- Email: [email protected]
Determining snow specific surface area from near-infrared reflectance measurements: Numerical study of the influence of grain shape
Cold Regions Science and Technology 56 (2009) 10–17
Contents lists available at ScienceDirect
Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Determining snow specific surface area from near-infrared reflectance measurements: Numerical study of the influence of grain shape G. Picard ⁎, L. Arnaud 1, F. Domine 1, M. Fily 1 Laboratoire de Glaciologie et Geophysique de l'Environnement, CNRS / Université Joseph Fourier - Grenoble, 54 rue Moliere, 38400 St Martin d'Heres, France
a r t i c l e
i n f o
Article history: Received 6 May 2008 Accepted 4 October 2008 Keywords: Snow Specific surface area Albedo Ray tracing
a b s t r a c t 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. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Snow is a complex porous medium whose geometry, or microstructure, can be revealed by microtomography (Pieritz et al., 2004). Even though micro-structure observations are highly valuable for “abinitio” 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
⁎ Corresponding author. Tel.: +33 476 82 42 53; fax: +33 476 82 42 01. E-mail address: [email protected] (G. Picard). 1 Tel.: +33 476 82 42 53; fax: +33 476 82 42 01. 0165-232X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2008.10.001
(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, 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
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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, 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 / (Vρice) 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
11
(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 bidirectional 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 planewave, 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
12
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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. 2. Snow ray tracing 2.1. SnowRat model description
Fig. 1. Configuration simulated by SnowRat.
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
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 view (Monte-Carlo variant), only one of the two rays is traced further (Kaempfer et al., 2007). The choice between transmission or reflexion is random and depends on the Fresnel coefficients that are interpreted as probabilities instead of as energy ratios. The selected ray carries all the energy of the incident ray. Our model is able to deal with both wave and photon variants but the later, being much more efficient, is used in what follows. When a ray is propagating in the ice, part of its energy is lost by absorption (note that Kaempfer et al. (2007) rely on a probabilistic interpretation of the absorption while we use energy, i.e. the Beer law). The rays are propagated until they definitely escape the medium or when their energy is negligible, below a tunable threshold. We record the energy escaping the medium as a function of the face (see Fig. 1) in order to calculate albedo, transmittance, and leakage by the sides. In addition, the energy outgoing from the upper face can be recorded as a function of ray direction to compute bidirectional reflectance but this is not used in the present study. The incident rays are assumed unpolarized and remain so during their travel through the snowpack. The reflection coefficient for unpolarized waves is the average of the Fresnel coefficients for both transverse magnetic and transverse
Fig. 2. Photographs of snow grains: a) rounded grains, b) cubic grains, c) cubic depth hoar and d) long hollow cylinder. White scale bars: 1 mm except for d), 30μm.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
13
electric modes. The same simplification is made in Kaempfer et al. (2007) and is valid for isotropic media. The ray-tracing approach is fundamental and exact (ab-initio computation) as long as the geometrical optics approximation is valid. This is the case for the ideal objects used in this study, since all their dimensions are much larger than the infrared wavelengths (≈1 µm). SnowRat models the following geometrical shapes: • Non-sticky spheres. The spheres are first distributed on a regular grid and then moved randomly several times. However, a movement is accepted only if the sphere do not intersect another one (Metropolis shuffling (Ding et al., 2001)). At the end, the spheres don't intersect each other and are not necessarily in contact. • Randomly distributed spheres. The spheres are distributed randomly in the box independently of the other spheres. Hence, spheres may intersect other spheres. This is a more realistic description of rounded grain snow (see Fig. 2a). • Cubes. Cubes (all the faces are squares) or cuboids (all the faces are rectangles) are found in natural snow (see Fig. 2b). They form when the prism faces develop, while hexagonal shapes are observed when basal faces develop. • Hexahedra (polyhedron with six faces). Natural crystals are not as perfect as cubes. Hexahedra in SnowRat (Fig. 3b) are cubes whose faces are tilted randomly (with a given maximum tilt angle) and are used to test the effect of parallel faces. They are not general hexahedra, just “irregular cubes”. • Cylinders. This shape is intermediate between the sphere (curved face) and the cube (parallel face) and is interesting from a theoretical point of view. It is not useful for representing real snow grains subjected to metamorphism but cylinders can form in clouds where they are usually hollow and elongated (Fig. 2d). • One-sheeted hyperboloids. It is a simple shape that has concave faces and may be used to model bonds between spherical grains (Fig. 3c). • Depth-hoar. This kind of crystal is frequent in seasonal arctic and alpine snows (Marbouty, 1980); (Sturm and Benson, 1997); (Domine et al., 2008). SnowRat models the cubic version using an aggregate of hollow cubes (Fig. 3a). It is found in nature (see Fig. 2c) but is less frequent than the hexagonal depth-hoar. From an optical point of view, it largely differs from a sphere as it is hollow and has parallel faces. All these shapes can be elongated along one axis by a factor called the aspect ratio (following Neshyba et al. (2003)). Note that cubes with aspect ratios different from one should be called cuboids but we keep the term of “cube” for the sake of simplicity. The present study does not consider hexagonal plates or columns that are frequent in falling snow but much less common in metamorphosed snow. 2.2. Simulation details In our simulations at 1310 nm, the rectangular box (Fig. 1) is 90 mm wide and 40 mm deep which ensures negligible leakage and transmission. The light source sends parallel rays on a 45 × 45 mm2 square zone, centered with respect to the box. The incidence is normal to the face. N = 10,000 rays are necessary to estimate albedo within 1% pffiffiffiffi (the uncertainty decreases as 1= N ). The number of grains in the box is calculated so that the density is about 300 kg m− 3 but this choice has no influence on the albedo, as expected from theory, under the condition that the box is large enough to prevent significant transmission or leakage. For a given simulation, all the grains have the same shape (chosen among spheres, cubes, etc). Their size is either fixed (the same for all the grains) or randomly distributed (with a gamma distribution). The grain position is either completely random (grains can then intersect each other) or random but with a nonstickiness condition (in the case of spheres only).
Fig. 3. a) Hollow cubic depth-hoar, b) hexahedron (the dotted line shows the initial cube from which the hexahedron is derived) and c) one-sheeted hyperboloids as modeled by SnowRat.
The SSA of the generated snow sample is estimated by stereology: one thousand straight lines are randomly drawn across the box. For each line, we compute the number n of intersections with the ice/air interface and the total length L of intersection between ice and the line. The sample SSA is then estimated by b2 × n / L N where b·N stands for the average over all the lines (Underwood, 1970). In the case of non-sticky spheres where SSA can be calculated geometrically, we found the stereological method to be accurate within 1% for 1000 lines. The ray tracing computation time highly increases with albedo which depends mainly on grain size and wavelength. It also depends on grain shape to a lesser extent. At 1310 nm, it is only 7 min for 800diameter spheres on a 3.2 Ghz Pentium and 7 h for 200 µm-diameter. 2.3. Consistency check In order to assess SnowRat accuracy, the model is compared to DISORT (Stamnes et al., 1988) for a semi-infinite medium. DISORT
14
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
solves the radiative transfer equation for plane-parallel media. Its input parameters include the phase function and the scattering and absorption coefficients. They are calculated from Mie theory to simulate non-sticky spheres. For the purpose of the comparison, SnowRat is run in similar conditions, i.e. the grains are non-sticky spheres and the box is as large as computationally possible (160 mm deep, 250 mm wide) in order to reduce the transmission through the snow sample. Fig. 4 shows the albedo as a function of wavelength for grain diameters of 200 and 800. The agreement between both models is nearly perfect for the wavelengths longer than 1030 nm, the albedos predicted by both models differ by no more than 0.01. At the wavelengths shorter than 1030 nm, the transmission through the sample explains the larger albedo difference. The difference reaches 0.03 at 850 nm. 3. 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 calculated from Eqs. (9), (8) and (41) in Kokhanovsky and Zege (2004): qffiffiffiffiffiffiffiffiffiffiffiffi 9 ω = exp − b γdopt 7
ð1Þ
γ = 2kni
ð2Þ
dopt =
6 ρice SSA
ð3Þ
where γ is the ice absorption coefficient calculated using ni the imaginary part of the refractive index, k the wave number related to the wavelength by k = 2π / λ, dopt is the optical diameter related to SSA by Eq. (3) and ρice is the ice density. b depends on the phase function and accounts for the shape: b = 4.53 for spheres, b = 3.62 for tetrahedral fractal grains. At 1310 nm, the agreement with SnowRat for spheres (Fig. 5c) is remarkable. The relationship for fractal grains is also very close to that for cubes.
