Snow albedo plays an important role in Earth's climate and hydrology. For example, a small (0.015 to 0.030) decrease in snow albedo over the Northern Hemisphere is twice as effective as a doubling of CO2 at raising global air temperature (Hansen and Nazarenko, 2004). Likewise, during the COVID-19 lockdowns, a cleaner snowpack, presumably from a reduction in anthropogenic emissions, prevented 6.6 km3 of snow/ice from melting in the Indus River basin (Bair et al., 2021a), more water than is stored in the largest reservoir in California. Yet, snow albedo is difficult to measure (Bair et al., 2018), especially in the mountains where lighting conditions vary dramatically. To understand Earth's climate and the effect humans have on it, an understanding of how snow surface topography affects snow albedo is imperative. The concepts of intrinsic and apparent albedos form the basis of this study. Intrinsic albedo is the bihemispherical reflectance (Nicodemus et al., 1977; Schaepman-Strub et al., 2006) of a substance independent of effects of roughness or topography. Apparent albedo is the ratio of the reflected divided by the incident radiation and may incorporate artifacts caused by roughness or topography. Here we use the term albedo to refer to a broadband albedo, covering the solar spectrum. Albedos covering a narrower spectral range are denoted with additional descriptors such as near-infrared albedo. Since the snow surface is rarely smooth, distinction between apparent and intrinsic albedo is an important consideration that is often ignored. For example, MODIS measurements of snow albedo that comprise the National Solar Radiation Database have been found to be positively biased because they fail to account for surface roughness (Gueymard et al., 2019). Both albedos should be studied, as apparent and intrinsic albedos have different uses. An apparent albedo should be used when modeling energy budgets (Bair et al., 2016), as it dictates how much shortwave radiation is absorbed by the surface. Intrinsic albedos are needed to understand changes in snow properties that affect albedo, such as changes in grain size and darkening from light-absorbing particles like soot or dust (Clarke and Noone, 1985; Jones, 1913; Warren, 2019).

Most snow albedo models follow approaches developed 4 decades ago, based on radiative transfer (Warren, 1982). These models provide intrinsic albedos controlled by illumination angle, water equivalent when snow is shallow, and grain-scale snow properties, which have included grain size (Wiscombe and Warren, 1980), grain shape (Libois et al., 2013), snow structure (Kaempfer et al., 2007), direct and indirect effects of light-absorbing particles (Picard et al., 2020; Skiles and Painter, 2019), and vertical heterogeneity (Zhou et al., 2003). Other efforts have focused on rapid calculation (Bair et al., 2019; Flanner et al., 2021; Gardner and Sharp, 2010) and inversion from remotely sensed imagery (Bair et al., 2021b; Nolin, 2010; Painter et al., 2012a). Weiser et al. (2016) present a correction for albedometers over snow where the underlying terrain is unknown, based on modeled or measured irradiance from nearby well-leveled radiometers, but not accounting for surface roughness. A shade endmember has been introduced to account for lighting differences across surfaces (Adams et al., 1986), thereby enabling the use of an apparent albedo for quantitative spectroscopy. These shade endmembers have proven successful when applied to snow cover mapping (Bair et al., 2021b; Nolin et al., 1993; Painter et al., 2003; Rosenthal and Dozier, 1996). Yet, the widely used albedo models cited above do not account for varying illumination within the field of view, meaning their results can be positively biased.

