# Compact Midwave Imaging System: Results from an Airborne Demonstration
## 4 Discussion
### Implications for Notional Spaceflight
#### 4.1.1 Accuracy
Supposing that the leader-follower time separation is \(T\), we consider the nadir looks as shown in Figure 21. The acquisition time difference can be solved for using the spacecraft (S/C) equation to eliminate it in the other, and after linearizing in the variable \(v_{x}\) and similarly for \(v_{y}\):  
\[(x_{n2}-x_{n1})=v_{x}.T, \tag{4}\]  
\[(y_{n2}-y_{n1})=v_{y}.T. \tag{5}\]  
Figure 21: Leader–follower matches enable unambiguous determination of the AT winds. Consecutive nadir acquisitions are shown, which is the simplest case. Equations (1), (3) and (4) can be used to calculate, for example, \(\left(x_{f2}-x_{n1}\right)=\left(x_{f2}-x_{n2}\right)+\left(x_{n2}-x_{n1}\right)\). The full systems of equations for the states \((h,\,v_{x})\) and \(v_{y}\) are therefore  
\[\left[ \begin{array}{c}x_{f1}-x_{n1}\\ x_{a1}-x_{n1}\\ x_{f2}-x_{n1}\\ x_{a2}-x_{n1}\\ x_{n2}-x_{n1}\end{array} \right]=\left[ \begin{array}{cc}tan(\alpha)&(H/V)tan( \alpha)\\ -tan(\alpha)&-(H/V)tan(\alpha)\\ tan(\alpha)&T+(H/V)tan(\alpha)\\ -tan(\alpha)&T-(H/V)tan(\alpha)\\ 0&T\end{array} \right]\left[ \begin{array}{c}h\\ v_{x}\end{array} \right]\!, \tag{6}\]  
\[\left[ \begin{array}{c}y_{f1}-y_{n1}\\ y_{a1}-y_{n1}\\ y_{f2}-y_{n1}\\ y_{a2}-y_{n1}\\ y_{n2}-y_{n1}\end{array} \right]=\left[ \begin{array}{c}(H/V)tan(\alpha)\\ -(H/V)tan(\alpha)\\ T+(H/V)tan(\alpha)\\ T-(H/V)tan(\alpha)\\ T\end{array} \right][v_{y}]. \tag{7}\]  
We can now estimate retrieval accuracy considering the geolocation accuracy expected from a low-cost CubeSat system flying at an International Space Station (ISS) altitude \(H=410\) km with \(V=7.7\) km/s. We will also assume \(\alpha=20^{\circ}\) and a XT field of view of \(\pm 25^{\circ}\), which is similar to the optical configuration of the air campaign. Using GPS allows for very accurate positioning of the spacecraft (\(\sim\)4 m) and time tagging of observations; therefore, attitude knowledge and calibration of the optics are likely the drivers in terms of geolocation accuracy. A pixel count of 1280 provides for a ground footprint of 150 m. At this resolution, pointing knowledge of 0.005\({}^{\circ}\) that is obtainable in a CubeSat translates to \(\sim\)40 m on the ground, which is about a quarter of a pixel. Matching also suffers from some uncertainty, which from our experience with GEO-LEO and GEO-GEO stereo winds can be expected to be \(\sim\)0.5 pixels. Therefore, a good design goal for the uncertainties in each \(\Delta x\) and \(\Delta y\) would be \(\sigma=\sqrt{2\cdot 0.25^{2}+0.5^{2}}\ \sim\ 0.6\) pixel (i.e., 100 m). Equation (5) are overdetermined, so they should be solved by least-squares. Designating the coefficient matrices in Equation (5) as \(M_{x}\) and \(M_{y}\), and assuming the uncertainties in each \(\Delta x\) and \(\Delta y\) are independent from each other enables an estimate of the retrieval accuracy. The covariances for the states are  
\[cov(h,\,v_{x})=\sigma^{2}\Big{(}M_{x}^{T}M_{x}\Big{)}^{-1}, \tag{8}\]  
\[cov\big{(}v_{y}\big{)}=\sigma^{2}\Big{(}M_{y}^{T}M_{y}\Big{)}^{-1}. \tag{9}\]  
Since the time \(T\) between leader and follower has not been specified, we vary that parametrically. Figure 22 plots the retrieval uncertainties (square-roots of the diagonal elements of the covariances) using this model as a function of \(T\). This solution assumes no cloud deformation with time. For certain cloud types such as small cumulus, the overall lifetime could be less than 10 min, so the reduction in uncertainly for winds may not always be realized for such cases. A trailing satellite separated by 15 min from the leader would be approximately one-sixth of an orbital period behind (at a 410-km altitude). Given the rapid decrease in retrieval uncertainty with increased distance between the leader and follower, the optimal separation is likely to be 5 min or less. For single-camera zero-wind retrievals, Equation (2) provides the systematic error, and the random error is estimated by coefficient matrices in the reduced system of equations  
\[\left[ \begin{array}{c}x_{f}-x_{n}\\ x_{a}-x_{n}\end{array}