The Delta–M Method: Rapid Yet Accurate Radiative Flux Calculations for Strongly Asymmetric Phase Functions

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  • 1 National Center for Atmospheric Research, Boulder, Colo. 80307
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Abstract

The delta–M method represents a natural extension of the recently proposed delta–Eddington approximation to all orders M of angular approximation. It relies essentially on matching the first 2M phase function moments and using a Dirac delta–function representation of forward scattering. Computed fluxes are remarkably accurate at very low orders M of approximation, even when the phase function is strongly asymmetric; thus the associated M × M matrix computations remain small and manageable. Flux is automatically conserved, making phase function “renormalization” unnecessary. Phase function truncation is effected in a much more attractive manner than in the past; furthermore, truncation tends to zero as M → ∞. Errors are shown to oscillate with (roughly) exponentially decreasing amplitude as M increases; which has the curious consequence that increasing M by small amounts does not necessarily reduce error. Mie computations associated with the δ–M method can be considerably reduced, based on a simple technique for phase function moment calculations proposed herein.

Abstract

The delta–M method represents a natural extension of the recently proposed delta–Eddington approximation to all orders M of angular approximation. It relies essentially on matching the first 2M phase function moments and using a Dirac delta–function representation of forward scattering. Computed fluxes are remarkably accurate at very low orders M of approximation, even when the phase function is strongly asymmetric; thus the associated M × M matrix computations remain small and manageable. Flux is automatically conserved, making phase function “renormalization” unnecessary. Phase function truncation is effected in a much more attractive manner than in the past; furthermore, truncation tends to zero as M → ∞. Errors are shown to oscillate with (roughly) exponentially decreasing amplitude as M increases; which has the curious consequence that increasing M by small amounts does not necessarily reduce error. Mie computations associated with the δ–M method can be considerably reduced, based on a simple technique for phase function moment calculations proposed herein.

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