Determining snow specific surface area from near-infrared reflectance measurements: Numerical study of the influence of grain shape

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Abstract

Determining the specific surface area of snow from reflectance measurements in the near infrared domain represents a promising technique to rapidly and quantitatively acquire snow stratigraphic profiles in the field. In this paper, we develop a ray tracing model that simulates the albedo of snowpacks composed of geometric crystals (spheres, cubes, cylinders, etc) and model simulations are exploited to study the influence of the grain shape on the SSA-albedo relationship. The results clearly show that the relationship depends on the grain shape at 1310 nm: Cubic (resp. cylindrical) grains reflect about 40% (resp. 20%) more than spherical grains at equal SSA. Depth-hoar modeled as a collection of hollow cubes is found to reflect exactly as much as cubes. None of the tested shapes (including concave and hollow shapes) reflects more than cubes. These results suggest that determining SSA from albedo measurement is uncertain when the snow grain shape is unknown. This uncertainty reaches ± 20% considering that spherical and cubic grains are the two extreme cases in terms of reflexion. This large value is probably over-pessimistic for practical applications as only perfect crystals are considered in this theoretical study and natural snow is always a mixture of curved and plane faces. Therefore, further experimental studies should focus on jointly measuring SSA and albedo in order to assess the influence of the grain shape (or snow type) on the SSA-albedo relationship in natural snows.

Introduction

Snow is a complex porous medium whose geometry, or micro-structure, can be revealed by microtomography (Pieritz et al., 2004). Even though micro-structure observations are highly valuable for “ab-initio” calculations of snow properties (e.g. thermal properties (Kaempfer et al., 2005) or optics (Kaempfer et al., 2007)), macroscopic variables describing synthetically the underlying geometry are still necessary for most applications. Density is the most important of these macroscopic variables not only because it is easily measured but also because many snow properties strongly depend on it, as for instance the thermal conductivity (Sturm et al., 1997). However, it is insufficient to fully characterize snow micro-structure and to determine other snow properties (e.g. the thermal conductivity varies by up to a factor of five for a given snow density (Sturm et al., 1997)). The specific surface area (SSA), i.e. the total surface area of the air/ice interface per unit of mass, is another macroscopic variable whose interest is growing. The SSA is obviously a relevant variable for air-snow chemistry (Domine and Shepson, 2002, Taillandier et al., 2006) and photochemistry (Domine et al., 2008, Grannas et al., 2007) as the adsorption of gases and many (photo)chemical reactions take place at the air/ice interface. While snow metamorphism also involves gas exchange (Sokratov, 2001), most snow physical models such as CROCUS (Brun et al., 1989, Brun et al., 1992), Sntherm (Jordan, 1991) or SNOWPACK (Lehning et al., 2002) ignore SSA. However, recent experimental and theoretical studies (Legagneux et al., 2004, Legagneux and Domine, 2005, Flanner and Zender, 2006, Taillandier et al., 2007) show that SSA evolution is related to metamorphism. In remote sensing and snow albedo modeling, SSA has long been used but under a different name and concept: the grain optical diameter. The optical diameter dopt (Warren, 1982, Nolin and Dozier, 1993, Nolin and Dozier, 2000) is unambiguously and mathematically related to the SSA (SSA = 6 / ρicedopt), but it carries the idea of independent grains whereas SSA only depends on the total surface area of the ice/air interface and on the total volume of ice.

The SSA of snow samples can be accurately measured by methane adsorption (Legagneux et al., 2002, Domine et al., 2007), by photography using coaxial reflected illumination and stereology principles (Narita, 1971, Alley, 1987, Arnaud et al., 1998) or derived from microtomography (Flin et al., 2004, Pieritz et al., 2004, Kerbrat et al., 2008). All these techniques are time-consuming and require the samples to be transported to a laboratory, which limits the number of samples that can be practically measured. The consequence is that SSA is not systematically measured during field campaigns, as opposed to density for instance and there are only a few studies of the temporal evolution of SSA in natural settings (Hanot and Domine, 1999, Cabanes et al., 2002, Cabanes et al., 2003, Taillandier et al., 2006, Taillandier et al., 2007). Likewise, the spatial distribution, both vertically and horizontally, has been little explored with these techniques (Domine and Shepson, 2002, Taillandier et al., 2006).

