# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 2 Materials and Methods
### Developing an Atmospheric Index, Atm-I, to Scale the Correction
Numerous S2 images obtained within two months of summer solstice were assembled that contained the target crop in arid to humid climates and across a range of atmospheric conditions from clear to hazy. The sampling period was chosen to reduce solar zenith angle effects. 4. One or more AOIs were chosen for each image that appeared clearly. For AOIs where haze was present, such haze was examined to assure that it was expressed evenly across the AOI. 5. AOIs were chosen to contain an index plot of the most vigorous expression of the alfalfa crop identified through examination of TOAR NDVI. Such plots were chosen as the index samples to serve as the dependent variable that would represent Atm-I. 6. Subplots of homogeneous cover were chosen around each index plot to represent various levels of vegetation response ranging from bare ground to DDV across many AOIs set in widely divergent environments. 7. The extent of each AOI containing index plots and subplots was restricted to within a diameter of about 5 km to further control variation in Atm-I. 8. Values of the four VNIR spectral bands were extracted from the index plots and associated subplots characterized through their maximum, minimum, and median values. 9. The lowest consistent TOAR blue band values of the target crop were chosen to represent the surrogate Atm-I of the index plots. 10. The extracted VNIR spectral data from the index plots and their associated subplots were pooled and then modeled by multiple linear regression and assuming a negative binomial distribution. Model iterations were tested to determine the best data combinations to predict the surrogate Atm-I blue band TOAR of the index plots from the paired TOAR spectral data. 11. The blue and red band reflectance levels were found to be highly significant for the model function, while the green and NIR bands were found to add little. The minimum values for the red and blue bands were chosen as the inputs to the final Atm-I model. Since a binomial distribution was assumed, the estimated value from the Atm-I model was based on a logarithmic scale; the estimated Atm-I level was obtained by applying the exponential function to the model value. Application of the Atm-I model is made using a non-overlapping grid to select the lowest values of the red and blue bands in each grid cell. The output forms a grayscale, with the brightness constituting the scalar Atm-I levels needed for e AC of the image. Minor smoothing is applied to the Atm-I grayscale image to control pixelation effects due to abrupt changes in estimated Atm-I levels between grid cells. An example Atm-I grayscale output is shown in Figure 2.  
Figure 2: An example 100 m resolution (10 \(\times\) 10 pixel grid cell) Atm-I grayscale for the 8-22-21 S2 tile over Lake Tahoe, CA, USA. At least some ground signal must remain for correction (exceeded in portions of this image). The Atm-I model was confirmed by comparison to observable patterns in the grayscale output for images with significant haze mapped using the pattern of blue reflectance over continuous areas of DDV (Figure 2). Because the Atm-I model works directly with the red and blue band responses across the scene, it produces a surrogate spatial estimate for what direct application of dark vegetation would produce were it present in the image. No attempt was made to interrelate Atm-I output to a similar measure used to estimate aerosol particle loading, aerosol optical depth (AOD), which ranges from 0.1 for \"clear\" atmospheric conditions to 1.0 representing \"very hazy\" conditions [28] and is derived using sun photometry [29]. In the CMAC application, the Atm-I model output represents the predicted blue band reflectance from DDV and is measured as a digital number--DN (reflectance \(\times\) 10,000)--that represents a correctable range from the high 700s (extremely clear) to over 1700 (features obscured in a dense haze). The abbreviation DN is used throughout to designate this rescaled reflectance.