# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix B
Further, the thermal environment of the thermocouple may not be exactly the same as experienced by the amplifier itself. For the purposes of this model, we assume all errors due to incomplete or erroneous knowledge of the true receiver noise power are 100% correlated with each other for a given track, as we assume that the errors evolve slowly compared to the timescales of interest. Therefore, the error correlation terms for \(C_{N}\) and \(P_{r}\) for any arbitrary CYGNSS samples \(x_{i}\) and \(x_{j}\) are the following:  
\[Ecorr_{C_{N}}(x_{i},x_{j})=Ecorr_{P_{r}}(x_{i},x_{j})=\left\{ \begin{matrix}1, \text{ }conditions\text{ }met\\ 0,\text{ }else\end{matrix} \right. \tag{10}\]  
The conditions that must be met are the following: \(x_{i}\) and \(x_{j}\) must share a CYGNSS receiver and a GPS transmitter and be observed within 10 min of each other. \(C\) is the measured parameter, the raw counts of power from a science observation near the region of the specular point. The analog-to-digital processing chain is the primary source of errors, such as quantization errors and non-common-mode interference. We assume that these error terms are 100% uncorrelated with each other; that is, for every sample, it can be treated as white noise. The error correlation term for \(C\) is then the following:  
\[Ecorr_{C}\left(x_{i},x_{j}\right)=\left\{ \begin{matrix}1,\text{ }x_{i}=x_{j}\\ 0,\text{ }x_{i}\
eq x_{j}\end{matrix} \right. \tag{11}\]  
The error term for counts measured during a blackbody sample \(C_{B}\) is a source of analytically defined correlated error. Every 10 min (earlier in the mission, every 1 min), the receiver is switched from the nadir science antenna to look at the onboard blackbody source for a period of 4-6 s. Science observation processing linearly interpolates the counts between the nearest blackbody looks. When errors are made in estimating \(C_{B}\), those errors are correlated linearly with all adjacent samples due to this interpolation. Correlation due to linear interpolation has an analytical form. Assuming a blackbody look happens at timesteps 0 and \(n\), then, the correlation between any two samples \(x_{i}\) and \(x_{j}\) at arbitrary timesteps \(i\) and \(j\) where \(0\leq i<j\leq n\) is the following:  
\[Ecorr_{C_{B}}\left(x_{i},x_{j}\right)=\frac{1}{\sigma_{i}\sigma_{j}}\left( \frac{(n-i)(n-j)}{n^{2}}+\frac{ij}{n^{2}}\right), \tag{12}\]  
where \(\sigma_{i}=\sqrt{\frac{(n-i)^{2}+i^{2}}{n^{2}}}\) and \(\sigma_{j}=\sqrt{\frac{(n-i)^{2}+i^{2}}{n^{2}}}\). The actual values of the sampled blackbodies do not matter, as the correlated error is simply a function of how far the samples are from the blackbody looks in time. Errors in \(P_{B}\) are due to misestimations of the blackbody's true noise power, which may be because the thermocouple is measuring incorrectly. We assume that the errors of this nature not only are slowly varying compared to the timescales of interest but, because of the marginal absolute magnitude, factor a negligible and unmeasurable amount in the overall correlated error structure. As such, the model ignores this term. The rolled-up correlated error model from the sources in \(P^{S}\) between any arbitrary samples \(i\) and \(j\) can be expressed as follows:  
\[K_{P^{S}}(i,j)=\left[ \begin{array}{c}\alpha\ \cdot E(C)^{2}\cdot Ecorr_{C}(i,j)+ \\ \beta\cdot E(C_{N})^{2}\cdot Ecorr_{C_{N}}(i,j)+\\ \beta\cdot E(P_{r})^{2}\cdot Ecorr_{P_{i}}(i,j)+\\ E(C_{B})^{2}\cdot Ecorr_{C_{B}}(i,j)\end{array} \right] \tag{10}\]  
Because this model estimates the correlated error in each term that calculates \(P^{S}\), this contribution to the overall correlated error is not multiplied by the rolled-up error magnitude \(E(P^{S})\). As such, we do not normalize this construction, as it will be normalized when combined with the other constituent terms in \(R_{mod}\). \(R_{P^{S}}\) contains two tuning parameters: \(\alpha\) is used to size uncorrelated white-noise error, and \(\beta\) is used to size the magnitude of totally correlated errors.