Snow Grain Size Research Articles

Snow depths and grain-size relationships with relevance for passive microwave studies

The application of passive microwave radiometry to the remote sensing of snow properties is based on the ratio of emitted to scattered portions of the upwelling radiation. Increased scattering is indicative of increased snow amount, i.e. the number of snow grains present. However, scattering is also directly proportional to snow grain-size for a given snow amount. Current snow cover retrieval algorithms produce inaccurate results when snow grain-sizes are unusually large. Therefore, it is necessary to characterize snow grain-size on a regional scale (and perhaps local scale in extreme situations) in order to adjust passive microwave algorithms. Preliminary analysis indicates that: (1) algorithms are not as sensitive to the presence of large grain-sizes as the initial theory had indicated; (2) standard deviation of grain-size diameters throughout the total snow cover may often be less than 0.5 mm, thus average grain-size data may often serve to characterize the detailed stratigraphy of the total snow cover; (3) conditions in subfreezing snow which produce grain-sizes that greatly exceed a mean diameter value of 1–2 mm result from snow cover/climate relationships which can be modelled/monitored on a regional scale. A preliminary method is investigated for selecting snow retrieval algorithms according to prevailing regional-scale grain-size.

Analysis of satellite-altimeter height measurements above continental ice sheets

AbstractIn order to estimate the accuracy of altimetric height measurements over ice sheets, an altimeter wave-form simulator has been developed, and different tracking methods have been tested. A large range of surface features, including large-scale and medium-scale features and micro-roughness have been taken into account for modeling of either surface-or volume-scattering. A large set of parameters affects the trailing edge of the radar wave form, so that re-tracking algorithms based on the detection of its leading edge provide better retrievals of the surface height than those based on the analysis of the whole wave form. A volume component is clearly present in the radar wave forms; its effect on the leading edge depends mostly on the snow grain-size (and therefore on the snow temperature) and on the pointing angle. However, on average, the induced error on the snow-surface height estimation should only be around 25 cm.

Analysis of satellite-altimeter height measurements above continental ice sheets

AbstractIn order to estimate the accuracy of altimetric height measurements over ice sheets, an altimeter wave-form simulator has been developed, and different tracking methods have been tested. A large range of surface features, including large-scale and medium-scale features and micro-roughness have been taken into account for modeling of either surface-or volume-scattering. A large set of parameters affects the trailing edge of the radar wave form, so that re-tracking algorithms based on the detection of its leading edge provide better retrievals of the surface height than those based on the analysis of the whole wave form. A volume component is clearly present in the radar wave forms; its effect on the leading edge depends mostly on the snow grain-size (and therefore on the snow temperature) and on the pointing angle. However, on average, the induced error on the snow-surface height estimation should only be around 25 cm.

Snow grain-size determination from Landsat imagery over Terre Adélie, Antarctica

A Landsat Thematic Mapper scene over Terre Adélie, Antarctica, is used to estimate the snow grain-size. New calibration coefficients are computed for the TM scene, using sea-water reflectance. Topographic effects are corrected with the visible band TM2. Atmospheric effects are taken into account. The snow theoretical reflectance is calculated with the Wiscombe and Warren (1980) model. Estimated snow grain-sizes are similar to those given in the literature.

Geographic and seasonal variations in the surface properties of the ice sheets by satellite-radar altimetry

AbstractGeosat-altimeter wave forms from the Greenland and Antarctic ice sheets are analyzed using an algorithm based upon a combined surface-and volume-scattering model. The results demonstrate that sub-surface volume-scattering occurs over major parts of the ice sheets. Quantitative estimates of geographic variations in the near-surface ice-sheet properties are derived by retrackingindividualaltimeter wave forms. The derived surface properties correlate with elevation, latitude and microwave brightness-temperature data. Specifically, the extinction coefficient of snow obtained by this method varies from 0.48 to 0.13 m−1over the latitudes from 65° to 72°N on the central part of the Greenland ice sheet and from 0.20 to 0.10 m−1over a section of Wilkes Land in East Antarctica where the elevation increases from 2550 to 3150 m.Analysis of passive-microwave data over East Antarctica shows that the brightness temperature increases with elevation as the extinction coefficient decreases. Larger snow grain-sizes occur at lower elevations of the ice sheet because of higher mean annual temperatures. The larger grain-sizes increase the extinction coefficient of snow and decrease the emitted energy (brightness temperature) from greater snow depths. The passive-microwave data are also used to determine the average number of melt d year−1(1979–87) for the central part of the Greenland ice sheet. For latitudes from 65° to 68.5° N, the average number of melt days decreases from 3.5 to 0.25 d year, whereas no melt events are observed for latitudes above 69°N over the 8 year period. Snow subjected to alternate melting and freezing has enhanced grain-sizes compared to that of dry snow. This accounts for the larger values and larger spatial variations ofkeon the Greenland ice sheet compared to East Antarctica, where surface temperatures are never high enough to cause surface melting.

