# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix E  
Errors in the zenith-specular ratio \(\zeta\) are defined as a function of specular incidence angle \(\theta_{inc}\), which is a function of the geometry of a given GPS transmitter, a CYGNSS receiver at any given sample time.  
\(\zeta\) is used to estimate GPS EIRP and is derived via the following, as described in [33]:  
\[\zeta\equiv\frac{EIRP_{z}}{EIRP_{S}}=\frac{EIRP(t,\theta_{z},\phi_{z})}{EIRP( t,\theta_{S},\phi_{S})}=\frac{P^{T}(t)G^{T}(\theta_{z},\phi_{z})}{P^{T}(t)G^{T}( \theta_{S},\phi_{S})}=\frac{G^{T}(\theta_{z},\phi_{z})}{G^{T}(\theta_{S},\phi _{S})}, \tag{33}\]  
where the angles are defined in the GPS reference frame. For specular geometries, the azimuthal angles in the zenith direction are nearly identical to the specular direction, so \(\phi_{z}=\phi_{S}=\phi\). In addition, the elevation angles in the GPS antenna reference frame \(\theta_{z}\) and \(\theta_{S}\) can be estimated from the angle of incidence of specular reflection from Earth \(\theta_{inc}\):  
\[\theta_{z}\cong\theta_{z}(\theta_{inc}) \tag{34a}\] \[\theta_{S}\cong\theta_{S}(\theta_{inc}) \tag{34b}\]  
As a result, \(\zeta\) can be expressed as a function of the specular incidence angle and azimuthal angle in the GPS antenna reference frame. While GPS antenna patterns are known to exhibit azimuthal dependence, this variation is less significant than the elevation angle, and CYGNSS uses the azimuthal average for its EIRP estimate:  
\[\zeta(\theta_{inc})\equiv\frac{1}{2\pi}\int_{0}^{2\pi}\frac{G_{T}(\theta_{z} (\theta_{inc}),\phi)}{G_{T}(\theta_{S}(\theta_{inc}),\phi)}d\phi \tag{35}\]  
The estimated correlated error in \(\zeta\), however, comes with two steps of this processing. First is the mapping of Earth scattering incidence angle \(\theta_{inc}\) to GPS antenna elevation angles \(\theta_{z}\) and \(\theta_{S}\) in Equations (34a) and (34b). This particular mapping is coarse, as even the high-fidelity-derived GPS antenna maps are plotted to 0.5-degree increments. Becausethe dynamic range of \(\theta_{z}\) only extends to about 15 degrees, that only leaves \(\sim\)30 data points to map the full dynamic range of scattering incidence angles.  
The second aspect has to do with the way in which \(\zeta\) is processed and generated and invokes the same logic as in the Filter Lemma. For every GPS satellite, a \(\zeta\) LUT is generated as a function of observation incidence angle \(\theta_{inc}\). To minimize discontinuities, a fourth-order power series is fit. We argue that this smoothing is the predominant source of correlated error structure. An example of this is demonstrated in Figure 12.  
While linear interpolation itself imparts some degree of error structure, we believe it is the most representative way to express 'raw' data in a continuous series for the purposes of exploring correlated error due to the power series smoothing. For each GPS PRN, we calculate the difference between these estimates:  
\[\Delta\zeta_{\phi}(\theta_{inc})=\left(\frac{G_{T}(\theta_{z}(\theta_{inc}), \phi)}{G_{T}(\theta_{S}(\theta_{inc}),\phi)}\right)_{interp}-\left(\frac{G_{T}( \theta_{z}(\theta_{inc}),\phi)}{G_{T}(\theta_{S}(\theta_{inc}),\phi)}\right)_{ smooth}\]  
Then, the error correlation is simply the following:  
\[Ecorr_{\zeta}(\theta_{i},\theta_{j})=corr\big{(}\Delta\zeta_{\phi}(\theta_{i} ),\ \Delta\zeta_{\phi}(\theta_{j})\big{)}, \tag{10}\]  
where \(\theta_{i}\) is the incidence angle of the observation at sample \(i\), and \(\theta_{j}\) is the incidence angle of the observation at sample \(j\). In practice, the correlation is computed by using each azimuthal cut as an instance and building a LUT of correlation as a function of incidence angles for samples \(i\) and \(j\). An example of this LUT for GPS PRN 2 is shown in Figure 10. The rolled-up correlated error for \(\zeta\) can then be expressed as follows:  
\[R_{\zeta}(i,j)=\left(\beta\cdot E(\zeta)^{2}\cdot Ecorr_{\zeta}(i,j)\right)^{ \frac{1}{2}}, \tag{10}\]  
with the same tuning parameter \(\beta\) as introduced in Appendix B.