# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 3 Results  
On a desktop PC (64 MB Ram, Windows 10, C++-based code), CMAC corrected the four VNIR bands of full S2 image tiles (~120 million image pixels) in an average of 91 s. The python-based desktop Sen2 Cor processed the same tiles in an average of about 20 min, though for the 13 bands included in S2 ''.SAFE'' image packages. _Analysis 1:_ Reno QIA statistics were extracted and analyzed from eighteen images. A summary table presented for the Reno QIA available in the Supplementary Materials provides CV% values that are noticeably elevated above CV% for Analysis 2, also available in the Supplementary Materials. This was due to the inclusion of three high Atm-I images that proved challenging for both methods. This conclusion can also be reached through the examination of Figure 10 in which seven curves are plotted showing the trend in correction as judged by Atm-I from unusually clean atmospheric conditions to the extreme haze from wildfire smoke. Scatter increased with Atm-I. To reduce the number of curves for clarification of trends responding to increasing Atm-I, data of similar Atm-I ranges were averaged (\(n=3\) or \(n=4\)) for each percentile of reflectance within each treatment. Sen2 Cor can be observed to fail at Atm-I levels of around 1345 and above. Data from the three extreme Atm-I images also show increased CMAC CV% as indicated in Figure 10 by the slight deviation from the statistical weight of the curves generated by results for Atm-I levels of 962 DN and lower. These calculations and the complete dataset are available for \"Reno.xlsx\" through a link in Supplementary Materials. As shown in Figure 10, CMAC surface reflectance estimates are roughly commensurate with Sen2 Cor for images with lower Atm-I--these curves are displayed in color. The average of the three unusually clear images with the lowest average Atm-I (813) displayed in red was taken to be the best \"surrogate\" approximation of true surface reflectance. Above Atm-I levels of approximately 2000 DN, the CMAC AC curves significantly diverge above the corresponding surrogate surface reflectance curves. However, this represents a net decrease in estimated surface reflectance. Though CMAC and Sen2 Cor AC methods are very different, both captured the trend of decreasing bright reflectance at higher Atm-I levels. Hypothetically, this divergence indicates an atmospheric phenomenon of diffuse shading by aerosol particles that reduces TOAR reflectance. As the CMAC Equation predicts, this results in a lower surface  
Figure 10: Reno QIA reflectance curves plotted for the four VNIR bands of S2 (rows). Colored curves were derived from \(n=3\) or \(n=4\) percentile averages for each band and treatment for the low-Atm-I images (clear-appearing, lacking haze). Legend values are Atm-I or average Atm-I. Curves in black are for single images that exceed Sen2 Cor AC capability. Though not as accurately, CMAC corrected the extremely high Atm-I curves for the visible bands. CMAC curves are tighter (more precise) than Sen2 Cor in all bands. reflectance estimate. This divergence is systematic, so it can potentially be adjusted by empirically fitted relationships for each band if the estimation of the true surface reflectance for brighter targets is desired and the added processing time is not problematic. As discussed later, the true reflectance of bright targets may not be of direct interest for most analyses. _Analysis 2:_ The six SoCal QIAs were evaluated to test AC performance within an Atm-I range expected to be encountered in typical operation. Figure 11 presents seven curves derived for the Rochester QIA, each being an \(n=4\) average calculated from within similar Atm-I ranges. Figure 11: Seven reflectance curves for the Rochester QIA by treatment (columns) and bands (rows). Each curve represents an average of four images with similar average Atm-I. Dispersion is notable for Sen2 Cor in the lower limb of visible bands where high precision is needed to support applications such as precision agriculture and AI feature extraction.