There is therefore a growing interest for rapid SSA measurements in the field, even at the expense of some loss of accuracy. Domine et al. (2006) have experimentally demonstrated that a monotonic relationship exists between snow SSA and its reflectance in the infrared. Optical methods have the potential to be used for rapid measurements of SSA. Near-infrared photography around 850 nm wavelength (Matzl and Schneebeli, 2006) or directional reflectance measurements using the band-integration method near 1030 nm (Painter et al., 2007, Nolin and Dozier, 2000) have been developed. In addition, we are currently developing new instruments using reflectance at 1310 nm where snow reflectance is more sensitive to SSA than at shorter wavelengths (Domine et al., 2006). The purpose of this paper is therefore to address the issue of accuracy of such systems from a theoretical standpoint using numerical simulations.

The theoretical basis to measure SSA using near-infrared reflection (either captured by photography, a spectrometer or other instruments) originates from the idea to represent snow by a collection of spheres with the same volume/surface area ratio (V/A) as the snow grains. This idea is found in Warren, 1982, Warren, 1984) and Grenfell and Warren (1999) and the references therein. The hypothesis states that the albedo of real snow can be calculated considering an idealized medium composed of spheres whose diameter is chosen so that the ratio between total ice volume and total grain surface area (V/A) is the same for both real and idealized media. The motivation for finding such an equivalence was to speed up calculations of snow albedo or transmittance using spheres (for which Mie theory is adequate) instead of complex-shaped grains that require difficult electromagnetic calculations. Rapid calculations are indeed useful in climate modeling or for remote sensing retrieval for instance.

SSA is simply related to V/A through SSA = A / (ice) where ρice is the density of ice. From theory, the relationship between V/A and albedo is smooth and monotonic in the case of spherical grains so that it can be inverted to estimate SSA from albedo measurements. However, theoretical studies have shown that in a wide range of wavelengths, grain shape influences the albedo at constant V/A (Grenfell and Warren, 1999, Leroux et al., 1999, Neshyba et al., 2003, Grenfell et al., 2005). In other terms, it means that the V/A equivalence is not strictly valid when considering different grain shapes. This represents a potential source of error that must be quantified to determine SSA from albedo measurements. The present paper is focused on this issue and gives a quantification of the uncertainty induced by variations in the grain shapes.

To treat this issue, an optical model that simulates in detail the trajectory of light from the source, through the snow sample and toward the detector is developed. This approach, known as ray tracing, applies the laws of geometrical optics (i.e. reflection and refraction) to compute the ray trajectory. The model needs the refraction index for each material (ice and air) and the position, shape, size and orientation of each individual grain. The size, position and number of grains are generated using a few input parameters (snow density, slab size, SSA, and spatial and size distribution functions).

Ray tracing is exact under the geometric optics approximation but it is a resource demanding method (much more than radiative transfer calculation for instance), particularly in the case of ice. Ice is indeed a weakly absorbing material at visible and near infrared wavelengths (Warren and Brandt, 2008). As a first consequence, the penetration depth is large relative to the grain size, which imposes to simulate a thick and large snowpack with millions of grains, each grain requiring memory to store its position, orientation and shape. This limitation can be avoided by using periodic boundaries. As a second consequence, each ray usually interacts with a large number of air/ice interfaces (typically about 30 in the near-infrared) before being significantly absorbed. Since each interface generates two rays (one reflected and one transmitted) the number of air/ice interactions to be computed is very large (the order of 230). Hence, millions of intersections must be calculated in the NIR and even more in the visible.

