the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Effect of small-scale snow surface roughness on snow albedo and reflectance

### Terhikki Manninen

### Kati Anttila

### Emmihenna Jääskeläinen

### Aku Riihelä

### Jouni Peltoniemi

### Petri Räisänen

### Panu Lahtinen

### Niilo Siljamo

### Laura Thölix

### Outi Meinander

### Anna Kontu

### Hanne Suokanerva

### Roberta Pirazzini

### Juha Suomalainen

### Teemu Hakala

### Sanna Kaasalainen

### Harri Kaartinen

### Antero Kukko

### Olivier Hautecoeur

### Jean-Louis Roujean

The primary goal of this paper is to present a model of snow surface albedo accounting for small-scale surface roughness effects. The model is based on photon recollision probability, and it can be combined with existing bulk volume albedo models, such as Two-streAm Radiative TransfEr in Snow (TARTES). The model is fed with in situ measurements of surface roughness from plate profile and laser scanner data, and it is evaluated by comparing the computed albedos with observations. It provides closer results to empirical values than volume-scattering-based albedo simulations alone. The impact of surface roughness on albedo increases with the progress of the melting season and is larger for larger solar zenith angles. In absolute terms, small-scale surface roughness can decrease the total albedo by up to about 0.1. As regards the bidirectional reflectance factor (BRF), it is found that surface roughness increases backward scattering especially for large solar zenith angle values.

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The global energy budget is affected by surface albedo, which describes the level of brightness of the surface. Due to its central role for climate, it has been defined as an essential climate variable (ECV) by the GCOS Secretariat (2006). The large areal coverage of seasonal snow, together with the high reflectivity of snow, contributes to the relevance of snow albedo on the global energy budget (Flanner et al., 2011; Mialon et al., 2005). The snow component is also important for the liveability of dry and cold areas for both humans and ecosystems by providing a source of meltwater in spring and shelter and insulation in winter. Changes in the duration of snow cover and snow type are vital for people and the ecology of these areas. Accurate large-scale monitoring of snow properties over large areas is only feasible in practice using satellite-data-based methods. Prior to that, it is required to obtain a detailed understanding of the reflectivity and scattering properties of snow.

The surface reflectivity of snow depends on grain size, shape and impurity content, which are the basic properties for handling the volume scattering of snow. Traditionally snow grain size is characterized by its largest diameter, whereas it has been demonstrated that the specific surface area (SSA) is a more appropriate variable to describe the scattering area per volume (Domine et al., 2012; Leppänen et al., 2015). Light attenuation within the snowpack is related to the density of the scattering elements per unit volume. In addition, layer structure, grain shape, anthropogenic and natural impurities (such as black carbon, dust and algae), and close-packing effects of snow grains affect scattering properties and thus the albedo of a snowpack (Warren and Wiscombe, 1980; Kokhanovsky and Zege, 2004; Aoki et al., 2011; Kokhanovsky, 2013; Libois et al., 2013; Libois et al., 2014; Komuro and Suzuki, 2015; Peltoniemi et al., 2015; Pirazzini et al., 2015; Räisänen et al., 2015; Cook et al., 2017; He et al., 2017, Kokhanovsky et al., 2018). Several models for the coupled mass and energy balances of snow on the ground have also been developed (Flanner and Zender, 2006; Essery, 2015). The decrease in snow albedo due to shadowing effects of larger-scale topography (Picard et al., 2020) and surface features such as sastrugi and crevasses have also been investigated from the point of view of measurement and modelling (Leroux and Fily, 1998; Warren et al., 1998; Zhuravleva and Kokhanovsky, 2011; Lhermitte et al., 2014). But smaller-scale (mm to 10 cm) surface roughness has so far received poor attention in snow albedo modelling. Very recently, a study was published about artificially generated surface roughness in the centimetre scale (Larue et al., 2020).

