## 3 Results
### Averaging Multiple Images  
The average of six individual images with wind directions \(-150^{\circ}\), \(-75^{\circ}\), \(0^{\circ}\), \(45^{\circ}\), \(60^{\circ}\), and \(112.5^{\circ}\) is shown in Figure 10. An ensemble of 30 simulations was run to estimate the accuracy and precision of this method. The estimated emissions across the ensemble are 13.143 \(\pm\) 0.566 Mt yr\({}^{-1}\) (4.35% precision), where the true simulated emissions were 13.0 Mt yr\({}^{-1}\). Observations from individual images with no averaging led to an emission estimate of 13.137 \(\pm\) 1.886 Mt yr\({}^{-1}\). The estimate from the averaged images represents better than a three times improvement in the precision of the emission estimate. This improvement was better than the statistical increase of \(\sqrt{6}\approx\) 2.5 that we would expect having six times as many observations. Figure 10 shows the averaged images compared to the theoretical XCO\({}_{2}\) enhancements using the estimated emission rate. We see the plume is significantly more defined in comparison to the background noise. In Figure 4 the plume was still distinguishable, but after averaging the observations the background noise nearly disappears and the plume is very prominent.  
The advantages of this method are further illustrated by considering a weaker 6 Mt yr\({}^{-1}\) source. The individual images are shown in Figure 11. For such a weak source, the plume is nearly indistinguishable from the background noise. After averaging the six images, the plume clearly stands out from the background noise (Figure 12). While not as obvious as the 13 Mt yr\({}^{-1}\) source, the plume was still discernible. Across an ensemble of 30 simulations, the emission rate was estimated as 6.2055 \(\pm\) 0.6918 Mt yr\({}^{-1}\), or 11.5% precision. Using individual images with no averaging, we estimated the emissions to be 6.1646 \(\pm\) 1.7639 Mt yr\({}^{-1}\), or 29.4% precision. By averaging images, we again achieve a nearly three times improvement in precision.  
Figure 10: XCO\({}_{2}\) observations relative to a 400 ppm background for one instance of the 30 member ensemble. (**a**) Average observations from 6 simulated images at varying wind directions for a 13 Mt yr\({}^{-1}\) source. (**b**) Fitted relative XCO\({}_{2}\) enhancements from the least squares fit. The label identifies that the estimated emission rate was 12.602 Mt CO\({}_{2}\) yr\({}^{-1}\) for this individual ensemble member.  
Figure 11: Simulated images of XCO\({}_{2}\) relative to a 400 ppm background as in Figure 4, for a 6 \(\mathrm{M}\mathrm{t}\mathrm{y}\mathrm{r}^{-1}\) source.  
Figure 12: XCO\({}_{2}\) observations relative to a 400 ppm background for one instance of the 30 member ensemble for a 6 \(\mathrm{M}\mathrm{t}\mathrm{y}\mathrm{r}^{-1}\) source. (**a**) Average observations from 6 simulated images at varying wind directions. (**b**) Fitted relative XCO\({}_{2}\) enhancements from the least squares fit. The label identifies that the estimated emission rate was 6.382 \(\mathrm{M}\mathrm{t}\mathrm{y}\mathrm{r}^{-1}\) for this ensemble member.  
An ensemble of 10 simulations was run with 16 wind directions, from which we estimate the emissions to be 5.923 \(\pm\) 0.407 Mt yr\({}^{-1}\), or 6.78% precision. The averaged estimate represents more than a two times improvement in accuracy and more than a four times improvement in precision when compared to estimates from individual images.