A physically based snow albedo model (PBSAM), which can be used in a general circulation model, is developed. PBSAM calculates broadband albedos and the solar heating profile in snowpack as functions of snow grain size and concentrations of snow impurities, black carbon and mineral dust, in snow with any layer structure and under any solar illumination condition. The model calculates the visible and near-infrared (NIR) albedos by dividing each broadband spectrum into several spectral subbands to simulate the change in spectral distribution of solar radiation in the broadband spectra at the snow surface and in the snowpack. PBSAM uses (1) the look-up table method for calculations of albedo and transmittance in spectral subbands for a homogeneous snow layer, (2) an “adding” method for calculating the effect of an inhomogeneous snow structure on albedo and transmittance, and (3) spectral weighting of radiative parameters to obtain the broadband values from the subbands. We confirmed that PBSAM can calculate the broadband albedos of single- and two-layer snow models with good accuracy by comparing them with those calculated by a spectrally detailed radiative transfer model (RTM). In addition, we used radiation budget measurements and snow pit data obtained during the two winters from 2007 to 2009 at Sapporo, Hokkaido, Japan, for simulation of the broadband albedos by PBSAM and compared the results with the in situ measurements. A five-layer snow model with one visible subband and three NIR subbands were necessary for accurate simulation. Comparison of solar heating profiles calculated by PBSAM with those calculated by the spectrally detailed RTM showed that PBSAM calculated accurate solar heating profiles when at least three subbands were used in both the visible and NIR bands.
 Snow and ice in the Arctic are presently undergoing drastic changes. The mass balance loss from the Greenland Ice Sheet during the period with good observation data increased significantly after the mid-1990s [Steffen et al., 2008]. A gravity survey by the satellite-borne Gravity Recovery and Climate Experiment confirmed that the rate of ice loss accelerated during from 2002 to 2006 [Chen et al., 2006], mainly as a result of high rates of ice loss in southern Greenland [Velicogna and Wahr, 2006]. Moreover, the decrease of the Arctic sea ice extent, recorded since 1978, accelerated from 1996 to 2006 [Comiso and Nishio, 2008]. The smallest ice extent, recorded in 2007, was smaller than any value predicted by climate models [Stroeve et al., 2007]. This abrupt melting of Arctic snow and ice has not been accurately simulated by many general circulation models (GCM). The remarkable albedo difference between snow covered and snow-free surfaces may have a large climatic impact, known as snow (ice) albedo feedback. The development of more accurate snow and albedo parameterizations should improve model estimates of climate sensitivity to albedo changes [Levis et al., 2007]. Although snow albedo is one of the most important parameters in climate studies, in many GCMs, it is parameterized empirically [e.g., Pedersen and Winther, 2005; Brun et al., 2008]. Furthermore, GCMs typically deposit all absorbed solar radiation in the top-most snow layer, and this unrealistic representation of snow–radiation interaction may bias climate predictions in snowy regions [Flanner and Zender, 2005]. Thus, a physically based snow albedo model for GCMs that simulates broadband albedo and the solar heating rate in the snowpack from snow physical parameters and illumination conditions with low computational cost is needed for more accurate climate simulations in the cryosphere [Aoki et al., 2003].
 Snow albedo is essentially a function of snow physical parameters and solar illumination. Snow physical parameters are snow grain size, snow impurity concentrations, liquid water content, and their layer structure in the snowpack, together with snow depth. When the snow depth is optically thin, the underlying surface albedo also affects the snow albedo. The near-infrared (NIR) albedo strongly depends on snow grain size [Wiscombe and Warren, 1980], whereas the visible albedo is more dependent on the concentrations of snow impurities such as black carbon (BC) and mineral dust [Warren and Wiscombe, 1980]. Under clear-sky conditions, snow albedo shows solar zenith angle dependence, whereas under cloudy conditions, this dependence is lost because the snow surface is illuminated diffusely. Because atmospheric constituents other than cloud cover such as aerosols and absorptive gases also change the direct to diffuse insolation ratio to the snow surface, they also affect the snow albedo [Aoki et al., 1999]. For the same snow conditions, the broadband snow albedo is higher under cloudy skies than under clear skies [e.g., Yamanouchi, 1983] because of the difference in the spectral distribution of the downward solar flux between clear and cloudy skies [Liljequist, 1956].
 The first physically based snow albedo model used in a GCM, developed by Marshall and Oglesby , takes into account the effects of snow grain size, snow impurity concentrations, the diffuse fraction of the downward solar flux, and the solar zenith angle and uses the snow albedo scheme of Wiscombe and Warren . The Fourth Assessment Report of the Intergovernmental Panel on Climate Change [Intergovernmental Panel on Climate Change (IPCC), 2007] estimated that the global annual mean radiative forcing due to “BC on snow” is +0.1 ± 0.1 W m−2, a value based mainly on estimates by Hansen and Nazarenko  and Hansen et al. , who used prescribed albedo reduction values from snowpack BC concentration data obtained in field experiments by Clarke and Noone  along with more recent measurement data. Jacobson  simulated the global snow albedo reduction due to BC using a GCM coupled with a spectrally detailed radiative transfer model (RTM) by Toon et al.  for snow and sea ice with inclusions of deposited BC and organic matter (OM). Subsequently, Flanner and Zender [2005, 2006] developed the two-stream, multilayer SNow, ICe, and Aerosol Radiation model (SNICAR) to calculate snow albedo and solar heating based on the models by Wiscombe and Warren  and Toon et al. . SNICAR was implemented as part of a land-surface model in a GCM to calculate snow microphysics and radiative properties. Flanner et al.  estimated the radiative forcing due to BC from different sources in snowpack and obtained surface radiative forcing values of around +0.05 W m−2 for a strong (1998) and weak (2001) boreal fire year. Furthermore, Flanner et al.  concluded that the effect on radiative forcing of the reduction of surface-incident solar energy (dimming) caused by atmospheric aerosols containing BC and OM is smaller than the effect of the reduction of snow albedo caused by deposition of such aerosols (darkening). Recently, Gardner and Sharp  proposed a physically based parameterization of snow and ice albedo based on spectrally detailed radiative transfer calculations. Their model calculates the broadband albedo as a function of the specific surface area of snow and ice instead of snow grain size, and of BC concentration, snow depth, solar zenith angle, and cloud optical thickness. Yasunari et al.  implemented a snow metamorphism model that incorporated an albedo parameterization developed by Yamazaki et al.  into the land-surface model of a GCM to examine the effects of BC and dust on albedo. Painter et al.  also estimated regional albedo reduction effects due to dustfall events in the San Juan Mountains, Colorado, USA during the springtime snowmelt season. Thus, the effects of snow impurities on the radiation budget at the snow surface have been recently investigated using improved snow albedo models and parameterizations. However, the albedo scheme used in these various models and parameterizations is still insufficiently validated.
 In the present study, we developed a physically based snow albedo model (PBSAM) for GCM to calculate the broadband albedos and solar heating profile of the snowpack as functions of snow grain size and mass concentrations of snow impurities for any snow layer structure and any solar illumination condition. Key differences of PBSAM from the other previous snow albedo models are: (1) the number of spectral subbands for the visible and near-infrared regions and the number of snow layers are tunable by which we can select a trade-off between the accuracy and computation time according to the host GCM; (2) the model uses a look-up table (LUT) method for radiation calculation and we can also modulate the computation time by changing the grid points in LUTs; and (3) PBSAM incorporates major factors affecting the broadband snow albedo and solar heating profile mentioned above, although there are very few similar models. We evaluated the accuracy of PBSAM by comparing PBSAM-calculated broadband albedos and solar heating profiles with those calculated by a spectrally detailed RTM for the atmosphere–snow system. In addition, we performed radiation budget measurements and snow pit work at Sapporo, Japan, during two winters and input the measured snow parameter values and solar radiation to PBSAM to simulate the visible and NIR albedos and solar heating profiles. We also validated the calculated albedos by using in situ measurement data.
 PBSAM is designed for use in a GCM. It can calculate broadband albedos of the snow surface and the solar heating (absorptivity) profile in snow layers from snow physical parameters and solar illumination conditions. The spectral regions for which radiative parameters are calculated are basically tunable depending on the radiation scheme of the GCM. Here, we employ the visible and NIR bands at the boundary wavelength λ = 0.7 μm, which is used in the Earth System Model of the Meteorological Research Institute [Yukimoto et al., 2011]. When PBSAM is used in a GCM, snow microphysics parameters, including snow grain size, affected by snow metamorphism processes are calculated with a snow metamorphism model (M. Niwano et al., manuscript in preparation, 2011) coupled with PBSAM.