Snow grain-size determination from Landsat imagery over Terre Adélie, Antarctica

A Landsat Thematic Mapper scene over Terre Adélie, Antarctica, is used to estimate the snow grain-size. New calibration coefficients are computed for the TM scene, using sea-water reflectance. Topographic effects are corrected with the visible band TM2. Atmospheric effects are taken into account. The snow theoretical reflectance is calculated with the Wiscombe and Warren (1980) model. Estimated snow grain-sizes are similar to those given in the literature.

Geographic and seasonal variations in the surface properties of the ice sheets by satellite-radar altimetry

AbstractGeosat-altimeter wave forms from the Greenland and Antarctic ice sheets are analyzed using an algorithm based upon a combined surface-and volume-scattering model. The results demonstrate that sub-surface volume-scattering occurs over major parts of the ice sheets. Quantitative estimates of geographic variations in the near-surface ice-sheet properties are derived by retracking individual altimeter wave forms. The derived surface properties correlate with elevation, latitude and microwave brightness-temperature data. Specifically, the extinction coefficient of snow obtained by this method varies from 0.48 to 0.13 m−1 over the latitudes from 65° to 72°N on the central part of the Greenland ice sheet and from 0.20 to 0.10 m−1 over a section of Wilkes Land in East Antarctica where the elevation increases from 2550 to 3150 m.Analysis of passive-microwave data over East Antarctica shows that the brightness temperature increases with elevation as the extinction coefficient decreases. Larger snow grain-sizes occur at lower elevations of the ice sheet because of higher mean annual temperatures. The larger grain-sizes increase the extinction coefficient of snow and decrease the emitted energy (brightness temperature) from greater snow depths. The passive-microwave data are also used to determine the average number of melt d year−1 (1979–87) for the central part of the Greenland ice sheet. For latitudes from 65° to 68.5° N, the average number of melt days decreases from 3.5 to 0.25 d year, whereas no melt events are observed for latitudes above 69°N over the 8 year period. Snow subjected to alternate melting and freezing has enhanced grain-sizes compared to that of dry snow. This accounts for the larger values and larger spatial variations of ke on the Greenland ice sheet compared to East Antarctica, where surface temperatures are never high enough to cause surface melting.

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

AbstractLaws of snow metamorphism have been introduced in a numerical model which simulates the evolution of temperature, density and liquid-water profiles of snow cover as a function of weather conditions.To establish these laws, the authors have summarized previous studies on temperature gradient and on wet-snow metamorphism and they have also conducted metamorphism experiments on dry or wet fresh-snow samples. An original formalism was developed to allow a description of snow with parameters evolving continuously throughout time.The introduction of laws of metamorphism has improved significantly the derivation of the settlement of internal layers and of snow-covered albedo, which depend on the simulated stratigraphy, i.e. the type and size of snow grains of different layers of the snow cover.The model was tested during a whole winter season without any re-initialization. Comparison between the simulated characteristics of the snow cover and the observations made in the field are described in detail. The model proved itself to be very efficient in simulating accurately the evolution of the snow-cover stratigraphy throughout the whole winter season.

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

AbstractLaws of snow metamorphism have been introduced in a numerical model which simulates the evolution of temperature, density and liquid-water profiles of snow cover as a function of weather conditions.To establish these laws, the authors have summarized previous studies on temperature gradient and on wet-snow metamorphism and they have also conducted metamorphism experiments on dry or wet fresh-snow samples. An original formalism was developed to allow a description of snow with parameters evolving continuously throughout time.The introduction of laws of metamorphism has improved significantly the derivation of the settlement of internal layers and of snow-covered albedo, which depend on the simulated stratigraphy, i.e. the type and size of snow grains of different layers of the snow cover.The model was tested during a whole winter season without any re-initialization. Comparison between the simulated characteristics of the snow cover and the observations made in the field are described in detail. The model proved itself to be very efficient in simulating accurately the evolution of the snow-cover stratigraphy throughout the whole winter season.

