suffers from a very low SNR due to the poor GNSS Effective Isotropical Radiated Power (EIRP), especially at spaceborne altitudes. Accordingly, a strong noise reduction is necessary for enabling meaningful geophysical information retrieval. This is accomplished by incoherently averaging subsequent single-snapshot DDMs, i.e., by temporal multilook, which reduces the thermal noise power by the factor $\sqrt{N}$, where $N=\frac{{T}_{i}}{{T}_{c}}$ is the number of averaged DDMs.

- The acquisition geometry, i.e., the values of $\theta $, ${\theta}_{s}$, and ${\varphi}_{s}$ are assigned as input parameters. This implies that the receive antenna beam is conceptually centered around the input (${\theta}_{s}$, ${\varphi}_{s}$) direction.
- The area sensed by the receive antenna beam is determined according to the half-power beamwidths (HPBWs) defined in Table 1. To this end, a surface points grid is generated according to the spherical Earth model previously detailed and assuming an initial spatial sampling step of 200 m and 10 m in the spaceborne and airborne configurations, respectively.
- The coordinates of the transmitter and the receiver, and, then, of the specular reflection point, are computed in the scattering reference frame by simple geometric considerations.
- Delay-Doppler coordinates (${f}_{d},\tau $) are evaluated for each surface point of the sensing region from the position and velocity of the transmitter and the receiver. The delay-Doppler coordinates of the reference frame origin ($\overline{{f}_{d}}$, $\overline{\tau}$) are computed as well.
- Surface points falling within the delay-Doppler resolution cell centered in ($\overline{{f}_{d}}$, $\overline{\tau}$), i.e., such that$$|{f}_{d}-\overline{{f}_{d}}|\le \mathsf{\Delta}{f}_{d}/2\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}|\tau -\overline{\tau}|\le \mathsf{\Delta}\tau /2$$
- Steps 2–5 are repeated until more than 10 surface points are identified as the scattering region. In each iteration, the spatial sampling step is halved to generate a more refined surface grid.

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

2-D | Two-dimensional |

A-PTSM | Anisotropic PTSM |

BA-PTSM | Bistatic A-PTSM |

DDM | Delay-Doppler map |

EIRP | Effective Isotropical Radiated Power |

EM | Electromagnetic |

GNSS | Global Navigation Satellite System |

GNSS-R | GNSS-Reflectometry |

GO | Geometrical Optics |

HPBW | Half-power beamwidth |

LHCP | Left-hand circular polarization |

NRCS | Normalized radar cross section |

Probability density function | |

PSD | Power Spectral Density |

PTSM | Polarimetric TSM |

RCS | Radar-cross section |

RHCP | Right-hand circular polarization |

RMS | Root-mean square |

SAR | Synthetic aperture radar |

SNR | Signal-to-noise ratio |

SPM | Small Perturbation Method |

SSA2 | Second-order small-slope approximation |

TSM | Two-scale model |

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**Figure 1.**Conventional (LHCP-forward-scattering, red box) vs. unconventional (RHCP-backscattering, green box) GNSS-R.

**Figure 3.**SNR in RL polarization for a spaceborne GNSS-R receiver obtained using BA-PTSM (solid lines) and GO (dashed lines) for wind speed 5 m/s (blue lines), 15 m/s (red lines), and 25 m/s (green lines). (

**a**) ${\varphi}_{s}={0}^{\circ}$; (

**b**) ${\varphi}_{s}={45}^{\circ}$; (

**c**) ${\varphi}_{s}={90}^{\circ}$; (

**d**) ${\varphi}_{s}={120}^{\circ}$; (

**e**) ${\varphi}_{s}={150}^{\circ}$; (

**f**) ${\varphi}_{s}={180}^{\circ}$.

**Figure 4.**SNR in RR polarization for a spaceborne GNSS-R receiver obtained using BA-PTSM (solid lines) and GO (dashed lines) for wind speed 5 m/s (blue lines), 15 m/s (red lines), and 25 m/s (green lines). (

