Base of the Landsat TM Histogram Digital Number (DN) Scatter Selection from Chavez (1988)
Atmospheric scatter (path radiance) removal has been based on the dark object subtraction (DOS; Chavez, 1988) method, where the assumption is made that for a satellite image with millions of pixels, there should be some pixels that have no reflectance. The reason there are not pixels with zero reflectance (for visible bands and perhaps the NIR band), is the result of atmospheric scatter (Chavez, 1988), erroneously increasing satellite sensor reflectance to higher values (shifting the entire histogram to the right). This assumption is then combined with "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). Notice below that Chavez (1988) did not select the lowest value in the image, but the value more at the base of the low end of the histogram (where an abrupt increase in frequencies begins; though it is possible that the lowest value may be the proper dark object). Keep in mind when viewing the Landsat TM histograms below, that Landsat 8 has much higher radiometric resolution (Landsat TM has an 8-bit range, while Landsat 8 has a 16-bit range when processed to Level-1 data, which result in thousands of more values). Also, Landsat 8 typically has a much longer statistical tail than Landsat TM. Chavez (1996) then calculated the base of the histogram scatter reflectance value to be .01 (1% as opposed to zero) by deducting .01 from the dark object reflectance (known as one-percent dark object reflectance), because of the "fact that very few targets on Earth's surface are absolute black, so an assumed one-percent minimum reflectance is better than zero percent" (Chavez, 1996).
Landsat TM histogram (Chavez, 1988) for bands 1 (blue), 2 (green), 3 (red), and 5 (NIR) showing digital numbers (DNs) selected by Chavez as dark objects (below histogram); it is important to note that lowest value is not selected from Chavez (though that is not to say the lowest value cannot be selected). The selected dark objects DNs will have a surface reflectance of .01 (1%).
Landsat TM band 1 (blue); DN 40 (frequency of 18) is selected as the DN scatter value.
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Landsat TM band 2 (green); DN 13 (frequency of 20) is selected as the DN scatter value.
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Landsat TM band 3 (red); DN 12 (frequency of 69) is selected as the DN scatter value.
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Landsat TM band 4 (NIR); DN 8 (frequency of 5) is selected as the DN scatter value.
Scatter for bandwidths longer than NIR are insignificant enough to ignore. Overall when there is more haze, the scatter amount becomes more proportionally equal between bands but the shorter wavelength bands always have more scatter in a non-opaque atmosphere. There should be a power relationship between band center wavelength and scatter (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.