Combined Henyey-Greenstein and Rayleigh phase function.

The phase function is an important parameter that affects the distribution of scattered radiation. In Rayleigh scattering, a scatterer is approximated by a dipole, and its phase function is analytically related to the scattering angle. For the Henyey-Greenstein (HG) approximation, the phase function preserves only the correct asymmetry factor (i.e., the first moment), which is essentially important for anisotropic scattering. When the HG function is applied to small particles, it produces a significant error in radiance. In addition, the HG function is applied only for an intensity radiative transfer. We develop a combined HG and Rayleigh (HG-Rayleigh) phase function. The HG phase function plays the role of modulator extending the application of the Rayleigh phase function for small asymmetry scattering. The HG-Rayleigh phase function guarantees the correct asymmetry factor and is valid for a polarization radiative transfer. It approaches the Rayleigh phase function for small particles. Thus the HG-Rayleigh phase function has wider applications for both intensity and polarimetric radiative transfers. For microwave radiative transfer modeling in this study, the largest errors in the brightness temperature calculations for weak asymmetry scattering are generally below 0.02 K by using the HG-Rayleigh phase function. The errors can be much larger, in the 1-3 K range, if the Rayleigh and HG functions are applied separately.