# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 3 Results  
### Bulk Behavior  
The constructed model \(R_{mod}\) is first tuned to match the overall behavior of \(\mathcal{R}[DD](\tau)\), as described in Section 2.3.4. With the appropriate tuning parameters, the bulk-modeled autocorrelation \(\widetilde{\mathcal{R}}_{mod}\) closely resembles \(\mathcal{R}[DD](\tau)\) in most important aspects, including the relative magnitude of the white noise component, the relative rate of decorrelation roll-off, and endpoint behavior at large lags. The final tuned \(\widetilde{\mathcal{R}}_{mod}\) is shown in Figure 3 with \(\mathcal{R}[SD_{obs}]\), \(\mathcal{R}[SD_{mod}]\), and \(\mathcal{R}[DD]\). The behavior in Figure 3 is worth discussing in detail. The single-differenced model generated \(\sigma^{o}_{mod}\) decays at a much slower rate than the single-differenced observations, and double-differenced data suggests that the errors of slight mismatch in observation location and time are generating errors on a fundamentally different scale than the instrument errors. Further, the fact that the double-differenced decorrelation is almost identical to the observation single-differencing suggests that single-differencing near-simultaneous observations are a reasonable approximation for bulk error correlation. Evaluating the double-difference trace in Figure 3 also reveals important qualities of the CYGNSS error correlation structure. First, the uncorrelated error component accounts for roughly 5% of the total system error. Therefore, assumptions that samples may be treated as independent with uncorrelated error are empirically refuted. Further, the error decorrelation roll-off is swift: about 50% of the error decorrelates within 7 s of observation. Using a flat-plane approximation of Earth, this corresponds to a distance of approximately 50 km, which is generally the scale of the gridding of modern global weather models but much coarser than the resolution of state-of-the-art regional models. Finally, the endpoint behavior at large lag times suggests a small, positive correlated error (\(\sim\)2%) at larger timescales. The statistical properties of the lag correlation become less stable at larger lags, but the existence of a long-timescale correlation is not surprising, considering all observations from a single receiver share a common electronics and processing chain. The fact that our tuned model \(R_{mod}\) can approximate the bulk error correlation structure \(\widetilde{\mathcal{R}}_{mod}\) with similar features as \(\mathcal{R}[DD]\) validates the assumption that the overall instrument-correlated error can be modeled from the fundamental components of the instrument observable: in our case, the radar-range equation (Equation (2)). Further, that \(R_{mod}\) can be generated between arbitrary samples suggests that an observation error correlation matrix \(\mathbf{R}\) can be generated dynamically from first principles given appropriate knowledge about the instrument and retrieval. The value of the chosen tuning parameters is also worth investigating. The untuned model is defined by having the parameters \(\alpha=\beta=\gamma=\delta=1\). Figure 3 demonstrates that the modification of tuning parameters can change the overall model behavior significantly to match observed behavior. Figure 3 further suggests that the untuned \(R_{mod}\) generally overestimates both the relative magnitudes of uncorrelated error (i.e., white noise, tuned by \(\alpha\)) and totally correlated error (i.e., endpoint correlation, tuned by \(\beta\)). The chosen parameters for tuning are shown in Table 2.  
Figure 3: Comparison of the bulk error autocorrelation behavior for all matched tracks. The single difference of modeled \(\sigma^{o}_{mod}\) (blue), the single difference of observed \(\sigma^{o}\) (green), the double-difference \(DD\) (red), the untuned bulk model error correlation \(\widetilde{\mathcal{R}}_{mod}\) (dotted purple), and the final tuned bulk model error correlation \(\widetilde{\mathcal{R}}_{mod}\) (orange) are shown.