## 2 Materials and Methods
### Derivation and Scaling of CMAC Surface Reflectance Retrieval Methods  
Field spectroscopy applied a portable Analytical Spectral Devices unit to initially sample vegetation spectra to understand the potential for a vegetation-based SR reference against which to measure atmospheric effects. The S2 blue band 2 reflectance was selected for representation of atmospheric effects because it retains sensitivity over a wide range of atmospheric aerosol content and because its SR remains remarkably consistent for continuous healthy canopies formed by many different plant species, thus offering a competent reference against which to judge atmospheric effects. Such maximal vegetation expression has been given the name \"dense dark vegetation\" (DDV) and has been applied to estimate atmospheric effects for AC of Landsat and S2 images [6; 9]. An initial concept of blue reflectance to serve as a reference was tested for images containing continuous DDV cover in the Amazon basin and during summer images in the intensively cropped regions of the American Midwest. Spectra of such cover were sampled with non-overlapping grids after applying an NIR threshold to first remove the low blue reflectance caused by water surfaces. This test application provided a sensitive portrayal of atmospheric effects where DDV is continuous; however, it was obviously of limited value where DDV was lacking.  
The requirement for DDV to be present in images was bypassed through the assembly of a model that uses VNIR band statistics as input. The first step in model assembly was to select alfalfa as a standard because it is a cultivated crop that reaches maximal canopy expression in a wide range of conditions, whether through irrigation in low-plant-cover arid climates or watered by rainfall in humid climates. This wide latitude of conditions supported the sampling of many types of adjacent plant cover to model atmospheric effects for the entire range of cover from deserts to continuous healthy canopies. The known spectral statistics of the highest-NDVI alfalfa fields were used as a surrogate ground truth estimate for blue band surface reflectance.  
To assemble the statistically based model, verdant alfalfa fields were identified on numerous mid-summer images under clear to hazy conditions across a range of arid to humid climates; these cultivated fields were treated as index plots. Adjacent plots were identified as subsamples in ranges of cover from none to continuous canopies of whatever vegetation was present, not necessarily alfalfa, nor cultivated. The spectral band values of the index plots and their subsamples were extracted, pooled and then used in regression modeling to predict blue band values of the index samples that were assumed from relationships generated during earlier steps of spectral canopy measurements.  
The model output of the measured atmospheric index is used in subsequent CMAC calculations to adjust the magnitude of the calculations that remove the atmospheric effect, thereby delivering SR. This scalar, the predicted blue-band DDV response to the atmospheric effect, is abbreviated \"Atm-I\", and in operation, is estimated by the model from extracted TOAR spectral data within spatially discrete, non-overlapping sampling grid cells arrayed across the image. The model provides spatially discrete Atm-I values that represent the lumped sum atmospheric effect applicable to many surfaces, including water bodies. The brightness of this grayscale is the driving variable for AC and scales the degree of correction necessary to reverse the lumped-sum atmospheric effect as shown by the example in Figure 1.