Fig. 4. Albedo versus wavelength for 200 µm-spherical non-sticky grains (two upper curves) and 800 µm-ones (two lowest curves) as predicted by DISORT (line without symbols) and SnowRat (with symbols).
4. 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 an SSA around 4 m2/kg. This error increases to 10% for SSA around 35 m2/kg and further increases for values larger than 35 m2/kg, typical of fresh fallen snow. The error depends on the SSA value because the SSA-albedo relationship is nonlinear. However, the non-linearity reduces at wavelengths longer than 1310 nm. This suggests that longer wavelengths may be more adequate to measure large SSAs (experimental evidences are given in (Domine et al., 2006)) but other aspects such as the shorter penetration depth must also be taken into consideration in designing new instruments. 4.1. On the shape dependence Fig. 5b shows the albedo-SSA relationship for different shapes computed with SnowRat Cubes, depth-hoar, tetrahedral fractal grains reflect about 30% more than spheres for a given SSA and cylinders reflect about 20% more. These values are roughly independent of the grain size, suggesting that shape is determinant. Simulations with a distribution of grain sizes (gamma distribution with shape factor of 5) give the same results as without size distribution. This result confirms that the V/A equivalence is valid for a given shape. The albedo-SSA relationship also highly depends on the aspect ratio as shown for cubes, cylinders and depth hoar in Fig. 5d. The albedo decreases with increasing aspect ratio for a given SSA. It implies for example that long cylinders (i.e. needles) reflect light more like spheres than like equidimensional cylinders (i.e. cylinders with aspect ratio of 1). More surprisingly, although equidimensional cubes and depth-hoar reflect as much as each other for a given V/A, the same shapes elongated with an aspect ratio of 4 reflect differently. Similar results have already been reported for hexagonal columns and plates (solid crystals (Leroux et al., 1999; Neshyba et al., 2003) (referred as NGW03) and hollow crystals (Grenfell et al., 2005)) (referred as GNW05). NGW03 calculated albedo at 1600 nm and found values around 0.10 for spheres and 0.13 for equidimensional prisms (both with the same SSA ≈ 35 kg/m2). This 30% difference is similar to our results. As discussed by NGW03, the variations of the albedo-SSA relationship with shape and aspect ratio may be explained by the g asymmetry factor. The asymmetry factor depends on the phase function and measures the ratio between forward and backward scattering for a single particle. Since the sphere tends to backscatter less than the equidimensional prism (the shape studied by NGW03), their g is higher. NGW03 present variations of g as a function of aspect ratio for prisms (see their Fig. 7) and compare them with g for spheres. The difference is maximal for an aspect ratio of one (i.e. equidimensional prism) and it decreases for aspect ratios larger or lower than one (i.e. long prisms and plates). g for prisms with aspect ratio around 14 is equal to that of spheres. These results are qualitatively similar to the variations of the albedo-SSA relationship presented here in Fig. 5b and d. However, quantitatively we find that cylinders with aspect ratio as low as 4 already behave nearly like spheres. NGW03 explain that two opposite geometrical factors influence g. The presence of sharp edges and plane faces intersecting at right angle tends to enhance backscattering with respect to spheres and thus reduces g. In contrast parallel faces transmit rays without change in direction, and thus increases g. The authors conclude that the first effect dominates because g for prisms is lower than for spheres.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
15
Fig. 5. Relationship between albedo and SSA for different grain shapes at 1310 nm and normal incidence. a) non-sticky spherical grains simulated by DISORT, SnowRat and calculated by (Kokhanovsky and Zege, 2004). b) non-sticky spheres, sticky spheres, cylinders, cubes simulated by SnowRat. c) Fractal grains calculated by (Kokhanovsky and Zege, 2004). Cubes and non-sticky spheres simulated by SnowRat. d) Cubes, cylinders and depth hoar with various aspect ratios simulated by SnowRat.
To investigate these effects we have modeled cubes whose faces were randomly tilted. This results in an hexahedron without right angles or parallel faces. We find that the albedo-SSA relationship for such hexahedra (curve not shown) is exactly the same as for perfect cubes (in the limit of numerical noise) which invalidates the hypothesis that the right angles (or parallel faces) influence the albedo-SSA relationship. In GNW05′s conclusion, the authors speculated that the difference with respect to spheres would be smaller for irregular crystals than for regular prisms but they also conclude that their speculation was not backed by their own study. Our results establish that irregular cubes reflect just like perfect cubes and not like spheres. The proportion of plane faces may be another factor influencing the albedo-SSA relationship. This hypothesis is supported by the observation that the albedo-SSA relationship for cylinders is between those for spheres and cubes. Indeed cylinders are geometrical intermediates between spheres (having no plane faces) and cubes (having solely plane faces). These shapes differ by their convexity (or their distribution of convexity). To test the influence of the convexity, we have modeled one-sheeted hyperboloids. This shape has negative convexity expect on the flat caps. Beyond theoretical aspects, onesheeted hyperboloids may be adequate to represent bonds between spherical grains. The simulations (not shown) report no difference with cylinders, hence weakening the hypothesis of an influence of the convexity. Finally, NGW03 hypothesized that hollow crystals may deviate from the V/A equivalence more strongly than solid cubes. This issue was addressed in GNW05 for hollow hexagonal columns but the
results do not back the former hypothesis. This is also confirmed here as we show that cubic depth-hoar behaves like solid cubes. In summary, all the shapes studied up to now to our knowledge are more reflective than spheres as long as their aspect ratio is close to one. The most reflective shapes are cubes and depth-hoar. In contrast, any shapes with a dimension much longer than the others reflect less than equidimensional crystals. Further theoretical work is definitely required to explain the geometrical factors influencing the observations made here, in NGW03 and in GNW05. 4.2. Estimating SSA from albedo measurements and uncertainty We distinguish two cases: 1) Assuming the grain shape is perfect and known, the albedo-SSA relationship is then unequivocal. Following Kokhanovsky and Zege's (2004) formalism summarized in Eqs. (1)–(3) at normal incidence, SSA can be calculated from albedo as follows: SSA = b2 kni
972 1 49ρice log2 ω
ð4Þ
The factor b2 depends on grain shape only, the factor kni on wavelength only and ω is the measured albedo. Table 1 summarizes b factors for different shapes. In the case of the shapes modeled by SnowRat, b is obtained by fitting Eq. (4) to the albedo-SSA relationship obtained by simulation.
