# Multi-LEO Satellite Stereo Winds
## Appendix A
We rearrange Equation (1) to isolate \(\overset{\rightarrow}{\epsilon}_{n}\) on the left-hand side:  
\[\overset{\rightarrow}{\epsilon}_{n}=\lambda\ \bigg{(}\overset{\rightarrow}{r}_{0}( \mathcal{O})-\overset{\rightarrow}{R}_{n}+\overset{\rightarrow}{\delta} \bigg{)}-\bigg{(}\overset{\rightarrow}{r}_{n}(\mathcal{O})-\overset{ \rightarrow}{R}_{n}\bigg{)}. \tag{2}\]  
Equation (2) is used to derive an equation for \(\lambda\) from the constraint \(\hat{\boldsymbol{z}}_{n}\cdot\overset{\rightarrow}{\epsilon}_{n}=0\),  
\[\lambda=\frac{\hat{\boldsymbol{z}}_{n}\cdot\bigg{(}\overset{\rightarrow}{r} _{n}(\mathcal{O})-\overset{\rightarrow}{R}_{n}\bigg{)}}{\hat{\boldsymbol{z}}_ {n}\cdot\bigg{(}\overset{\rightarrow}{r}_{0}(\mathcal{O})-\overset{ \rightarrow}{R}_{n}+\overset{\rightarrow}{\delta}\bigg{)}}. \tag{3}\]  
Equation (3) is nonlinear in \(\overset{\rightarrow}{\delta}\), but given a guess \(\overset{\rightarrow}{\delta}_{0}\) for the value of \(\overset{\rightarrow}{\delta}\), we can linearize it in a Taylor series:  
\[\lambda\ \cong\frac{\hat{\boldsymbol{z}}_{n}\cdot\bigg{(}\overset{\rightarrow}{r }_{n}(\mathcal{O})-\overset{\rightarrow}{R}_{n}\bigg{)}}{\hat{\boldsymbol{z}} _{n}\cdot\bigg{(}\overset{\rightarrow}{r}_{0}(\mathcal{O})-\overset{ \rightarrow}{R}_{n}+\overset{\rightarrow}{\delta}_{0}\bigg{)}}-\frac{\hat{ \boldsymbol{z}}_{n}\cdot\bigg{(}\overset{\rightarrow}{r}_{n}(\mathcal{O})- \overset{\rightarrow}{R}_{n}\bigg{)}}{\left(\hat{\boldsymbol{z}}_{n}\cdot \bigg{(}\overset{\rightarrow}{r}_{0}(\mathcal{O})-\overset{\rightarrow}{R}_ {n}+\overset{\rightarrow}{\delta}_{0}\bigg{)}\right)^{2}}\hat{\boldsymbol{z}}_ {n}\cdot\bigg{(}\overset{\rightarrow}{\delta}-\overset{\rightarrow}{\delta}_ {0}\bigg{)}. \tag{4}\]  
Substituting \(\lambda\) from Equation (4) into (2) and discarding terms higher than linear in \(\overset{\rightarrow}{\delta}-\overset{\rightarrow}{\delta}_{0}\) gives an approximate linear measurement model:  
\[\overset{\rightarrow}{\epsilon}_{n}=A_{n}\bigg{(}\overset{\rightarrow}{ \delta}_{0}\bigg{)}\cdot\bigg{(}\overset{\rightarrow}{\delta}-\overset{ \rightarrow}{\delta}_{0}\bigg{)}-\overset{\rightarrow}{b}_{n}\bigg{(} \overset{\rightarrow}{\delta}_{0}\bigg{)}. \tag{5}\]  
The matrix \(A_{n}\bigg{(}\overset{\rightarrow}{\delta}_{0}\bigg{)}\) and vector \(\overset{\rightarrow}{b}_{n}\bigg{(}\overset{\rightarrow}{\delta}_{0}\bigg{)}\) are defined accordingly (with \(I\) as the identity matrix):  
\[A_{n}\bigg{(}\overset{\rightarrow}{\delta}_{0}\bigg\[\lambda_{0}=\frac{\hat{z}_{n}\cdot\left(\overset{\rightarrow}{r}_{n}(\mathcal{O})- \overset{\rightarrow}{R}_{n}\right)}{\hat{z}_{n}\cdot\left(\overset{ \rightarrow}{r}_{0}(\mathcal{O})-\overset{\rightarrow}{R}_{n}+\overset{ \rightarrow}{\delta}_{0}\right)}. \tag{10}\]  
We next parameterize \(\overset{\rightarrow}{\delta}(t)=h\frac{\overset{\rightarrow}{r}_{0}( \mathcal{O})-\overset{\rightarrow}{R}_{0}}{\left(\overset{\rightarrow}{r}_{0}( \mathcal{O})-\overset{\rightarrow}{R}_{0}\right)\hat{z}_{0}}+\overset{ \rightarrow}{V}\cdot(t-t_{0})\) so that at \(t=t_{0}\), \(\overset{\rightarrow}{\delta}(t_{0})\) is parallel to the line-of-sight from the apparent location in the reference scene to the reference satellite with the height being the projection onto the local vertical \(\hat{z}_{0}\) and further require that the wind velocity \(\overset{\rightarrow}{V}\) is perpendicular to \(\hat{z}_{0}\) (i.e., \(\overset{\rightarrow}{V}=V_{u}\hat{u}_{0}+V_{v}\hat{v}_{0}\) for vectors in the local tangent plane \(\hat{u}_{0}\) and \(\hat{v}_{0}\) directed towards the east and north, respectively).