# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix D
Observe that convolution operations are commutative and, further, that the cross-correlation can be written as a convolution by exploiting its symmetry:  
\[K(t)*D(t)[n]=K(-t)*D(t)[n] \tag{103}\]Therefore, to evaluate the correlated error imposed by kernel \(K(t)\),  
\[F(t)\star F(t) = \{K(t)\ast D(t)\}\ast\{K(t)\ast D(t)\}\] \[= K(-t)\ast D(-t)\ast K(t)\ast D(t)\] \[= \{K(-t)\ast K(t)\}\ast\{D(-t)\ast D(t)\}\] \[= \{K(t)\star K(t)\}\ast\{D(t)\star D(t)\}\]  
If the arbitrary signal \(D(t)\) happens to be white noise, it is completely uncorrelated, and its autocorrelation collapses to a Dirac delta function centered at \(t=0\). Therefore, the entire structure of the correlated error is from the filter itself:  
\[F(t)\star F(t)=K(t)\star K(t) \tag{28}\]  
\(\square\)  
For the purposes of this work's error model, we have no knowledge of the potential correlated structure in the actual errors in the gain pattern. The nadir antenna patterns are updated after applying a 6-degree boxcar averaging filter in both the azimuthal and elevation in the spacecraft coordinate frame and then an additional 10-degree two-dimensional smoothing window. We assume that zenith antenna patterns use a similar post-processing technique during their generation. These filtering kernels act like low-pass filters. All correlated structure on scales \(\sim\)5 degrees and smaller and uncorrelated error will be strongly influenced by the filtering process, and the correlated structure can be estimated from the Filter Lemma. This model assumes that there is no residual larger-scale structure in correlated error in the antenna gain patterns. The generated filter kernel, which applies to both the nadir and zenith antenna patterns, can be shown in Figure 11. The correlated error is a function of how close any two observations are with respect to the relevant antenna gain pattern coordinates. Figure 11: The filtering kernel used to smooth nadir and zenith antenna gain patterns. This kernel imposes correlated error structure onto the antenna gain patterns. The coordinate system should be read as distance in the relevant antenna reference frame. Therefore, if two observations are nearby in the antenna pattern, they will have strongly correlated errors. However, if two observations are far apart in the pattern, the correlated structure decays. For any arbitrary samples \(i\) and \(j\), we compute the gain pattern coordinates \((\theta_{i},\phi_{i})\) and \((\theta_{j},\phi_{j})\) in the relevant antenna reference frame. To retrieve how correlated the error is, we compute the distance between the two observations in the reference frame:  
\[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]0 \[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]1  
The error correlation function is computed via a LUT of the filter kernel K:  
\[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]2 \[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]3  
This error correlation holds if the samples \(i\) and \(j\) share the same antenna and are on the same spacecraft. If they are on separate antennas or spacecrafts, the correlated error is zero. The rolled-up correlated errors for the gain patterns can be expressed as follows:  
\[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]4 \[K(t)*D(t)[n]=\sum_{t=-\infty}^{\infty}K(t)D(n+t)=\sum_{t=-\infty}^{\infty}K(t -n)D(t), \tag{102}\]5  
where two new tuning parameters have been introduced. \(\gamma\) is used to the tune the overall magnitude of the correlated error from these components, and \(\delta\) is used to scale the decorrelation roll-off rate as the samples spread in antenna coordinates.