2.1.1. Input Parameters
 PBSAM is applicable to any snowpack layer structure and any snow depth. The input parameters used to define the modeled snow layer structure are snow grain radius, mass concentrations of BC and mineral dust, and snow water equivalent (SWE) in each snow layer, and the underlying surface albedo. Atmospheric conditions, such as clear or cloudy skies, can change solar illumination conditions of the snow surface, thus affecting the albedo [Aoki et al., 1999]. The input parameters characterizing the solar illumination conditions are solar zenith angle, relative fractions of global solar radiation in the visible and NIR bands, and the diffuse fraction in each spectrum. The absolute values of global solar radiation are necessary to calculate the solar heating profile in the snowpack.
2.1.2. Look-Up Tables
 The PBSAM algorithm is based on the look-up table (LUT) method. The spectral albedos and transmittances for homogeneous snow are calculated off-line with a spectrally detailed RTM [Aoki et al., 1999, 2000, 2003] for wide ranges of snow parameter values and solar illumination conditions. The spectrally integrated values in broadbands or spectral subbands are stored in LUTs; two types of albedo, αBS and αWS, and two types of flux transmittance (hereafter, “transmittance,” unless otherwise stated), tBS and tWS, are stored for each spectral band, where the superscripts “BS” and “WS” denote “black sky” and “white sky,” respectively, and refer to the direct and diffuse components of insolation (Table 1). These parameters are functions of the effective snow grain radius re, a snow impurity factor (SIF; which expresses the effects of BC and dust as a single parameter), the SWE, and the solar zenith angle θ0. The albedo and transmittance with diffuse insolation are independent of θ0. Thus, the LUTs consist of αi,BS (re, SIFi, SWE, θ0), αi,WS (re, SIFi, SWE), ti,BS (re, SIFi, SWE,θ0), and ti,WS (re, SIFi, SWE), where the superscript i denotes the spectral broadband subband or the broadband itself. Since the spectral distribution of solar radiation changes in snowpack, it is necessary to take into account the effect of a change in the spectral distribution within the broadband to accurately calculate the broadband albedo and solar heating profile in snowpack. We therefore introduced a method for calculating the broadband radiative parameters by dividing the broadband into several subbands and then spectrally integrating them, and LUTs containing numerical data for several subbands in both the visible and in the NIR bands were calculated. Hence, the superscript i, in general, corresponds to a spectral subband or broadband. We employed the 1–5 subbands in the visible (V) and NIR (N) bands (indicated by V1N1–V5N5; Table 2) and determined the spectral boundaries of the subbands mainly from the spectral distribution of solar radiation absorbed in different snow layers (see section 6). Although we used the same number of subbands in both the visible and NIR bands, it would be possible to use different numbers of subbands (e.g., V2N4). The physical and chemical properties used to calculate the mass absorption cross sections (MAC) of BC and dust shown in Table 2 are explained in section 2.3.
Table 1. Parameters in the PBSAM Look-Up Tables (LUTs)
Range of Parameter
Seven grid points.
Seventeen grid points.
The value of SIFi depends on the spectral subband i (see Table 2). We used the values of SIFi in LUTs calculated by equation (1) only for the range of cBC.
Table 2. Mass Absorption Cross Sections (MAC) kai,BC for Hydrophilic BC, kai,dust for Hydrophilic Dust in Spectral Subbands i, and Subband Weight at θ0 = 60° for Clear and Cloudy Skiesa
Black Carbon MAC (m2g−1)
Clear-Sky Subband Weight at 0 = 60°
For MAC, numbers after “V” and “N” (e.g., “V5N5”) in the subheads denote the number of subbands in the visible (V) and near-infrared (N) bands, respectively. For example, in the V3N3 model, the spectral regions of subband i = 1, 2, and 3 in the visible band are 0.200–0.550, 0.550–0.625, and 0.625–0.700 μm, respectively, and the corresponding values of kai,BC are 1.322E+01, 1.039E+01, and 9.197E+00, and the spectral regions of subband i = 1, 2, and 3 in the near-infrared band are 0.700–0.950, 0.950–1.400, and 1.400–3.000 μm, respectively, and the corresponding values of kai,BC are 7.364E+00, 4.914E+00, and 2.721E+00.
see entry above
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see entry above
Dust MAC (m2g−1)
Cloudy-Sky Subband Weight at 0 = 60°
see entry above
see entry above
see entry above
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see entry above
2.1.3. Snow Impurity Factor
 Snow impurity factor (SIF) was introduced to express the effects of different light-absorbing snow impurities by a single parameter, which also depends on the spectral subband. For subband i, SIFi of BC and mineral dust is given by
where kai,BC and kai,dust are the MACs of BC and dust, respectively, for subband i (Table 2), and cBC and cdust are the mass concentrations of BC and dust, respectively, in snow, which are input parameters to PBSAM. SIF is interpreted as a composite MAC of snow impurities per unit snow mass. In the actual LUT calculations, we used only BC as a snow impurity because the range of albedo variation depending on BC is wider than that depending on dust. The input dust concentration is converted into an “equivalent” quantity of BC with equation (1).
2.1.4. “Adding” Method
 In general, the snowpack is vertically inhomogeneous, and sometimes it is optically thin. The effects of these snowpack characteristics on albedo and transmittance are calculated by using the “adding” method. This numerical technique is often used for radiative transfer calculations in the atmosphere [e.g., Lacis and Hansen, 1974] and snow on sea ice [e.g., Briegleb and Light, 2007], but we employed a simplified adding method in the present study. In the original method, the product of reflection (or transmission) functions is usually integrated over the hemispherical solid angle, whereas we simply use the scalar product. Direct solar radiation is usually treated explicitly in the adding procedure, whereas we incorporate it into the reflection and transmission functions for each snow layer. When the snowpack consists of the top layer a and bottom layer b (Figure 1), black sky albedo αa+bi,BS and black sky transmittance ta+bi,BS of the composite snow layer a + b are calculated from multiple reflections between the two snow layers as follows:
where the superscript “*WS” refers to diffuse illumination from below. White sky albedo αa+bi,WS and white sky transmittance ta+bi,WS of the two snow layers are obtained by replacing all instances of “BS” in equation (2) with “WS.” In the case of illumination from below, the white sky albedo αa+bi,*WS and white sky transmittance ta+bi,*WS of the two snow layers are expressed as
Here, we assumed that αi,*WS = αi,WS and ti,*WS = ti,WS when the snowpack consists of a single layer.
 When the snowpack consists of three or more layers, the adding procedure is first applied to the top two layers. Then, by repeating the adding procedure using the composite layer determined by the previous iteration of the adding procedure and the next layer beneath it, the albedo and transmittance of any snow layer structure can be calculated. In general, the effect of the underlying surface albedo αUSi,WS is calculated by setting as αbi,WS = αUSi,WS in the final adding calculation.
2.1.5. Subband to Broadband Calculation
 In PBSAM, the values of αi,BS, αi,WS, ti,BS, and ti,WS for subband i and each snow layer are calculated by interpolation from the grid point values in the LUTs using the input snow parameters of each snow layer. The values of αi,BS, αi,WS, ti,BS, and ti,WS for the overall snow thickness are calculated by the adding procedure. The broadband albedos (transmittances) for BS and WS are obtained by spectral integration of αi,BS and αi,WS (ti,BS and ti,WS) using the subband weights wi,BS and wi,WS, respectively. For example, the black sky albedo in the visible band is calculated by
where V is the number of subbands (= 1–5) in the visible band (Table 2). The subband weights wi,BS and wi,WS are obtained by spectral integration of the spectral downward solar flux calculated with a spectrally detailed RTM at higher spectral resolution (25 nm: we refer to this spectral resolution as “narrowband” to distinguish from “subband” for discussion of the weight) for one assumed snow condition under typical clear or cloudy sky condition. These snow and atmospheric conditions are described in section 2.2. We used the subband weights as a function of θ0 for both clear and cloudy conditions. The clear-sky (cloudy-sky) subband weights are almost constant at θ0 < 80° for both the visible and NIR regions and change by 20% (30%) at maximum when θ0 is close to 90°. The values of subband weights at θ0 = 60° are listed in Table 2.
