# Closed-Form Method for Atmospheric Correction (CMAC) of SmallSat Data Using Scene Statistics
## 2 Materials and Methods
### Forward Scatter Effects and Calibration  
Two parameters per band are needed to apply the CMAC Equation: the TDL slope and offset. Through calibration, these parameters can be predicted from Atm-I for reversing the TOAR snapshot that captured the result from atmospheric degradation of the original surface reflectance. Each smallsat spectral band requires calibration to accommodate differing sensor responses to Atm-I, thereby customizing CMAC for that smallsat's sensor responses. A calibration target is an obvious choice for a precise and convenient establishment of the bright and dark points of the TDL, provided that special concerns are first addressed in the design of such targets. Through R&D analyses, an additional atmospheric phenomenon became apparent that results from the interactive effects of ground-target reflectance and atmospheric aerosol particles. As Atm-I increases, the higher reflective energy of bright targets illuminates aerosols from below, thus brightening and scattering the Atm-I responses captured by the imager. This phenomenon affects the relationship of slope and offset for the application of the CMAC Equation and may prevent the use of existing targets that combine adjacent dark and bright panels in a checkerboard pattern (Figure 6). Due to forward scatter effects, such checkerboard targets can provide measurable TOAR only when (1) employed under the cleanest of atmospheric conditions, (2) the panels are physically separated, and (3) the panels are large in relation to the imager's spatial resolution. Forward scatter affects calibration curves beyond what is captured by the conceptual model in Figure 3; fortunately, this can be accounted for intrinsically through calibration. An appropriate calibration target with accompanying precise and recently measured ground truth was not available; therefore, to provide for CMAC development, slope and offset were calibrated vicariously through image comparisons between clear and hazy conditions, with additional guidance from ground truth data collected by ASD field spectrometer. CMAC results for this calibration compared well with Sen2 Cor results developed for the same images. This low Atm-I calibration was then adjusted sequentially for higher Atm-I by matching the surface reflectance estimates of the higher to the lower  
Figure 6: Salon de Provence, France region: a calibration target (arrows) in S2 TOAR regional images 16 June 2021 under light haze (**a**,**d**) and 8 March 2021 under moderate haze from wildfire smoke (**b**,**e**). A Google Earth image (**c**) of the target shows the 30 m \(\times\) 30 m black and white panels. Atm-I image through the selection of appropriate slopes and offsets. The slopes and offsets derived in this manner were recorded for each measured Atm-I on multiple clear to hazy images to establish the calibration relationships for surface reflectance estimation across the range of Atm-I that can be encountered operationally. During previous method proofing, CMAC v1.1 was tested on multiple Sentinel 2 images from diverse environments that included low spectral diversity such as oceans and deserts as well as images degraded from haze due to smoke, fog and, to a more limited extent, dust. CMAC was found to correct all images and remove profound levels of haze, thus demonstrating applicability in all environments and at least to first order, doing so irrespective of aerosol type. This testing also indicated that the corrected VNIR bands are correct relationally and therefore represent close estimates of surface reflectance. To verify the mathematics of the correction for this investigation, CMAC v1.1 (Advanced Remote Sensing, Inc., Hartford, SD, USA) was tested against Sen2 Cor v2.11 (European Space Agency, Paris, France). Performance was judged by estimation of VNIR surface reflectance for seven AOIs exhibiting consistent reflectance throughout the summer.