16
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
Table 1 b value for various shapes to be used in Eq. (4) for estimating SSA from albedo measurement when grain shape is known b from SnowRat b after Kokhanovsky and Zege (2004) Non-sticky sphere Random sphere Cubes Cylinder Depth hoar Cuboid (aspect ratio 4) Long cylinder (aspect ratio 4) Tetrahedral fractal
4.44 4.07 3.39 3.67 3.16 4.5 4.83
4.53
3.62
2) When the grain shape is unknown, the dependence of the albedoSSA relationship on grain shape translates into an uncertainty on SSA. The important conclusion of the last section is that no crystal reflects more than cubes. This gives the upper albedo limit for the possible albedo-SSA relationship range. The other limit is less strict since very long shapes (i.e. with large aspect ratios) may reflect less than spheres. However, we consider at this point that spheres give the lower limit, relying on the fact that snowpacks only composed of long shapes are rare, expect perhaps in fresh fallen snow and in some types of depth hoar (Sturm and Benson, 1997). With these two limits, the maximum relative error shape on SSA retrieved from a perfect albedo measurement ω can be estimated by: shape =
SSAspheres ðωÞ − SSAcubes ðωÞ SSAspheres ðωÞ + SSAcubes ðωÞ
ð5Þ
where SSAX(ω) is the SSA-albedo relationship obtained for shape X with SnowRat. For SSA in the range 4–35 m2/kg, we find that the maximal relative error ranges between 20 and 25%. It means that for a given albedo measurement ω, the true SSA of the snowpack lies in the range SS˜A ± 0.25 × SS˜A where SS˜A is the SSA estimated in the middle of the range and calculated by: h i ˜ = SSAspheres ðωÞ + SSAcubes ðωÞ =2 SSA
ð6Þ
This result agrees with Kokhanovsky and Zege (2004) who found that using the albedo-SSA relationship established for spheres in the case of tetrahedral fractal grains results in underestimating the snow grain size by 40% (and thus overestimating the SSA by 40%). This is equivalent to a ± 20% error. This maximal error gives extreme bounds and is probably pessimistic since snow composed uniquely of perfect cubes, spheres or others are unlikely to be found in nature. Some averaging should result in a shorter range of albedo values for a given SSA. Note that SS˜A can also be estimated by combining Eqs. (4) and (6). The combinationq has the same form as Eq. (4) with the shape factor b˜ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2cubes + b2spheres ˜ calculated as b = ≈3:95. 2 5. 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 (spheres, cubes, cubic depth-hoar, cylinders, hyperboloids, irregular cubes and tetrahedral fractal grains). We tested both size distribution and less regular shapes (cubes whose faces were tilted randomly) and found minor differences with respect to single size distribution or perfect shapes. However, natural metamorphosed snow is always composed of a mixture of shapes. Facets and sharp angles appear in dry snow subjected to high
temperature gradient metamorphism (Marbouty, 1980; Colbeck, 1983). These shapes are caused by rapid crystal growth. However, there are always other parts of snow crystals that have to sublimate to provide the vapor needed for the growth of faceted shapes. These sublimating regions are rounded (Nelson, 1998) so that rounded shapes always coexist with facets in metamorphosed snow. Likewise, snow evolving under low temperature gradient conditions also shows some degree of faceting (Domine et al., 2003). Even in refrozen snow, we have frequently observed faceted shapes, both by optical and electron microscopy (Domine, unpublished results), so that metamorphosed snow is always a mixture of rounded and faceted shapes. We therefore speculate that the accuracy of SSA determination from infrared albedo may in fact be better than inferred from our theoretical consideration, but a significant amount of experimental work is required to test and quantify this suggestion. The analytical albedo-SSA relationship derived from the simple model established by Kokhanovsky and Zege (2004) agrees very well with our computationally intensive ray tracing simulations. That analytical formulation is very attractive because the most important factors influencing snow albedo (incident angle, wavelength, shape factor, SSA/ optical diameter) are taken into account and are independent of each other. In particular the shape factor b is independent of the wavelength and of the incidence angle. The values of b presented in Table 1 are therefore general and can be applied at other wavelengths in the visible and near infra-red domain, as well as at any incidence angles or with diffuse radiations. Hence, Kokhanovsky and Zege's (2004) formulation can provide the theoretical albedo-SSA relationship for a broad spectral range as in the case of the infrared photography technique developed by Matzl and Schneebeli (2006) (the spectral range is 850–1000 nm and depends on the camera sensor and filter sensitivities). Kokhanovsky and Zege's (2004) formulation can also provide accurate spectral albedo, accounting for the zenith angle dependence, to climate modeling at nearly no computation cost, at least in the case of thick snowpacks. The theoretical predictions of the present paper need experimental support that is not yet available. The bottleneck is the availability of joint measurements of hemispherical spectral albedo and SSA with accuracy better than 20% for snow crystals having a wide range of shapes. In numerous studies where snow albedo measurements were conducted (e.g. (Nakamura et al., 2001); (Tanikawa et al., 2006)), the grain size was measured by visual inspection which is an imprecise method and does not allow to measure the optical diameter (or the SSA). Nolin and Dozier (2000)) relate the reflectance around 1030 nm to grain size but only some of their grain size measurements are actually optical diameters derived by stereological analysis of grain images. Nevertheless, while they found a general good agreement between measured and optically estimated grain diameters, the dispersion is as large as 150% for the smallest grains (i.e. highest SSA). Hence, to our knowledge, no published measurements with accuracy better than 20%, are available to allow a correct detection of the grain shape influence. Such measurements are urgent and would provide information on 1) which model shape best approximates the different types of real snow, and therefore warrant further theoretical work 2) the b factor values for different snow type to be use in climate models or in remote sensing and 3) the range of SSA-albedo relationships that limits the accuracy of SSA retrieval from albedo when the grain shape is unknown. Even if our estimate of the inaccuracy due to unknown grain shape (20%–25%) is large and may be reduced by future experimental investigations, the grain shape will likely remain a problem for accurately retrieving SSA from albedo measurements. An obvious solution is to acquire grain shapes systematically in parallel with the albedo measurements and to apply the correcting b shape factor, determined theoretically or empirically. However an automatic and adequate determination of the b shape factor would be preferable. We hypothesize that measuring bi-directional reflectance (at least at a few selected angles and with coarse angular resolution) instead of only the albedo could provide the information necessary to estimate the b shape factor and then improve the SSA retrieval.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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 Alley, R.B., 1987. Texture of polar firn for remote sensing. Ann. Glaciol. 9, 1–4. Arnaud, L., Gay, M., Barnola, J., Duval, P., 1998. Imaging of firn and bubbly ice in coaxial reflected light: a new technique for the characterization of these porous media. Ann. Glaciol. 44, 326–332. Brun, E., David, P., Sudul, M., Brugnot, G., 1992. A numerical model to simulate snowcover stratigraphy for operational avalanche forecasting. J. Glaciol. 38 (128), 13–22. Brun, E., Martin, E., Simon, V., Gendre, C., Coléou, C., 1989. An energy and mass model of snow cover suitable for operational avalanche forecasting. J. Glaciol. 35, 333–342. Bänninger, D., Bourgeois, S., Matzl, M., Schneebeli, M., 2008. Reflectance modelling for real snow structures using a beam tracing model. Sensors 3482–3496. Cabanes, A., Legagneux, L., Domine, F., 2002. Evolution of the specific surface area and of crystal morphology of Arctic fresh snow during the ALERT 2000 campaign. Atmos. Environ. 36, 2767–2777. Cabanes, A., Legagneux, L., Domine, F., 2003. Rate of evolution of the specific surface area of surface snow layers. Environ. technolTechnol. 37, 661–666. Colbeck, S.C., 1983. Theory of metamorphism of dry snow. J. Geophys. Res. 88, 5475–5482. Ding, K.H., Tsang, L., Shih, S.E., 2001. Monte Carlo simulations of particle positions for densely packed multispecies sticky particles. Microw. Opt. Tech. Lett. 30 (3), 187–192. Domine, F., Albert, M., Huthwelker, T., Jacobi, H.W., Kokhanovsky, A.A., Lehning, M., Picard, G., Simpson, W.R., 2008. Snow physics as relevant to snow photochemistry. Atmos. Chem. Phys. 8 (2), 171–208. URL http://www.atmos-chem-phys.net/8/171/ 2008/. Domine, F., Lauzier, T., Cabanes, A., Legagneux, L., Kuhs, W., Techmer, K., Heinrichs, T., 2003. Snow metamorphism as revealed by scanning electron microscopy. Microsc. Res. Tech. 62, 33–48. Domine, F., Salvatori, R., Legagneux, L., Salzano, R., Fily, M., Casacchia, R., 2006. Correlation between the specific surface area and the short wave infrared (SWIR) reflectance of snow. Cold Reg. Sci. Technol. 46 (1), 60–68. Domine, F., Shepson, P.B., 2002. Air–snow interactions and atmospheric chemistry. Science 297, 1506–1510. Domine, F., Taillandier, A.S., Simpson, W., 2007. A parameterization of the specific surface area of snow in models of snowpack evolution, based on 345 measurements. J. Geophys. Res. 112 (F02031). Flanner, M.G., Zender, C.S., 2006. Linking snowpack microphysics and albedo evolution. J. Geophys. Res. 111 (D10), 12208–+. Flin, F., Brzoska, J., Lesaffre, B., Coléou, C., Pieritz, R., 2004. Three-dimensional geometric measurements of snow microstructural evolution under isothermal conditions. Ann. Glaciol. 38, 39–44. Grannas, A.M., Jones, A.E., Dibb, J., Ammann, M., Anastasio, C., Beine, H.J., Bergin, M., Bottenheim, J., Boxe, C.S., Carver, G., Chen, G., Crawford, J.H., Domine, F., Frey, M.M., Guzmán, M.I., Heard, D.E., Helmig, D., Hoffmann, M.R., Honrath, R.E., Huey, L.G., Hutterli, M., Jacobi, H.W., Klán, P., Lefer, B., McConnell, J., Plane, J., Sander, R., Savarino, J., Shepson, P.B., Simpson, W.R., Sodeau, J.R., von Glasow, R., Weller, R., Wolff, E.W., Zhu, T., 2007. An overview of snow photochemistry: evidence, mechanisms and impacts. Atmos. Chem. Phys. 7, 4329–4373. Grenfell, T.C., Neshyba, S.P., Warren, S.G., 2005. 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. 110 (D9) 17203–+. Grenfell, T.C., Warren, S.G., 1999. Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation. J. Geophys. Res. 104, 31697–31710. Hanot, L., Domine, F., 1999. Evolution of the surface area of a snow layer. Environ. Sci. Technol. 33, 4250–4255. Jordan, R., 1991. A one-dimensional temperature model for a snow cover: technical documentation for sntherm.89. Technical Report Special Report 91-16, U. S. Army Cold Regions Research and Engineering Laboratory, Hanover, N. H.
