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  • Author or Editor: V. L. Galinsky x
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V. L. Galinsky


The forward discrete ordinate method (DOM) has been reformulated to include effects of a weak inhomogeneity of a medium. The modification is based on an expansion of the direct beam source term. This treatment of the source term is similar to the gradient correction (GC) method, presented in the first part of the paper for the diffusion approximation. The same requirement on the scales of variations applies to the current method: length of horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. The modification has another important advantage in that it permits obtaining particular solutions for both infrared and direct beam radiative transfer in one computational step.

An effective algorithm for solving inverse radiative transfer problems has been developed following the above reformulation. This algorithm modifies the DOM using Newton’s iterative scheme in order to find a solution of the inverse problem. The algorithm convergence rate is very high. Typically, two to three iterations are enough in order to obtain a solution with sufficiently high accuracy. The algorithm can be applied to the plane-parallel radiative transfer as well as to the GC method.

The combination of the GC approach and the effective inverse algorithm creates an extremely useful and efficient tool for extraction of cloud fields from satellite imagery. It allows the algorithm to use radiance data with multiple views of the same pixel as an input and produce correct output even for large solar zenith or satellite view angles, when an independent pixel approximation fails.

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V. L. Galinsky
V. Ramanathan


The solution of the three-dimensional radiative transfer equation in weakly horizontally inhomogeneous medium has been obtained in the diffusion approximation using the expansion of the three-dimensional delta-Eddington approximation. The solution approach, referred as the gradient correction (GC) method, expands the horizontal fluxes and the source function in terms of the horizontal gradient of the extinction coefficient and/or the cloud-top boundary. In the transfer equation, only the zeroth- and first-order gradient terms are retained and hence the following limitations apply. First, the length of the horizontal variations of optical properties of the medium should be large in comparison to the mean radiative transport length. Second, the ratio of the vertical to horizontal scales should be small enough so that fluxes from boundaries may be neglected.

Since there are no restrictions to the amplitude of the optical properties variations, this method may even be applicable to a medium with strong horizontal variations of optical properties, as long as scales of the variations are large enough in comparison to the radiative transport length. The analytical solutions are in excellent agreement with the more accurate numerical solutions. The solution also shows the solar zenith angle dependence of the albedo, similar to that observed in analyses of satellite imagery.

The GC approach may be useful as a fast and computationally inexpensive method both for the correction of the independent pixel approximation used for extraction of cloud fields from satellite imagery and possibly for the calculation of the radiation fluxes in climate models.

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