 The albedo (transmittance) for subband i in LUTs is calculated beforehand by spectral integration of the spectral narrowband albedo (transmittance) with the spectral narrowband weight of the spectral narrowband downward solar flux within subband i. The narrowband weight is calculated with a spectrally detailed RTM for the same snow and atmospheric conditions as described above. For example, the black sky albedo αi,BS for subband i is calculated by
where Fdir (λ) is the direct component of spectral narrowband downward flux, αBS (λ) is the spectral narrowband black sky albedo, and λ1 and λ2 indicate the spectral domain of subband i.
2.1.6. Effect of Illumination Conditions
 Snow albedo, in general, depends on θ0 under a clear sky and becomes independent of θ0 under a cloudy sky [Wiscombe and Warren, 1980; Aoki et al., 1999]. Hence, snow albedo changes with the illumination condition. This effect is calculated in PBSAM by using the diffuse fraction of global solar radiation for each of visible and NIR band, and the final broadband albedo and transmittance are calculated by
where the input parameters to PBSAM fVISdiff and fNIRdiff are the diffuse fractions of global solar radiation in the visible and NIR bands. Albedo and transmittance for the entire shortwave (SW) spectrum are calculated by
where FVIS, FNIR, and FSW (= FVIS + FNIR) are global solar radiation in the visible, NIR, and SW bands, respectively. These values are input parameters to PBSAM.
2.1.7. Solar Heating Rate in the Snowpack
 PBSAM can calculate solar heating (absorptivity) in each snow layer as well as albedo and transmittance. The absorptivity aj of a given snow layer j is calculated from the downward and upward components of solar flux at the upper and lower boundaries of the snow layer j as follows:
where d and u are the ratios of downward and upward solar flux components, respectively, in the snowpack relative to the downward solar flux at the snow surface, and the subscripts UB and LB denote the upper and lower boundaries, respectively, of snow layer j. Thus, uUB=top = α and dLB=bottom = t.
 When the snowpack consists of two layers, the values of da+bi,BS and ua+bi,BS at the boundary between the two snow layers are calculated from the albedo and transmittance in each snow layer by the adding method, given by equation (2):
The values of da+bi,WS and ua+bi,WS are calculated by replacing all “BS” in equation (9) with “WS.” Similarly, the values of di,BS, ui,BS, di,WS, and ui,WS at any given boundary in all snow layers are calculated by changing the boundary for the final adding procedure iteration. For example, to calculate the values of d and u at the boundary between the third and fourth snow layers in a five-layer snow model, we apply the adding method to (1) the upper three layers, (2) the lower two layers and the underlying surface, and (3) the two composite snow layers obtained in steps (1) and (2). To calculate the values of d and u at the other boundaries, we repeat the adding procedure for the appropriate combinations of the upper and lower snow layers [e.g., Lacis and Hansen, 1974]. The last absorptivity needed is aj,SW, which is calculated from di,BS, ui,BS, di,WS, and ui,WS by subband to broadband calculation, as shown in equations (4)–(7).
 The solar heating rate H(z) is generally described by the vertical gradient of the net solar flux ∂Fnet (z)/∂z in the snow, where z is the depth from the snow surface, which is positive in the downward direction, as follows:
where ρ(z) is the snow density and Cp is the specific heat capacity of ice. The vertical gradient of the net solar flux in a given snow layer j with thickness dzj is approximated by
where Fj↓ and Fj↑ are the downward and upward solar flux, respectively, and the subscripts UB and LB are the upper and lower boundaries of the layer, respectively. Multiplying both sides of equation (8) for the shortwave spectrum by FSW, we obtain
From equations (10) and (13), the solar radiative heating in a given homogeneous snow layer j is obtained as follows:
where ρj is the snow density, SWEj is the snow water equivalent, and FSW is the downward solar radiation at the snow surface. The specific heat capacity of ice is a function of the snow temperature Tj as follows: Cp (Tj) = 7.8 × 10−3Tj + 2.117(kJ kg−1 °C−1). In equation (14), FSW, SWEj, and Tj are input parameters to PBSAM, and aj,SW is calculated by PBSAM.
2.2. Exact Reference Model
 In the present study, we used a spectrally detailed RTM to calculate the subband albedo and transmittance values to be stored in LUTs for PBSAM. We also used the spectrally detailed RTM as an exact reference model for validating PBSAM results for the broadband albedos (see section 3) and the solar heating profile in the snowpack (see section 6.1). The detailed specifications of the model have been described by Aoki et al. [1999, 2000, 2003]. In the present study, we employed the refractive index of ice revised by Warren and Brandt , in which the imaginary part at the wavelengths 0.2 < λ < 0.5 μm is much lower than that in the previous data set compiled by Warren . This change means that albedos of pure snow with a semi-infinite snow depth are high, nearly 1.0 at 0.2 < λ < 0.5 μm. The spectral resolution of the spectrally detailed RTM used is 25 nm.
 To make the LUTs, the spectral black sky albedo and transmittance were calculated with the spectrally detailed RTM for wide ranges of snow grain size, BC mass concentration (only BC was used to make the LUTs), SWE in a single snow layer, and solar zenith angles (Table 1). The ranges of four variables, that is the size of LUT, are 7 grid points for 20 ≤ re ≤ 2000 μm, 17 grid points for SIF (as BC: 0 ≤ cBC ≤ 100 ppmw), 13 grid points for 0.1 ≤ SWE ≤ 5000 kg m−3, and 16 grid points for 0 ≤ θ0 ≤ 89.9°. The black sky albedo αi,BS and transmittance ti,BS values for the subbands in the LUTs were calculated by spectral integration (see equation (5)) of the spectral albedo αBS (λ) and the spectral transmittance tBS (λ), respectively, using the spectral narrowband weights of downward solar flux within the spectral subbands. The spectral narrowband weights were calculated for typical snow conditions under typical clear and cloudy sky conditions. We assumed the snow conditions re = 100 μm and cBC = 0.01 ppmw with semi-infinite snow depth. The clear sky condition was simulated by the model atmosphere of “midlatitude winter” [Anderson et al., 1986], including the “continental average” aerosol model [Hess et al., 1998] with optical thickness τa = 0.05 at λ = 0.5 μm. The cloudy sky condition is the same as the clear condition except that the overcast was simulated by the “continental clean” cumulus cloud model [Hess et al., 1998] with optical thickness τc = 10 at λ = 0.5 μm. The white sky albedo αi,WS and transmittance ti,WS for the subbands were calculated in the same manner as by Wiscombe and Warren  by assuming isotropic incident radiation and integrating the black sky albedo αi,BS (μ0) and transmittance ti,BS (μ0), respectively, over all angles of incidence as follows:
where μ0 = cos(θ0). The values of αi,WS and ti,WS are independent of θ0.
2.3. Optical Properties of Snow Impurities
 In PBSAM, we consider the snow impurities BC and mineral dust, and their MACs are important optical parameters associated with light absorption by snow impurities. For BC, we basically calculated optical parameters similar in concept to those given by Flanner et al. . We can choose hydrophobic or hydrophilic optical properties for both BC and dust according to the hygroscopic properties of aerosols calculated in our host aerosol transport model (Model of Aerosol Species IN the Global AtmospheRe, MASINGAR) [Tanaka et al., 2003; Tanaka and Chiba, 2005]. We calculated hydrophilic optical properties with a coated sphere [Toon and Ackerman, 1981], in which the core is BC or dust, and the shell is water, and hydrophobic properties for dry sphere with Mie theory. As the refractive index of BC we employed the spectrally independent value nBC = 1.85 − 0.71i, which is an intermediate value among those recommended by Bond and Bergstrom . For dust, we used the spectral refractive index of dust aerosols measured in the Taklimakan Desert in China by Aoki et al. . We employed lognormal size distributions for BC and dust, with mode radii of 0.046 and 0.5 μm, respectively, for the core particles, and geometric standard deviations of 1.5 and 2.2, respectively. The core radius of BC was determined such that the MAC of hydrophobic BC was close to 7.5 m2 g−1 at λ = 0.55 μm. For dust, we employed the size distribution used in the “mineral-transported” model by Hess et al. . We assumed a material density of 1.5 g cm−3 for BC and 2.6 g cm−3 for dust. The shell/core ratio used for hydrophilic particles was 1.9 for BC [Chin et al., 2002] and 1.27 for dust [Yamazaki et al., 2007]. The absorption enhancement (change of MAC) caused by the water coating was 1.5 and 1.4 times for BC and dust, respectively, at λ = 0.55 μm. Figure 2 depicts the spectral variations of the MAC per dry material weight of BC and dust. BC and dust particles as snow impurities were assumed to be externally mixed with snow particles.
