# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## 3 Results
### Dynamic Correlated Error Estimation and Impact of Tuning  
The model \(R_{mod}\) can produce plausible dynamics, suggesting the broader importance of a realistic dynamic correlated error model. Figure 5 plots two nearby tracks captured nearly simultaneously by the same receiver (but different GPS transmitters). Because the correlated error \(R_{mod}\) can be generated for any arbitrary observation using the bottoms-up model, the dynamics of how instrument errors decorrelate can be explored. To illustrate, we choose an arbitrary exemplar index (i = 129) to demonstrate that this calculation can be performed for any sample within a track. If the observations were treated as completely independent without any correlated error, the matrix in Figure 5a would contain non-zero elements exclusively along the main diagonal. However, we observe several structural elements. The 'pixelation' pattern is largely a result of the \(R_{\xi}\) term and originates from the coarse mapping of the estimated GPS antenna gain pattern as a function of scattering incidence \(\theta_{inc}\). This phenomenon is explored in depth in Appendix D.  
Figure 5: Generated untuned and tuned \(R_{mod}\) for two nearby tracks captured by the same CYGNSS receiver, each with three different representations. (**a**) Untuned \(R_{mod}\) as represented by sample index. The two tracks are concatenated in a vector, and the modeled error correlation is calculated for each combination of indices. The green vertical line represents an exemplar index (i = 129) along the track, where the correlated error is estimated for every other observation in the neighborhood. (**b**) Untuned \(R_{mod}\) for the same exemplar index as represented by the physical location of sample acquisition. (**c**) Untuned \(R_{mod}\) for the same exemplar index represented by time of acquisition. The traces for the two different tracks are plotted in different colors. The error correlation along the same track is in blue, while the error correlation for the adjacent track is in red. (**d**) Tuned \(R_{mod}\) as represented by sample index for the same two tracks as in (**a**). (**e**) Tuned \(R_{mod}\) for the same tracks represented in the physical location. (**f**) Tuned \(R_{mod}\) for the same exemplar index represented by the time of acquisition. Note that for both the untuned and tuned cases, the correlated error for the adjacent track is non-zero but generally very small compared to the error correlation along the track. The smooth decorrelation roll-off is from the \(R_{G^{R}}\) and \(R_{G^{Z}}\) components of the model and represent the direct and reflected signals moving about the nadir and zenith antenna patterns in CYGNSS receivers as the observation is collected along the track. We note that the model allows for cross-track correlation where the two different tracks share a CYGNSS receiver but have a different GPS transmitter. In both the untuned and tuned cases, these tracks appear to have uncorrelated cross-track error. There are residual correlations between the tracks when the scattering geometry is such that the two tracks share similar incidence angles or are near similar antenna coordinates, in addition to the shared receiver noise at lag \(\tau=0\). We can also determine \(R_{mod}\) for our matchup tracks where CYGNSS observes similar track geometries with the same GPS transmitter but with two different receivers. The correlated structure for one of the tracks in the previous example (but matched by different receivers) is shown in Figure 6.  
The model \(R_{mod}\) allows for correlated error between two different receivers that share a GPS transmitter. We note that observations sharing the same GPS transmitter may contain correlated error due to the correlated misestimation of GPS transmit power. This may be an incomplete articulation of the full cross-receiver error correlation. For example, CYGNSS assets in similar orbital regimes may experience similar thermal and environmental conditions that cause correlated errors during our timescales of interest.