Stereology approach to snow optics Aleksey V. Malinka Institute of Physics National Academy of Sciences of Belarus

Introduction  Common approach to model light scattering in snow:  snow grains as independent scatterers of different shapes:  Hexagonal prisms columns and plates smooth or rough surface  Cylinders  Cubes and cuboids  Fractals  Spheres  Real snow is a kind of stochastic medium with grains of different irregular shapes

Stereology. Chord length distribution E.C. Pielou, Biometrics 20, (1964) P. Switzer, Ann. Math. Statist 36, (1965) P. Debye et al J. of Appl. Phys. 28 (6), (1957)

Stereology. Main relationships  Any convex particle  Volume fraction  Specific surface area per unit volume  Specific surface area per unit mass

Debye model Debye developed the model of light scattering in random mixture, based on Rayleigh-Gans approximation (1 st Born) The requirement is Not the case of snow! On the other hand, the approach of geometrical optics is applicable to snow in the visible and near IR

Geometrical optics Absorption coefficient of ice 1.Absorbed energy along the first chord T - Fresnel transmission, θ – incident angle 2.Along the second chord … and so on… After averaging and summarizing: T diff – Fresnel transmission for diffuse light

Snow layer albedo Crosses – T.C. Grenfell et al., “Reflection of solar radiation by the Antarctic snow…” J. Geophys. Res., Curve – modeling (snow layer optical thickness – 180)

Scattering phase function  Legendre polynomials expansion  Polarization can also be easily included

Phase function Hexagonal prisms with rough surface (Yang P., K.N. Liou Geometric–optics–integral–equation method for light scattering by nonspherical ice crystals. Appl. Opt., 35, 6568–6584) In comparison to the random mixture model

Polarization. Element 22

Polarization. Element 33

Polarization. Element 44

Polarization. Element 12

Polarization. Element 34 The feature of the random mixture is that the polarization matrix element 34 is always equal to zero – circular polarization cannot be born in the process of Fresnel reflection/refraction!

BRDF of 500nm M. Dumont, O. Brissaud, G. Picard, B. Schmitt, J.-C. Gallet, and Y. Arnaud, “High-accuracy measurements of snow BRDF…”, Atmos. Chem. Phys., 10, 2507–2520, 2010 Curve – theoretical modeling for random mixture Circles – experiment Normal incidence

BRDF of 500nm Oblique incidence

BRDF of 500nm Oblique incidence

Conclusion  Stereological approach is an alternative to the shape- based approach  Optical properties of snow are completely determined by and easily related to the chord distribution  Chord length distribution describes all possible sizes  For pure random mixture the chord distribution is exponential  Exact shape of the distribution does not affect the optical properties in the range of weak absorption  Main size parameter is the mean chord length, straightforwardly related to SSA