# Classifying Inundation in a Tropical Wetlands Complex with GNSS-R
## 2 Theoretical Background
### Measurement Definition  
The CYGNSS receivers acquire GNSS-R measurements that are processed as delay-Doppler maps (DDMs) [36]. A DDM is the result of the cross-correlation between the reflected GPS signal and a replica of the pseudo-random noise (PRN) code of the emitting GPS satellite. This cross-correlation is performed in the on-board receiver for different delay lags (in steps of 0.25 chips) and Doppler frequency offsets (in steps of 500 Hz), capturing the signal power coming from each delay and Doppler cell, i.e., forming a map of the reflected power in delay and Doppler frequency coordinates. The DDM equation was first formulated by [PERSON] [36], and has been recently reformulated by [PERSON] [37] to account for the coherent component of an area with very low surface roughness (e.g., an ocean surface with very calm winds) and to describe the contributions to scattering processes associated with varying amounts of coherent and non-coherent scattering. Previous studies have observed coherent reflections from sea ice [38], rivers [39], and wetlands [10; 30; 32]. In particular, the authors in Reference [38] performed a phase altimetry study using TDS-1 measurements to detect sea ice thickness, making use of the coherency of the returns over ice. A rough surface scatters an incident signal in all directions, and intensity and phase vary with scattering angle. This gives rise to incoherent scattering. Conversely, a perfectly smooth surface reflects the incident signal in the specular direction, preserving the phase of the incident wave, resulting in coherent scattering. Areas with standing water such as lakes, rivers, and inundated wetlands are characterized by relatively low surface roughness, which gives rise to a forward scattered field dominated by coherent scattering. In the case of inundated wetlands, where emergent vegetation is present, the forward scattered signal is still highly coherent, because of the influence of the standing water, but there is also an incoherent component added from the GPS signal's interaction with the vegetation. The SNR observed over inundated wetlands is stronger than that for non-inundated vegetation and bare soil regions. Hence, we assumed in this study that the forward scattered signal for inundated wetlands is dominated mainly by coherent scattering. The coherent component of the reflected power over inundated wetlands is obtained from References [37; 40; 41] as:  
\[P_{coh}=\Gamma\frac{P_{TX}\lambda^{2}G_{TX}G_{RX}}{(4\pi)^{2}(R_{TX}+\ R_{RX}) ^{2}}\, \tag{1}\]  
where \(\Gamma\) is the Fresnel reflectivity of the surface, \(P_{TX}\) is the power transmitted by the GPS satellite, \(\lambda\) is the wavelength of the GPS signal (19 cm), \(G_{TX}\) is the gain of the transmitter antenna in the direction of the scattering target, \(G_{RX}\) is the gain of the receiver antenna in the direction of the scattering target, \(R_{TX}\) is the distance between the surface point and the transmitter, and \(R_{RX}\) is the distance between the receiver and the surface point. In areas of bare soil and non-inundated vegetation, the relative degree of coherent and incoherent scattering is difficult to define. However, since our focus here is inundated wetlands, we assume, in terms of calibration, that the coherent component dominates all measurements within the Pacaya-Samiria wetland complex study area. This implies that, for the measurements of interest, the scattering from the first Fresnel zone is the major contributor to the total power. A Fresnel zone is the area from where the reflected power scatters, and is defined as ellipses of which the semi-minor and semi-major axes are a function of the distances between the transmitter, the receiver, and the specular point, as well as the wavelength of the scattered signal. The rougher the surface, the greater the number of Fresnel zones from which the signal scatters.