17
Kaempfer, T.U., Hopkins, M.A., Perovich, D.K., 2007. A three-dimensional microstructurebased photon-tracking model of radiative transfer in snow. J. Geophys. Res. 112 (D11), 24113–+. Kaempfer, T.U., Schneebeli, M., Sokratov, S.A., 2005. A microstructural approach to model heat transfer in snow. Geophys. Res. Lett. 32, 21503–+. Kerbrat, M., Pinzer, B., Huthwelker, T., Gäggeler, H.W., Ammann, M., Schneebeli, M., 2008. Measuring the specific surface area of snow with x-ray tomography and gas adsorption: comparison and implications for surface smoothness. Atmos. Chem. Phys. 8 (5), 1261–1275. URL http://www.atmos-chem-phys.net/8/1261/2008/. Kokhanovsky, A.A., Zege, E.P., 2004. Scattering optics of snow. Appl. Opt. 43, 1589–1602. Legagneux, L., Cabanes, A., Domine, F., 2002. Measurement of the specific surface area of 176 snow samples using methane adsorption at 77 K. J. Geophys. Res. 107 (D17), 4335. Legagneux, L., Domine, F., 2005. A mean-field model of the isothermal metamorphism of dry snow. J. Geophys. Res. 110 (F04011). Legagneux, L., Taillandier, A.S., Domine, F., 2004. Grain growth theories and the isothermal evolution of the specific surface area of snow. J. Appl. Phys. 95, 6175–6184. Lehning, M., Bartelt, P., Brown, B., Fierz, C., Satyawali, P., 2002. A physical SNOWPACK model for the Swiss avalanche warning part II: snow microstructure. Cold Reg. Sci. Technol. 35, 147–167. Leroux, C., Lenoble, J., Brogniez, G., Hovenier, J., De Haan, J., 1999. A model for the bidirectional polarized reflectance of snow. J. Quant. Spectrosc. Radiat. Transfer 61 (13), 273–285. Marbouty, D., 1980. An experimental study of temperature gradient metamorphism. J. Glaciol. 26, 303–312. Matzl, M., Schneebeli, M., 2006. Measuring specific surface area of snow by nearinfrared photography. J. Glaciol. 52 (7), 558–564. URL http://www.ingentaconnect. com/content/igsoc/jog/2006/00000052/00000179/art00007. Nakamura, T., Abe, O., Hasegawa, T., Tamura, R., Ohta, T., 2001. Spectral reflectance of snow with a known grain-size distribution in successive metamorphism. Cold Reg. Sci. Technol. 32 (1), 13–26. Narita, H., 1971. Specific surface of deposited snow II. Low Temp. Sci. A29, 69–81. Nelson, J., 1998. Sublimation of ice crystals. J. Atmos. Sci. 55, 910–919. Neshyba, S.P., Grenfell, T.C., Warren, S.G., 2003. Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: 2. Hexagonal columns and plates. J. Geophys. Res. 108, 4448–+. Nolin, A., Dozier, J., 1993. Estimating snow grain size using AVIRIS data. Remote Sens. Environ. 44 (2–3), 231–238. Nolin, A., Dozier, J., 2000. A hyperspectral method for remotely sensing the grain size of snow. Remote Sens. Environ. 74 (2), 207–216. Painter, T.H., Molotch, N., Cassidy, M., Flanner, M., Steffen, K., 2007. Contact spectroscopy for the determination of stratigraphy of snow grain size. J. Glaciol. 53, 121–127. Pieritz, R., Brzoska, J.B., Flin, F., Lesaffre, C., Coléou, B., 2004. From snow X-ray microtomograph raw volume data to micromechanics modeling: first results. Ann. Glaciol. 38 (1), 52–58. Sokratov, S., 2001. Parameters influencing the recrystallization rate of snow. Cold Reg. Sci. Technol. 33, 263–274. Stamnes, K., Tsay, S.C., Jayaweera, K., Wiscombe, W., 1988. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt. 27, 2502–2509. Sturm, M., Benson, C., 1997. Vapor transport, grain growth and depth hoar development in the subarctic snow. J. Glaciol. 43, 42–59. Sturm, M., Holmgren, J., König, M., Morris, K., 1997. The thermal conductivity of seasonal snow. J. Glaciol. 43, 26–41. Taillandier, A.S., Domine, F., Simpson, W., Sturm, M., Douglas, T., 2007. The rate of decrease of the specific surface area of dry snow: isothermal versus temperature gradient conditions. J. Geophys. Res. 112 (F03003). Taillandier, A.S., Domine, F., Simpson, W., Sturm, M., Douglas, T., Severin, K., 2006. Evolution of the snow area index (SAI) of the subarctic snowpack in central Alaska over a whole season. consequences for the air to snow transfer of pollutants. Environ. Sci. Technol. 40, 7521–7527. Tanikawa, T., Aoki, T., Hori, M., Hachikubo, A., Abe, O., Aniya, M., 2006. Monte Carlo simulations of spectral albedo for artificial snowpacks composed of spherical and nonspherical particles. Appl. Opt. 45, 5310–5319. Underwood, E.E., 1970. Quantitative Stereology. Addison Wesley, Reading, Massachusetts. Warren, S.G., 1982. Optical properties of snow (Paper 1R1505). Rev. Geophys. Space Phys. 20, 67–+. Warren, S.G., 1984. Optical constants of ice from the ultraviolet to the microwave. Appl. Opt. 23, 1206–1225. Warren, S.G., Brandt, R.E., 2008. Optical constants of ice from the ultraviolet to the microwave: a revised compilation. J. Geophys. Res. 113 (D14220).