 The values of kai,BC and kai,dust used to calculate SIFi by equation (1), shown in Table 2, were obtained by spectral integration of the hydrophilic aerosol curves shown in Figure 2 expressed by equations of the same form as equation (5). The spectral narrowband weight is the spectral downward solar flux within spectral subband i for clear sky condition as described in section 2.2. However, the obtained MACs kai,BC (θ0) and kai,dust (θ0) are a function of θ0. For the sake of simplicity, we obtained the θ0-independent values kai,BC and kai,dust by θ0 integration using equation (15).
3. Validation of Broadband Albedos Calculated by PBSAM Using the Exact Reference Model
3.1. Single-Layer Snow Model
 We compared broadband albedos calculated by PBSAM with those calculated by the spectrally detailed RTM (exact reference model) as a function of the SWE of pure snow (Figure 3a) and the BC concentration (Figure 3b) in snow. Since the plotted albedos were calculated for a single-layer homogeneous snow model, the PBSAM albedos are just interpolated values from the LUT, and the adding procedure was not used. The albedos calculated by the exact reference model were used to calculate the LUTs. As a result, the two sets of albedos agree almost completely even in the V1N1 case. The albedos for re = 50 μm increase faster with SWE at small SWE values than those for re = 1000 μm (Figure 3a) because the penetration depth in snow composed of small grains is shallower and thus less sensitive to the effect of the underlying surface albedo than snow composed of large grains. In Figure 3a, the visible and near-infrared albedos converge to 0.3 at SWE = 0.1 kg m−2, although snow is more absorptive for the near-infrared region than for the visible region. This is because the visible transmittance (0.68) is larger than that for the near-infrared (0.60), whereas the visible absorptivity (3.9 × 10−5) is much smaller than that for the near-infrared (0.11). The rate of albedo reduction with increasing BC concentrations is larger for re = 1000 μm than for re = 50 μm (Figure 3b) because, as Warren and Wiscombe  explain, radiation penetrates deeper in more coarsely grained snow and encounters more absorbing materials before it can reemerge from the snowpack.
 We also compared the broadband albedos as a function of dust concentration calculated by PBSAM with those determined by the exact reference model (Figure 4). PBSAM uses a LUT calculated by assuming BC as the only snow impurity (section 2.1.3). In PBSAM, the input dust concentration is transformed to SIF by equation (1), and the albedo in the LUT is interpolated using the SIF value. Figure 4, thus, demonstrates the validity of using SIF values for dust. In the case of re = 50 μm, the albedos calculated by PBSAM agree well with the exact albedos in the cdust range shown in Figure 4, whereas in the case of re = 1000 μm, the albedos by PBSAM are underestimated for cdust > 100 ppmw. If the BC concentration were to increase without limit, the albedo would approach a value close to zero. However, in the same situation, the albedo for dust would converge on a higher value closer to that of the desert surface than the BC value. Thus, the use of SIF for dust by PBSAM is restricted. If the only snow impurity is known to be dust, calculation of accurate albedos would require use of LUTs calculated using dust instead of BC.
3.2. Two-Layer Snow Model
 We next compared broadband albedos of a two-layer snowpack calculated by PBSAM with those obtained with the exact reference model (Figure 5). In the top layer, SWE (Figure 5a) or the BC concentration (Figure 5b) was allowed to vary, and the bottom layer was considered pure snow with semi-infinite thickness. These results demonstrated the effect of the number of spectral subbands on the accuracy of albedos calculated by PBSAM. The shortwave and NIR albedos calculated by PBSAM using V1N1 were underestimated for small SWE (Figure 5a) and BC concentrations (Figure 5b), when the optical thickness of the top layer was low. This tendency was particularly marked in the case of the NIR albedo. The visible albedos calculated using small numbers of subbands mostly agree with the exact reference model results. For larger numbers of subbands, the albedos by PBSAM (V3N3) generally agree well with the exact reference model albedos in the case of both SWE and BC concentrations (Figures 3a and 3b). This difference is because the energy contribution to the reflected radiation (thus, the albedo) from the light transmitted through the top layer and reflected by the bottom layer and transmitted upward back through the top layer (α1 + α2 + … in Figure 1), is not precisely calculated by PBSAM for V1N1 when the top layer has low optical thickness. The albedo for the bottom layer used in the adding procedure for each subband is calculated by spectral integration of the spectral albedo within the subband using the spectral narrowband weights of the spectral downward solar radiation under the condition that the snow layer is just below the atmosphere and stored in LUT. Thus, the effect of a change in the spectral distribution of downward solar radiation within the subband in the bottom layer of snow is not taken into account in the case of V1N1, whereas it is accurately accounted for when the number of subbands is increased. We attribute the remarkable underestimate in the NIR albedo compared with the visible albedo in the case of V1N1 to the larger spectral variations of albedo and transmittance in the NIR spectrum than in the visible spectrum. When the number of subbands is increased further such as to V4N4 or V5N5 (not shown), the visible and NIR albedos are both almost the same as those obtained with V3N3. These results suggest that the number of subbands must be at least V1N3 to ensure the accuracy of broadband albedos.
4. Radiation and Snow Observations at Sapporo
4.1. Observation Site and Instrumentation
 Radiation budget observations and snow pit work were carried out at Sapporo, Hokkaido, Japan, for validation of PBSAM. The observation site was the meteorological observation field (43°04′56″N, 141°20′30″E, 15 m a.s.l.) of the Institute of Low Temperature Science, Hokkaido University, located in an urban area of Sapporo City [Aoki et al., 2006, 2007a]. Our field measurements of year-round surface meteorological parameters, including the detailed radiation budget and snow pit information during the snow covered season, have been performed since September 2003 and continue at present (December 2010). The measured components relevant to the present study are as follows: Basic surface meteorological parameters are measured by an automatic weather station (AWS). For the radiation budget observations, downward and upward (ground-reflected) solar flux densities are measured with pyranometers (CMP21, Kipp & Zonen, Delft, Netherlands) in the shortwave band (λ = 0.305–2.8 μm) and the NIR band (λ = 0.708–2.8 μm) with an RG715 cut-off filter dome, where the value of 0.708 μm at the lower end in the NIR band is the actual wavelength measured when the transmittance of the cutoff filter dome of the pyranometer is 0.5. The flux densities in the visible band (λ = 0.305–0.708 μm) are obtained by subtracting the NIR flux densities from the shortwave flux densities. The direct component of solar flux is measured with a pyrheliometer (CH1, Kipp & Zonen) mounted on a solar tracker, and the diffuse component with a pyranometer (CM21, Kipp & Zonen) mounted on a solar tracker equipped with a shadow ball to shade the direct solar beam. The direct and diffuse components of only the shortwave band are measured; no data for the diffuse fractions of the visible or NIR bands are obtained. Therefore, in this study we assumed those values to be the same as the diffuse fraction of the shortwave band. The global solar radiation measured as the sum of direct and diffuse components is generally more accurate than that measured with only a pyranometer. However, in this study we used the latter to calculate the albedo, because the solar tracker is occasionally inoperable because of wet heavy snow. All radiation components are sampled every 10 s, and 1 min averaged values are stored in a data logger. For analyses of albedos and meteorological components in this study, the daily 30 min averaged values from 1131 to 1200 LT around local solar noon were used to minimize the effect of mounting flame shadows cast by the measurement apparatus on the snow surface. For the present study, we analyzed data collected during the two winters from December 2007 to March 2009.
 The measured albedo, that is, the simple ratio of the measured upward flux to the measured downward flux, may contain some errors owing to the influence of the mounting frame of the pyranometers, shadows cast by the measurement apparatus on the snow surface, dirt on the glass dome of the pyranometers, etc. Among these, we can estimate the influence of the mounting frame on the measured upward flux. The mounting frame blocks 3.56% of the upward flux in the case of isotropic upward radiation. The frame is made up of steel pipes, and its surfaces reflect a part of both the upward and downward illuminating solar radiation with weak specular reflection. We measured the spectral reflectance of the steel pipe surface under diffuse illumination with a spectrometer (FieldSpecFR, ASD Co. Ltd., CO, USA, λ = 0.35–2.5 μm). All surfaces of the mounting frame viewed from the pyranometers reflect both the upward (snow-reflected) and downward solar radiation. Using these spectral reflectance data, we can calculate the bias error contained in the measured broadband albedo. However, the correction value depends on the solar illumination conditions (mainly sky conditions, but weakly, snow conditions), that is, the spectral distribution of downward flux. Since it is difficult to simulate illumination conditions for all albedo measurements, we set the correction value by assuming the following sky and snow conditions. The downward solar fluxes were calculated with the spectrally detailed RTM for the atmosphere–snow system under the same clear sky conditions as were used to calculate the albedos for the spectral subbands from the spectral albedos, described in section 2.2, except that we chose cBC = 0.2 ppmw. We chose this value of cBC as close to the median value measured at Sapporo (see Table 3 and section 4.2). Under this condition, the spectrally integrated diffuse reflectance mt of the mounting frame surface was 0.34 in both the visible and NIR bands, where the reflectance outside of the spectral range of the FieldSpecFR spectrometer is assumed to be the same as the reflectance at each end of the spectrum. The corrected albedo snow of the measured albedo obs can be expressed as
where f = 0.0356 is the fraction of the upward flux blocked by the mounting frame and u is the contribution ratio of upward flux reflected by the mounting frame viewed from the pyranometer.
