# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 2 Materials and Methods
### Verification Techniques
#### 2.3.4 Model Tuning  
The constructed correlated error model \(R_{mod}(i,j)\) is designed to estimate the correlated error between arbitrary samples \(i\) and \(j\). Validating the correlated error with empirical matches is challenging since the empirical error correlation can only be estimated in a broader, statistical sense. To create an appropriate comparison, we introduce the _modeled_ error autocorrelation \(\widetilde{\rho}_{mod}\), which is a function of lag \(\tau\):  
\[\widetilde{\rho}_{mod}(\tau)=\frac{1}{N-\tau}\sum_{i=1}^{N-\tau}{R_{mod}(i,i+ \tau)}, \tag{12}\]  
where \(R_{mod}\) is calculated using Equation (7), \(\tau\) is the lag, and \(N\) is the total number of lags observed. The quantity \(\widetilde{\rho}_{mod}\) is reasonably comparable to the autocorrelation \(\rho(\tau)\) of observed error for a single track of samples. A similar analog can be made for the _bulk  
Figure 2: A display of model (blue) and observed (green) single-differenced matchup data for a single CYGNSS track, with the double-differenced data (red) overlaid. Note that both the model and observed single differences are similarly high early in the track, suggesting that the differences in the observations may be because the samples are observing fundamentally different targets, captured in \(SD_{mod}\). The double-difference accounts for these types of errors, and for the entire track, the double-difference is quite stable. The sparsity of data is caused by quality control parameters, such as when the CYGNSS observatory is performing its onboard calibration procedure, which occurs once every minute. Because the two matched observatories do not have synchronized calibration clocks, the matched dataset typically flags out two of these cadences for every minute of sampling. modeled_ error autocorrelation \(\widetilde{\mathcal{R}}_{mod}(\tau)\), which estimates the autocorrelation behavior of the error model across \(M\) tracks:  
\[\widetilde{\mathcal{R}}_{mod}(\tau)=\frac{1}{M}{\sum_{j=1}^{M}}\widetilde{ \rho}_{mod,j}\left(\tau\right)\,, \tag{13}\]  
where \(\widetilde{\rho}_{mod,j}\left(\tau\right)\) is the modeled autocorrelation for track \(j\) at lag \(\tau\). A parametrized version of \(R_{mod}\) is constructed using the following form:  
\[R_{mod}(\alpha,\beta,\gamma,\delta)|_{i,j}=\frac{1}{\mathcal{N}}\left[R_{ps} (\alpha,\beta)+R_{p^{Z}}(\alpha,\beta)+R_{G^{R}}(\gamma,\delta)+R_{G^{Z}}( \gamma,\delta)+R_{\zeta}(\beta)\ \right]\,, \tag{14}\]  
where the following is true:  
* \(\alpha\) represents the relative magnitude of the white noise component of the error, which decorrelates at \(\tau=1\);
* \(\beta\) represents the relative magnitude long-decay pedestal or any residual correlated errors at the edge of our timescales of interest;
* \(\gamma\) represents the relative magnitude of the correlated error caused by the terms \(R_{G^{R}}\) and \(R_{G^{Z}}\), which exhibit smooth decay as samples spread apart when projected through the nadir and zenith antenna coordinates, respectively [see Appendix D for an in-depth discussion]; and
* \(\delta\) represents the relative decorrelation roll-off in terms \(R_{G^{R}}\) and \(R_{G^{Z}}\). With an appropriate benchmark, the tuning parameters \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are iterated such that \(\widetilde{\mathcal{R}}_{mod}\) matches a target signal. The tuning parameters are applied such that they can be modified to change specific characteristics of the modeled behavior:  
The parameters modify \(R_{mod}\) directly and are described in depth in Appendices B-E. The target for matching the modeled autocorrelation \(\widetilde{\mathcal{R}}_{mod}\) is the bulk autocorrelation of the double-differenced data DD:  
\[\mathcal{R}[DD](\tau)\cong\widetilde{\mathcal{R}}_{mod}(\tau) \tag{15}\]  
Because this is an under-determined system, there is no unique solution to optimizing the tuning parameters. Further, these parameters all relate to one another, and changing one will impact the others.