Contents lists available at ScienceDirect
Cold Regions Science and Technology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c o l d r e g i o n s
Determining snow specific surface area from near-infrared reflectance measurements: Numerical study of the influence of grain shape G. Picard ⁎, L. Arnaud 1, F. Domine 1, M. Fily 1 Laboratoire de Glaciologie et Geophysique de l'Environnement, CNRS / Université Joseph Fourier - Grenoble, 54 rue Moliere, 38400 St Martin d'Heres, France
a r t i c l e
i n f o
Article history: Received 6 May 2008 Accepted 4 October 2008 Keywords: Snow Specific surface area Albedo Ray tracing
a b s t r a c t 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. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Snow is a complex porous medium whose geometry, or microstructure, can be revealed by microtomography (Pieritz et al., 2004). Even though micro-structure observations are highly valuable for “abinitio” 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
⁎ Corresponding author. Tel.: +33 476 82 42 53; fax: +33 476 82 42 01. E-mail address: [email protected] (G. Picard). 1 Tel.: +33 476 82 42 53; fax: +33 476 82 42 01. 0165-232X/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.coldregions.2008.10.001
(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, 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
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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, 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 / (Vρice) 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
11
(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 bidirectional 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 planewave, 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
12
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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. 2. Snow ray tracing 2.1. SnowRat model description
Fig. 1. Configuration simulated by SnowRat.
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
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 view (Monte-Carlo variant), only one of the two rays is traced further (Kaempfer et al., 2007). The choice between transmission or reflexion is random and depends on the Fresnel coefficients that are interpreted as probabilities instead of as energy ratios. The selected ray carries all the energy of the incident ray. Our model is able to deal with both wave and photon variants but the later, being much more efficient, is used in what follows. When a ray is propagating in the ice, part of its energy is lost by absorption (note that Kaempfer et al. (2007) rely on a probabilistic interpretation of the absorption while we use energy, i.e. the Beer law). The rays are propagated until they definitely escape the medium or when their energy is negligible, below a tunable threshold. We record the energy escaping the medium as a function of the face (see Fig. 1) in order to calculate albedo, transmittance, and leakage by the sides. In addition, the energy outgoing from the upper face can be recorded as a function of ray direction to compute bidirectional reflectance but this is not used in the present study. The incident rays are assumed unpolarized and remain so during their travel through the snowpack. The reflection coefficient for unpolarized waves is the average of the Fresnel coefficients for both transverse magnetic and transverse
Fig. 2. Photographs of snow grains: a) rounded grains, b) cubic grains, c) cubic depth hoar and d) long hollow cylinder. White scale bars: 1 mm except for d), 30μm.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
13
electric modes. The same simplification is made in Kaempfer et al. (2007) and is valid for isotropic media. The ray-tracing approach is fundamental and exact (ab-initio computation) as long as the geometrical optics approximation is valid. This is the case for the ideal objects used in this study, since all their dimensions are much larger than the infrared wavelengths (≈1 µm). SnowRat models the following geometrical shapes: • Non-sticky spheres. The spheres are first distributed on a regular grid and then moved randomly several times. However, a movement is accepted only if the sphere do not intersect another one (Metropolis shuffling (Ding et al., 2001)). At the end, the spheres don't intersect each other and are not necessarily in contact. • Randomly distributed spheres. The spheres are distributed randomly in the box independently of the other spheres. Hence, spheres may intersect other spheres. This is a more realistic description of rounded grain snow (see Fig. 2a). • Cubes. Cubes (all the faces are squares) or cuboids (all the faces are rectangles) are found in natural snow (see Fig. 2b). They form when the prism faces develop, while hexagonal shapes are observed when basal faces develop. • Hexahedra (polyhedron with six faces). Natural crystals are not as perfect as cubes. Hexahedra in SnowRat (Fig. 3b) are cubes whose faces are tilted randomly (with a given maximum tilt angle) and are used to test the effect of parallel faces. They are not general hexahedra, just “irregular cubes”. • Cylinders. This shape is intermediate between the sphere (curved face) and the cube (parallel face) and is interesting from a theoretical point of view. It is not useful for representing real snow grains subjected to metamorphism but cylinders can form in clouds where they are usually hollow and elongated (Fig. 2d). • One-sheeted hyperboloids. It is a simple shape that has concave faces and may be used to model bonds between spherical grains (Fig. 3c). • Depth-hoar. This kind of crystal is frequent in seasonal arctic and alpine snows (Marbouty, 1980); (Sturm and Benson, 1997); (Domine et al., 2008). SnowRat models the cubic version using an aggregate of hollow cubes (Fig. 3a). It is found in nature (see Fig. 2c) but is less frequent than the hexagonal depth-hoar. From an optical point of view, it largely differs from a sphere as it is hollow and has parallel faces. All these shapes can be elongated along one axis by a factor called the aspect ratio (following Neshyba et al. (2003)). Note that cubes with aspect ratios different from one should be called cuboids but we keep the term of “cube” for the sake of simplicity. The present study does not consider hexagonal plates or columns that are frequent in falling snow but much less common in metamorphosed snow. 2.2. Simulation details In our simulations at 1310 nm, the rectangular box (Fig. 1) is 90 mm wide and 40 mm deep which ensures negligible leakage and transmission. The light source sends parallel rays on a 45 × 45 mm2 square zone, centered with respect to the box. The incidence is normal to the face. N = 10,000 rays are necessary to estimate albedo within 1% pffiffiffiffi (the uncertainty decreases as 1= N ). The number of grains in the box is calculated so that the density is about 300 kg m− 3 but this choice has no influence on the albedo, as expected from theory, under the condition that the box is large enough to prevent significant transmission or leakage. For a given simulation, all the grains have the same shape (chosen among spheres, cubes, etc). Their size is either fixed (the same for all the grains) or randomly distributed (with a gamma distribution). The grain position is either completely random (grains can then intersect each other) or random but with a nonstickiness condition (in the case of spheres only).
Fig. 3. a) Hollow cubic depth-hoar, b) hexahedron (the dotted line shows the initial cube from which the hexahedron is derived) and c) one-sheeted hyperboloids as modeled by SnowRat.
The SSA of the generated snow sample is estimated by stereology: one thousand straight lines are randomly drawn across the box. For each line, we compute the number n of intersections with the ice/air interface and the total length L of intersection between ice and the line. The sample SSA is then estimated by b2 × n / L N where b·N stands for the average over all the lines (Underwood, 1970). In the case of non-sticky spheres where SSA can be calculated geometrically, we found the stereological method to be accurate within 1% for 1000 lines. The ray tracing computation time highly increases with albedo which depends mainly on grain size and wavelength. It also depends on grain shape to a lesser extent. At 1310 nm, it is only 7 min for 800diameter spheres on a 3.2 Ghz Pentium and 7 h for 200 µm-diameter. 2.3. Consistency check In order to assess SnowRat accuracy, the model is compared to DISORT (Stamnes et al., 1988) for a semi-infinite medium. DISORT
14
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
solves the radiative transfer equation for plane-parallel media. Its input parameters include the phase function and the scattering and absorption coefficients. They are calculated from Mie theory to simulate non-sticky spheres. For the purpose of the comparison, SnowRat is run in similar conditions, i.e. the grains are non-sticky spheres and the box is as large as computationally possible (160 mm deep, 250 mm wide) in order to reduce the transmission through the snow sample. Fig. 4 shows the albedo as a function of wavelength for grain diameters of 200 and 800. The agreement between both models is nearly perfect for the wavelengths longer than 1030 nm, the albedos predicted by both models differ by no more than 0.01. At the wavelengths shorter than 1030 nm, the transmission through the sample explains the larger albedo difference. The difference reaches 0.03 at 850 nm. 3. 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 calculated from Eqs. (9), (8) and (41) in Kokhanovsky and Zege (2004): qffiffiffiffiffiffiffiffiffiffiffiffi 9 ω = exp − b γdopt 7
ð1Þ
γ = 2kni
ð2Þ
dopt =
6 ρice SSA
ð3Þ
where γ is the ice absorption coefficient calculated using ni the imaginary part of the refractive index, k the wave number related to the wavelength by k = 2π / λ, dopt is the optical diameter related to SSA by Eq. (3) and ρice is the ice density. b depends on the phase function and accounts for the shape: b = 4.53 for spheres, b = 3.62 for tetrahedral fractal grains. At 1310 nm, the agreement with SnowRat for spheres (Fig. 5c) is remarkable. The relationship for fractal grains is also very close to that for cubes.