Table 3. Median and Range (Minimum–Maximum) of Mass Concentrations of Snow Impurities in Two Snow Sampling Layer Thicknesses Measured at Sapporo in Each Winter, 2007–2008 and 2008–2009, and in Both Winters Combineda
Snow Depth (cm)
Snow Impurity Concentration (ppmw)
2007–2009 (Two Winters)
Units are ppmw (micrograms of impurities per gram of snow). EC, elemental carbon; OC, organic carbon.
 If we assume u = 1.0 (all mounting frame surfaces viewed from the pyranometers reflect only upward radiation) and albedo ranges of obs = 0.75–0.95 (VIS), 0.60–0.80 (NIR), and 0.70–0.90 (SW), the broadband albedos are underestimated by 0.018–0.023 (VIS), 0.014–0.019 (NIR), and 0.017–0.022 (SW), whereas if we assume u = 0.0 (all mounting frame surfaces viewed from the pyranometers reflect downward radiation) and the same albedo ranges as above, the broadband albedos are underestimated by 0.015–0.023 (VIS), 0.010–0.017 (NIR), and 0.013–0.021 (SW). The contribution from the upward flux reflected by the mounting frame surface should be larger than that from the downward flux because the frame surface is a weak specular reflector. We thus calculated all broadband correction values for the measured albedos by assuming u = 0.8. All measured albedos given hereafter are corrected values, and we used these corrected values for validation of the calculated albedos (see section 5).
 Sapporo is located on the Sea of Japan side of Hokkaido, and cold northwesterly winds off the Asian Continent bring frequent snowfalls. Figures 6a and 6b depict air temperature, snow or ground surface temperature, and snow depth during the two winters of our study. The second winter was warmer and the snow depth was less compared with the first winter. In particular, in January and February of the second year, some warm air advections with rainfall and related snow depth reductions were observed on 18, 23, and 29 January and on 14 February 2009. Snow melting occurred mainly in March in both years. We hereafter refer to January and February as the “accumulation season” and to March as the “melting season.”
4.2. Snow Pit Work and Snow Impurity Measurements
 Snow pit work was performed around 1100 local time (LT) twice a week to measure snow grain shape, temperature, density, and hardness of all snow layers, and snow grain size of each snow layer from the surface to d = 10 cm depth. In addition, snow samples were collected from two types of snow layer thickness, d = 0–2 and 0–10 cm. Snow grain size was estimated using a handheld lens with a scale of 10 μm resolution. We defined the snow grain size as one half of the branch width of the dendrites or as one half of the width of the narrower part of broken crystals (“r2” of Aoki et al. ). This dimension corresponds to the optically equivalent snow grain radius [Aoki et al., 2000, 2003]. We estimated the minimum, mode, and maximum values of r2 in each snow layer. The averaged values of measured snow grain mode sizes in d = 0–2 and 2–10 cm during the two winters are shown in Figures 6c and 6d. During the accumulation season of the first winter, grain sizes were small, whereas during the accumulation season of the second winter, grain size sometimes increased synchronizing with temperature rises (Figure 6b). Hereafter, all measured snow grain sizes mentioned are mode values unless otherwise stated.
 The snow samples were collected in stainless steel containers in winter 2007–2008 and in dust-free plastic bags in winter 2008–2009. The mass concentrations of elemental carbon (EC) and dust in the snow samples from the first winter have been reported by Kuchiki et al. . We used the same method to measure EC and dust mass concentrations in the second winter. Briefly, the collected snow samples were transported to the Meteorological Research Institute in Tsukuba, Japan, and stored in a freezer at −18°C until analysis. To extract snow impurities, the snow samples were melted and filtered through a Silver Membrane filter (Sterlitech Corp., Kent, WA, USA) with a pore size of 0.45 μm, and the total mass concentration of snow impurities (cTOT) was estimated from the difference in the filter weight before and after filtering [Aoki et al., 2003, 2007b]. To obtain the mass concentration of EC (cEC), a representative portion of the filter was punched out and analyzed with the Lab OC-EC Aerosol Analyzer (Sunset Laboratory Inc., Tigard, OR, USA) by the thermal optical reflectance method [Chow et al., 1993]. We adopted the Interagency Monitoring of Protected Visual Environments (IMPROVE) thermal evolution protocol [Chow et al., 2001] for this measurement. The instrument was calibrated using a standard solution of sucrose, and replicate analyses of the standard showed good agreement within 3% [Kuchiki et al., 2009].
 The Lab OC-EC Aerosol Analyzer used can measure the concentration of organic carbon (OC) (cOC) as well as that of EC. We determined the dust concentration as cdust = cTOT − (cEC + cOC). The values of cdust may be overestimated, however, because the cOC value is the concentration of only the carbon molecules contained in organic matter in the snow samples. However, we visually confirmed that dust particles were the major constituent by volume of the snow impurities on the filters collected at Sapporo. The cTOT values were much higher than the sum of cEC and cOC, suggesting that dust accounted for most of the mass of snow impurities at Sapporo. Thus, it is likely that the overestimate of cdust was very small. The measured cdust and cEC values for d = 0–2 cm and 0–10 cm are shown in Figures 6c and 6d. Dust concentrations were a few orders of magnitude higher than EC concentrations. The median and range of the measured mass concentrations of dust, EC, and OC in the two snow layer thicknesses are listed in Table 3. The median EC (OC) concentration during the two winters was 0.215 (0.233) ppmw in d = 0–2 cm and 0.183 (0.183) ppmw in d = 0–10 cm. These values are much higher than in situ measurement results obtained in the Arctic. For example, median BC or EC concentrations of 7 ppbw in Arctic sea ice in 2005 [Perovich et al., 2009], 0–80.8 ppbw (average, 8.7 ppbw) around Svalbard in 2007 [Forsström et al., 2009], and 7–39 (<21 ppbw except in Russia) in wide areas of the Arctic during 2005–2009 [Doherty et al., 2010] have been reported. The input parameter to PBSAM on snow impurities, other than cdust, is cBC. According to field measurements of atmospheric BC and EC in Asian outflow, reported by Miyazaki et al. , the measured concentrations of these aerosol components agree with each other within 2%. We thus assumed cBC = cEC as the input parameter to PBSAM.
5. Validation of Broadband Albedos Simulated by PBSAM Using the Measurements at Sapporo
5.1. Broadband Albedo Variations During the Two Winters
Figure 7 depicts the broadband albedos measured at Sapporo during the two winters and those simulated by PBSAM. The simulated albedos shown in Figures 7a and 7b were calculated using a single-layer snow model (1L) with V1N1 subband numbers, and those shown in Figures 7c and 7d were calculated using a five-layer snow model (5L) with V5N5 subband numbers. In the 1L model, we used the SWE measured in the entire snowpack and values of the other snow parameters measured in the top layer only. Even in the 1L-V1N1 simulation, the daily variation of albedo was successfully simulated for each broadband. The steep albedo drops at the end of February 2008, the beginning of January 2009, and twice in late January 2009 are well reproduced by the simulation. The main cause of those albedo reductions was increase in air temperature and the last case was accompanied by rainfall. On the other hand, some albedos (especially the visible albedos) in the melting season are underestimated. In the 5L model, the measured snow grain size and impurity concentrations (BC and dust) are given for the top three model layers (d = 0–2, 2–5, and 5–10 cm). However, we assumed the measured impurity concentration of d = 0–10 cm for both the d = 2–5 and d = 5–10 cm model layers because we measured impurities only in two layer thicknesses d = 0–2 and 0–10 cm. Since we did not measure the snow grain size and impurities in the lower two model layer depths (d = 10–20 and 20 cm to the bottom), as mentioned in section 4.2, the typical grain sizes were assumed based on the measured snow grain shapes, and the snow impurity concentrations were assumed to be the same as those in the model layers d = 2–5 and 5–10 cm. The assumed snow gain sizes are r2 = 50 μm for decomposing and fragmented precipitation particles, 100 μm for rounded grains, 500 μm for faceted crystals, 1000 μm for depth hoar or melt forms (granular snow) with T < −0.2°C, and 2000 μm for melt forms with T ≥ −0.2°C. The sizes of all precipitation particles (new snow) were measured. These snow shapes we employed are in accordance with the snow classification by Fierz et al. . Values of the other snow parameters (SWE and snow temperature) were obtained by in situ measurements. The underlying surface albedos were determined by averaging the measured values before and after the snow covered period.
