# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 2 Materials and Methods
### Translating an Observation into the CMAC Equation  
The final step in the CMAC workflow reverts TOAR to an estimate of surface reflectance by removing the atmospheric effect measured as Atm-I. This workflow is based on a graphical conceptual model representing the effect from a single Atm-I level as a line in cartesian space defined by surface reflectance and TOAR. Development of this approach began by comparing reflectance between clear and hazy conditions for a given AOI. For images with essentially the same surface reflectance, higher atmospheric aerosol increased dark reflectance from backscatter and decreased bright reflectance through attenuation. This observation was further translated into a graphic conceptual model through inversion of the empirical line method [30], which was adjusted further by placing surface reflectance in both axes (Figure 3). For a single level of Atm-I, this formulation represents the upward reflectance deviation for dark targets and the attenuation of reflectance for bright targets as a single TOAR deviation line (TDL). All VNIR bands were found to conform to this conceptual model.  
Precedence for the CMAC conceptual model can be found in Figure 2 of a paper by [PERSON] and [PERSON] [31] and is reproduced here in Figure 4. The linear relationships are similar to those shown in Figure 3; however, the y-axis in Figure 4 is functionally TOAR. In the context of AC, the parameter L represents radiance that necessarily includes sensor response that is the basis for RadTran applications. Including radiance complicates the calculation of surface reflectance. In the CMAC automated workflow, calibration removes the central importance of true radiance for AC.  
Figure 3: CMAC conceptual model illustrated as a dashed line expressing the effect upon any pixel, dark to bright, from a single level of atmospheric aerosol. SR is surface reflectance. The TDL crosses the x-axis at the axis point.  
The TDL of Figure 3 represents the deviation from surface reflectance caused by the imposed atmospheric effects measured by Atm-I. As Atm-I increases and backscatter and attenuation increase, displacing the intercept upward and steepening the slope. This conceptual model was translated into the CMAC Equation, which converts TOAR to surface reflectance for each pixel differentially across the image according to the slope (m) and the offset (b) for the TDL imposed by the overlying atmosphere. The conversion is rapid due to the efficiency of the closed-form CMAC rather than the less-efficient alternative of iteration and potentially less accurate lookup tables employed in RadTran-based methods [32; 33].  
CMAC Equation: SR = (TOAR - b)/(m + 1)  
The conceptual model and the observations supporting it underscore that the CMAC approach rescales the TOAR to surface reflectance. These changes narrow the reflectance window, a behavior that can be measured for increasing levels of Atm-I (Figure 5). Given the behavior of reflectance to increasing atmospheric aerosol concentration, the often-used dark object subtraction method [3; 4] subtracts the lowest reflectance over dark targets (e.g., water bodies) from the entire reflectance distribution. This does little to correct the reflectance distribution while introducing additional error.  
Figure 4: Figure 2 reproduced from [PERSON] and Kaufmann [31].  
Figure 5: Data extracted from S2 images from 2021 over an area of interest with consistent reflectance in Reno, Nevada that experienced wide swings of aerosol concentration from regional wildfire smoke. The application of such invariant locations is described further in Section 2.4 below. DN refers to reflectance scaled by 10,000.