## 3 Results
### Revisit Rate
(**b**) Difference between emissions with weekly and hourly factors removed and mean emissions, normalized by the mean emissions. Figure 17: Quantile-quantile plot of relative hourly emission data with squared correlation coefficient \(r^{2}\). Red line shows the expected distribution if the residuals are perfectly normally distributed. (**a**) Relative emissions with no hourly factor adjustment. (**b**) Relative emissions with hourly and weekly cycles removed. We used a statistical approach based on a theoretical Normal distribution to determine the number of revisits needed to constrain annual emissions from an individual power plant. Treating the data as Normally distributed, we compute the number of samples required to achieve a desired width of 95% confidence interval. If \(N\) is the number of revisits, \(\bar{y}\) is the mean emission estimate and \(\sigma\) is the uncertainty in each estimate, then the 95% confidence interval is \(\bar{y}\pm\bar{y}\delta\), where  
\[\delta=1.96\frac{\sigma}{\sqrt{N}}. \tag{4}\]  
Hence, for a desired precision \(\delta\), the number of required revisits is  
\[N=\left(1.96\frac{\sigma}{\delta}\right)^{2}. \tag{5}\]  
We compute the total uncertainty \(\sigma\) as a combination of the underlying standard deviation of the power plant emissions, and the uncertainty in each estimate, treating each estimate as an independent sample. The standard deviation in the daily relative emissions as computed from the 50 sources is \(s_{d}=0.286784\), and the standard deviation in the hourly relative emissions is 0.3100965. [PERSON] et al. (2014) estimated US power plant emissions to within 1%, 4%, and 17% of EPA reported daily values. Their method absorbs any daily variability in power plant emissions in their sample uncertainty. From this small sample size, it was not clear what the precision and accuracy limits for a single estimate should be. Furthermore, preliminary results with OCO-2 version 8 and 9 data and the same power plants show improvements to both the precision and accuracy of emission estimates. Since we have now removed the daily variability, we may assume a 10% uncertainty on each individual emission estimate (\(\epsilon_{est}=0.10\)), which was very likely achievable with minor improvements to the data and methods. The total uncertainty is therefore  
\[\sigma_{h}=\sqrt{s_{h}^{2}+\epsilon_{est}^{2}}\approx 0.3258. \tag{6}\]  
To constrain annual emissions within at least 15%, we need \(N=19\) overpasses. To constrain annual emissions within 20% would need only 11 overpasses, but to constrain annual emissions within 10% would need 41 overpasses. If the individual estimate uncertainty can be reduced to 5%, we still require 38 overpasses. Alternatively, we may absorb any daily emission variation into the sample uncertainty as in [PERSON] et al. (2014) and used the standard deviation of daily emissions to compute the required number of overpasses. Increasing the sample uncertainty to \(\epsilon_{est}=0.15\) to absorb the daily emission variation, we compute \(\sigma_{d}=\sqrt{s_{d}^{2}+\epsilon_{est}^{2}}\). Following Equation (5), we require 18 overpasses to constrain annual emissions within 15%. [PERSON] et al. (2013) attempted to address the question of uncertainties for annual emission estimates from one CarbonSat satellite or a constellation, using hourly data from 187 U.S. power plants (\(\geq\)5 Mt CO\({}_{2}\) yr\({}^{-1}\)). They found that with one satellite, the accuracy on the annual estimate (or sampling error) was \(\sim\)4.9% or better for 50% of the power plants and \(\sim\)12.4% or better for 90% of the power plants. This assumed a 500 km wide swath, resulting in about 50 overpasses of a midlatitude power plant. The fraction of these overpasses that would be sufficiently cloud-free varied significantly according to local climate conditions, but clear-sky probability was expected to be \(\sim\)30-70% for the majority of the continental U.S. based on MODIS.