HIRIS — Eos instrument with high spectral and spatial resolution

The High-Resolution Imaging Spectrometer (HIRIS) is a JPL-facility instrument designed for NASA's Earth Observing System (Eos). It will have 10-nm wide spectral bands from 0.4 to 2.5 μm at 30-m spatial resolution over a 24-km swath. The spectral resolution allows identification of many minerals in rocks and soils, important algal pigments in oceans and inland waters, spectral changes associated with plant canopy biochemistry, composition of atmospheric aerosols, and grain size of snow and its contamination by absorbing impurities. The bands will have 12-bit quantization over a dynamic range suitable for bright targets, such as snow. For targets of low brightness, such as water bodies, image-motion compensation will allow gains up to a factor of 8 to increase signal-to-noise ratios. The sensor will be able to point ±26° crosstrack and +52°/−30° downtrack. In the 705-km orbit altitude proposed for Eos, the crosstrack pointing capability will allow 3–4 views during a 16-day revisit cycle.

Absorption of Solar Radiation at the Antarctic Snow Surface (Abstract)

Solar radiation incident on, and reflected by, the snow surface was measured near the South Pole as a function of wavelength, angle, and distance from the station. The objectives of the study were: (1) to observe spectral albedos of snow across the solar spectrum, (2) to obtain depth profiles of snow-grain radius in order to construct theoretical models of spectral albedo for pure snow, (3) to document the extent and degree of soot pollution due to station activities and to assess whether it could invalidate solar-radiation measurements made close to large stations, and (4) to obtain the spectral distribution of incident solar radiation at the Antarctic surface for various cloud conditions, in order to test radiation models of the Antarctic atmosphere.Spectral albedo, measured under diffuse lighting conditions (overcast cloud) on many days, repeatedly agreed with the results of theoretical models which predicted values approaching unity in the visible and found grain-size to be the most important variable controlling snow albedo in the near-infra-red. A representative albedo curve is shown in Figure 1.The visible albedo values were found to be 98–99% and were relatively insensitive to grain-size. (These results disagree with the only previous measurements of Antarctic snow albedo which had good spectral resolution: those of Kuhn and Siogas. Their maximum albedo was only about 90% in the visible.) The near-infra-red albedo, however, varied substantially among the experiments, due to day-to-day variations in snow grain-size, caused by precipitation and wind drifting. The experimental points in the figure match theoretical calculations for grain radius less than 50 μm at wavelengths beyond 1.5 μm, and 50–100 μm for shorter wavelengths. At the shorter wavelengths the light penetrates more deeply into the snow, so the albedo is sensitive to grains beneath the surface, whereas at the longer wavelengths the albedo is influenced only by the grains very close to the surface. The observed albedos can thus be explained by an increase in grain-size with depth.In order that our measurements would be representative of large areas, we were concerned to avoid possible effects of pollution from the station. We collected samples from the top 20 cm of snow, melted and filtered them, and analyzed the filters. The conclusion is that the pollution is very minor. Just 500 m up-wind of the station there is normally less than 1 ng of carbon per gram of snow (1 ppb). Even down-wind of the station the carbon content did not exceed 3 ppb. For snow grain-sizes typical of Antarctica, our models predict that 15 ppb carbon would reduce snow albedo by only 1% at the most sensitive wavelength. Thus we reject our earlier suggestion that the low visible albedos of Kuhn and Siogas were due to impurities in the snow and now favor other explanations.

