# Multi-LEO Satellite Stereo Winds
## Appendix A  
The math model of the retrieval algorithm for the multi-LEO case is adapted from the MISR-GEO [30] and MODIS-GEO [31] models. Each satellite image is assumed to have been remapped into common coordinates (e.g., a polar stereographic projection) so that disparities can be easily found by pattern matching. It is applicable to any constellation of satellites (including only GEO satellites) and sensors with and without multi-angle capabilities. The reduced solution order (three versus the five variables of our other models [28; 29; 30; 31]) allows a solution to be found with fewer observations. Consider a moving object \(\mathcal{O}\) (e.g., the cloud shown in Figure 1) and let \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\) be where it appears to rest on the surface (or ellipsoid if imagery is registered to the ellipsoid rather than the terrain surface) as viewed from satellite/look \(n\) at the respective observation times \(t_{n}\). Reserve \(n=0\) to designate the reference scene from which we draw the template. The true position vector for the object \(\mathcal{O}\) would be \(\overset{\rightarrow}{r}(\mathcal{O})=\overset{\rightarrow}{r}_{0}(\mathcal{ O})+\overset{\rightarrow}{\delta}(t)\), where \(\overset{\rightarrow}{\delta}(t)\) is the adjustment needed to position \(\mathcal{O}\) in 3D space off the surface (or ellipsoid) at time \(t\). We seek to discover \(\overset{\rightarrow}{\delta}\), for which the component in the direction of the local vertical would be the height assignment and the trajectory of \(\overset{\rightarrow}{\delta}\) that is parallel to the surface of the ellipsoid would represent the wind. Consider a single disparity pairing for which the matching algorithm has found the coordinates for \(\mathcal{O}\) as it would appear from satellite/look \(n\) were it to sit on the surface (or ellipsoid). The line-of-sight to \(\mathcal{O}\) for satellite/look \(n\) starts at the position vector for the satellite (\(\overset{\rightarrow}{R}_{n}\)) and runs parallel to the vector touching the apparent location of \(\mathcal{O}\) on the surface (or ellipsoid) \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\) and must run through the 3D location of the object \(\overset{\rightarrow}{r}(\mathcal{O})=\overset{\rightarrow}{r}_{0}(\mathcal{O })+\overset{\rightarrow}{\delta}\). Expressed mathematically, there must be some multiplier \(\lambda^{-1}\) that makes this equation hold:  
\[\overset{\rightarrow}{r}_{0}(\mathcal{O})+\overset{\rightarrow}{\delta}= \lambda^{-1}\bigg{(}\overset{\rightarrow}{r}_{n}(\mathcal{O})-\overset{ \rightarrow}{R}_{n}+\overset{\rightarrow}{\epsilon}_{n}\bigg{)}+\overset{ \rightarrow}{R}_{n}. \tag{1}\]  
The apparent position of \(\mathcal{O}\) on the surface (or ellipsoid) \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\) could not have been determined perfectly, so we have introduced a vector \(\overset{\rightarrow}{\epsilon}_{n}\) in Equation (1) to make equality hold that we can interpret as deriving from an error or uncertainty in the disparity measurement. Normally, \(\overset{\rightarrow}{\epsilon}_{n}\) should be small. All imagery has been mapped into common coordinates, the point of correspondence is found where the template best matches, the coordinates of the match inverted to determine the latitude and longitude, and finally the apparent position vector (\(\overset{\rightarrow}{r}_{n}(\mathcal{O})\)) is calculated. In doing so, \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\) can only rest on the surface (or ellipsoid), so we can consider small deviations in the calculations represented by \(\overset{\rightarrow}{\epsilon}_{n}\) as sitting in the tangent plane at \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\); i.e., \(\hat{\boldsymbol{z}}_{n}\overset{\rightarrow}{\epsilon}_{n}=0\), where \(\hat{\boldsymbol{z}}_{n}\) is taken to be the local vertical at \(\overset{\rightarrow}{r}_{n}(\mathcal{O})\).