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Kaufman 1982
Kaufman, Y.J. (1982). Solution of the equation of radiative transfer for remote sensing over nonuniform surface reflectivity. Journal of Geophysical Research 87: doi: 10.1029/JC087iC06p04137. issn: 0148-0227.

A fast yet accurate approximate solution has been developed for radiative transfer in an atmosphere bounded by a surface with nonuniform reflectivity for the purpose of remotely sensing aerosol characteristics (optical thickness and scale height). This solution is derived for the upward radiation at zenith for scattering and absorbing atmospheres and with surface reflectivity in the form of a 'two-halves' field (two halves of field of a different reflectivity such as a sea shore). Any vertical distribution of absorbers and scatterers can be used. The resultant solution is used to develop expressions for the atmospheric line spread function and the atmospheric Modulation Transfer Function. This function can be applied to calculate the upward radiance for any surface reflectivity. The results of calculations are compared with numerical results calculated by the Monte Carlo method. The present results are shown to fit the Monte Carlo calculations almost exactly. This comparison shows that the present fast solution can be used for any surface reflectivity when aerosol optical thickness &tgr;A?1, and from the near UV to the near IR spectrum. The solution is also compared with high resolution satellite data (80 m resolution) obtained from Landsat 1 images of the African coast taken during ground measurements of the aerosol characteristics. The comparison shows that the present approximation can predict the measured radiances over a nonuniform surface reflectivity. It is shown that the horizontal range of the adjacency effects (the range at which the high reflectivity area affects the zenith radiance over the low reflectivity area) are proportionally dependent on the average elevation of the aerosols while independent of their exact vertical profile. In addition, this range is not very sensitive to the aerosol size distribution, optical thickness, and surface reflectance. Thus, this range can be used to retrieve the scale height of aerosols from zenith radiances detected from space.

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Journal of Geophysical Research
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American Geophysical Union
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