Features that affect snow roughness include suncups (ablation hollows), penitentes, and wind-formed features like ripples, sastrugi, and dunes (Filhol and Sturm, 2015). Because of their topographic variation in solar exposure, all of these roughness features can significantly affect apparent albedo. Matthes (1934) described “suncups” as having “a honeycombed appearance, the surface being pitted with deep cell-like hollows”. However, Rhodes et al. (1987) use the term “ablation hollows” to describe these features as they are not always caused by solar radiation. Instead Rhodes et al. (1987) find that the presence of impurities on the snow surface governs the formation of ablation hollows, growing in direct sunlight for relatively clean snow and decaying in dirty snow (Lliboutry, 1964). This hypothesis was confirmed with a field experiment where an ash-covered snowfield on Mount Olympus from the Mount Saint Helens eruption was cleared. After 2 weeks, the ash-free area had developed larger ablation hollows than the rest of the ash-covered snowfield (Rhodes et al., 1987). Observations of penitentes go back to Darwin (1845, Ch. XV). Penitentes are columns of snow that point at the sun and are thought to be sublimation features (Betterton, 2001). Penitentes can be much larger than ablation hollows, with measured heights over 2 m (Lhermitte et al., 2014). Ripples, sastrugi, and dunes are formed by wind erosion whose orientation varies with the direction of the prevailing winds (Filhol and Sturm, 2015; Seligman, 1936). Warren et al. (1998) report that sastrugi can reduce albedo by altering the angle of incidence for direct solar radiation and by trapping photons through multiple reflections.

Several studies have attempted to model the reflectance of roughness features with simple shapes (Carroll, 1982; Leroux and Fily, 1998; Zhuravleva and Kokhanovsky, 2011), with more recent studies employing ray tracing of three-dimensional surface models (Larue et al., 2020; Manninen et al., 2021). A few studies have focused on the surface roughness and the implications for remote sensing by incorporating multiple viewing geometries (Corbett and Su, 2015; Kuchiki et al., 2011; Lyapustin et al., 2010; Nolin and Payne, 2007) or by measuring spatial variability within a satellite sensor pixel (Wright et al., 2014). These approaches are well-suited toward expansive high-latitude snowpacks but ill-suited towards dynamic midlatitude snowpacks with mixed pixels where the snow cover can change between satellite overpasses. The consensus in the literature is that roughness features can lower the snow albedo by up to 0.40, but decreases of a few percent are more common. To our knowledge, none of these studies have tracked the snow surface topography throughout a snow season, nor have they examined the effects of snow surface topography on spectral mixture analysis.

An example of ablation hollows mapped by the laser scanner is shown in Fig. 5a, b. In situ albedos from CUES from the water year 2021 are shown in Fig. 6: uncorrected αuncorrected, planar-corrected αplanar, and intrinsic αintrinsic are based on the spatial calculations.

In situ and remotely sensed albedos on Mammoth Mountain from the water year 2021 are shown in Fig. 7. An unadjusted (i.e., not adjusted for shade or trees) fractional snow-covered area (fsca), estimated with SPIReS (Bair et al., 2021b), from a nearby target pixel is also shown. The high fsca confirms that mixed (snow and non-snow) pixel effects are minimal. An estimate of the broadband pixel albedo measured by Sentinel 2A/B (S2) is also shown, as described in Sect. 2.3. Finally, the surface roughness (in degrees, divided by 30 for scale) is plotted, also described in Sect. 2.2.

In Fig. 6, the intrinsic albedo is usually greater than the uncorrected or planar-corrected albedo, agreeing with previous work over more limited timespans (e.g., Larue et al., 2020; Lhermitte et al., 2014; Manninen et al., 2021). The largest planar corrections appear in winter, when the planar sloped surface facing away from the sun receives the lowest irradiance relative to a flat surface. The spatial corrections are more nuanced because they involve the solar geometry and the roughness of the surface. As the days get longer in the spring, the solar zenith angle is smaller, but the rougher surface causes more variability in the view factor and illumination on each slope.

Warren et al. (1998) posited two mechanisms for albedo reduction caused by surface roughness: reduction of effective illumination angle and photon trapping. The difference αuncorrected-αintrinsic characterizes the combined contribution. In this study covering 110 d of the water year 2021 snow season, the differences amounted to -0.056 in the broadband albedo and -0.052 in the near infrared. Larue et al. (2020) estimate a decrease in spectral albedo at 1000 nm of -0.02 to -0.03 for low-SSA (specific surface area, i.e., large grain size) snow, but in the snow studied here with extensive ablation hollows, the magnitudes are greater. The difference Iinternal0-∑Iesc-Iesc0 characterizes photon trapping, which accounts for a mean of 4 % of the lost broadband radiation and 5 % of the loss in the near infrared. In the late spring when the snow surface was quite rough, these losses exceeded 20 %. These result follows from Warren et al. (1998), who state that intermediate snow albedos will be most impacted by photon trapping.

