## 3 Results
### Revisit Rate  
It is well known that anthropogenic CO\({}_{2}\) emissions exhibit seasonal and diurnal variability [11; 12], with the day-night difference for U.S. power plants previously estimated to be \(\sim\)9% [13]. In this section we study the diurnal and seasonal variability using 2016-2018 daily emissions data and 2018 hourly emissions data for the 50 largest US power plants (based on 2017 reported emissions). We compute the mean weekly and hourly emission factors, and use these factors to remove the seasonal and daily cycles to achieve a true estimate of the variability in US power plant emissions. Mean weekly and hourly cycles were computed and then removed from the data. The data for each plant was normalized to its mean emissions for the time period studied, resulting in the dimensionless daily and hourly relative emissions. The relative emissions for all 50 plants are aggregated and binned by week of year and by hour of day. The last eight days (nine days for the leap year 2016) of the year were grouped together in the last weekly bin of the year, allowing us to compute 52 weekly emission factors. The mean weekly and hourly relative emissions were computed, and then normalized so that the sum of the weekly factors was 52 and the sum of the hourly factors was 24. This last normalization step ensures the mean emissions remain unchanged when removing the weekly and hourly cycles. Figure 13 shows the weekly emission scale factors, with error bars showing the standard deviation in the binned weekly emissions. no periodicity is evident in the mean residual emission rates. Finally, we can estimate the standard deviation of daily power plant emissions. Computing the daily standard deviation of the residuals, we find \(s_{d}=0.286784\). We similarly investigate the hourly variability in power plant emissions. Figure 15 shows the hourly scale factors derived from all 50 sources. Hourly scale factors were derived from hourly emissions data after using the weekly scale factors shown in Figure 13 to remove any seasonal dependence. In this way, the hourly scale factors exclude any seasonal component. However, we note that in general, daily energy demand will have a different distribution throughout the day in different seasons. For example, the length of the overnight minimum in power plant emissions might depend on the number of daylight hours, or the shape of the midday peak may depend on air conditioning demand. These factors are excluded, and any seasonality is relegated to the weekly scale factors. Figure 16 shows the hourly emissions from all 50 sources for all of 2018. Notice the strong seasonal cycle in the top panel. After removing the weekly and hourly cycles, the residuals in the bottom panel appear much more randomly distributed about zero. Figure 14: Quantile-quantile plot of relative daily 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 weekly factor adjustment. (**b**) Relative emissions with weekly cycle removed. Figure 15: Hourly scale factors for 50 highest emitting US power plants, using 2018 data. Error bars show the standard deviation in the binned hourly relative emissions. In the same way as daily emissions, we compare the hourly emissions to a Normal distribution in Figure 17. We note in particular that the fit of a Normal distribution is not very good before removing hourly and weekly cycles. The quantiles disagree noticeably across most of the Normal quantiles in addition to in the tails. After adjusting the emissions by the hourly and weekly scale factors, the fit improves significantly. The second panel shows that the Normal approximation is good for this dataset. Computing the standard deviation of the residuals, we find the hourly standard deviation is \(s_{h}=0.310096\). Figure 16: 2D histogram of relative hourly power plant emissions for 2018. (**a**) Relative emission data from all 50 plants, with no adjustments.