# Response to Variations in River Flowrate by a Spaceborne GNSS-R River Width Estimator
## Materials and Methods
### Theoretical Basis of the High-Spatial-Resolution Land/Water Mapping Capability  
Measurement of small inland water bodies with CYGNSS is possible due to the high-spatial-resolution capability of bistatic radar over coherent scattering scenes with low roughness (i.e., inland water bodies), which allows for delineation of water bodies that are much smaller than the nominal footprint of the data. The signal strength received by a GNSS-R bistatic radar from a rough, incoherently scattering surface is given by [5] as  
\[P_{R}^{inc}(\tau,f)=\frac{P_{T}\lambda^{2}}{\left(4\pi\right)^{3}}\iint\frac{ G_{T}G_{R}\sigma_{0}\Lambda^{2}(\tau-\tau^{\prime})S^{2}(f-f^{\prime})}{ \left|\tau_{T}-\overline{\tau}\right|^{2}\left|\tilde{r}_{R}-\overline{\tau} \right|^{2}}d\overline{r}, \tag{1}\]  
and the signal strength received from a smooth, coherently scattering surface filling the first Fresnel zone is given by  
\[P_{R}^{coh}=\frac{P_{T}\lambda^{2}G_{R}G_{T}}{\left(r_{R}+r_{T}\right)^{2}(4 \pi)^{2}}\big{|}\Re\big{|}^{2}Z_{f}e^{-4\omega_{0}^{2}h^{2}\ \cos^{2}\ \theta} \tag{2}\]  
where:  
\((\tau,f)=\) delay and Doppler coordinates in the delay Doppler map (DDM)  
\(P_{T}=\) transmitted power  
\(G_{R}=\) receive antenna gain  
\(G_{T}=\) transmit antenna gain  
\(\Lambda=\) lag correlation function due to the correlation with a local replica of the GPS pseudorandom noise (PRN) code  
\(S=\) Doppler filter response due to selective filtering of the Doppler-shifted received signal  
\(\overline{r_{T}},\overline{r_{R}}=\) vector distances from the specular point to the transmitter and receiver, respectively  
\(\sigma_{0}=\) normalized scattering cross section of the rough surface (BRCS)  
\(\lambda=\) signal wavelength (19 cm)  
\(\overline{r}=\) vector distance from a point on the Earth's surface to the specular point  
\(Z_{f}=\) term to account for area/shape of smooth surface region affecting first Fresnel zone  
\(\Re=\) surface Fresnel reflectivity for incidence angle \(\theta\)  
\(k_{0}=2\pi/\lambda\); electromagnetic wavenumber  
\(h=\) surface rms height For heterogeneous conditions containing both smooth and rough surfaces within the region bounded by the lag correlation and Doppler filter responses, the net received signal strength is an area-weighted composite of the two contributions. The relative strengths of the two components are largely determined by their differing dependencies on the propagation distances, \(\overline{r_{R}}\) and \(\overline{r_{T}}\), in Equations (1) and (2). The two distances vary from sample to sample, depending on the measurement geometry of the GPS transmitter, the specular point, and the GNSS-R bistatic radar receiver. The variation of the two distances for CYGNSS throughout a typical 24-hour period is shown in Figure 1, where it can be seen that \(\overline{r_{R}}\), the distance from specular point to the receiver, varies between \(\sim\)500 and 1200 km, and \(\overline{r_{T}}\), the distance from GPS transmitter to specular point, varies between \(\sim\)19,500 and 24,000 km. Using Equations (1) and (2), we can compute a first-order approximation of the relative strength of signals received from coherent and incoherent portions of a heterogeneous scene as  
\[\frac{P_{R}^{coh}}{P_{R}^{inc}}\propto\frac{(\overline{r_{T}}+\overline{r_{R}} )^{-2}}{(\overline{r_{T}}\overline{r_{R}})^{-2}A_{inc}}, \tag{3}\]  
where \(A_{inc}\) is the scattering area of the incoherent region bounded by the lag correlation and Doppler filter response functions. For the GPS L1 C/A signal used by CYGNSS, \(A_{inc}\) is \(\sim\) 15 \(\times\) 15 km\({}^{2}\). Variations in both \(\overline{r_{R}}\) and \(\overline{r_{T}}\) are well parameterized by the incidence angle of the propagating signal at the specular reflection point. This is illustrated in Figure 2a,b, which respectively plot the incidence angle vs. \(\overline{r_{R}}\) and vs. the relative signal strength between coherent and incoherent scatterers for the same 24-h collection of data.