 Although the albedos simulated by the 1L-V1N1 and 5L-V5N5 models (Figure 7) are similar, they differ in some details. For example, in January 2008 and February 2009, some high visible albedos (higher than 0.95) calculated by the 1L-V1N1 model are higher than those calculated by the 5L-V5N5 model. Grain size of new snow is small, but that of older snow is larger, depending on the elapsed time after the last snowfall, because snow grains grow with an evolution of snow metamorphism. As a result, measured grain sizes are generally small in the top snow layer and larger in lower layers. Thus, the use of the snow grain size measured in the top layer in the 1L model causes the simulated albedo to be overestimated (Figures 7a and 7b). This tendency toward overestimation is lessened by use of the 5L-V5N5 model (Figures 7c and 7d). The results are compared statistically in section 5.2. In Figures 7c and 7d, the error bars show the simulated albedo range based on the measured minimum and maximum snow grain sizes. Most measured shortwave and NIR albedos fall within the simulated range, but some visible albedos fall outside the range indicated by the error bars. This difference is because the visible albedo change corresponding to the variation in snow grain size is smaller than the NIR albedo change. Possible causes of the discrepancy in albedos between simulations and measurements are (1) PBSAM faultiness; (2) inappropriately modeled snow layers structure (e.g., number of layers and depths of layer boundaries); (3) the assumption that the diffuse fractions of the visible and NIR bands are the same as the measured diffuse fraction of the shortwave radiation; (4) errors in the measured snow grain size and snow impurity concentrations; and (5) errors in the albedo measurements. The measured values mentioned in items 3 and 4 are used for input parameters to PBSAM. As a factor in (1), PBSAM calculates the effect of difference in F(λ) within the subband between clear and cloudy conditions on the subband albedos by F(λ) calculated for assumed typical clear and cloudy conditions. There are also uncertainties in the optical properties of snow impurities in PBSAM. These factors could cause the errors of broadband albedos. As for item 3, we estimated the possible errors in broadband albedos calculated by PBSAM based on the simulations for spectral diffuse fractions in the visible and near-infrared bands under clear condition with a spectrally detailed RTM. The error is nothing under overcast condition but reaches a maximum under clear condition because of difference in Rayleigh scattering between the visible and near-infrared bands. The maximum error estimated was 0.005 (visible), 0.013 (near-infrared), and 0.006 (shortwave) under clear sky. However, the ratio of days with measured shortwave diffuse fraction less than 0.5 was only 20% at Sapporo during two winters from 2007 to 2009. Therefore, the finally estimated possible errors (RMSE) due to (3) were 0.001 (visible), 0.005 (near-infrared), and 0.003 (shortwave) during the two winters. The data plotted in Figure 7 have already been corrected for the major factor in item 5, that is the influence of the mounting frame of the pyranometers. Next, we investigate the effect of the number of modeled snow layers and the number of spectral subbands used in PBSAM on the accuracy of the calculated albedos.
5.2. Comparison of Broadband Albedos Between Measurements and Simulations
 We compared the measured broadband albedos with their simulations for both the 1L-V1N1 and 5L-V5N5 models (Figure 8). Even the 1L-V1N1 model simulation results are highly correlated (R = 0.893–0.939) with the measurements in all three spectra, but the correlations are higher in the case of the 5L-V5N5 simulations (R = 0.901–0.948). The root mean square errors (RMSE) are also improved from the 1L-V1N1 to 5L-V5N5 results. The slopes of the regression lines for the NIR albedos are close to 1.0. The slopes for the visible albedos are larger than 1.0 in both 1L-V1N1 and 5L-V5N5 results, although the deviation is somewhat less in the case of the 5L-V5N5 results. Deviations of the slope from the 1:1 line are due mainly to the underestimation of simulated visible albedo values, as described in section 5.1. The dashed 1:1 line is mainly within the error bars in Figure 8b (determined in the same way as those shown in Figures 7c and 7d), except in the case of some visible albedos.
 To investigate the effects of the number of modeled snow layers and the number of spectral subbands on the accuracy of simulated albedos, RMSEs and correlation coefficients calculated using of 1L, 3L, and 5L snow layer models with subbands varying in number from V1N1 to V5N5 are summarized in Figure 9. In the case of the 1L model, RMSE and R are almost independent of the number of subbands. This is because PBSAM does not take into account the effect of spectral changes of solar radiation within the snowpack (described in section 3.2; see Figure 5) in calculations using the 1L model.
 RMSE decreases with the number of modeled snow layers, except for 3L-V1N1, 3L-V2N2, and 5L-V1N1 for the NIR band, and 3L-V1N1 for the shortwave band (Figure 9a). For the visible band, the effect of multiple snow layers on RMSE is clear, but the effect of the number of spectral subbands on RMSE is very small. The latter is attributable to the small spectral variation of albedo in the visible region (discussed in section 3.2; see Figure 5). As a result, even the V1 model is applicable to visible albedo simulation. However, a multilayer model, at least 3L or preferably 5L, is necessary for visible albedo simulation. For the NIR band, some RMSE values calculated by the 3L and 5L models are worse (larger) than those calculated by the 1L model. We have already seen (section 3.2) that the albedo calculated using a single spectral subband and a two-layer snow model is underestimated, and the underestimation is substantial in the case of the NIR band. Some of the worse NIR band RMSE values can be explained in the same way. When the number of modeled snow layers is fixed, the NIR band RMSE values obtained by using three or more subbands (N3 to N5) are almost the same. When three or more subbands (N3 to N5) are used, the RMSE values calculated with the 5L model are better than those calculated with the 1L or 3L model. Therefore, accurate albedos in all three broadbands can be expected by PBSAM using 5L-V1N3. However, this result is derived using our model configuration of snow layer structure and spectral subbands. Therefore, the numbers of layers and subbands represented by 5L-V1N3 (especially 5L) are not universally valid for other albedo model configurations.
 The values of the correlation coefficient R tend to increase with both the number of modeled snow layers and the number of subbands (Figure 9b). As in the case of the RMSEs, some values of R for the NIR and shortwave bands calculated by the 3L model are worse (lower) than those calculated by the 1L model, and for the same reasons. There is no significant change in R for larger numbers of subbands than V1 and N3. Thus, as for RMSE, a high correlation between measured albedos and their simulations is also expected from the 5L-V1N3 model.
 The RMSE, mean error (ME), and R values obtained from the simulations by PBSAM using 5L-V3N3 are shown in Table 4 for each winter and for the two winters combined. These results are almost the same as those for 5L-V5N5 and represent the best performance of PBSAM that is, RMSE = 0.047, ME = 0.005, and R = 0.932 for shortwave band in case of the two winters combined.
Table 4. Root-Mean-Square Error (RMSE), Mean Error (ME = Simulations – Measurements), and Correlation Coefficient (R) Between the Broadband Albedos Simulated by PBSAM Using the 5L-V3N3 Model and in Situ Measurements at Sapporo in Each Winter, 2007–2008 and 2008–2009, and in Both Winters Combined
2007–2009 (Two Winters)
6. Solar Heating in the Snowpack
6.1. Validation of the Solar Heating Profile Using the Exact Reference Model
 In this section, we discuss the solar heating profiles calculated by the following three RTMs: (1) a spectrally detailed RTM (exact reference model), (2) PBSAM, and (3) the model of Brandt and Warren . The first model, which is a modified version of an RTM for the atmosphere–snow system [Aoki et al., 1999, 2000, 2003] (section 2.2), calculates the vertical profiles of solar heating for any vertical resolution. PBSAM is the same as that described in the previous sections. The model of Brandt and Warren  is the two-stream approximation proposed by Schlatter , which they used to calculate the heating rate in the snowpack with high spectral resolution. To compare these models, we chose to use atmospheric and snow conditions in Antarctica in accordance with Brandt and Warren , where the snow grain radius (r) is 100 μm, snow density ρ = 380 kg m−3, and global solar radiation (Fdn) is 400 W m−2.