The intrinsic albedo is generally greater than the planar-corrected albedo, showing that the planar correction that has been performed in previous research (Bair et al., 2018; Painter et al., 2012b) accounts for surface slope but not for roughness. But the planar correction is useful as the difference between the planar-corrected and the intrinsic albedo quantifies the impact of sub-slope surface roughness at this location. This difference implies that in areas where the average surface topography is accurately quantified (e.g., over 0.5–1.0 km pixels), a terrain-corrected (adjusted to level) surface reflectance can be used in a spectral mixture model in with a shade endmember to decrease uncertainty in impurity estimates. However, for sensors with finer resolution (e.g., ≤ 30 m), caution is advised with terrain corrections. If ground control points are not available, as in the case of many remote parts of the world, vertical errors in high-resolution elevation products approach the pixel size (Gottwald et al., 2017; Rodríguez et al., 2006; Shean et al., 2016). These errors are compounded when computing gradients (i.e., slope and aspect) needed for terrain corrections. These errors are especially noticeable for sharp features such as ridgelines. Thus, a shade endmember without any terrain correction may produce the most accurate results for these locations.

Narrow-to-broadband albedo conversions confirm that the apparent albedo is being seen from space. As surface roughness increases to its maximum during melt, albedo falls rapidly. This period coincides with the time of year when snow becomes dirtiest on the surface, as the albedo is no longer being refreshed with new snowfall. Thus, the darkening effects of surface roughness occur simultaneously with the build-up of impurities (Betterton, 2001; Rhodes et al., 1987), which presents a challenge for remote sensing. However, because impurities only affect visible through near-infrared snow albedo, and snow grain size only affects albedo in the nIR/SWIR, while shadowing affects the entire broadband spectrum, an instrument with sufficient spectral resolution and accuracy should be able to discriminate between the causes of darkening.

To test this hypothesis, SPIReS was run on S2 imagery with dirty snow endmembers and with a clean snow assumption. The resulting grain size and impurity concentration estimates were then used in the updated broadband snow albedo that now accounts for shade. Because the pixel is close to fully snow covered, this estimated albedo should be comparable to the narrow-to-broadband conversions shown in Fig. 7. The uncorrected albedo measured at CUES from Fig. 7 is plotted along with these two model runs (Fig. 8). With overlapping error bars for each scene, the resulting albedos are indistinguishable within measured error (Bair et al., 2021b). In the clean-snow run, the dust endmember is swapped for the shade endmember (Table 2). In situ spectroscopic measurements (also in Table 2) provide some validation but also illustrate the wide spatial variability of the snow surface just across the CUES study area.

Importantly, the spectroscopic measurements show that, when used in a model, there is a consistent ability to discriminate between darkening caused by impurities and by shade. For example, despite the high spatial variability, neither the shade endmember nor the dust concentration is zero in any of the solutions. An example of dirty snow with a shaded solution is shown in Fig. 9.

Because the snow surface is rarely flat or level, shade needs to be accounted for, even when using measurements taken from a field spectrometer. Thus, shade needs to be included in snow albedo models, which often use lookup tables for rapid processing. Figure 10 shows the results of radiative transfer simulations to illustrate the effect of shade on the difference between intrinsic and apparent albedo.

There is a positive relationship: as fshade increases, the difference between intrinsic and apparent albedo increases, but the scatter also increases. A simple adjustment is not possible; instead the lookup tables and albedo model presented in Bair et al. (2019) have been updated to include fshade. The new albedo model estimates an apparent albedo as 10αapparent=frg,μ,Z,LAPname,δ,1-fshade, where αapparent is the apparent albedo over three wavelength ranges (broadband, near-infrared, and visible), rg is the grain radius in micrometers, μ is the cosine of the solar zenith angle, Z is the surface elevation in kilometers, LAPname is the type of light-absorbing particles (dust or soot), and δ is the LAP concentration. Other properties such as an assumed midlatitude winter atmosphere are unchanged from Bair et al. (2019).

