Atmospheric Scatter Background
(See Landsat & Sentinel-2 Conversion to SR Tutorials drop-down menu for step-by-step conversion to surface reflectance based on image-based atmospheric correction)
The path of solar radiation to Earth then to a satellite sensor involves many interactions. The first graphic below (Jenson, 2007) shows various paths of incoming solar radiation (see descriptions below graphic).
Ls is the total radiance at the sensor, which includes erroneous additive path radiance, Lp, of Path 2 and Path 4. Path 2 is diffuse irradiance that can include Rayleigh scattering and Mie scattering. Path 4 is reflectance from neighboring areas which is adjacency radiation. Lt is the total radiance from the actual surface/target of interest. Path 1 is irradiance that was reduce very little before striking the surface. Path 3 is irradiance that has been scattered downward onto the study area. Path 5 is radiation that has been reflected from a neighboring area, then scattered down onto the study area.
Overall, the result of the incoming solar radiation interactions is to erroneously add to Top of Atmosphere (TOA) reflectance (as is shown in the simplified diagram below). (TOA reflectance is the ratio of total radiance at the sensor [Ls above] to total incoming radiation at the top of the atmosphere where a satellite sensor is located). Ultimately, the goal of satellite imagery atmospheric correction and conversion to surface reflectance is to retrieve surface reflectance values similar to values measured at the surface (where a device can accurately and more simply measure the ratio of incoming radiation to the amount reflected). Fortunately, even with all the atmospheric radiation paths and interactions, a relatively simple image-based method to calculate surface reflectance quite accurately has been developed (Chavez, 1996). The COST method (opens to tutorial on this website) was developed by Chavez and is a Dark Object Subtraction (DOS) method researched with Landsat 5 and can also be applied to Landsat 7. GIS Ag Maps developed a DOS method customized for Landsat 8 and Sentinel-2, that is even easier to apply than the original COST method (Chavez did not research Landsat 8 or Sentinel-2 imagery).
Image-based atmospheric correction removes/deducts Path 2 (above) scatter/path radiance. Path 2 radiance is the predominant additive scatter and erroneously increases visible and near infrared (NIR) reflectance (though it increases NIR much less than visible reflectance). Path 2 can include Rayleigh scatter which is the scattering of light by molecules (particles much smaller than wavelengths of light), where smaller wavelengths scatter much more than larger wavelengths and is the reason why the sky is blue on a clear day (blue light has smaller wavelengths than green or red), as well as Mie scatter (which can includes haze) and is the scattering of light by particles with a similar size to the light wavelengths; Mie is more of a larger wavelength scatter than Rayleigh scatter. Clearer days have less overall scatter but a higher ratio of blue light to red light scatter (because clearer days have more Rayleigh scatter); while, conversely, hazier days have more overall scatter but a smaller ratio of blue light to red light scatter (because hazier days have more Mie scattter which results in more larger wavelengths being scattered). Atmospheric scatter reflectance is deducted from Top of Atmosphere (TOA) reflectance in order to calculate surface reflectance. Near infrared radiation (not shown below) scatters a very small amount, though is should still be deducted (it is the longest wavelengths that should be deducted; see Relative Scatter Calculator page for specific values).
An image-based method to convert to surface reflectance (to account for the erroneous increase in reflectance at the satellite sensor due to atmospheric scatter) is called dark object subtraction (DOS) and was developed by Chavez (1988). The theory behind DOS is that in a satellite image scene with tens of millions of pixel (which satellite imagery commonly has), there should be some pixels that have zero reflectance; the reason there is not (in visible and NIR bands) is due to atmospheric scattering erroneously increasing reflectance values. DOS does not consider secondary scattering into shadowed areas (Chavez, 1996). Chavez (1996) then deducted .01 from the established scatter reflectance value (so that the established dark object/value had a surface reflectance of .01) because of "the fact that very few targets on the Earth's surface are absolute black, so an assumed one-percent minimum reflectance is better than zero percent" (Chavez, 1996).
Blue scatters the most (includes Rayleigh scattering), followed by green, red, then NIR (scatter amounts for bandwidths larger than NIR are very negligible at best, and do not need to be deducted). A clearer atmosphere has less overall scatter and a higher ratio of Rayleigh scatter (which has a higher ratio of smaller to longer wavelengths than a hazier atmosphere); conversely, a hazier atmosphere has more overall scatter and more equal scatter between bands (Chavez, 1988). There should be a power relationship between wavelength and scatter amount that changes for different atmospheric conditions, which is shown in the diagram below from Chavez (1988).
References
Chavez, P.S., Jr. 1996. Image-based atmospheric corrections–revisited and improved. Photogrammetric Engineering and Remote Sensing 62(9): pp.1025-1036.
Chavez, P.S., Jr. 1988. An improved dark-object subtraction technique for atmospheric scattering correction of multispectral data. Remote Sensing of Environment 24: pp.459-479.
Jensen, J. R. (2007). Remote sensing of the environment: An earth resource perspective (2nd ed.). Upper Saddle River, NJ: Pearson Education, Inc.