# Classifying Inundation in a Tropical Wetlands Complex with GNSS-R
## 2 Theoretical Background
### Measurement Definition
We assumed that the scattering area over inundated wetlands is the size of the first Fresnel zone, the ellipse of which has semi-major and semi-minor axes defined respectively in Equations (2) and (3) [42; 43]:  
\[a=\ \frac{1}{\cos(\theta_{inc})}\bigg{(}\frac{nR_{TX}R_{RX}\lambda}{R_{TX}+\ R_{RX}} \bigg{)}^{0.5}, \tag{2}\]  
\[b=\bigg{(}n\frac{R_{TX}R_{RX}\lambda}{R_{TX}+\ R_{RX}}\bigg{)}^{0.5}, \tag{3}\]  
where \(n=1\) defines the first Fresnel zone and \(\theta_{inc}\) is the incidence angle of the reflection, defined as the angle between the normal vector at the specular point and the vector between the specular point and the receiver. For non-inundated vegetated areas or open non-inundated areas, the number of Fresnel zones that contribute to the total power increases, and further \(n>1\) would need to be considered to perform a rigorous calibration of those signals. The semi-major (\(a\)) and semi-minor (\(b\)) axes of the first Fresnel zone (\(n=1\)) define the resolution of the CYGNSS measurement along the transmitter-to-receiver direction and its perpendicular direction, respectively. The \(\theta_{inc}\) of the forward scattered signals received by CYGNSS can vary between \(0^{\circ}\) and \(70^{\circ}\), giving rise to variation in the size of the first Fresnel zone. Figure 1 shows the incidence angle dependence of the size of those two axes. The GNSS-R signals received by CYGNSS are integrated incoherently over a 1 second time window. As a consequence, the first Fresnel zone elongates on the along-track direction due to the distance traveled by the specular point during this integration time (\(\sim\)6 km). Depending on the orientation of the first Fresnel zone ellipse with respect to the along-track direction, the size of the final spatial resolution varies. We selected the case where the semi-major axis aligns with the along-track direction, knowing that the semi-major axis dimension varies from 600 m to 2.8 km for incidence angles below \(70^{\circ}\) (Figure 1) and that the final size of the ellipse after integration would be 6.6 km and 8.8 km in the along-track dimension. For the same example, the cross-track size would not suffer any elongation and its size would correspond to that of the semi-minor of the first Fresnel zone, varying from 600 m to 1 km depending on incidence angle. The spatial resolution of CYGNSS measurements in that particular case is a variable value between a minimum of 600 m \(\times\) 6.6 km and a maximum of 1 km \(\times\) 8.8 km,considering the whole range of incidence angles. In the opposite case, where the semi-minor axis aligns with the along-track direction, the final spatial resolution after integration would be a variable value between a minimum of 600 m \(\times\) 6.6 km and a maximum of 2.8 km \(\times\) 7 km, considering the whole range of incidence angles. In order to obtain the GPS scattering area, ellipses were generated based on the semi-major and semi-minor first Fresnel zone dimensions elongated to 6 km in the along-track direction for each single reflection. These ellipses were then mapped onto a fixed 500 m \(\times\) 500 m grid, and any overlapping areas were averaged. Figure 2 shows an example of the two different gridding strategies considered in this study. The 7 km \(\times\) 7 km grid contains the average power of all reflection signals with specular points falling within each cell, regardless of the size of the scattering area. In the case of the 7 km \(\times\) 7 km fixed grid in Figure 2a, multiple reflections falling within a single grid cell are averaged using a drop-in-the-bucket averaging scheme. In Figure 2b, the 500 m \(\times\) 500 m grid is fine enough to map the shape of the first Fresnel zone ellipses for each measurement.