# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 3 Results
### Estimation Error  
As noted in Figure 10, AC results for the lowest Atm-I (clear) images are expected to be closest to true surface reflectance since the correction requires minimal adjustment for its estimation. An assumption that the surface reflectance estimates of CMAC and Sen2 Cor for low Atm-I images are surrogates for true surface reflectance offers a means to examine the propagation of error in each AC method. Following this logic, % error was estimated using the average of the three lowest Atm-I images for the 28 images of each SoCal QIA and used as surrogate true surface reflectance values. These calculations were made per the 22 cumulative percentile steps so that the statistical examination could further define the relationship of the error to reflectance brightness. Because each of the QIAs was chosen for its quasi-invariant reflectance, AC results for the collection of QIA images are expected to yield reflectance values that remain similarly invariant. Examination of the values for each image indicated that there was no apparent relationship with time for either method. For CMAC, there was also no striking relationship for the magnitude of surface reflectance estimates with Atm-I. For Sen2 Cor, however, a trend for surface reflectance error increasing as Atm-I increased was readily discernable. Error estimates were calculated for CMAC and Sen2 Cor for each QIA from their individual surrogate true surface reflectance (CMAC with CMAC, Sen2 Cor with Sen2 Cor). The values for all six QIAs were combined to robustly represent the error distribution for each band (\(n=3696\): 6 QIAs \(\times\) 28 images \(\times\) 22 percentiles). The error estimates were ranked from lowest (under-estimation error) to highest (over-estimation error) and plotted to compare CMAC to Sen2 Cor per band for the entire SoCal dataset (Figure 13). These calculations, as well as the trend in the data output, can be traced in the individual downloadable QIA spreadsheets that fed statistics to the downloadable available in Supplementary Materials as \"SoCal%Error.xlsx\". The ranked percent error curves for CMAC and Sen2 Cor in Figure 13 agreed well through most of the bandwise % error distributions with the notable exception for the higher ranked values in the visible bands. The under-estimation error was well-constrained and roughly equivalent between the two methods. The upward deviating curves represent an over-estimation error that was much greater for Sen2 Cor, especially at shorter band wavelengths--the blue band was more severely affected than the green band. The estimated error was the lowest for NIR 8A. Over-estimation of visible-band reflectance in Figure 13 was confirmed to be the origin of the greater statistical dispersion of low reflectance observed in the plots of Figure 11 and in the CV% tables and the \"raw data\" across the six Excel files available through Supplementary Materials. The data in the summary tables show this trend but under-represent the severity of Sen2 Cor instability at higher Atm-I because the most widely divergent values were masked by averaging with the majority of comparatively low Atm-I images in the larger image pool. Some Sen2 Cor CV% values exceeded 100% for individual images with higher Atm-I, a trend that can be seen in the individual downloadable QIA spreadsheets. The trend for overestimation errors in visible bands is evident in Figure 13 for Sen2 Cor. This can be viewed as instability. In contrast to visible bands, NIR 8A error estimates by CMAC and Sen2 Cor closely agree. The CMAC % error curves have roughly equivalent shapes and magnitudes. In Figure 14, the estimates for the visible bands plot nearly atop one another. The agreement of CMAC error curves indicates greater stability in VNIR band correction. Toward understanding the range of Atm-I that can be corrected accurately by both methods, the low, dark end of reflectance was explored further using averages of the low reflectance % error values as an index to compare to the median QIA Atm-I measured for each image.