Absorption of Solar Radiation at the Antarctic Snow Surface (Abstract)

Solar radiation incident on, and reflected by, the snow surface was measured near the South Pole as a function of wavelength, angle, and distance from the station. The objectives of the study were: (1) to observe spectral albedos of snow across the solar spectrum, (2) to obtain depth profiles of snow-grain radius in order to construct theoretical models of spectral albedo for pure snow, (3) to document the extent and degree of soot pollution due to station activities and to assess whether it could invalidate solar-radiation measurements made close to large stations, and (4) to obtain the spectral distribution of incident solar radiation at the Antarctic surface for various cloud conditions, in order to test radiation models of the Antarctic atmosphere. Spectral albedo, measured under diffuse lighting conditions (overcast cloud) on many days, repeatedly agreed with the results of theoretical models which predicted values approaching unity in the visible and found grain-size to be the most important variable controlling snow albedo in the near-infra-red. A representative albedo curve is shown in Figure 1.The visible albedo values were found to be 98–99% and were relatively insensitive to grain-size. (These results disagree with the only previous measurements of Antarctic snow albedo which had good spectral resolution: those of Kuhn and Siogas. Their maximum albedo was only about 90% in the visible.) The near-infra-red albedo, however, varied substantially among the experiments, due to day-to-day variations in snow grain-size, caused by precipitation and wind drifting. The experimental points in the figure match theoretical calculations for grain radius less than 50 μm at wavelengths beyond 1.5 μm, and 50–100 μm for shorter wavelengths. At the shorter wavelengths the light penetrates more deeply into the snow, so the albedo is sensitive to grains beneath the surface, whereas at the longer wavelengths the albedo is influenced only by the grains very close to the surface. The observed albedos can thus be explained by an increase in grain-size with depth. In order that our measurements would be representative of large areas, we were concerned to avoid possible effects of pollution from the station. We collected samples from the top 20 cm of snow, melted and filtered them, and analyzed the filters. The conclusion is that the pollution is very minor. Just 500 m up-wind of the station there is normally less than 1 ng of carbon per gram of snow (1 ppb). Even down-wind of the station the carbon content did not exceed 3 ppb. For snow grain-sizes typical of Antarctica, our models predict that 15 ppb carbon would reduce snow albedo by only 1% at the most sensitive wavelength. Thus we reject our earlier suggestion that the low visible albedos of Kuhn and Siogas were due to impurities in the snow and now favor other explanations.

Infrared Measurement Of Free-Water Content And Grain Size Of Snow

A method has been developed to measure the free-water content and grain size of snow by use of near-infrared methods. It requires no direct physical sampling and uses reflectances at three optimally selected wavelengths. The method is based on two basic phenomena: first, snow reflection as a multiple-scattering phenomenon is strongly dependent on the grain size and the spectral variations in the ice absorption coefficient, and second, owing to differences between the absorption coefficients of ice and water, the reflection spectrum of dry snow differs from that of wet snow. Hemispherically integrated spectral reflectance has been measured in the range between 600 and 2000 nm for several types of snow including metamorphic states from newly fallen to melted-refrozen and densities from 0.10 to 0.45 g/cm3. The effect on reflectance of up to 20 vol % free water has been studied. The reflectance of snow has also been analyzed theoretically using an approximate solution of the radiative transfer equation. The experiments and theoretical calculations show that the free-water content and average grain diameter can be derived from reflectance measurements made at 1030, 1260, and 1370 nm. From measurements made so far it is estimated that an accuracy of ± 1.5 vol % can be achieved in the measurement of the free-water content and of ± 0.2 mm can be achieved in the measurement of the average grain size.