A time series of intrinsic and apparent snow albedos over a season at a sub-alpine site were presented. In situ albedo measurements were compared to those from a spaceborne multispectral sensor. The multispectral measurements and those from a field spectrometer were used in a spectral mixture model. As expected and consistent with other studies, the results show that intrinsic albedo is consistently greater than apparent albedo. Both albedos decrease rapidly as ablation hollows form during melt, combining effects of build-up of impurities on the surface and increasing roughness.

There are several conclusions with implications for remote sensing, but also in situ measurement of snow albedo. For multispectral sensors, darkening effects from snow surface roughness are significant and can easily be confused with those from impurities. In contrast, measurements from a field spectrometer have sufficient spectral resolution and accuracy to distinguish between the two effects. A spectral mixture model run on spectra obtained at a study site confirms significant darkening at the snow surface, simultaneously occurring from roughness and impurities, with wide variation spatially. In turn, a spectral mixture model was used with Sentinel 2A/B multispectral imagery assuming a clean snowpack and a dirty snowpack. Both model runs were able to match measured snow albedo with plausible solutions, but the clean snow model used the shade endmember in place of the dust endmember.

The 0.056 difference between intrinsic and apparent albedo is equivalent to the decrease in broadband albedo caused by 63 ppm dust for typical snow in spring. If the surface topography is known to the point where a plane can be fit, the difference between the intrinsic and planar-corrected albedo (mean of 0.022) could be used instead, equivalent to darkening by around 22 ppm dust. Thus, to improve uncertainty in impurity estimates, a terrain correction used in conjunction with a shade endmember in a spectral mixture model can be used for moderate resolution sensors (e.g., 0.4–1 km), but caution is advised for terrain corrections at finer resolutions (≤ 30 m) owing to elevation model errors. Generally, impurity estimates from multispectral sensors are only distinguishable from surface roughness effects for relatively dirty snow. Likewise, for a multispectral sensor, mixed pixels can be spectrally inseparable from pixels containing only dirty snow. Thus, only pixels with high snow fraction should be used for impurity estimates from a multispectral sensor (Bair et al., 2021b; Painter et al., 2012a). These conclusions were also reached by Warren (2013), but for black carbon on the snow surface in the Arctic.

This study emphasizes the difficulties in modeling lighting conditions on the snow surface. Because of these difficulties, a recommendation is to always use a shade endmember in unmixing models, even for in situ spectroscopic measurements. Likewise, snow albedo models should produce apparent albedos by accounting for the shade fraction. To this end, lookup tables and code have been revised to account for shade. The apparent albedo produced should be used in energy balance models where intrinsic albedos have been previously used.

In this study, albedos were used rather than directional reflectance quantities. The justifications are the use of nadir-looking instruments with measurements taken midday; that as surface roughness increases, so does backscattering, thereby counteracting the forward scattering in snow; and that ablation hollows, the largest surface roughness features observed, have no preferred orientation, unlike sastrugi or penitentes. These factors reduce the importance of angular effects. But the most compelling justification is that for snow, the average or sub-pixel-scale snow surface topography is usually unknown, so the directional factors cannot be accurately computed.

Future work could focus on testing these findings in other snow climates with different surface roughness features, mainly formed by wind (Filhol and Sturm, 2015). The findings about discrimination between darkening from surface roughness and impurities as well as detection limits for impurities from multispectral sensors require further testing. For example, results from dirtier snowpacks should be examined, although the size of the ablation hollows will be reduced (Lliboutry, 1964; Rhodes et al., 1987). These findings highlight the need for hyperspectral measurements of snow from aerial and spaceborne sensors. The NASA Earth Observing-1 Hyperion was promising in this regard, but lack of coverage, repeat passes, or a surface reflectance product limited utility. The upcoming NASA Surface Biology and Geology (SBG) and ESA Copernicus Hyperspectral Imaging Mission for the Environment (CHIME) spaceborne spectrometers may offer chances to test these findings using spectroscopic measurements from space.