# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 3 Results
### Estimation Error
For this final test, the absolute values of percent error were averaged for the 0.1, 1, 3, 5, 10, 15 and 20 th percentiles to form an index, one per image, to compare to the range of the recorded median image Atm-I. The % error values were combined for all six QIAs and plotted (Figure 15). The corresponding reflectance range encompasses up to 2000 DN for NIR 8a and about 1000 to 1200 DN for visible bands. These calculations can be viewed in Supplementary Materials as \"SoCalLowRef%ErrorDistr.xlsx\". Figure 14: CMAC estimated percent error plotted together for all four VNIR bands. Figure 13: Percent error distribution for CMAC and Sen2 Cor plotted according to the rank for the 1 st through 3696 th estimated error values. Because the suite of 28 images for each QIA expressed different Atm-I values, the test in Figure 15 provided 168 unique independent values for comparison to the low reflectance % error index in each band. To place the low reflectance measured over the SoCal QIAs into context, the reflectance level for the visible bands measured at the 0.1 cumulative percentile for five of the six QIAs corresponds to the approximate reflectance typical of the dense dark vegetation used to calibrate the Atm-I model. Hence, a percentile range from 0.1% to 20% is considered to be a valid range for examination of how the low reflectance limb of the surface reflectance distribution is impacted by an error that would affect image utility whether for precision agriculture or AI feature extraction. Figure 15 illustrates a marked decrease in Sen2 Cor's visible band stability as Atm-I increases above about 900 DN. CMAC exhibits a much shallower error trend as Atm-I increases, starting at an Atm-I level of approximately 850 DN. Figure 15 demonstrates that CMAC outperforms Sen2 Cor accuracy for AC in the low reflectance portion of visible bands for the most common range of Atm-I that will be encountered in routine AC. As noted in Figure 10, CMAC can still provide useful results even at \"extreme\" levels of Atm-I exceeding 1700 DN, well beyond the atmospheric effects that can be corrected by Sen2 Cor. These graphics form a milestone for judging future R&D efforts to enhance CMAC accuracy. As mentioned earlier, CMAC v1.1 has been tested on images from diverse environments around the Earth. While these tests demonstrate that CMAC can clear images with profound haze, there has been no direct means to confirm that the cleared images represent true surface reflectance. The data developed for this investigation have provided such direct confirmation. Returning to Figure 10, the most extreme image haze evaluated in this paper, graphed for the Atm-I = 1743 image, also provided results close to surrogate surface reflectance that was confirmed to provide image clearing (Figure 16). Hence, when corrected by CMAC, images that have heavy haze cleared from the image can provide results that are close to the true surface reflectance. This is a valuable result that will help push the upper limit of correction to even higher Atm-I levels. Figure 15: Percent error in Sen2 Cor low surface reflectance estimates rapidly increase with Atm-I in all bands except NIR 8A. CMAC error shows a more gradual trend of increased error with increasing Atm-I level. Figure 16: A clip from the S2 Reno image whose data are shown in Figure 10 statistics for Atm-I = 1743 and as the 22-8-2021 statistics in Supplementary Materials as “Reno QIA Curves.xlsx”. As in all other image displays in this paper, these examples are screenshots from QGIS display. (**a**) TOAR representation made from the full tile stretch. A full tile image stretch can be taken to visually represent the degraded TOAR mathematics of the image. The alternative, a clipped image stretch, does not appropriately represent the unbiased mathematics of the image that confronts AI and other machine analysis; such clip stretches may visibly clear some haze (but are typically accompanied by color balance problems).