Effect of Surface Roughness on Remote Sensing of Snow Albedo

Narrow field-of-view sensors on satellites monitoring solar radiation measure the reflected radiance in a particular direction. For climatic studies of the Earth’s radiation budget, the albedo is needed, which is the integral of the upward radiance over all angles divided by the downward irradiance. In order to infer the albedo from a radiance measurement at only one angle, it is necessary to know a priori the distribution of reflected radiation with angle, i.e. the bi-directional reflectance-distribution function (BRDF). The BRDF is a function of four angles: solar zenith and azimuth, and satellite zenith and azimuth. For areal or temporal averages on many natural surfaces, only three angles are needed to describe the function, because only the difference between the two azimuths is important, not their individual values. This assumption was made when developing empirical BRDFs from Nimbus-satellite data for use in the Earth Radiation Budget Experiment (ERBE). However, in large areas of the polar regions, all four angles are needed, because the sastrugi are oriented parallel to a prevailing wind direction. The BRDF shows a forward peak when the solar beam is along the direction of the sastrugi, and an enhanced backward peak when it is perpendicular. Averaging over all solar azimuths (relative to the sastrugi azimuth) causes back-scattering to be averaged together with forward-scattering. The conclusion of the ERBE analysis, that snow is the most nearly isotropic of all Earth surfaces, is therefore at least partly a spurious result of this averaging.Measurements of the BRDF were carried out from a 23 m tower at the South Pole during January and February at 900 nm wavelength for varying azimuths between the Sun and the sastrugi fabric. The wavelength was selected near the midpoint of the solar-energy spectrum but where scattered sky radiation is negligible. Measurements were made with 10° field of view at 15° intervals in viewing zenith and azimuth angles throughout the day, at intervals of 1 h (15° of solar azimuth). For BRDF normalized such that its angular average is unity, the principal features of the results include a forward-scattering peak with a value of about five together with a side- and back-scattering lobe of 1.1 to 1.3. Variations in solar azimuth produced a skewness in BRDF which was approximately consistent with enhanced scattering at the specular angle with respect to the solar azimuth and the orientation of the principal fabric of the sastrugi pattern. The angularly averaged pattern was remarkably similar to the results of Taylor and Stowe even though their values were integrated over wavelength and were made through the atmosphere. Our studies thus suggest that, for mid- to late summer, the Taylor and Stowe results require only small corrections for sastrugi effects. This is not, however, expected to be true from sunrise through late November.Spectral albedos showed values at visible wavelengths of 0.97 to 0.99 which agree very well with the model calculations of Wiscombe and Warren using our observed mean snow grain-sizes. Albedos for wavelengths above 1400 nm were higher than model predictions, indicating that the depth dependence of grain-size must be included in the analysis.This research was supported by National Science Foundation grant DPP-83–16220.

Effect of Surface Roughness on Remote Sensing of Snow Albedo

Narrow field-of-view sensors on satellites monitoring solar radiation measure the reflected radiance in a particular direction. For climatic studies of the Earth’s radiation budget, the albedo is needed, which is the integral of the upward radiance over all angles divided by the downward irradiance. In order to infer the albedo from a radiance measurement at only one angle, it is necessary to know a priori the distribution of reflected radiation with angle, i.e. the bi-directional reflectance-distribution function (BRDF). The BRDF is a function of four angles: solar zenith and azimuth, and satellite zenith and azimuth. For areal or temporal averages on many natural surfaces, only three angles are needed to describe the function, because only the difference between the two azimuths is important, not their individual values. This assumption was made when developing empirical BRDFs from Nimbus-satellite data for use in the Earth Radiation Budget Experiment (ERBE). However, in large areas of the polar regions, all four angles are needed, because the sastrugi are oriented parallel to a prevailing wind direction. The BRDF shows a forward peak when the solar beam is along the direction of the sastrugi, and an enhanced backward peak when it is perpendicular. Averaging over all solar azimuths (relative to the sastrugi azimuth) causes back-scattering to be averaged together with forward-scattering. The conclusion of the ERBE analysis, that snow is the most nearly isotropic of all Earth surfaces, is therefore at least partly a spurious result of this averaging. Measurements of the BRDF were carried out from a 23 m tower at the South Pole during January and February at 900 nm wavelength for varying azimuths between the Sun and the sastrugi fabric. The wavelength was selected near the midpoint of the solar-energy spectrum but where scattered sky radiation is negligible. Measurements were made with 10° field of view at 15° intervals in viewing zenith and azimuth angles throughout the day, at intervals of 1 h (15° of solar azimuth). For BRDF normalized such that its angular average is unity, the principal features of the results include a forward-scattering peak with a value of about five together with a side- and back-scattering lobe of 1.1 to 1.3. Variations in solar azimuth produced a skewness in BRDF which was approximately consistent with enhanced scattering at the specular angle with respect to the solar azimuth and the orientation of the principal fabric of the sastrugi pattern. The angularly averaged pattern was remarkably similar to the results of Taylor and Stowe even though their values were integrated over wavelength and were made through the atmosphere. Our studies thus suggest that, for mid- to late summer, the Taylor and Stowe results require only small corrections for sastrugi effects. This is not, however, expected to be true from sunrise through late November. Spectral albedos showed values at visible wavelengths of 0.97 to 0.99 which agree very well with the model calculations of Wiscombe and Warren using our observed mean snow grain-sizes. Albedos for wavelengths above 1400 nm were higher than model predictions, indicating that the depth dependence of grain-size must be included in the analysis. This research was supported by National Science Foundation grant DPP-83–16220.