Snow grain size can also be related to micro-scale surface roughness. Initially snow surface is formed by falling snowflakes, which attach to the surface at first contact instead of being arranged according to the positions of minimum energy (Löwe et al., 2007). Surface crystals are rearranged and shaped by the winds near the surface through saltation, which is the transport of snow in periodic contact with and directly above the snow surface. This process is governed by both the atmospheric shear forces and the moving snow particles (Pomeroy and Gray, 1990). The wind both breaks the particles into smaller pieces and helps the grains grow mass from the air moisture (Armstrong and Brun, 2008). These atmosphere–surface interactions create some links between local small-scale surface roughness and the grain size properties of the topmost layers in the snowpack. Moreover, the physical processes governing the snow grain metamorphism (temperature gradient, absorption of solar radiation, water vapour diffusion, liquid water formation) also affect the stickiness and, thus, the aggregation of grains (Löwe et al., 2007), which is associated with the formation of millimetre- to centimetre-scale surface roughness.

If the surface were completely isotropic, the surface albedo might in many cases be well explained using only the grain size as a descriptor of the snowpack of sufficient thickness to be semi-infinite from the scattering point of view. But typically, the surface structure slopes and snow properties influenced by wind are not identical in the windward and leeward sides (Sommer et al., 2018). This means that in clear-sky conditions the albedo will not necessarily be the same for azimuthally opposite viewing directions, when the saltation effect is marked. In addition, hoar frost formation depends more on the air temperature and humidity than the grain size of the existing snowpack. All in all, despite the dominant character of the snow grain size to the scattering from a snowpack, the small-scale surface roughness has also a role independent of the snow grain size that should be paid attention to. This study focuses on the effect of surface roughness on snow albedo.

Here, a method taking into account the small-scale surface roughness in addition to the normal bulk volume scattering is developed for the black-sky (directional–hemispherical reflectance, DHR), white-sky (bihemispherical reflectance, BHR, in isotropic diffuse illumination) and blue-sky albedo (bihemispherical reflectance, BHR, in ambient illumination) (Lucht et al., 2000; Schaepman-Strub et al., 2006). The main points of the model are described in Sect. 3.2, and detailed equations are derived in Appendix A. The Two-streAm Radiative TransfEr in Snow (TARTES) snow model is used to simulate the albedo of a smooth snowpack (Warren, 1984; Warren and Brandt, 2008; Kokhanovsky and Zege, 2004; Baldridge et al., 2009; Libois et al., 2013; Libois et al., 2014; Picard et al., 2016).

The rough snowpack albedo model is tested with measurements carried out during the Snow Reflectance Transition Experiment (SNORTEX) campaign (Roujean et al., 2010; Manninen and Roujean, 2014) in Sodankylä, Finnish Lapland, in March–April 2009 and in March 2010 augmented with operational albedo measurements that the Finnish Meteorological Institute (FMI) carries out in the Arctic Space Centre of FMI in Tähtelä, Sodankylä. The physical properties of snow during the campaign were measured from snow pit profiles. The modelled albedo is compared with measured albedo values in diffuse and clear-sky cases. The diverse snow measurements are briefly described in Sect. 2, and more details are available in the given references. The high-resolution surface roughness profiles obtained using a scaled plate (Sect. 2.1) were also analysed with ray tracing calculations to obtain the directional scattering characteristics related to the small-scale surface roughness. The bidirectional reflectance factor (BRF) thus obtained was compared to empirical BRFs provided by Finnish Geodetic Institute Field Goniospectrometer (FIGIFIGO) measurements (Peltoniemi et al., 2005, 2015, 2014; Sect. 2.8). The varying role of the small-scale roughness from midwinter conditions throughout the melting season is demonstrated in Sect. 4.

