# Geolocation, Calibration and Surface Resolution of CYGNSS GNSS-R Land Observations
## 2 Geolocating GNSS Reflections from Land
### Local Surface Snell Reflection Criteria  
The final check on the land surface geo-location is purely geometrical and is not linked to the signal peak in the received GNSS-R DDM. This check is often the most robust as it is sometimes the case that the CYGNSS-observed delay and Doppler values can be difficult to estimate with certainty due to low receiver antenna gain or adverse Earth surface properties. a reference frame to facilitate the calculation of the incidence and scattering angles from a local surface was defined and is illustrated in Figure 1. With respect to this figure, the following points are defined: S is the center pixel in a 3 by 3 group on the local surface grid, and S1, S2, S3 and S4 are points at pixels directly to the north, south, east and west, respectively, spaced typically at 1 km. The points T and R are the GNSS transmitter and instrument receiver satellite, respectively, while \(P_{1}\) and \(P_{2}\) are the line-of-sight paths from the transmitter to a given surface point and from the surface point to the receiver respectively. The \(\theta_{i}\), \(\phi_{i}\), \(\theta_{r}\), and \(\phi_{r}\) angles represent the incidence and scattering polar and azimuthal angles, respectively, and are the variables to be calculated at every local surface point (\(\vec{S}\) ).  
In the example in Figure 1, all 9 points are shown above the WGS84 reference ellipsoid (shown in green). From the 5 local reference points, incidence and scattering angles can be calculated based on the incoming line-of-sight rays to the center pixel from the transmitter and receiver. We define local east, north, up (ENU) unit vectors from the center reference pixel, S, as follows:  
\[\hat{E}_{r}=\frac{(\vec{S3}-\vec{S4})}{\left|\vec{S3}-\vec{S4}\right|},\hat{N }_{r}=\frac{(\vec{S1}-\vec{S2})}{\left|\vec{S1}-\vec{S2}\right|},\hat{U}_{r}= \hat{E}\times\hat{N} \tag{12}\]  
Subsequently, the projected line-of-sight ray paths from the transmitter to the center surface pixel from the center pixel to the receiver (\(P_{1}\) and \(P_{2}\), respectively, in Figure 1) are projected to the local surface ENU surface reference frame directions:  
\[P_{1,E}^{{}^{\prime}} =(\vec{S}-\vec{T})\cdot\hat{E}_{r}\] (13) \[P_{1,N}^{{}^{\prime}} =(\vec{S}-\vec{T})\cdot\hat{N}_{r}\] \[P_{1,U}^{{}^{\  
From these unit vectors it is possible to calculate the polar and azimuth angles shown in Figure 1 as  
\[\theta_{i}=atan(P_{1,U}^{\prime},\sqrt{(P_{1,E}^{\prime})^{2}+(P_{1,N}^{\prime})} ^{2}) \tag{15}\]  
\[\theta_{r}=atan(P_{2,U}^{\prime},\sqrt{(P_{2,E}^{\prime})^{2}+(P_{2,N}^{\prime})} ^{2}) \tag{16}\]  
\[\phi_{i}=atan(P_{1,N}^{\prime},P_{1,E}^{\prime}) \tag{17}\]  
\[\phi_{r}=atan(P_{2,N}^{\prime},P_{2,E}^{\prime}) \tag{18}\]  
We are interested in the difference between the calculated angles and an ideal forward specular reflection from the given surface point. The resulting angle differences from a specular reflecting geometry are then  
\[\delta\theta=\theta_{i}-\theta_{r} \tag{19}\]  
\[\delta\phi=atan(\sin{(\phi_{r}-(\phi_{i}+\pi))},\cos{(\phi_{r}-(\phi_{i}+\pi) )}) \tag{20}\]  
We define the total angular error metric as the sum of the absolute errors in the azimuth and elevation angles relative to that of an ideal specular reflection,  
\[\delta\Psi=|\delta\theta|+|\delta\phi| \tag{21}\]