# An Instrument Error Correlation Model for Global Navigation Satellite System Reflectometry
## Appendix A  
The construction of \(R_{mod}\) makes a few important assumptions that are worth examining in depth. First, we will describe the general mathematical framework. In general, if \(X\) and \(Y\) are random vectors of length \(N\), then the covariance matrices of \(X\) and \(Y\) are constructed:  
\[K_{XX}=\langle X-\langle X\rangle\rangle\langle X-\langle X\rangle\rangle^{T}, \tag{11a}\] \[K_{YY}=\langle Y-\langle Y\rangle\rangle\langle Y-\langle Y\rangle\rangle^{T}, \tag{11b}\]  
where \(K_{XX}\) and \(K_{YY}\) are the \(N\)-by-\(N\) covariance matrices of \(X\) and \(Y\), respectively, the bracket operator denotes the expectation operation, and the superscript \(T\) denotes the transpose vector. Notably, the covariance matrix is symmetric and positive semi-definite. The covariance matrices can also be constructed in the following manner:  
\[K_{XX}=S_{X}R_{XX}S_{X}, \tag{12}\]  
where \(S_{X}\) is a diagonal \(N\)-by-\(N\) matrix with the standard deviations of \(X_{i}\):  
\[S_{X}= \begin{bmatrix}\sigma_{X1}&0&0\\ 0&\ddots&0\\ 0&0&\sigma_{XN}\end{bmatrix} , \tag{13}\]  
and \(R_{XX}\) is the correlation matrix with correlation coefficients \(\rho_{ij}\):  
\[R_{XX}= \begin{bmatrix}1&\rho_{X1,X2}&\cdots&\rho_{X1,XN}\\ \rho_{X2,X1}&1&&\\ \vdots&&\ddots&&\\ \rho_{XN,X1}&&&1\end{bmatrix}  \tag{14}\]As standard, \(-1\leq\rho_{i,j}\leq 1\), and the diagonals all must equal 1. The _cross_-covariance matrices can also be defined in a similar fashion:  
\[K_{XY} =\langle X-\langle X\rangle\rangle\langle Y-\langle Y\rangle\rangle^{T}, \tag{10a}\] \[K_{YX} =\langle Y-\langle Y\rangle\rangle\langle X-\langle X\rangle\rangle^{T}, \tag{10b}\]  
where \(K_{XY}\) and \(K_{YX}\) are the cross-covariance matrices and are not generally identical. The error magnitudes in Table 1 are calculated using a standard error propagation technique as highlighted in [PERSON] et al. (2015). \[E_{n}=\left|\frac{\partial F}{\partial n}\right|\Delta n \tag{11}\]  
\(E_{n}\) is the error magnitude of term \(n\), \(\Delta n\) is the estimated 1-sigma dynamic range of \(n\) and \(F\) is an arbitrary function. We can propagate these errors as the sums of covariance matrices with the derivation below, inspired by Chapter 9 in [PERSON] (2015). For an arbitrary function \(F\) with random vector arguments X and Y, \(F(X,Y)\), where \(X=(X_{1},X_{2},\ldots,X_{N})\) and \(\mathrm{Y}=(Y_{1},Y_{2},\ldots,Y_{N})\), we can approximate F by its first order Taylor Series expansion about the mean value, assuming that the errors are generally small compared to the arguments:  
\[F_{i}=F(X_{i},Y_{i}) \tag{12}\]  
\[F_{i}\approx F(\langle X\rangle,\langle Y\rangle)+\frac{\partial F}{\partial X }(X_{i}-\langle X\rangle)+\frac{\partial F}{\partial Y}(Y_{i}-\langle Y\rangle) \tag{13}\]  
Noting that \(\langle F\rangle=F(\langle X\rangle,\langle Y\rangle)\), and using \(F\approx F(\langle X\rangle,\langle Y\rangle)+\frac{\partial F}{\partial X}( X-\langle X\rangle)+\frac{\partial F}{\partial Y}(Y-\langle Y\rangle)\), then,  
\[K_{FF}=\langle F-\langle F\rangle\rangle\langle F-\langle F\rangle\rangle^{T}, \tag{14}\]  
and expanding using Equations (10), (12), and (13),  
\[