# First Evaluation of Topography on GNSS-R: An Empirical Study Based on a Digital Elevation Model
## 3 Data and Methodology
### Delay Doppler Maps Parameters  
CyGNSS Level 1 Science Data Record Version 2.1 [15; 16; 12] was selected for this study. Original data were first filtered out using an equivalent \"CyGNSS overall quality flag\" over land surfaces [35]. Reflected delay waveforms \(\text{WF}_{\text{r,raw}}\) were obtained from the original DDMs \(\left\langle\left|\text{Y}_{\text{r,topo}}(\tau,\text{f})\right|^{2}\right\rangle\) at zero Doppler frequency:  
\[\text{WF}_{\text{r,raw}}=\left\langle\left|\text{Y}_{\text{r,topo}}(\tau, \text{f}=0)\right|^{2}\right\rangle. \tag{15}\]  
Figure 2: Global Multi-resolution Terrain Elevation Data 2010 (GMTED2010) model-based topographic parameters. (**a**) Topographic position index (TPI) (m), (**b**) terrain ruggedness index (TRI) (m), (**c**) vector ruggedness measure (VRM) (deg), (**d**) Slope \(\beta\) (deg), (**e**) tangential curvature (TC) (rad/m), and (**f**) profile curvature (PC) (rad/m). Parameters were displayed over the Cyclone Global Navigation Satellite System (CyGNSS) coverage over land surfaces.  
The delay bin resolution of the original 17 lag waveforms \(\mathrm{WF_{r,raw}}\) was \(\sim\) 0.2552 GPS C/A chips. A re-sampling and interpolation of the resulting 1700 lag waveforms was performed using a spline method to increase the accuracy of the waveforms, before applying the algorithms to extract the observables.  
The trailing edge (TE) width was defined as the lag difference between the 70% power threshold of the high resolution waveforms \(\mathrm{WF_{r,threshold}}\), and that corresponding to the maximum power of the waveforms \(\mathrm{WF_{r,peak}}\)[9]:  
\[\mathrm{TE}=\tau_{\mathrm{WF_{r,threshold}}}-\tau_{\mathrm{WF_{r,peak}}}. \tag{16}\]  
Different power thresholds were tested. The 50% power threshold was found to cut off some lags, when this threshold was out of the original 17 lag waveforms. On the other hand, the 90% power threshold provided a lower dynamic range as compared to the 70%. The incoherent scattering term \(\sigma^{\mathrm{incoh,0}}\) was the main contribution to \(\mathrm{WF_{r,threshold}}\), while both the coherent \(\sigma^{\mathrm{coh,0}}\) and the incoherent \(\sigma^{\mathrm{incoh,0}}\) scattering terms significantly contributed to the peak power of the waveforms \(\mathrm{WF_{r,peak}}\).  
The reflectivity \(\Gamma\) was estimated as the ratio of the reflected \(\mathrm{WF_{r,peak}}\) and the direct \(\mathrm{WF_{d,peak}}\) power waveforms' peaks, after compensating for the noise power floor and the antennas' gain patterns as a function of the elevation angle \(\theta_{\mathrm{e}}\) as in [10]:  
\[\Gamma=\mathrm{WF_{r,peak}}/\mathrm{WF_{d,peak}}. \tag{17}\]  
The antennas' gain patterns were compensated versus the gain at the corresponding boresight direction, and the difference of both antennas at boresight [10]. The nominal mission lifetime was 2 years. This milestone was achieved in March 2019. At present, the mission is operating nominally, and the automatic gain control (AGC) was disabled from August 2018 [12]. This fact improves the accuracy in the estimation of \(\Gamma\) using Equation (17). On the other hand, \(\Gamma\) could also be inverted from scattering models. This approach relies on the assumption that the scattering is totally coherent [35] or totally incoherent [36]. The assumption in [35] is valid over smooth surface areas with little vegetation. The assumption in [36] is valid if inland water bodies are removed. However, in a more general scenario, the total scattered electromagnetic field is composed of both a coherent and an incoherent contribution [18] in different proportions depending on the dielectric and geometrical properties of the scattering medium, and the directions of incoming and outgoing electromagnetic waves. Both, the coherent \(\sigma^{\mathrm{coh,0}}\) and the incoherent \(\sigma^{\mathrm{incoh,0}}\) scattering terms contribute to the peak power of the waveforms. This was the main motivation to use Equation (17) in this work.