**a**) ${\varphi}_{s}={0}^{\circ}$; (

**b**) ${\varphi}_{s}={45}^{\circ}$; (

**c**) ${\varphi}_{s}={90}^{\circ}$; (

**d**) ${\varphi}_{s}={120}^{\circ}$; (

**e**) ${\varphi}_{s}={150}^{\circ}$; (

**f**) ${\varphi}_{s}={180}^{\circ}$.

**Figure 5.**SNR in RL polarization for an airborne GNSS-R receiver obtained using BA-PTSM (solid lines) and GO (dashed lines) for wind speed 5 m/s (blue lines), 15 m/s (red lines), and 25 m/s (green lines). (

**a**) ${\varphi}_{s}={0}^{\circ}$; (

**b**) ${\varphi}_{s}={45}^{\circ}$; (

**c**) ${\varphi}_{s}={90}^{\circ}$; (

**d**) ${\varphi}_{s}={120}^{\circ}$; (

**e**) ${\varphi}_{s}={150}^{\circ}$; (

**f**) ${\varphi}_{s}={180}^{\circ}$.

**Figure 6.**SNR in RR polarization for an airborne GNSS-R receiver obtained using BA-PTSM (solid lines) and GO (dashed lines) for wind speed 5 m/s (blue lines), 15 m/s (red lines), and 25 m/s (green lines). (

**a**) ${\varphi}_{s}={0}^{\circ}$; (

**b**) ${\varphi}_{s}={45}^{\circ}$; (

**c**) ${\varphi}_{s}={90}^{\circ}$; (

**d**) ${\varphi}_{s}={120}^{\circ}$; (

**e**) ${\varphi}_{s}={150}^{\circ}$; (

**f**) ${\varphi}_{s}={180}^{\circ}$.

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

${P}_{t}$ | Transmitted power | 26.61 W |

${G}_{t}$ | Transmitting antenna gain | 13 dBi |

${G}_{r}$ | Receiving antenna gain | 13.3 dBi (spaceborne) |

15.05 dBi (airborne) | ||

$HPB{W}_{al}$ | Receiver HPBW along-track | ${60}^{\circ}$ |

$HPB{W}_{ac}$ | Receiver HPBW across-track | ${30}^{\circ}$ |

$\lambda $ | GNSS wavelength | 0.19 m |

$\theta $ | Viewing angle | ${30}^{\circ}$ |

${\theta}_{s}$ | Zenith scattering angle | Ranging in $[{0}^{\circ},{85}^{\circ}]$ |

${\varphi}_{s}$ | Azimuth scattering angle | Ranging in $[{0}^{\circ},{180}^{\circ}]$ |

${h}_{t}$ | Transmitter altitude | 20,200 km |

${h}_{r}$ | Receiver altitude | 540 km (spaceborne) |

10 km (airborne) | ||

${\sigma}_{sea}^{0}$ | Sea surface NRCS | Evaluated via BA-PTSM |

${T}_{a}$ | Receiving antenna noise temperature | 99.4 K |

${T}_{e}$ | Receiver noise temperature | 374.35 K (spaceborne) |

161.23 K (airborne) | ||

${T}_{i}$ | Incoherent integration time | 1 s |

${T}_{c}$ | Coherent integration time | 1 ms (spaceborne) |

10 ms (airborne) | ||

${\tau}_{c}$ | Chip length | 977.52 ns |

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**MDPI and ACS Style**

Di Martino, G.; Di Simone, A.; Iodice, A.; Riccio, D.
Link Budget Analysis for GNSS-R Sea Surface Return in Arbitrary Acquisition Geometries Using BA-PTSM. *Remote Sens.* **2022**, *14*, 520.
https://doi.org/10.3390/rs14030520

**AMA Style**

Di Martino G, Di Simone A, Iodice A, Riccio D.
Link Budget Analysis for GNSS-R Sea Surface Return in Arbitrary Acquisition Geometries Using BA-PTSM. *Remote Sensing*. 2022; 14(3):520.
https://doi.org/10.3390/rs14030520

**Chicago/Turabian Style**

Di Martino, Gerardo, Alessio Di Simone, Antonio Iodice, and Daniele Riccio.
2022. "Link Budget Analysis for GNSS-R Sea Surface Return in Arbitrary Acquisition Geometries Using BA-PTSM" *Remote Sensing* 14, no. 3: 520.
https://doi.org/10.3390/rs14030520