Fig. 4. Albedo versus wavelength for 200 µm-spherical non-sticky grains (two upper curves) and 800 µm-ones (two lowest curves) as predicted by DISORT (line without symbols) and SnowRat (with symbols).
4. 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 an SSA around 4 m2/kg. This error increases to 10% for SSA around 35 m2/kg and further increases for values larger than 35 m2/kg, typical of fresh fallen snow. The error depends on the SSA value because the SSA-albedo relationship is nonlinear. However, the non-linearity reduces at wavelengths longer than 1310 nm. This suggests that longer wavelengths may be more adequate to measure large SSAs (experimental evidences are given in (Domine et al., 2006)) but other aspects such as the shorter penetration depth must also be taken into consideration in designing new instruments. 4.1. On the shape dependence Fig. 5b shows the albedo-SSA relationship for different shapes computed with SnowRat Cubes, depth-hoar, tetrahedral fractal grains reflect about 30% more than spheres for a given SSA and cylinders reflect about 20% more. These values are roughly independent of the grain size, suggesting that shape is determinant. Simulations with a distribution of grain sizes (gamma distribution with shape factor of 5) give the same results as without size distribution. This result confirms that the V/A equivalence is valid for a given shape. The albedo-SSA relationship also highly depends on the aspect ratio as shown for cubes, cylinders and depth hoar in Fig. 5d. The albedo decreases with increasing aspect ratio for a given SSA. It implies for example that long cylinders (i.e. needles) reflect light more like spheres than like equidimensional cylinders (i.e. cylinders with aspect ratio of 1). More surprisingly, although equidimensional cubes and depth-hoar reflect as much as each other for a given V/A, the same shapes elongated with an aspect ratio of 4 reflect differently. Similar results have already been reported for hexagonal columns and plates (solid crystals (Leroux et al., 1999; Neshyba et al., 2003) (referred as NGW03) and hollow crystals (Grenfell et al., 2005)) (referred as GNW05). NGW03 calculated albedo at 1600 nm and found values around 0.10 for spheres and 0.13 for equidimensional prisms (both with the same SSA ≈ 35 kg/m2). This 30% difference is similar to our results. As discussed by NGW03, the variations of the albedo-SSA relationship with shape and aspect ratio may be explained by the g asymmetry factor. The asymmetry factor depends on the phase function and measures the ratio between forward and backward scattering for a single particle. Since the sphere tends to backscatter less than the equidimensional prism (the shape studied by NGW03), their g is higher. NGW03 present variations of g as a function of aspect ratio for prisms (see their Fig. 7) and compare them with g for spheres. The difference is maximal for an aspect ratio of one (i.e. equidimensional prism) and it decreases for aspect ratios larger or lower than one (i.e. long prisms and plates). g for prisms with aspect ratio around 14 is equal to that of spheres. These results are qualitatively similar to the variations of the albedo-SSA relationship presented here in Fig. 5b and d. However, quantitatively we find that cylinders with aspect ratio as low as 4 already behave nearly like spheres. NGW03 explain that two opposite geometrical factors influence g. The presence of sharp edges and plane faces intersecting at right angle tends to enhance backscattering with respect to spheres and thus reduces g. In contrast parallel faces transmit rays without change in direction, and thus increases g. The authors conclude that the first effect dominates because g for prisms is lower than for spheres.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
15
Fig. 5. Relationship between albedo and SSA for different grain shapes at 1310 nm and normal incidence. a) non-sticky spherical grains simulated by DISORT, SnowRat and calculated by (Kokhanovsky and Zege, 2004). b) non-sticky spheres, sticky spheres, cylinders, cubes simulated by SnowRat. c) Fractal grains calculated by (Kokhanovsky and Zege, 2004). Cubes and non-sticky spheres simulated by SnowRat. d) Cubes, cylinders and depth hoar with various aspect ratios simulated by SnowRat.
To investigate these effects we have modeled cubes whose faces were randomly tilted. This results in an hexahedron without right angles or parallel faces. We find that the albedo-SSA relationship for such hexahedra (curve not shown) is exactly the same as for perfect cubes (in the limit of numerical noise) which invalidates the hypothesis that the right angles (or parallel faces) influence the albedo-SSA relationship. In GNW05′s conclusion, the authors speculated that the difference with respect to spheres would be smaller for irregular crystals than for regular prisms but they also conclude that their speculation was not backed by their own study. Our results establish that irregular cubes reflect just like perfect cubes and not like spheres. The proportion of plane faces may be another factor influencing the albedo-SSA relationship. This hypothesis is supported by the observation that the albedo-SSA relationship for cylinders is between those for spheres and cubes. Indeed cylinders are geometrical intermediates between spheres (having no plane faces) and cubes (having solely plane faces). These shapes differ by their convexity (or their distribution of convexity). To test the influence of the convexity, we have modeled one-sheeted hyperboloids. This shape has negative convexity expect on the flat caps. Beyond theoretical aspects, onesheeted hyperboloids may be adequate to represent bonds between spherical grains. The simulations (not shown) report no difference with cylinders, hence weakening the hypothesis of an influence of the convexity. Finally, NGW03 hypothesized that hollow crystals may deviate from the V/A equivalence more strongly than solid cubes. This issue was addressed in GNW05 for hollow hexagonal columns but the
results do not back the former hypothesis. This is also confirmed here as we show that cubic depth-hoar behaves like solid cubes. In summary, all the shapes studied up to now to our knowledge are more reflective than spheres as long as their aspect ratio is close to one. The most reflective shapes are cubes and depth-hoar. In contrast, any shapes with a dimension much longer than the others reflect less than equidimensional crystals. Further theoretical work is definitely required to explain the geometrical factors influencing the observations made here, in NGW03 and in GNW05. 4.2. Estimating SSA from albedo measurements and uncertainty We distinguish two cases: 1) Assuming the grain shape is perfect and known, the albedo-SSA relationship is then unequivocal. Following Kokhanovsky and Zege's (2004) formalism summarized in Eqs. (1)–(3) at normal incidence, SSA can be calculated from albedo as follows: SSA = b2 kni
972 1 49ρice log2 ω
ð4Þ
The factor b2 depends on grain shape only, the factor kni on wavelength only and ω is the measured albedo. Table 1 summarizes b factors for different shapes. In the case of the shapes modeled by SnowRat, b is obtained by fitting Eq. (4) to the albedo-SSA relationship obtained by simulation.
