# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix A
\begin{array}{l}K_{\mathrm{FF}}=\left\langle\frac{\partial F}{\partial X}( X-\langle X\rangle)+\frac{\partial F}{\partial Y}(Y-\langle Y\rangle)\right\rangle \left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle)+\frac{\partial F }{\partial Y}(Y-\langle Y\rangle)\right\rangle^{T}\\ =\left[\left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle)\right\rangle +\left\langle\frac{\partial F}{\partial Y}(Y-\langle Y\rangle)\right\rangle \right]\left[\left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle) \right\rangle+\left\langle\frac{\partial F}{\partial Y}(Y-\langle Y\rangle) \right\rangle\right]^{T}\\ =\left[\left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle)\right\rangle \right]\left[\left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle) \right\rangle\right]^{T}+\left[\left\langle\frac{\partial F}{\partial Y}(Y- \langle Y\rangle)\right\rangle\right]\left[\left\langle\frac{\partial F}{ \partial Y}(Y-\langle Y\rangle)\right\rangle\right]^{T}\\ \quad+\left[\left\langle\frac{\partial F}{\partial X}(X-\langle X\rangle) \right\rangle\right]\left[\left\langle\frac{\partial F}{\partial Y}(Y- \langle Y\rangle)\right\rangle\right]^{T}+\left[\left\langle\frac{\partial F }{\partial Y}(Y-\langle Y\rangle)\right\rangle\right]\left[\left\langle\frac{ \partial F}{\partial X}(X-\langle X\rangle)\right\rangle\right]^{T}\\ =\left(\frac{\partial F}{\partial X}\right)^{2}K_{XX}+\left(\frac{\partial F}{ \partial Y}\right)^{2}K_{YY}+\left(\frac{\partial F}{\partial X}\right)\left( \frac{\partial F}{\partial Y}\right)K_{XY}+\left(\frac{\partial F}{\partial X }\right)\left(\frac{\partial F}{\partial Y}\right)K_{YX}\ \#\\ \end{array}  \tag{15}\]  
Substituting \(E_{n}^{2}=\left(\frac{\partial F}{\partial n}\right)^{2}\sigma_{n}^{2}\) from Equation (11), and noting the decomposition in Equation (10),  
\[K_{FF}=E_{x}^{2}R_{XX}+E_{y}^{2}R_{YY}+E_{x}E_{y}R_{XY}+E_{x}E_{y}R_{YX} \tag{16}\]  
The construction of \(R_{mod}\) in Equation (7) ignores the cross-correlation between component terms, i.e., we assert \(R_{XY}=R_{YX}=0\). This is for two main reasons:  
1. The terms \(K_{n}\), as described in Appendices B-E, are not constructed from random variables but rather through analytic specification to emulate the expected correlated behavior. We generally have insufficient knowledge to measure or estimate the cross-correlation _between_ error components. Instead, this model simply estimates the cross-correlation of error _within_ individual components, which are then added independently. 2. Any residual cross-correlation between error components can be tuned per our tuning parameters. This formulation assumes that each component term of the error \(E_{n}\) comes from a wide-sense stationary distribution where the error magnitudes are treated as constants. This means that for each component of \(K_{FF}\), and noting Equation (A2),  
\[\left(\frac{\partial F}{\partial X}\right)^{2}\!K_{XX}=\left(\frac{\partial F}{ \partial X}\right)^{2}\!S_{X}R_{XX}S_{X}=\left(\frac{\partial F}{\partial X} \right)^{2}\!\sigma_{\mathrm{X}}^{2}R_{XX}=E_{x}^{2}R_{XX}\] (A12)  
For our error correlation model \(R_{mod}\), we simply construct \(R_{XX}\) analytically for each term, for which we adopt the nomenclature \(Ecorr\) to emphasize that it is an error correlation function. Therefore, from Equation (5) (reproduced here),  
\[R_{mod}(i,j)=\frac{1}{\mathcal{N}}\!\sum_{n}K_{n}(i,j),\]  
where \(K_{n}(i,j)=E_{n}^{2}\cdot Ecorr_{n}(i,j)\). For this model, \(F\) is drawn from Equation (4) (reproduced here):  
\[\sigma^{o}=\frac{P\!\!\!^{g}4\pi L^{atm}L^{Z}G^{Z}}{P^{Z}\xi^{E}G^{R}R^{tot} \bar{A}},\]  
where we use the parameters with the largest error magnitude, \(\sigma^{o}=F(P\!\!\!^{g},P\!\!\!^{Z},G^{Z},G^{R},\xi^{E})\).