Figure 10a shows the spectral variations of downward solar radiation, snow albedos, and solar radiation absorbed by snow calculated with the spectrally detailed RTM for pure homogeneous snow with re = 50, 100, 200, and 1000 μm. These illumination and snow conditions are used to calculate the solar heating profiles hereafter. The Fdn(λ) (downward solar radiation) curves must differ slightly depending on the albedo α(λ) due to the multiple reflections between snow surface and atmosphere. However, we compared solar heating profiles under different snow conditions but fixed illumination using the Fdn(λ) value calculated for pure snow of re = 100 μm. The solar zenith angle (θ0 = 68.4°) was determined so that the spectrally integrated downward solar flux Fdn equaled the value of 400 W m−2. The spectral distributions of Fabs(λ) (= Fdn (λ)[1 − α (λ)]) have a peak at around λ = 1.0–1.5 μm, depending on snow grain size. We also prepared Fdn(λ), α (λ), and Fabs(λ) to examine the effect of snow impurities on the solar heating profile for BC-contaminated snow with cBC = 0.2 ppmw (Figure 10b). This value of cBC is based on the median EC concentrations measured at Sapporo (0.215 ppmw for d = 0–2 cm and 0.183 ppmw for d = 0–10 cm), and the Fdn(λ) value used is the same as that used for pure snow (Figure 10a). The effect of BC on the spectral albedo was substantial in the visible region, and the spectral distributions of Fabs(λ) increase at λ < 0.8 μm (Figure 10b) compared with those for pure snow case (Figure 10a).
 The solar radiation absorbed by each layer (d) of pure homogeneous snow calculated by the spectrally detailed RTM for re = 50 and 1000 μm is shown in Figures 11a and 11b. These snow grain sizes correspond to that of new snow and old melting snow (granular snow), respectively [Wiscombe and Warren, 1980]. The upper borders of the colored area in Figures 11a and 11b are the same as the Fabs(λ) curves for re = 50 and 1000 μm, respectively, shown in Figure 10a (red curves). In the case of re = 50 μm (Figure 11a), most solar radiation is absorbed by the topmost layer d = 0–1 mm at λ > 1.4 μm. For λ ≤ 1.4 μm, much solar radiation is absorbed by the relatively deeper layer d = 1–10 cm, and the energy fraction absorbed below d = 10 cm is very small. For re = 1000 μm (Figure 11b), the relative fraction of solar energy absorbed by the deeper layers increases at all wavelengths because of the increased light penetration depth. In particular, a part of the solar radiation is absorbed by the snow layers deeper than d = 10 cm at λ < 1.0 μm and deeper than d = 1 m at around λ = 0.5 μm, at which light absorption by ice is relatively weak. From Figures 11a and 11b, we can estimate the fraction of solar radiation absorbed by each snow layer when re = 50 μm (1000 μm) to be 69.1% (26.4%) for d = 0–1 mm, 23.6% (40.4%) for d = 1–10 mm, 6.7% (25.7%) for d = 1–10 cm, and 0.6% (7.4%) below d = 10 cm. At the snow surface, the energy fraction absorbed by new snow (re = 50 μm) is larger than that absorbed by old melting snow (re = 1000 μm), whereas the reverse is true for layers deeper than d = 1 mm. However, we should note that the reversal depth of two profiles might change depending on the solar zenith angle, snow impurity concentration, and illumination condition.
Figures 11c and 11d show the solar radiation absorbed by each snow layer in the case of snow contaminated with BC (cBC = 0.2 ppmw). BC in snow mainly reduces the albedos at λ < 0.8 μm, as shown in Figure 10b. As expected, total absorbed solar radiation by BC-contaminated snow increases at those wavelengths (compare Figures 11c and 11d with Figures 11a and 11b), particularly when re = 1000 μm (Figure 11d), because the albedo reduction caused by snow impurities is enhanced by larger snow grains [Wiscombe and Warren, 1980]. In any case, most solar radiation is absorbed by the top layers, within d = 10 mm at λ > 1.4 μm. The solid vertical line in Figure 11d indicates the boundary between the visible and NIR bands, and the dashed vertical lines are the boundaries of spectral subbands for V5N5. The subband boundaries in the NIR band were determined so that the spectral fraction of solar radiation absorbed by each snow layer (each colored area) would vary spectrally as little as possible within each subband. In contrast, the boundaries of the five subbands in the visible band are at almost constant spectral intervals, because the spectral variation of the visible solar radiation absorbed by each snow layer varies strongly with snow grain size and snow impurity concentrations (Figure 11).
 Since the vertical profile of solar radiation absorbed by snow strongly depends on wavelength, a certain number of subbands are needed for PBSAM to calculate the solar heating rate accurately. To determine the appropriate number of subbands, we compared the solar radiation absorbed per unit snow thickness (i.e., the solar heating profile) in the top 10 cm of pure homogeneous snow for four different snow grain sizes. Figure 12 depicts the solar heating profiles calculated by the spectrally detailed RTM (exact reference model), PBSAM, and the model of Brandt and Warren . The results calculated by PBSAM using the spectral subbands V1N1, V1N5, and V5N5 are shown in Figure 12. In the case of V1N1, the solar heating profiles by PBSAM do not agree with those calculated by the exact reference model for any re. In these PBSAM calculations, we used a different vertical resolution of the heating profile (dzj in the right-hand side of equation (11)), that is dz = 1 mm for d = 0–1 cm and dz = 1 cm for d = 1–10 cm, in order to determine the detailed profile structure near the surface. The heating profiles calculated by PBSAM using V1N1 change discontinuously between d = 1.0 and 1.5 cm, indicating that the calculated result depends on the modeled snow layer thickness. In the case of V1N5 (Figure 12b), the solar heating profiles for re = 1000 μm calculated by PBSAM agree well with those calculated by the exact reference model, whereas those calculated by PBSAM for re = 50, 100, and 200 μm overestimate values in the snow layer d = 1–10 cm. The PBSAM results showed good agreement with the exact reference model with V5N5 (Figure 12c) for all re. We also tested other combinations of visible and NIR subband numbers and found that at least V3N3 was needed to maintain the accuracy of the calculated solar heating profile. We also compared calculated solar heating profiles between PBSAM and the exact reference model in contaminated snow (cBC = 0.05 and 0.2 ppmw) (not shown) and obtained similar results with regard to subband numbers.
 The solar heating profile calculated by Brandt and Warren  for snow grain radius r = 100 μm is also plotted in Figure 12 (dashed curve). Their profile mostly overlaps the profile calculated by the exact reference model for re = 200 μm (solid curve). Brandt and Warren  used the same single scattering parameters as were used by Wiscombe and Warren , calculated by Mie theory for monodisperse spherical ice grains and an effective snow grain radius also of 100 μm. As a result, there is a bias difference between our result and that of Brandt and Warren . The possible reasons for this bias are differences in the radiative transfer model employed, the size distribution of snow grains, and refractive index of ice. PBSAM uses F(λ) within the subbands calculated with a spectrally detailed RTM for assumed typical clear and cloudy conditions. This also could cause the errors of solar heating profiles. The tendency of snow grain size dependence of our solar heating profile is similar to that reported by Flanner and Zender , who showed that 20–45% of solar absorption by deep snowpacks, depending on snow grain size, occurs more than 2 cm beneath the surface under the conditions on the Tibetan Plateau.
 To examine the dependence of the solar heating profile on the BC concentration, we calculated profiles by PBSAM for pure snow and for different BC concentrations in a deep snowpack (Figure 13). In the surface layer d < 2 cm, absorption increased as the BC concentration increase, whereas the situation was reversed in the lower layers, d > 20 cm. This phenomenon is the same as the shading effect by light-absorbing aerosols in the atmosphere. Thus, we can say that BC contamination heats the top layer and cools the lower layers of a snowpack.
