# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 2 Materials and Methods  
### The CYGNSS Observatory  
The CYGNSS main observable is normalized bistatic radar cross section (NBRCS), \(\sigma^{o}\). This is derived from the radar-range equation [25]:  
\[\sigma^{o}=\frac{P^{g}(4\pi)^{3}L^{atm}}{P^{T}G^{T}\lambda^{2}G^{R}R^{tot}\bar{A }}\, \tag{2}\]  
where \(P^{g}\) is the calibrated power received by the nadir science antenna; \(L^{atm}\) are the losses due to attenuation in the atmosphere; \(P^{T}\) is the power of the transmitter, which, in this case, is a GPS satellite; \(G^{T}\) is the antenna gain of the GPS satellite in the direction of the Earth specular point; \(\lambda\) is the wavelength of the GPS signal, which, for CYGNSS, is the GPS L1 signal at 1575 MHz; \(G^{R}\) is the gain of the CYGNSS nadir science antenna in the direction of the Earth specular point; \(R^{tot}\) is the total range loss of the transmission via the Earth specular point; and \(\bar{A}\) is the effective scattering area of the specular point. NBRCS is directly related to the mean square slope (MSS) of ocean wave spectra via [25; 26]:  
\[\sigma^{o}(\theta)=\frac{\left|\Re(\theta)\right|^{2}}{MSS}\, \tag{3}\]  
where \(\theta\) is the angle of incidence of the scattering geometry and \(\Re(\theta)\) is the Fresnel electric field reflection coefficient for the ocean surface at the specular point. CYGNSS retrieves wind speed from \(\sigma^{o}\) via an empirical Geophysical Model Function (GMF) [27] that is processed operationally via a stored look-up table (LUT). In practice, the un-normalized bistatic radar cross section \(\sigma\) is calculated for every delay and Doppler, the native coordinates of the delay-Doppler Mapping Instrument (DDMI). \(\sigma^{o}\) is calculated from the average of \(\sigma\) over a 3-by-5 bin window centered at the specular point and is divided by the effective area \(\bar{A}\) corresponding to the surface area bounded by the 3-by-5 window [m\({}^{2}\)/m\({}^{2}\)]. After the launch of the CYGNSS mission, it became clear that uncertainty in the GPS effective isotropic radiated power (EIRP) would significantly impact calibration quality [28; 29; 30]. The EIRP is simply the product \(P^{T}G^{T}\) from Equation (2). Errors are present in both terms: not only is there general uncertainty with the actual (versus published) transmit power of GPS satellites, but newer generations of the constellation operate with a flexible power mode and dynamically change transmit power levels across their respective orbits [31]. Further, only a subset of the GPS antenna patterns was published [32], and they were not sufficiently detailed to constrain the error in EIRP for CYGNSS calibration. To remedy this, [PERSON] et al. developed a novel calibration technique that uses the onboard zenith CYGNSS antenna and empirically derived relationships from measured GPS antenna patterns to estimate GPS EIRP in the direction of the Earth's specular point at the time of observation [33]. This new technique has been adopted in the CYGNSS Level 1 calibration algorithm since version 3.0 and modifies the radar equation to become the following [34]:  
\[\sigma^{o}=\frac{P^{g}4\pi L^{atm}L^{Z}G^{Z}}{P^{Z}\xi^{R}GR^{tot}\bar{A}}\, \tag{4}\]  
where all the terms are the same as in Equation (2), with the additions of \(L^{Z}\), the transmission range loss from the GPS source to the CYGNSS zenith antenna (written as a loss in the numerator to distinguish from the \(R^{tot}\) range losses); the CYGNSS zenith antenna gain \(G^{Z}\) in the direction of the GPS satellite; the power received from the CYGNSS zenith receiver \(P^{Z}\); and the specular-zenith ratio \(\zeta^{E}\) for the GPS EIRP as described in [33]. Each of the terms in Equation (4) can potentially be a source of correlated error, but for practical reasons, this study only evaluates those sources with the largest error magnitude and, therefore, the largest potential utility for future data assimilation users.