A Single Parameter to Characterize Anisotropy of Single Scattering: Asymmetry Parameter vs. Mean Scattering Angle

A Single Parameter to Characterize Anisotropy of Single Scattering: Asymmetry Parameter vs. Mean Scattering Angle

A. Radkevich

Science Systems and Applications, Inc. 1 Enterprise Pkwy, Hampton, VA, 23666

Asymmetry parameter g i.e. mean cosine of scattering angle, <cos γ> is widely use by radiative transfer community for characterization of the degree of anisotropy of single scattering. The greater it is the more anisotropic single scattering is implied to be. Asymmetry parameter also has clear physical sense: difference (1 - g) defines (along with scattering efficiency) radiative pressure on the scattering center. This property, however, cannot be used as a measure of the scattering anisotropy. Asymmetry parameter is the first Legendre moment of a phase function. Anisotropy of the phase function is characterized by all moments rather than the first one. In many cases completely different habit mixtures returns the same or very close to each other asymmetry parameters, see, e.g. Levi et al. (2009). The same situation takes place in the case of monodisperse particle. For example, ice sphere with diameter of 80 μm gives g = 0.8865 at 0.5 μm and g = 0.8847 at 1.0 μm. So, the question is: is there another single parameter that can characterize anisotropy of single scattering and distinguish cases with close g?

Wiscombe and Grams (1976) and Remizovich and Radkevich (1996) showed an interesting relation between backscatter fraction, β, i.e. integral of the azimuth integrated phase function over forward hemisphere with respect to the zenith angle of the direction of incidence and over backward hemisphere with respect to the zenith angle of the scattering direction, and mean scattering angle <γ>: β = <γ>/π. Those papers also showed that <γ> can be presented as a weighted sum of all Legendre moments of odd orders of the phase function. This is because only Legendre polynomials of odd orders are responsible for disbalance in probability of scattering between forward and backward hemispheres. Thus, <γ> can be used as alternative characteristic of anisotropy of single scattering: greater <γ> indicates less anisotropic scattering.

Mean scattering angle was calculated in this work for a set of mono- and poly- disperse particles as well as for some aerosol models used in the MODIS aerosol retrieval algorithm (Levi et al. 2009) and some ice cloud bulk models described by Baum et al. (2014). It was shown that mean scattering angle allows to distinguish different phase functions better than asymmetry parameter g. For example, in the above mentioned case of ice spheres <γ> = 16.7° at 0.5 μm and <γ> = 17.0° at 1.0 μm, i.e. the difference of <γ> between these wavelengths is 1.6% while the difference of g is only 0.2%.

REFERENCES

Baum, B. A., P. Yang, A. J. Heymsfield, A. Bansemer, A. Merrelli, C. Schmitt, and C. Wang, (2014), Submitted to J. Quant. Spectrosc. Radiant. Transfer, Special Issue ELS-XIV.

Levy, R.C., L.A. Remer, D. Tanre, S. Mattoo, and Y.J. Kaufman, (2009), "Algorithm for Remote Sensing of Tropospheric Aerosol from MODIS for Collection 005: Revision 2", available at http://modis-atmos.gsfc.nasa.gov/_docs/ATBD_MOD04_C005_rev2.pdf

Remizovich, V.S. and A.V. Radkevich, (1996), Laser Physics, vol. 6, No. 4, pp. 679–694.

Wiscombe, W.J. and G.W. Grams, (1976), J. Atmos. Sci., vol. 33, pp. 2440 - 2451.