Atmospheric Correction of Satellite Images - Integrating GIS, Remote Sensing, and Mathematical Modelling for Surface Water Quality Management - page 216

Where:
is the pixel surface reflectance
is an average surface reflectance for the pixel and a surrounding region
S is the spherical albedo of the atmosphere
La is the radiance back scattered by the atmosphere
A and B are coefficients that depend on atmospheric and geometric conditions but not on the surface.
Each of these variables depends on the spectral channel; the wavelength index has been omitted for
simplicity. The first term in Equation (1) c orresponds to radiance that is reflected from the surface and
travels directly into the sensor, while the second term corresponds to radiance from the surface that is
scattered by the atmosphere into the sensor. The distinction between r and re accounts for the
adjacency effect (spatial mixing of radiance among nearby pixels) caused by atmospheric scattering.
To ignore the adjacency effect correction, set re = r. However, this correction can result in significant
reflectance errors at short wavelengths, especially under hazy conditions and when strong contrasts
occur among the materials in the scene.
The values of A, B, S and La are determined from MODTRAN4 calculations that use the viewing and
solar angles and the mean surface elevation of the measurement, and they assume a certain model
atmosphere, aerosol type, and visible range. The values of A, B, S and La are strongly dependent on
the water vapor column amount, which is generally not well known and may vary across the scene. To
account for unknown and variable column water vapour, the MODTRAN4 calculations are looped
over a series of different column amounts, then selected wavelength channels of the image are
analyzed to retrieve an estimated amount for each pixel. Specifically, radiance averages are gathered
for two sets of channels: an absorption set cantered at a water band (typically 1130 nm) and a
reference set of channels taken from just outside the band. A lookup table for retrieving the water
vapor from these radiances is constructed. For images that do not contain bands in the appropriate
wavelength positions to support water retrieval (for example, Landsat or SPOT), the column water
vapor amount is determined by the user-selected atmospheric mode.
After the water retrieval is performed, Equation (1) is solved for the pixel surface reflectances in all of
the sensor channels. The solution method involves computing a spatially averaged radiance image Le,
from which the spatially averaged reflectance re is estimated using the approximate equation (2) :
Spatial averaging is performed using a point-spread function that describes the relative contributions
to the pixel radiance from points on the ground at different distances from the direct line of sight. For
accurate results, cloud-containing pixels must be removed prior to averaging. The cloudy pixels are
found using a combination of brightness, band ratio, and water vapour tests, as described by Matthew
et al. (2000).
The FLAASH model includes a method for retrieving an estimated aerosol/haze amount from selected
dark land pixels in the scene. The method is based on observations by Kaufman et al. (1997) of a
nearly fixed ratio between the reflectances for such pixels at 660 nm and 2100 nm. FLAASH retrieves
the aerosol amount by iterating Equations (1) and (2) over a series of visible ranges, for example, 17
km to 200 km. For each visible range, it retrieves the scene-average 660 nm and 2100 nm reflectances
for the dark pixels, and it interpolates the best estimate of the visible range by matching the ratio to the
average ratio of ~0.45 that was observed by Kaufman et al. (1997). Using this visible range estimate,
FLAASH performs a second and final MODTRAN4 calculation loop over water.