16
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
Table 1 b value for various shapes to be used in Eq. (4) for estimating SSA from albedo measurement when grain shape is known b from SnowRat b after Kokhanovsky and Zege (2004) Non-sticky sphere Random sphere Cubes Cylinder Depth hoar Cuboid (aspect ratio 4) Long cylinder (aspect ratio 4) Tetrahedral fractal
4.44 4.07 3.39 3.67 3.16 4.5 4.83
4.53
3.62
2) When the grain shape is unknown, the dependence of the albedoSSA relationship on grain shape translates into an uncertainty on SSA. The important conclusion of the last section is that no crystal reflects more than cubes. This gives the upper albedo limit for the possible albedo-SSA relationship range. The other limit is less strict since very long shapes (i.e. with large aspect ratios) may reflect less than spheres. However, we consider at this point that spheres give the lower limit, relying on the fact that snowpacks only composed of long shapes are rare, expect perhaps in fresh fallen snow and in some types of depth hoar (Sturm and Benson, 1997). With these two limits, the maximum relative error shape on SSA retrieved from a perfect albedo measurement ω can be estimated by: shape =
SSAspheres ðωÞ − SSAcubes ðωÞ SSAspheres ðωÞ + SSAcubes ðωÞ
ð5Þ
where SSAX(ω) is the SSA-albedo relationship obtained for shape X with SnowRat. For SSA in the range 4–35 m2/kg, we find that the maximal relative error ranges between 20 and 25%. It means that for a given albedo measurement ω, the true SSA of the snowpack lies in the range SS˜A ± 0.25 × SS˜A where SS˜A is the SSA estimated in the middle of the range and calculated by: h i ˜ = SSAspheres ðωÞ + SSAcubes ðωÞ =2 SSA
ð6Þ
This result agrees with Kokhanovsky and Zege (2004) who found that using the albedo-SSA relationship established for spheres in the case of tetrahedral fractal grains results in underestimating the snow grain size by 40% (and thus overestimating the SSA by 40%). This is equivalent to a ± 20% error. This maximal error gives extreme bounds and is probably pessimistic since snow composed uniquely of perfect cubes, spheres or others are unlikely to be found in nature. Some averaging should result in a shorter range of albedo values for a given SSA. Note that SS˜A can also be estimated by combining Eqs. (4) and (6). The combinationq has the same form as Eq. (4) with the shape factor b˜ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b2cubes + b2spheres ˜ calculated as b = ≈3:95. 2 5. 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 (spheres, cubes, cubic depth-hoar, cylinders, hyperboloids, irregular cubes and tetrahedral fractal grains). We tested both size distribution and less regular shapes (cubes whose faces were tilted randomly) and found minor differences with respect to single size distribution or perfect shapes. However, natural metamorphosed snow is always composed of a mixture of shapes. Facets and sharp angles appear in dry snow subjected to high
temperature gradient metamorphism (Marbouty, 1980; Colbeck, 1983). These shapes are caused by rapid crystal growth. However, there are always other parts of snow crystals that have to sublimate to provide the vapor needed for the growth of faceted shapes. These sublimating regions are rounded (Nelson, 1998) so that rounded shapes always coexist with facets in metamorphosed snow. Likewise, snow evolving under low temperature gradient conditions also shows some degree of faceting (Domine et al., 2003). Even in refrozen snow, we have frequently observed faceted shapes, both by optical and electron microscopy (Domine, unpublished results), so that metamorphosed snow is always a mixture of rounded and faceted shapes. We therefore speculate that the accuracy of SSA determination from infrared albedo may in fact be better than inferred from our theoretical consideration, but a significant amount of experimental work is required to test and quantify this suggestion. The analytical albedo-SSA relationship derived from the simple model established by Kokhanovsky and Zege (2004) agrees very well with our computationally intensive ray tracing simulations. That analytical formulation is very attractive because the most important factors influencing snow albedo (incident angle, wavelength, shape factor, SSA/ optical diameter) are taken into account and are independent of each other. In particular the shape factor b is independent of the wavelength and of the incidence angle. The values of b presented in Table 1 are therefore general and can be applied at other wavelengths in the visible and near infra-red domain, as well as at any incidence angles or with diffuse radiations. Hence, Kokhanovsky and Zege's (2004) formulation can provide the theoretical albedo-SSA relationship for a broad spectral range as in the case of the infrared photography technique developed by Matzl and Schneebeli (2006) (the spectral range is 850–1000 nm and depends on the camera sensor and filter sensitivities). Kokhanovsky and Zege's (2004) formulation can also provide accurate spectral albedo, accounting for the zenith angle dependence, to climate modeling at nearly no computation cost, at least in the case of thick snowpacks. The theoretical predictions of the present paper need experimental support that is not yet available. The bottleneck is the availability of joint measurements of hemispherical spectral albedo and SSA with accuracy better than 20% for snow crystals having a wide range of shapes. In numerous studies where snow albedo measurements were conducted (e.g. (Nakamura et al., 2001); (Tanikawa et al., 2006)), the grain size was measured by visual inspection which is an imprecise method and does not allow to measure the optical diameter (or the SSA). Nolin and Dozier (2000)) relate the reflectance around 1030 nm to grain size but only some of their grain size measurements are actually optical diameters derived by stereological analysis of grain images. Nevertheless, while they found a general good agreement between measured and optically estimated grain diameters, the dispersion is as large as 150% for the smallest grains (i.e. highest SSA). Hence, to our knowledge, no published measurements with accuracy better than 20%, are available to allow a correct detection of the grain shape influence. Such measurements are urgent and would provide information on 1) which model shape best approximates the different types of real snow, and therefore warrant further theoretical work 2) the b factor values for different snow type to be use in climate models or in remote sensing and 3) the range of SSA-albedo relationships that limits the accuracy of SSA retrieval from albedo when the grain shape is unknown. Even if our estimate of the inaccuracy due to unknown grain shape (20%–25%) is large and may be reduced by future experimental investigations, the grain shape will likely remain a problem for accurately retrieving SSA from albedo measurements. An obvious solution is to acquire grain shapes systematically in parallel with the albedo measurements and to apply the correcting b shape factor, determined theoretically or empirically. However an automatic and adequate determination of the b shape factor would be preferable. We hypothesize that measuring bi-directional reflectance (at least at a few selected angles and with coarse angular resolution) instead of only the albedo could provide the information necessary to estimate the b shape factor and then improve the SSA retrieval.
G. Picard et al. / Cold Regions Science and Technology 56 (2009) 10–17
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 Alley, R.B., 1987. Texture of polar firn for remote sensing. Ann. Glaciol. 9, 1–4. Arnaud, L., Gay, M., Barnola, J., Duval, P., 1998. Imaging of firn and bubbly ice in coaxial reflected light: a new technique for the characterization of these porous media. Ann. Glaciol. 44, 326–332. Brun, E., David, P., Sudul, M., Brugnot, G., 1992. A numerical model to simulate snowcover stratigraphy for operational avalanche forecasting. J. Glaciol. 38 (128), 13–22. Brun, E., Martin, E., Simon, V., Gendre, C., Coléou, C., 1989. An energy and mass model of snow cover suitable for operational avalanche forecasting. J. Glaciol. 35, 333–342. Bänninger, D., Bourgeois, S., Matzl, M., Schneebeli, M., 2008. Reflectance modelling for real snow structures using a beam tracing model. Sensors 3482–3496. Cabanes, A., Legagneux, L., Domine, F., 2002. Evolution of the specific surface area and of crystal morphology of Arctic fresh snow during the ALERT 2000 campaign. Atmos. Environ. 36, 2767–2777. Cabanes, A., Legagneux, L., Domine, F., 2003. Rate of evolution of the specific surface area of surface snow layers. Environ. technolTechnol. 37, 661–666. Colbeck, S.C., 1983. Theory of metamorphism of dry snow. J. Geophys. Res. 88, 5475–5482. Ding, K.H., Tsang, L., Shih, S.E., 2001. Monte Carlo simulations of particle positions for densely packed multispecies sticky particles. Microw. Opt. Tech. Lett. 30 (3), 187–192. Domine, F., Albert, M., Huthwelker, T., Jacobi, H.W., Kokhanovsky, A.A., Lehning, M., Picard, G., Simpson, W.R., 2008. Snow physics as relevant to snow photochemistry. Atmos. Chem. Phys. 8 (2), 171–208. URL http://www.atmos-chem-phys.net/8/171/ 2008/. Domine, F., Lauzier, T., Cabanes, A., Legagneux, L., Kuhs, W., Techmer, K., Heinrichs, T., 2003. Snow metamorphism as revealed by scanning electron microscopy. Microsc. Res. Tech. 62, 33–48. Domine, F., Salvatori, R., Legagneux, L., Salzano, R., Fily, M., Casacchia, R., 2006. Correlation between the specific surface area and the short wave infrared (SWIR) reflectance of snow. Cold Reg. Sci. Technol. 46 (1), 60–68. Domine, F., Shepson, P.B., 2002. Air–snow interactions and atmospheric chemistry. Science 297, 1506–1510. Domine, F., Taillandier, A.S., Simpson, W., 2007. A parameterization of the specific surface area of snow in models of snowpack evolution, based on 345 measurements. J. Geophys. Res. 112 (F02031). Flanner, M.G., Zender, C.S., 2006. Linking snowpack microphysics and albedo evolution. J. Geophys. Res. 111 (D10), 12208–+. Flin, F., Brzoska, J., Lesaffre, B., Coléou, C., Pieritz, R., 2004. Three-dimensional geometric measurements of snow microstructural evolution under isothermal conditions. Ann. Glaciol. 38, 39–44. Grannas, A.M., Jones, A.E., Dibb, J., Ammann, M., Anastasio, C., Beine, H.J., Bergin, M., Bottenheim, J., Boxe, C.S., Carver, G., Chen, G., Crawford, J.H., Domine, F., Frey, M.M., Guzmán, M.I., Heard, D.E., Helmig, D., Hoffmann, M.R., Honrath, R.E., Huey, L.G., Hutterli, M., Jacobi, H.W., Klán, P., Lefer, B., McConnell, J., Plane, J., Sander, R., Savarino, J., Shepson, P.B., Simpson, W.R., Sodeau, J.R., von Glasow, R., Weller, R., Wolff, E.W., Zhu, T., 2007. An overview of snow photochemistry: evidence, mechanisms and impacts. Atmos. Chem. Phys. 7, 4329–4373. Grenfell, T.C., Neshyba, S.P., Warren, S.G., 2005. 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. 110 (D9) 17203–+. Grenfell, T.C., Warren, S.G., 1999. Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation. J. Geophys. Res. 104, 31697–31710. Hanot, L., Domine, F., 1999. Evolution of the surface area of a snow layer. Environ. Sci. Technol. 33, 4250–4255. Jordan, R., 1991. A one-dimensional temperature model for a snow cover: technical documentation for sntherm.89. Technical Report Special Report 91-16, U. S. Army Cold Regions Research and Engineering Laboratory, Hanover, N. H.
