# First Evaluation of Topography on GNSS-R: An Empirical Study Based on a Digital Elevation Model
## 2 Theoretical Background  
CyGNSS allows sampling of the Earth's surface along 32 tracks simultaneously, within a wide range of satellites' elevation angles \(\theta_{\mathrm{e}}\)- [20; 90]\({}^{\circ}\) over the tropical latitudinal band Latitude \(\sim\) [\(-\)40, 40]\({}^{\circ}\). Geophysical parameters retrieval from CyGNSS requires that the GNSS-R payload cross-correlatesdirectly the collected signals against a known replica of the GPS L1 C/A code, after compensating for the Doppler shift. Consequently, 32 complex reflected DDMs,Y\({}_{\mathrm{r}}\), are generated on board at a time, using a short coherent integration time,T\({}_{\mathrm{c}}\)\(\sim\) 1 ms. Complex DDMs, Y\({}_{\mathrm{r}}\), are incoherently averaged along \(\sim\) 1 s to finally generate the so-called power DDMs. Power DDMs, \(\left\langle\left|Y_{\mathrm{r}}(\tau,\mathrm{f})\right|^{2}\right\rangle\), can be modelled using geometrical and scattering related parameters as follows [17]:  
\[\left\langle\left|Y_{\mathrm{r}}(\tau,\mathrm{f})\right|^{2}\right\rangle= \frac{\mathrm{T}_{\mathrm{c}}^{2}\mathrm{P}_{\mathrm{T}}\lambda^{2}}{\left(4 \pi\right)^{3}}\iint\frac{\mathrm{G}_{\mathrm{T}}\mathrm{G}_{\mathrm{R}}\left| \chi(\tau-\left(\mathrm{R}_{\mathrm{T}}+\mathrm{R}_{\mathrm{R}}\right)/c, \mathrm{f}-\mathrm{f}_{\mathrm{c}})\right|^{2}\sigma^{0}}{\mathrm{R}_{ \mathrm{T}}^{2}\mathrm{R}_{\mathrm{R}}^{2}}\mathrm{d}^{2}\overline{\rho}, \tag{1}\]  
where P\({}_{\mathrm{T}}\) is the power of the right HCP(RHCP) transmitted signals, \(\lambda\)\(\sim\) 19 cm is the wavelength of the signals at L1, G\({}_{\mathrm{T}}\) and G\({}_{\mathrm{R}}\) are the transmitting and receiving antenna gains, R\({}_{\mathrm{T}}\) and R\({}_{\mathrm{R}}\) are the ranges from the transmitter and the receiver to the specular point, respectively, \(\chi\) is the Woodward ambiguity function (WAF), \(\tau\) is the delay of the signal from the transmitter to the receiver, f is the Doppler shift of the reflected signal, f\({}_{\mathrm{c}}\) is aimed to compensate the Doppler shift of the signal, \(\sigma^{0}\) is the bistatic scattering coefficient at LHCP, and \(\overline{\rho}\) is the positioning vector of the scattering point. The bistatic scattering coefficient can be defined as follows [18]:  
\[\sigma^{0}=\sigma^{\mathrm{coh,0}}+\sigma^{\mathrm{incoh,0}}, \tag{2}\]  
where \(\sigma^{\mathrm{coh,0}}\) and \(\sigma^{\mathrm{incoh,0}}\) are the coherent and the incoherent scattering terms respectively. Consequently, DDMs consist on a sum of two terms as follows [19; 20; 21]:  
\[\left\langle\left|Y_{\mathrm{r}}(\tau,\mathrm{f})\right|^{2}\right\rangle= \left\langle\left|Y_{\mathrm{r,coh}}(\tau,\mathrm{f})\right|^{2}\right\rangle +\left\langle\left|Y_{\mathrm{r,incoh}}(\tau,\mathrm{f})\right|^{2}\right\rangle. \tag{3}\]  
\(\left\langle\left|Y_{\mathrm{r,coh}}(\tau,\mathrm{f})\right|^{2}\right\rangle\) accounts for coherent reflections from the surface, while \(\left\langle\left|Y_{\mathrm{r,incoh}}(\tau,\mathrm{f})\right|^{2}\right\rangle\) is responsible for the diffuse scattering. \(\left\langle\left|Y_{\mathrm{r,coh}}(\tau,\mathrm{f})\right|^{2}\right\rangle\) can be expressed analytically as follows [10]:  
\[\left\langle\left|Y_{\mathrm{r,coh}}(\tau,\mathrm{f})\right|^{2}\right\rangle= \frac{\mathrm{T}_{\mathrm{c}}^{2}\mathrm{P}_{\mathrm{T}}\lambda^{2}\mathrm{G}_ {\mathrm{T}}\mathrm{G}_{\mathrm{R}}\left|\chi(\tau,\mathrm{f})\right|^{2}}{ \left(4\pi\right)^{2}\left(\mathrm{R}_{\mathrm{T}}+\mathrm{R}_{\mathrm{R}} \right)^{2}}\left|\mathrm{R}(\theta_{\mathrm{e}})\right|^{2}\gamma\exp(-(2 \mathrm{k}\sigma\sin\theta_{\mathrm{e}})^{2}), \tag{4}\]  
where R is the LHCP Fresnel reflection coefficient, k is the signal angular wavenumber, \(\sigma\) is the surface height standard deviation (related to small-scale surface roughness), and \(\gamma\) is the transmissivity of the vegetation. \(\left\langle\left|Y_{\mathrm{r,incoh}}(\tau,\mathrm{f})\right|^{2}\right\rangle\) includes contributions from vegetation, small-scale surface roughness, and topography.