# Circular Regression Applied to GNSS-R Phase Altimetry
## 1 Introduction  
Global Navigation Satellite Systems-Reflectometry (GNSS-R) is a passive bi-static radar technique for Earth observation using GNSS signals as signals of opportunity. This remote sensing technique uses the L-band signal coming directly from a GNSS satellite and the signal reflected by the Earth in order to characterize the reflecting surface. Several applications have been developed, such as ocean altimetry [1], the characterization of sea surface roughness [2] and soil moisture estimation [3]. Using GNSS signals for Earth observation offers multiple advantages: precision of the location and dating of the retrieved data, low sensibility to atmospheric perturbations and global coverage by several existing constellations, implying the availability of numerous reflections observations at the same time. In this work, the application concerns the design of a low altitude GNSS radar system for altimetry of water surfaces in coastal areas. The altimetry measurement is derived from the path difference between the direct and reflected signals. This path difference can be estimated from the carrier-to-noise ratio observations [4; 5; 6; 7; 8], the code observations [9; 10; 11; 12; 13; 14] or the phase observations [15; 16]. The GNSS signal path can be measured with the observation of phase delay between the direct and reflected signals. This phase delay provides a range measurement with an ambiguity corresponding to the signal wavelength. Many methods are proposed in the published works to solve this ambiguity [14; 17]. Ambiguity resolution is a complicated problem that is still studied nowadays. The technique developed in this article allows this problemto be skipped. The phase difference between the direct and the reflected signals evolves linearly with the satellite elevation sine [18; 19]. The slope of this evolution provides an observation of the height between the receiving antennas and the reflecting surface. In this article, we observe the phase difference with a receiver similar to the one proposed in [20]. The phase delay is obtained by correlating the reflected signal and a clean replica of the direct signal [19; 21]. This approach can be considered as conventional GNSS-R (cGNSS-R) with master/slave channel configuration [18; 22]. The observations of phase delay are noisy. In the published works, long coherent integration [23] and phase unwrapping are used in order to improve the estimation process. In [23; 24], a linear interpolation of the unwrapped phase is used to derive an accurate estimate of the height. In [19], an iterative unwrapping circular technique is proposed. This approach aims at improving the unwrapping process by smoothing the angular observations with a circular fusion filter. None of these methods can be used in the case of sets of measurements including long data losses. Unlike the code measurements, the carrier phase measurements are angular data. Angular data are defined in an interval of length \(2\pi\) (\([-\pi,\pi]\) or \([0,2\pi]\)), which corresponds, in terms of range, to the wavelength of the observed GNSS signal [25; 26]. Consequently, the classical estimation techniques of the linear domain provide biased results for circular data due to the occurrence of transitions between the interval bounds over time. In this context, techniques defined in the linear domain (for example, using the Gaussian distribution) are inapplicable for statistical processing. It was shown in [27] that the additive noise on the phase delay of a GNSS signal follows a von Mises distribution. This distribution (also called circular normal distribution) is widely used for the modeling of angles in a Bayesian framework [28; 29]. Applied to angular observations, the von Mises distribution plays the same role as the normal distribution for observations defined on the \(\mathbb{R}\) line. Detailed overviews of circular statistics can be found in [30; 31; 32].