6.2. Solar Heating in the Snowpack at Sapporo Simulated by PBSAM
 Using the snow pit and radiation data measured at Sapporo that we used for albedo simulation by PBSAM (Figure 7), we calculated the solar radiation absorbed in modeled snow layers by PBSAM. Figure 14 depicts the albedo and absorptivities of snow layers and the underlying ground simulated by PBSAM at Sapporo on the days that snow pit work was performed. Nearly all absorption of solar radiation by snow occurs in the top layer (d = 0–2 cm) in the accumulation season, whereas absorption occurs in the second (d = 2–5 cm) and third layers (d = 5–10 cm) as well in the melting season. This increase of absorption depth in the melting season is due to the light penetration depth being deeper because of the relatively large snow grain size then, compared with the small snow grain size in the accumulation season. Even in the accumulation season, considerable absorption by the second and third layers occurs at two times in late January 2009. On those days, larger snow grain sizes were measured in both the d = 0–2 cm and 2–10 cm snow layers (Figure 6b). In general, when the melting season starts, the snow grain size increases and the snow temperature approaches 0°C in the overall snowpack [Aoki et al., 2003, 2007a], with positive feedback between the snow temperature increase and snow grain growth. The feedback effect is enhanced by solar heating of the lower snow layers because of their larger snow grain size. This result supports the importance of solar heating of the subsurface snow layers, as emphasized by Flanner and Zender .
7. Summary and Outlook
 Although the global annual mean radiative forcing caused by BC on snow is small (+0.1 ± 0.1 W m−2) [IPCC, 2007], locally, values can be much higher [Flanner et al., 2007; McConnell et al., 2007]. On the other hand, drastic melting of snow and ice has been occurring recently in Greenland [Steffen et al., 2008] and in the Arctic sea [Comiso and Nishio, 2008]. There are still large uncertainties regarding the causal link between snow pollution by BC and snow and ice melting in the Arctic. Possible causes of snow albedo reduction other than snow impurities include snow grain growth associated with temperature increases, decreased snow depth, increased rainfall, and changes in solar illumination conditions. Moreover, some feedback mechanisms also exist between those parameters and snow albedo. Thus, we need to develop a physically based snow albedo model coupled with a snow metamorphism model and implement them into a GCM that can calculate the emission, transport, and deposition of light-absorbing aerosols. Then, using this GCM, the impacts of snow impurities and snow grain size change on albedo and climate should be evaluated.
 In the present study, PBSAM, which can be used in a GCM, was developed to calculate visible and NIR albedos and the solar heating profile in a snowpack from snow physical parameters and solar illumination conditions. The model can be applied to any layer structure and any snow depth. The input snow parameters to PBSAM are vertical profiles of snow grain size, mass concentrations of BC and dust, SWE, and snow temperature. The input parameters of solar illumination conditions are solar zenith angle, downward solar flux in the visible and NIR bands, their diffuse fractions, and the underlying surface albedos. The model is based on the LUT method for albedo and transmittance in spectral subbands. The numerical tables are calculated by a spectrally detailed RTM for the atmosphere–snow system as functions of snow grain size, SIF, SWE, and solar zenith angle. SIF is defined as the sum of the products of MACs and mass concentrations of snow impurities so that different light absorbing snow impurities can be represented by a single parameter. This method has the defect that the calculated albedo is systematically underestimated when the dust concentration is more than 100 ppmw.
 In general, snowpacks are vertically inhomogeneous, and they are sometimes optically thin. Hence, it is necessary to divide the overall snowpack into several homogeneous layers to calculate accurate albedo and solar heating rate values. The effects on albedo and transmittance in the overall snowpack are calculated using an “adding” method in PBSAM. Since light absorption by ice varies substantially depending on wavelength, the spectral distribution of solar radiation after absorption and scattering by snow grains varies within the visible and NIR spectra. When a multilayer model is used in PBSAM, the changes in the spectral distribution of solar radiation cannot be simulated accurately by monochromatic broadband calculations. We thus divided the broadbands into several spectral subbands. Broadband albedo and transmittance were obtained by subband to broadband calculation by spectral weighting after first using the LUT method to calculate the values in the spectral subbands for each homogeneous snow layer and applying the adding method to account for the effects of an inhomogeneous snow structure. Comparing the broadband albedos calculated by PBSAM using a two-layer snow model and the different spectral subbands with those calculated by a spectrally detailed RTM, we found that one visible subband and three NIR subbands are necessary to obtain results of sufficient accuracy in the visible and NIR albedos.
 We performed radiation budget measurements and snow pit work during two winters from 2007 to 2009 at Sapporo, Hokkaido, Japan. Using these data, the broadband albedos were simulated by PBSAM and compared with the measurements. RMSEs and the correlation coefficients between simulations and measurements for different numbers of modeled snow layers (1, 3, and 5) and different numbers of subbands (1–5) in the visible and NIR bands in PBSAM were compared. We found that a five-layer snow model with one visible subband and three NIR subbands were necessary to obtain accurate results. The best accuracy (RMSE) of albedos from the comparisons was 0.050, 0.057, and 0.047 for the visible, NIR, and shortwave bands, respectively.
 When the solar heating profile in the snowpack was calculated by PBSAM, both the number of spectral subbands as well as the albedo calculations were important, because the depth of the snow layer absorbing the penetrated solar radiation varied markedly depending on the wavelength. We first demonstrated the solar heating profile in a snowpack by simulations with a spectrally detailed RTM for homogeneous snow of ρ = 380 kg m−3 under Antarctic snow conditions. We showed that most NIR radiation at λ > 1.4 μm is absorbed by the topmost layer within d = 10 mm, whereas most visible and shorter NIR radiation is absorbed by relatively deeper layers. This drastic spectral transition of the absorptive layer causes an error of solar heating profile calculated by PBSAM when the number of spectral subbands employed is small. We then calculated the solar heating profiles in snowpack for different snow grain sizes by PBSAM and compared them with those calculated by the spectrally detailed RTM. We found that for accuracy, more than three subbands were necessary in both the visible and NIR bands. We also found from the solar heating profiles in snowpack contaminated with BC that BC concentration heats the top layers and cools the lower layers.
 Using the radiation budget and snow pit data measured at Sapporo, the absorptivities in the modeled five snow layers were calculated by PBSAM. The topmost layer (d = 0–2 cm) absorbs most of the snow-absorbed solar energy in the accumulation season, but absorption occurs in the top three layers (d = 0–10 cm) in the melting season. This difference is because the light penetration depth is relatively deep in the melting season owing to the relatively large snow grain size then compared with that in the accumulation season. This effect enhances the positive feedback between the snow temperature increase and snow grain growth.
 Finally, we describe future issues to be addressed in snow albedo studies. Abrupt changes of snow and ice are occurring in the Arctic, and the predictions made by GCMs for the cryosphere are insufficient, particularly with regard to snow albedo variations related to snow layer structures, including the snow impurities and snow grain size. To improve the snow albedo scheme used in GCMs, it is necessary to develop or improve a physically based snow albedo model for use in combination with a snow metamorphism scheme employed in a detailed snowpack model such as SNOWPACK [Lehning et al., 2002] or CROCUS [Brun et al., 1992], whereas there are still a few combined models [Flanner and Zender, 2006; Yasunari et al., 2011]. In those models and in PBSAM, described here, the following issues need to be addressed: (1) optical properties of the presently used light-absorbing snow impurities (BC and dust) should be validated, (2) brown carbon or organic carbon should be introduced as additional snow impurities, (3) a model of microbial activities [e.g., Takeuchi et al., 2003, 2005] on glaciers and ice sheets should be developed, (4) the snow metamorphism model should be improved to calculate the optically equivalent snow grain size, and (5) the global applicability of the models should be validated.
 We thank Teppei Yasunari, Yuki Sawada, Kou Shimoyama, Tetsuo Sueyoshi, Junko Mori, Masahiro Takahashi, Tomoyasu Kuno, Hayato Oka, Taro Nakai, Kazuhiro Okuzawa, Tsutomu Watanabe, Shun Tsutaki, Chusei Fujiwara, Kohei Ohtomo, Niyi Sunmonu, Masaki Okuda, Tatsuya Nakayama, Hirokazu Hirano, and Etsuko Tanaka for snow pit work during two winters in the Institute of Low Temperature Science of Hokkaido University. We also thank Masae Igosaki for supporting the laboratory measurements of snow impurities. We appreciate three anonymous reviewers for their helpful comments. This study was supported in part by (1) the Experimental Research Fund for Global Environment Conservation, the Ministry of the Environment of Japan, (2) the Grant for Joint Research Program, the Institute of Low Temperature Science, Hokkaido University, and (3) the GCOM-C/SGLI Mission, the Japan Aerospace Exploration Agency (JAXA).