# Compact Midwave Imaging System: Results from an Airborne Demonstration
## 4 Discussion
### Implications for Notional Spaceflight
#### 4.1.1 Accuracy  
Some very simple geometric relationships can help us estimate stereo wind retrieval accuracies and scale them from an airplane to a spacecraft (our retrieval software using a full fidelity and nonlinear model). Such estimates can be confirmed from the retrieval process, which outputs the three retrieval states (two horizontal wind vector components and the height) along with their 3 \(\times\) 3 covariance matrix. The latter represents the uncertainties in the states with modeled errors in the input disparities. Figure 20 shows an idealized configuration with a single CMIS flying over the locally flat Earth surface (or a tangent plane to the WGS-84 ellipsoid). We assume that CMIS flies at a constant altitude \(H\) and ground speed \(V\) along the \(x\)-axis. The three stereo line arrays are perpendicular to the direction of motion and parallel to the \(y\)-axis (into the paper). Consider only those points directly under the flight track (\(y=0\)). A cloud feature seen by each line array is assigned apparent geographic coordinates \((x_{n},\ y_{n})\) in each of the three looks (\(n=\) forward, nadir, aft) at acquisition times \(t_{n}\). The angle \(\alpha\) is fixed by the CMIS camera design. \(\angle AB^{\prime}C^{\prime}\) as shown in Figure 20 allows for the solution for the acquisition time difference, which is substituted into the tangent formula for \(\angle ABC\). In a spacecraft application, the wind speed and height are typically much less than the ground speed and altitude of the spacecraft (i.e., \(h\ll H\) and \(v_{x}\ll V\)), which leads to an approximately linear relationship between the states  
\[\left(x_{f}-x_{n}\right)=v_{x}\cdot\frac{H}{V}tan(\alpha)+h\cdot tan(\alpha). \tag{1}\]  
A nearly identical formula applies to the aft-nadir combination. Since it has the same coefficients as Equation (1), the height \(h\) and AT velocity \(v_{x}\) cannot be decoupled and an assumption must be made about one of these states; however, using the aft-nadir combination does reduce the uncertainty in the retrieval. If the AT velocity is assumed zero ('zero wind' retrieval) when it is not, we can see that this will bias the retrieved height by  
\[\Delta h=\frac{H}{V}v_{x}. \tag{2}\]  
Figure 20: Simplified CMIS configuration allows linear design rules to be derived using elementary geometry. Non-simultaneous forward, nadir, and aft looks are shown for a single camera. Forward and aft coordinates for a cloud feature seen in the nadir look are given at the matched locations in the forward- and aft-looks. The XT wind \(v_{y}\) can be similarly written as a linear function of only the XT coordinate difference by eliminating the time difference in the same manner,  
\[\left(y_{f}-y_{n}\right)=v_{y}.\frac{H}{V}tan(\alpha). \tag{3}\]  
A second acquisition of the tracked feature from another platform will provide the additional information to allow the decoupling of \(h\) and \(v_{x}\). This could be an acquisition from a geostationary (GEO) satellite or another low-Earth orbiter (LEO). The GEO-LEO combination has been extensively discussed in [PERSON] et al. (2018) and demonstrated with MISR and GOES-R to achieve better than 200 m height and 0.5 m/s retrieval accuracies. It offers the advantage of allowing the effective transfer of Image Navigation and Registration (INR)--or geolocation--knowledge from the presumably high accuracy GEO to the more modest accuracy LEO using a 'bundle adjustment'. The bundle adjustment is possible because of the overdetermination of the dual-satellite retrievals. Of particular interest for this paper is the case where a second CMIS follows the first. Matches would be made from the nadir look of the leading satellite into up to five other looks (leader-forward, leader-aft, follower-forward, follower-nadir, and follower-aft). This configuration enables a constellation of CMIS sensors to cover areas without GEO coverage, including the poles, for a full global capability.