This conventional and deterministic approach of the ray tracing can be replaced by a Monte-Carlo approach also known as photon tracing. It consists in following a single ray (= photon) instead of two at each intersection. The choice to follow the reflected or the transmitted ray is random with the probability given by the Fresnel reflexion coefficients. Hence, the number of interfaces to compute remains small for each incoming ray (the algorithm complexity is linear) but as a counterpart, the convergence is slow (a common issue shared by most Monte Carlo methods) and a large number of incoming rays must be traced to achieve a reasonable accuracy. In the case of snow, our tests show that the Monte-Carlo approach is more efficient than the deterministic approach and is used throughout this paper.

Recently, another ray tracing model for snow has been implemented and used to compute total reflectance, transmittance, and bi-directional reflectance in a large range of wavelengths (Kaempfer et al., 2007). It is very similar to the one presented here. Both models only differ in the shapes they are able to treat. Kaempfer et al.'s (2007) model a smaller number of ideal shapes (spheres and cylinders) than ours, but is able to model real snow whose micro-structure has been acquired by microtomography. To overcome the computational issues of the “full” ray tracing approach, a different approach has been taken by other authors (e.g. (Neshyba et al., 2003, Tanikawa et al., 2006)). Their approach mixes ray tracing to compute the phase function of single ice-body and radiative transfer theory to deduce multiple scattering albedo. While it is much more efficient than the full ray tracing, it is limited to configurations for which the radiative transfer equation can be solved, typically plane-parallel media, incident plane-wave, and the detector must be far from the medium. It is therefore adequate for climate models or remote sensing but is unable to treat particular measurement configurations or real snow structure. Another approach mixing ray-tracing on 2D sections of real snow structure acquired by microtomography and radiative transfer to extrapolate from 2D to 3D has been applied and validated with optical measurements (Bänninger et al., 2008).

Beside the numerical approaches, Kokhanovsky and Zege (2004) have studied the influence of grain shape on albedo by establishing a simple analytical solution of the radiative transfer using adequate approximations for snow. The originality of their work is that it does not prescribe the phase function, which depends on the grain shape. Hence, the influence of grain shape can be tracked analytically. In the present paper, we test their formulation against our ray-tracing calculations and obtain a very good agreement.

Snow optical properties have been studied in numerous works by considering grains with geometrically perfect shapes. Furthermore, in two recent studies, the optical properties of natural snow samples imaged by microtomography have been calculated (Kaempfer et al., 2007, Bänninger et al., 2008). That approach is promising but current computational issues limit its accuracy to address the question of the relationship between snow type and optical properties. To make the computation possible, it is either performed on 3D samples at a low resolution (170 µm) (Kaempfer et al., 2007) or on 2D sections at a much finer resolution (10 or 18 µm) (Bänninger et al., 2008). In addition, the sample size is limited (a few millimeters in both studies) and is much smaller than the typical penetration length in the near-infrared (a few centimeters).

Furthermore, the discretization of the sample volume into voxels significantly degrades the shape of the crystal surface. Since the vector normal to the surface is required to calculate the reflection and transmission directions, the reconstruction of the real surface then requires sophisticated methods and tuning. Because of all these limitations, the grain shapes seen by the photons in the model are distorted with respect to initially imaged samples. This currently prevents the use of these calculations to study the influence of grain shape.

Hence, we consider here geometrically perfect shapes that are present in snow, but natural snow micro-structure is always more complex than our modeling. We show that grain shape has an influence on snow albedo and quantify this effect. In the conclusion, we discuss the implications for real snow. The present work is motivated by the validation of a new instrument to measure snow albedo at 1310 nm but addresses more general questions related to the retrieval of SSA from albedo measurements. The first section describes the SnowRat model, the second presents Kokhanovsky and Zege's (2004) formulation. The third section addresses the major question of grain shape and V/A equivalence. The last section concludes on the error caused by variations in grain shapes. Unless otherwise specified, all the simulations are performed at 1310 nm, the wavelength used in our instrument. The range of SSA (0–35 m2/kg) explored here is representative of snow already submitted to significant metamorphism (Domine et al., 2007). Only dry or refrozen snows are considered, i.e. no liquid water is present.