Towards the Definition of Optimum Sensor Specifications for Microwave Remote Sensing of Snow

The interaction of nicrowaves with snow strongly depends on parameters such as snow wetness and the size and structure of snow grains. Therefore microwave radiometry and scatterometry are excellent tools for remote sensing of the snowcover. Multifrequency radiometry can be used to classify snow as was shown with ground-based measurements of the period April-June 1977 at a high altitude Alpine test site. The continuation of the measurement program yielded data of 3 additional snow seasons with widely varying snow conditions, therefore the present information has become representative for alpine regions. Relationships between the brightness temperature and the water equivalents show a similar variation with snow type as in other snow regions, so that the range of validity of our data set is not restricted to the Alps. The problem of discriminating regions of wet snow from snow-free land is found to be solvable with microwave scatterometry. Two cluster analyses in factorial spaces of both the ground truth and the microwave data sets demonstrate the potential of microwave sensors to classify snow which is a prerequisite for snow algorithms retrieving hydrologic parameters. The results are used to define sensor specifications with optimum sensitivity for microwave remote sensing of snow.

Two-Stream Theory of Reflectance of Snow

Spectral reflectance of snow under diffuse illumination is studied using the two-stream approximation of the radiative transfer equation. The scattering and absorption parameters of the radiative transfer equation-the single scattering albedo, the optical depth, and the integrated phase function are obtained from the grain size and density of snow. Analytical expressions for the intensity within the snowpack, the reflectance, and the asymptotic flux extinction coefficient, are given. Good agreement is shown between the theory and available experimental data on visible and near-infrared reflectance, and the asymptotic flux extinction coefficient. The theory may also be used to explain the observed effect of aging on the snow reflectance.

An analysis of water flow in dry snow

The equations describing water movement in a dry snow cover are derived, and examples of flow through ripe, refrozen, and fresh snows are given. The grain size of snow has a large effect on the timing of water discharge. Water is retained by dry snow to raise its temperature and satisfy the irreducible water saturation. These requirements delay and reduce runoff following rain on dry snow.

Physical Properties of the Snow Cover on Mt. Tateyama in Central Honshu, Japan

Mt. Tateyama and its vicinity are in the region of heavy snowfall in Japan, but the snow cover of this mountainous area has not been investigated before. The purpose of the present work was to clarify the characteristics of snow cover on the Murodo flat (2450 m above sea level) in Mt. Tateyama. Vertical distributions of density, Kinoshita's hardness, temperature and texture of snow were observed once a month through 3 winters from 1972 to 1975. The Murodo flat was covered with snow from November to July and the water equivalent of snow cover reached a maximum of about 250 cm, in the last decade of April. From November to March a major portion of snow cover consisted of fine-grained compact snow. Maximum values of density and hardness were recorded in March as 0.51 g/cm3 and 25 kg/cm2, respectively. The relation between hardness R and density ρ is expressed roughly as R = 4004.6ρ kg/cm2. These values of hardness are about two times larger than those observed in Sapporo and Toikanbetsu of Hokkaido. The relation between density and compactive viscosity coefficient η derived from density distribution and accumulation rate is expressed by η =C exp (kρ) g·day/cm2. k is 20.2 for all snow covers, while C is in the range from 3.3 to 1.0, and the maximum and minimum correspond to the colder period (February) and warmer period (March) respectively. During the thawing season after April the average values of grain size and hardness of wet granular snow were about 2 mm in diameter and 5 kg/cm2 respectively and remained unchanged, while the density increased with time and reached 0.65 g/cm3 at the maximum early in July.

Quantitative Stereological Analysis of Grain Bonds in Snow

A section-plane preparation technique and an operational definition of a snow grain bond are developed to allow the quantitative analysis of bonding in snow structure. Values for three-dimensional grain size, grain-bond size and number density, and related bonding measurements are presented for equi-temperature metamorphosed snow. These results from mutually orthogonal planes within a given snow block show that the assumptions of randomness and isotropy of grain and grain-bond location and orientation, necessary for stereological analysis from one plane, are satisfied to within ± 10% even after 30% uniaxial plastic deformation of the snow block. The idealization of a grain bond as a circular plane disk yields self-consistent results. The number of bonds per grain cannot be accurately determined from two-dimensional studies due to variations in the shape and size of snow grains within a given sample.