# Multi-LEO Satellite Stereo Winds
## 2 Materials and Methods
### Stereo Wind Retrievals  
In the GEO-GEO application, we designate one of a multi-temporal triplet of GEO satellite scenes as the reference and extract all pattern templates from it. The Level-1 products from operational GEO environmental satellites, such as GOES, Himawari, and Meteosat, are geo-registered to a reference ellipsoid (e.g., WGS-84 or GRS-80) so that points on the terrain surface will generally show a disparity with a component that is in proportion to the terrain height above the ellipsoid when feature matching between satellites. In the multi-LEO case, the LEO Level-1 products may not be geo-registered to the ellipsoid but instead registered to the terrain surface. There are two important differences between this multi-LEO stereo-winds study and our previous work [28; 29; 30; 31]. One is that the multi-LEO wind retrievals derive heights with respect to the terrain surface rather than the ellipsoid. Another difference is the level of overdetermination in the retrieval process associated with the multi-LEO measurements. Overdetermination helps to better constrain the retrieval problem and provide a more robust data quality control. The wind retrieval process is fundamentally a fit to model parameters and each disparity brings two scalar pieces of information. If there are \(N\) scenes, including the reference one, there are \(N-1\) such disparities or \(2(N-1)\) pieces of information. The GEO-GEO legacy retrieval model solves for five parameters: two horizontal wind vector components and a position adjustment with one vertical and two horizontal components. The vertical position adjustment represents the wind vector height and the horizontal position adjustments absorb the parallax shift of the feature, as seen in the reference scene plus any geo-registration errors. Therefore, the GEO-GEO model must have 2(\(N-1\)) > 5 to be overdetermined. This is easily achieved in a GEO-GEO configuration, with three repetitions of a scene from the reference satellite and two repetitions taken from the other satellite for \(N=5\). Overdetermination all but assures that the residual disparities after the retrieval at each site are not exactly zero. This enables retrieval quality screening based on the residual statistics. Those retrievals for which the residuals are anomalous with respect to the ensemble statistics can be flagged as suspect as their disparities have not been accurately interpreted by the retrieval model in terms of the model states. On the other hand, the LEO-GEO case may have less overdetermination. If the LEO sensor is MODIS or VIIRS, then three GEO scenes and one LEO scene count for \(N=4\). This case is only overdetermined in the five-state model by a single scalar observation and the residuals, which sit in a tangent plane on the ellipsoid, tend to elongate in directions perpendicular to the parallax. Moreover, working with a pair or triplet of LEO sensors such as MODIS or VIIRS (\(N<4\)) clearly provides insufficient information from which to retrieve the five states, while being sufficient for non-stereo winds. To admit the triple-LEO case, we have reformulated the retrieval problem (Appendix A) to only have three states: two wind velocity vector components and the height. The horizontal position for the wind vector is constrained so that the wind vector 3D-position within the atmosphere must lie on the line-of-site between the reference satellite and the center of the feature template on the terrain surface as indicated in Figure 1. In this formulation, we need 2(\(N-1\)) > 3 to be overdetermined and therefore no fewer than three satellites are required for a stereo retrieval. Such triple-LEO configurations can have the ground tracks of all LEO satellites generally parallel to each other (tandem) or one of the satellite ground tracks crossing the other two.