## 2 Methods
### Averaging Multiple Images  
With a noise strength of 1 ppm at 2 \(\times\) 2 km\({}^{2}\) resolution, the enhancements from power plants with emissions less than about 5 Mt CO\({}_{2}\) yr\({}^{-1}\) are indistinguishable from the background noise even for the smallest pixel sizes. For larger pixels, the plume was even harder to distinguish. To account for weak sources of SO\({}_{2}\), [PERSON] et al. (2014) averaged many individual overpasses of SO\({}_{2}\) from the Ozone Monitoring Instrument (OMI) to achieve a higher effective resolution and precision than any individual OMI overpass. In this way they were able to isolate power plant plumes that are not detectable in any individual overpasses. We investigate a similar technique for simulated XCO\({}_{2}\) observations in this study.  
For the purpose of this method, medium-sized 4 \(\times\) 4 km\({}^{2}\) square pixels were used with 0.5 ppm noise. The wind speed was \(u=4\) m s\({}^{-1}\), and the corresponding stability parameter is \(a=156.0\) m. Enhancements were simulated on a 60 km \(\times\) 60 km domain at 50 m resolution with a medium size 13 Mt yr\({}^{-1}\) source in the center of the domain for six different wind directions (Figure 3). The observations for each of the six images were first averaged to the 4 \(\times\) 4 km\({}^{2}\) pixel size, and the 0.5 ppm noise was added to simulate observation by such an instrument. The observations were then rotated back so that the plumes are all aligned. At this stage, the 4 km \(\times\) 4 km grids for each wind direction are misaligned. To average the enhancements, the observations for each direction must be interpolated onto a uniform grid. We interpolate the enhancements onto the original 60 km \(\times\) 60 km grid at 50 m resolution using a nearest neighbour method. Pixels in the corners of the regular grid that were not covered by the rotated grids use the nearest value of the rotated grid. Figure 4 shows the enhancements before and after being interpolated onto the regular grid. Finally, the observations across the images were averaged, resulting in a clear plume outline with reduced background noise.  
Figure 4: Simulated noisy XCO\({}_{2}\) observations relative to a 400 ppm background for 4 \(\times\) 4 km\({}^{2}\) pixels after rotating to align the plumes and interpolating the rotated grids back onto a regular grid aligned with the plume. Columns 1 and 3 show the images at their 4 km \(\times\) 4 km pixel resolution immediately after rotating to align the plumes. Columns 2 and 4 show the images after interpolating onto a higher resolution (50 m) regular grid using a nearest neighbour method.  
Figure 3: Simulated images of XCO\({}_{2}\) concentrations from a 13 Mt yr\({}^{-1}\) source relative to a constant 400 ppm background for six different wind directions. Angles are measured counter clockwise from the positive \(x\)-axis, or eastward. Panel labels identify the angle used for each panel.  
To more accurately estimate the source emission rates, the emission sensitivity function \(\alpha\) (Equation (3)) must be computed in the same way as the simulated observations. It is computed for each wind direction, averaged to the pixel size, rotated back to 0\({}^{\circ}\), interpolated onto the regular grid, and the samples are averaged to determine the final value.