## 1. Introduction

[2] Radiative characterization of snowpack is of importance to determine the overall energy/water balance between ground and atmosphere in snow covered regions [*Wiscombe and Warren*, 1980; *Marks and Dozier*, 1992; *Marks et al.*, 1998; *Aoki et al.*, 2000; *Flanner and Zender*, 2006; *Warren et al.*, 2006; *Kuipers Munneke et al.*, 2009; *Gardner and Sharp*, 2010], for climate models investigating surface-atmosphere energy balances [*Douville et al.*, 1995; *Roesch et al.*, 2002; *Jacobson*, 2004], and for remote sensing applications to map snow cover and to estimate snow water equivalents [*Foster and Rango*, 1989; *Nolin and Dozier*, 2000].

[3] One of the key elements in these applications has been to adequately account for the microstructure of snow in the determination of its optical properties, as snow accumulated on the ground generally consists of a complex 3D microstructure of ice and air [*Good*, 1987; *Dominé et al.*, 2003; *Flin et al.*, 2004; *Schneebeli and Sokratov*, 2004; *Fierz et al.*, 2009]. A common approach is to consider an optical equivalent grain size (OED) derived from the specific surface area (SSA, the air-ice phase boundary per total snow volume) of the actual 3D snow microstructure [*Wiscombe and Warren*, 1980; *Matzl and Schneebeli*, 2006; *Painter et al.*, 2007; *Gallet et al.*, 2009]. However, the accurate determination of OED is challenging and its estimations from visual micrograph examination lead to significantly smaller grain sizes than those estimated by traditional methods (e.g., mean major axis length of the broken snow structure estimated by hand-lens measurements) [*Aoki et al.*, 2000]. Additionally, grain sizes do not correlate with OED for complex snow structures such as rounded snow, facets, and polycrystals [*Painter et al.*, 2007]. Moreover, calculated reflectance for non-spherical, specific grain shaped ice particles of snow (e.g., cubes, cylinders, or hexagonal plates) can differ substantially from that obtained for the respective OED spheres at visible and near infrared (NIR) wavelengths [*Xie et al.*, 2006; *Picard et al.*, 2009]. In contrast, *Gallet et al.* [2009] found minor effects of grain sizes on reflectance measurements in natural snow and *Grenfell and Warren* [1999]reported successful determination of direction-independent radiative properties of a random collection of infinitely long cylinders by using OED spheres and same density as the snow sample measured. These partially contradicting findings indicate the need for in-depth pore-level analysis of the interaction of radiation with snow and for quantifying the error introduced in snow's radiative properties by morphological simplifications, which evidently vary with different snow types.

[4] The widely used DISORT model [*Stamnes et al.*, 1988] for radiative heat transfer in porous media solves a homogenized radiative transfer equation (RTE) with apparent radiative properties, which account for the snow microstructure by assuming a collection of independently scattering dilute (OED) spheres with the same volume fraction as the original snow sample. Thus, by applying a homogenized approach, DISORT, simplifies the snow morphology and neglects the two-phase nature of snow. DISORT has been applied to various investigations related to snow physics [e.g.,*Fily et al.*, 1997; *Jin and Simpson*, 1999; *Glendinning and Morris*, 1999; *Nolin and Dozier*, 2000; *Xie et al.*, 2006; *Painter and Dozer*, 2004; *Gallet et al.*, 2009]. However, the error introduced by the morphological and radiative transfer simplifications, which evidently vary with the type of snow, has remained for the most part unknown. DISORT results have not yet been compared to pore-level radiative transfer simulations where the exact 3D snow microstructure and the multiphase nature of snow have been accounted for explicitly. Hence, previous studies are of limited use for understanding the influence of snow morphology on its radiative properties.

[5] A first approach to account for the complex morphology of snow was introduced in tomography-based studies [*Kaempfer et al.*, 2007; *Bänniger et al.*, 2008], where computed tomography (CT) images of the snow microstructure were used as input for the reflectance and transmittance determination by a phenomenological ray-tracing approach. Recently, a more advanced phenomenological multiphase heat transfer model has been proposed for the radiative characterization of multiphase media in the limit of geometrical optics, assuming independent scattering and neglecting diffraction [*Lipiński et al.*, 2010a, 2010b]. The model is derived by applying volume-averaging theory to RTEs valid within each phase and applying the appropriate phase boundary conditions. The volume-averaged RTEs are based on the energy balance inside the 2-phase media, while the RTE valid for a single phase can be derived – under certain assumptions – directly from Maxwell's equations for discrete random media [*Mishchenko*, 2008; *Mishchenko et al.*, 2011]. The volume averaged RTEs have, so far, not been tested against Maxwell's equations. The volume averaged RTEs have been used to determine the radiative characteristics of reticulate porous ceramics [*Petrasch et al.*, 2007; *Haussener et al.*, 2010a; *Haussener and Steinfeld*, 2012] and packed beds [*Haussener et al.*, 2009, 2010b] by direct pore-level simulations (DPLS, also called direct numerical simulations) on the CT scans and, partially, validated by estimation of the radiative properties by spectroscopic measurements. The novel phenomenological multiphase heat transfer model allows investigating the macroscopic, surface radiative properties and, in addition to previous common snow radiation models, investigating detailed discrete-scale radiative properties and heat transfer at the pore-level scale of complex snowpacks in the limit of geometrical optics, assuming independent scattering and neglecting diffraction.

[6] Here, we apply a multiscale approach to snow (pure, without soot or other impurities), consisting of: (*i*) CT of snow samples to obtain the 3D geometrical representation of their porous microstructure ; (*ii*) use of the CT-determined digital 3D geometry (microstructure) in DPLS for solving the RTEs by collision-based Monte Carlo ray-tracing at the pore scale; (*iii*) extraction of the effective (volumetric) radiative properties of the porous medium, namely: the extinction coefficients, the scattering coefficients, and the scattering phase functions; (*iv*) incorporation of the effective radiative properties in the solution of the volume-averaged RTEs at the continuum scale to determine (surface) radiative properties such as overall reflectance and transmittance of an infinite snow slab. This CT-based Monte Carlo methodology is referred to as the “CTMC” model. Its results can be considered as approaching the exact solution within the limits of the numerical truncation error, the accuracy of geometrical representation (i.e., statistical variations and CT resolution), and the simplifications in the radiation model (geometric optics, independent scattering, neglecting diffraction). The reflectance and transmittance obtained by CTMC simulations are compared to those obtained by DISORT, which solves a homogenized RTE and simplifies the snow structure by OED independent spheres, and to those obtained by a multiphase Monte Carlo approach also applied to the simplified OED snow structure. This allows quantitatively investigating the influence of snow morphology and radiation model (homogenized versus multiphase approach) on the radiative properties. Calculations are carried out for 4 cm-thick infinite (in the other two dimensions) snow slabs of five characteristic snow types. As an example, in-depth analysis of volumetric radiative heat transfer at the pore-level scale (discrete-scale) is given to highlight the possibility of using CTMC to investigate a completely new set of research problems. Transmittance calculations are compared with measured values for seven additional snow types at NIR wavelengths to evaluate the different modeling approaches.