# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix B  
The term \(P\!\!\!^{g}\) represents calibrated received power from GPS signals reflected from the Earth's surface. \(P\!\!\!^{g}\) is calibrated both from pre-launch characterizations as well as on an on-orbit blackbody at a known temperature, which, as of 2022, takes a reading every 10 min to re-compute gain that may have changed due to the dynamic thermal environment in orbit [35]. For CYGNSS, \(P^{g}\) is computed by the Level 1A algorithm:  
\[pg=\frac{C-C_{N}}{G}, \tag{15}\]  
where \(C\) is the raw counts at delay-Doppler bins, where a scattered signal is present and is the measured parameter for an observation, \(C_{N}\) is an estimate of the background noise without any scattered signal, and \(G\) is the receiver gain in units of counts/watt. In current processing, \(C_{N}\) is an average of the raw counts at in the delay-Doppler map at coordinates at lower delay values than the specular point [25]. Further, gain \(G\) is measured in orbit by performing readings from an onboard blackbody at a known temperature and calibrated via the following:  
\[G=\frac{C_{B}}{P_{B}+P_{r}}, \tag{16}\]  
where \(C_{B}\) is the counts measured while looking at the blackbody, \(P_{B}\) is the power in watts from the blackbody as estimated from a thermocouple located near the receiver's low noise amplifier, and \(P_{r}\) is the receiver noise power in watts estimated from a pre-launch parameterization. Equations (15) and (16) can be combined:  
\[P^{g}=\frac{(C-C_{N})}{C_{B}}(P_{B}+P_{r}), \tag{17}\]  
The magnitude of the errors for the components calculating \(P^{g}\) is displayed in Table 1 and has a similar calculation as before. \(C_{B}\), \(P_{B}\), and \(P_{r}\) all vary with temperature, and the instrument gain \(G\) will fluctuate as the satellite enters different thermal conditions in orbit. The most significant errors will occur just as the satellite crosses the terminator. At that point, CYGNSS will go from a nearly steady-state thermal environment, such as approximately half an orbit of illumination or eclipse, and then quickly enter the opposite state. The fraction of orbit spent illuminated is determined by the orbit beta angle, which varies on scales of weeks to months. The dominant error term in the Level 1A algorithm is \(C_{N}\), which also varies with temperature. The calibration sequence is designed to correct for this, and we assume that the errors vary slowly with the timescale of interest, which is defined to be on orders of seconds to minutes. Therefore, all errors from \(C_{N}\) are assumed to be 100% correlated in time within a given track of CYGNSS observations, i.e., when a series of samples adjacent in space and time share a GPS transmitter and a CYGNSS receiver. \(C_{N}\) is also very sensitive to radio-frequency interference (RFI), which will present as non-physical signals above the specular point in a delay-Doppler map. We do not aim to model the complex phenomenologies of RFI in this work and assume there are no correlated error structures from RFI. \begin{table}
\begin{tabular}{c c} \hline
**Error Term** & **Error Magnitude [dB]** \\ \hline \(E(C_{N})\) & 0.14 [36] \\ \(E(P_{r})\) & 0.14 [36] \\ \(E(C)\) & 0.10 [36] \\ \(E(C_{B})\) & 0.07 [35] \\ \(E(P_{B})\) & \textless{}0.04 [36] \\ \hline \end{tabular}
\end{table} 
Table 1: The magnitudes of 1-sigma errors for each term in Equation (17). The shaded rows indicate that the absolute magnitude is negligible compared to the dominant error terms and is neglected in the construction of the error model in this work. Errors in \(P_{r}\) occur because of a variety of reasons. The low noise amplifiers were all characterized on the ground prior to launch to establish the relationship of the noise figure with respect to temperature. The values of this relationship were stored in a look-up table (LUT) for processing science data. However, as the amplifiers age, the noise floor characteristics may have evolved, producing errors in this mapping.