Section snippets

SnowRat model description

The ray tracing approach consists in following rays of light. Rays are randomly emitted from different points of the source (or in different directions) and their trajectory through the medium is computed following the simple and fundamental laws of refraction and reflection. From a wave point of view (deterministic variant), when a ray hits an interface, its energy is split into a transmitted ray and a reflected ray with proportions given by the Fresnel coefficients. From a photon point of

Analytical solution of radiative transfer in snow

Kokhanovsky and Zege (2004) established a simple analytical solution for the radiative transfer equation with approximations adapted to snow in the UV, visible, and NIR. The authors applied their calculation to spheres and fractal grains made of multi-size tetrahedrons (called tetrahedral fractal hereinafter). We only consider here illumination at normal incidence but their formulation handles any incidence angle as well as diffuse radiations.

The albedo ω under normal illumination can be

Relationship between albedo and SSA

The relationship between albedo and SSA can be easily obtained with DISORT for spherical grains. Fig. 5a shows this relationship at 1310 nm (see (Domine et al., 2006) for other wavelengths). The relationship is monotonic indicating that SSA could be unambiguously estimated from perfect albedo measurements. However, accurate albedo measurements are difficult in field conditions. To give an idea, considering a 5% relative error in albedo measurements, SSA can be estimated with an error of 5% for

Conclusion

We compute snow albedo-SSA relationships for various grain shapes using ray tracing simulations and conclude that for a given albedo, SSA can vary in a range of ± 20% depending solely on the grain shape. In the context of estimating SSA from albedo measurements, this translates into ± 20% error on SSA when the grain shape is unknown. One may argue that this error is pessimistic because all the grains have the same size and shape in a given simulation and the shapes are geometrically perfect

Acknowledgments

We are grateful to F. Roch for administrating the 168-processor cluster at the Observatoire des Sciences de l'Univers de Grenoble (OSUG) where the simulations were run. Fig. 1a and b were obtained with the help of Marion Bisiaux. Fig. 1d was obtained using the scanning electron microscope of GZG/University of Göttingen, Germany with the help of Kirsten Techmer, Till Heinrichs and Werner F. Kuhs.

References (51)

  • BrunE. et al.

    A numerical model to simulate snow-cover stratigraphy for operational avalanche forecasting

    J. Glaciol.

    (1992)
  • BrunE. et al.

    An energy and mass model of snow cover suitable for operational avalanche forecasting

    J. Glaciol.

    (1989)
  • BänningerD. et al.

    Reflectance modelling for real snow structures using a beam tracing model

    Sensors

    (2008)
  • CabanesA. et al.

    Rate of evolution of the specific surface area of surface snow layers

    Environ. technolTechnol.

    (2003)
  • ColbeckS.C.

    Theory of metamorphism of dry snow

    J. Geophys. Res.

    (1983)
  • DingK.H. et al.

    Monte Carlo simulations of particle positions for densely packed multispecies sticky particles

    Microw. Opt. Tech. Lett.

    (2001)
  • DomineF. et al.

    Snow physics as relevant to snow photochemistry

    Atmos. Chem. Phys.

    (2008)
  • DomineF. et al.

    Snow metamorphism as revealed by scanning electron microscopy

    Microsc. Res. Tech.

    (2003)
  • DomineF. et al.

    Air–snow interactions and atmospheric chemistry

    Science

    (2002)
  • DomineF. et al.

    A parameterization of the specific surface area of snow in models of snowpack evolution, based on 345 measurements

    J. Geophys. Res.

    (2007)
  • FlannerM.G. et al.

    Linking snowpack microphysics and albedo evolution

    J. Geophys. Res.

    (2006)
  • FlinF. et al.

    Three-dimensional geometric measurements of snow microstructural evolution under isothermal conditions

    Ann. Glaciol.

    (2004)
  • GrannasA.M. et al.

    An overview of snow photochemistry: evidence, mechanisms and impacts

    Atmos. Chem. Phys.

    (2007)
  • GrenfellT.C. et al.

    Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: 3. Hollow columns and plates

    J. Geophys. Res.

    (2005)
  • GrenfellT.C. et al.

    Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation

    J. Geophys. Res.

    (1999)
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