## 2 Methods
### Gaussian Plume Model  
XCO\({}_{2}\) enhancements are simulated using a vertically-integrated Guassian plume model as in [PERSON] et al. [4] that is based on a slightly modified set of equations from those used by [PERSON] et al. [5]. The total column averaged enhancement above the background concentration is computed as  
\[V(x,y)=\frac{F}{\sqrt{2\pi}\sigma_{y}(x)u}e^{-\frac{1}{2}\left(\frac{y}{\sigma _{y}(x)}\right)^{2}}, \tag{1}\]  
where  
\[\sigma_{y}(x)=a\left(\frac{x}{x_{0}}\right)^{\gamma}. \tag{2}\]\(V(x,y)\) is the CO\({}_{2}\) vertical column density in grams per square meter (g m\({}^{-2}\)) at coordinates \((x,y)\), where \(x\) is the distance in meters downwind of the point source, and \(y\) is the distance in meters perpendicular to the wind. \(F\) is the source emission rate in grams per second (g s\({}^{-1}\)), \(u\) is the wind speed in meters per second (m s\({}^{-1}\)), \(a\) is the atmospheric stability parameter as determined from the Pasquill-Gifford stability class [7], \(x_{0}=1000\) m is the characteristic length of the across wind diffusion, and \(\gamma=0.894\) is a constant. \(\sigma_{y}(x)\) determines the rate of diffusion perpendicular to the wind. The atmospheric stability parameter \(a\) may be more accurately described as the half-width of the plume at \(x=x_{0}\), however, we follow the convention of the literature. These equations assume the motion in \(x\) is dominated by advection due to the wind, the flow is not turbulent at the scales it is modelled at, and there are no boundary layer effects.  
Enhancements are computed on the 50 m resolution grid for sources with annual emissions of 6, 13, and 25 Mt CO\({}_{2}\) yr\({}^{-1}\). The largest fossil fuel burning power plants in the world can emit \(\sim\)30 Mt CO\({}_{2}\) yr\({}^{-1}\). Those in the range of 10-20 Mt CO\({}_{2}\) yr\({}^{-1}\) can be considered mid-sized with the lowest emitting power plant for which emissions were quantified in [PERSON] et al. [4] emitting 10 Mt CO\({}_{2}\) yr\({}^{-1}\). Past studies suggest that the 6 Mt yr\({}^{-1}\) source should be difficult to estimate with a difficult to discern plume, the 13 Mt yr\({}^{-1}\) source should usually be able to be estimated with a weak but visible plume, and the 25 Mt yr\({}^{-1}\) source should be easy to estimate with a strongly discernible plume [4; 5].  
Figure 1: Model and simulated column-averaged CO\({}_{2}\) mole fraction (XCO\({}_{2}\)) observations relative to a constant 400 ppm background for 3 source emission rates and 4 pixel sizes. The first column shows the simulated Gaussian plume at 50 m resolution for each power plant emission rate against a clean background. Columns 2, 3, 4, and 5 show the simulated observations averaged to pixels of size 2 \(\times\) 2 km\({}^{2}\), 4 \(\times\) 4 km\({}^{2}\), 7 \(\times\) 7 km\({}^{2}\), and 10 \(\times\) 10 km\({}^{2}\), respectively. The top row shows the simulated observations for a source with emission rate 6 Mt yr\({}^{-1}\), the middle row shows observations for a 13 Mt yr\({}^{-1}\) source, and the bottom row shows observations for a 25 Mt yr\({}^{-1}\) source.  
For all simulations unless stated otherwise, the model parameters were: \(u=4\) m s\({}^{-1}\), \(a=156.0\) m, \(\theta=45^{\circ}\) for square pixels and \(\theta=0^{\circ}\) for rectangular pixels, background XCO\({}_{2}=400\) ppm, and source emission rates are \(F=6,13,25\) Mt yr\({}^{-1}\). Noise strength for the 2 km square pixels is 1 ppm, or 0.25%.