17
Kaempfer, T.U., Hopkins, M.A., Perovich, D.K., 2007. A three-dimensional microstructurebased photon-tracking model of radiative transfer in snow. J. Geophys. Res. 112 (D11), 24113–+. Kaempfer, T.U., Schneebeli, M., Sokratov, S.A., 2005. A microstructural approach to model heat transfer in snow. Geophys. Res. Lett. 32, 21503–+. Kerbrat, M., Pinzer, B., Huthwelker, T., Gäggeler, H.W., Ammann, M., Schneebeli, M., 2008. Measuring the specific surface area of snow with x-ray tomography and gas adsorption: comparison and implications for surface smoothness. Atmos. Chem. Phys. 8 (5), 1261–1275. URL http://www.atmos-chem-phys.net/8/1261/2008/. Kokhanovsky, A.A., Zege, E.P., 2004. Scattering optics of snow. Appl. Opt. 43, 1589–1602. Legagneux, L., Cabanes, A., Domine, F., 2002. Measurement of the specific surface area of 176 snow samples using methane adsorption at 77 K. J. Geophys. Res. 107 (D17), 4335. Legagneux, L., Domine, F., 2005. A mean-field model of the isothermal metamorphism of dry snow. J. Geophys. Res. 110 (F04011). Legagneux, L., Taillandier, A.S., Domine, F., 2004. Grain growth theories and the isothermal evolution of the specific surface area of snow. J. Appl. Phys. 95, 6175–6184. Lehning, M., Bartelt, P., Brown, B., Fierz, C., Satyawali, P., 2002. A physical SNOWPACK model for the Swiss avalanche warning part II: snow microstructure. Cold Reg. Sci. Technol. 35, 147–167. Leroux, C., Lenoble, J., Brogniez, G., Hovenier, J., De Haan, J., 1999. A model for the bidirectional polarized reflectance of snow. J. Quant. Spectrosc. Radiat. Transfer 61 (13), 273–285. Marbouty, D., 1980. An experimental study of temperature gradient metamorphism. J. Glaciol. 26, 303–312. Matzl, M., Schneebeli, M., 2006. Measuring specific surface area of snow by nearinfrared photography. J. Glaciol. 52 (7), 558–564. URL http://www.ingentaconnect. com/content/igsoc/jog/2006/00000052/00000179/art00007. Nakamura, T., Abe, O., Hasegawa, T., Tamura, R., Ohta, T., 2001. Spectral reflectance of snow with a known grain-size distribution in successive metamorphism. Cold Reg. Sci. Technol. 32 (1), 13–26. Narita, H., 1971. Specific surface of deposited snow II. Low Temp. Sci. A29, 69–81. Nelson, J., 1998. Sublimation of ice crystals. J. Atmos. Sci. 55, 910–919. Neshyba, S.P., Grenfell, T.C., Warren, S.G., 2003. Representation of a nonspherical ice particle by a collection of independent spheres for scattering and absorption of radiation: 2. Hexagonal columns and plates. J. Geophys. Res. 108, 4448–+. Nolin, A., Dozier, J., 1993. Estimating snow grain size using AVIRIS data. Remote Sens. Environ. 44 (2–3), 231–238. Nolin, A., Dozier, J., 2000. A hyperspectral method for remotely sensing the grain size of snow. Remote Sens. Environ. 74 (2), 207–216. Painter, T.H., Molotch, N., Cassidy, M., Flanner, M., Steffen, K., 2007. Contact spectroscopy for the determination of stratigraphy of snow grain size. J. Glaciol. 53, 121–127. Pieritz, R., Brzoska, J.B., Flin, F., Lesaffre, C., Coléou, B., 2004. From snow X-ray microtomograph raw volume data to micromechanics modeling: first results. Ann. Glaciol. 38 (1), 52–58. Sokratov, S., 2001. Parameters influencing the recrystallization rate of snow. Cold Reg. Sci. Technol. 33, 263–274. Stamnes, K., Tsay, S.C., Jayaweera, K., Wiscombe, W., 1988. Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media. Appl. Opt. 27, 2502–2509. Sturm, M., Benson, C., 1997. Vapor transport, grain growth and depth hoar development in the subarctic snow. J. Glaciol. 43, 42–59. Sturm, M., Holmgren, J., König, M., Morris, K., 1997. The thermal conductivity of seasonal snow. J. Glaciol. 43, 26–41. Taillandier, A.S., Domine, F., Simpson, W., Sturm, M., Douglas, T., 2007. The rate of decrease of the specific surface area of dry snow: isothermal versus temperature gradient conditions. J. Geophys. Res. 112 (F03003). Taillandier, A.S., Domine, F., Simpson, W., Sturm, M., Douglas, T., Severin, K., 2006. Evolution of the snow area index (SAI) of the subarctic snowpack in central Alaska over a whole season. consequences for the air to snow transfer of pollutants. Environ. Sci. Technol. 40, 7521–7527. Tanikawa, T., Aoki, T., Hori, M., Hachikubo, A., Abe, O., Aniya, M., 2006. Monte Carlo simulations of spectral albedo for artificial snowpacks composed of spherical and nonspherical particles. Appl. Opt. 45, 5310–5319. Underwood, E.E., 1970. Quantitative Stereology. Addison Wesley, Reading, Massachusetts. Warren, S.G., 1982. Optical properties of snow (Paper 1R1505). Rev. Geophys. Space Phys. 20, 67–+. Warren, S.G., 1984. Optical constants of ice from the ultraviolet to the microwave. Appl. Opt. 23, 1206–1225. Warren, S.G., Brandt, R.E., 2008. Optical constants of ice from the ultraviolet to the microwave: a revised compilation. J. Geophys. Res. 113 (D14220).