## 2.1 Test area

Diverse properties of snow were measured in Sodankylä, northern Finland, in March and April 2009 and in March 2010 in an area of about 10 km × 10 km
(Fig. 1, Manninen and Roujean, 2014). Every day,
measurements were made at about half a dozen test sites in one land cover
type (either forest or open areas, with the latter being typically aapa mire).
The last (first) test site of the day was in 2009 (2010) in the NorSEN mast
area (67.3621^{∘} N, 26.63445^{∘} E), which is located in
similar terrain about 550 m from the place, where FMI conducts operational
surface albedo measurements (67.36664^{∘} N, 26.628253^{∘} E
downward; 67.36695^{∘} N, 26.62973^{∘} E reflected). Hence,
the operational albedo values should be representative for the time series
of the snow pit measurements at the NorSEN mast.

## 2.2 Grain size and density profiles of snowpack

Measurements of snow depth, total density, water equivalent (SWE), humidity
profile, temperature profile, grain size profile, surface roughness and
surface impurity content were carried out at snow pits located in
Sodankylä in an area with corner coordinates (67.36^{∘} N,
26.63^{∘} E; 67.45^{∘} N, 26.86^{∘} E) in March and
April 2009 and in March 2010 (Manninen and Roujean, 2014). In addition, crystal size photos of the snow layers, surface roughness photos and photos
of the top surface impurities were taken. In this study we concentrate on
the values measured in 2009. The air temperature in March was mostly below
0 ^{∘}C, whereas in April it was above
0 ^{∘}C almost all the time (Table 1). Hence, April represents
the melting season and March is still midwinter. This is also clear from the
increase in median density of the snowpack and the decrease in median snow
water equivalent value from March to April. About 40 snow pit measurement
points were located in a larger area (67.42^{∘} N, 26.04^{∘} E; 67.85^{∘} N, 26.91^{∘} E), where the maximum measured snow
depth was 0.92 m in March and 0.76 m in April. The total density varied in
the range 180–320 kg m^{−3} in March and in the range 270–570 kg m^{−3} in April. The corresponding variation ranges for the snow water
equivalent were 0.020–0.250 and 0.034–0.239 m, but in April there
was plain water in several places in the snowpack. Hence, the area covered
by the snow pit measurements represents the local variation of snow
properties to a large extent.

The traditional snow grain size (the largest dimension of the snow grains, Fierz et al., 2009) was visually estimated using graded plates, collecting the snow crystals from 10 cm thick snow layers from the bottom to the top of the snowpack. For each analysed sample of snow crystals, in addition to the typical value of the largest grain dimension also its minimum and maximum value were provided. The measured snow grain sizes differ from the optically equivalent snow grain size (Mätzler, 1997; Neshyba et al., 2003). To partly compensate for this, the minima of the largest grain diameter were applied in the radiative transfer calculations as the effective diameter. Although this causes some uncertainty in the interpretation of the computed absolute albedo values (particularly for the cases of fresh snow), it has much less impact on the derived effect of small-scale surface roughness on snow albedo. The density profile of the snowpack was measured for the same layer structure using the snow fork (Toikka, 1992). The variation range of the grain size and density is shown for the surface layer and for the whole snowpack in Table 2.

## 2.3 Surface roughness from plate measurements

The surface roughness up to 1 m scale was measured in March and April 2009 and in March 2010 by taking photos of a graded plate placed perpendicularly in the snowpack (Fig. 2). The snow surface profiles were automatically calculated from the photos using an image processing technique and the scale at the edge of the plate (Manninen et al., 2012). Control points at the scales were used both for the removal of the barrel distortion of the camera optics and transformation of the pixel coordinates to millimetres with photogrammetric methods. The plate surface roughness measurements were carried out at the same sites as the snow pit measurements (Sect. 2.2). At each site, profiles were measured in two perpendicular directions with a 1 m interval along 50 to 100 m distance.

The surface profiles were used to derive the root mean square (rms) height
and correlation and their distance dependence (Keller et al., 1987; Church,
1988). Details of the multiscale roughness theory are described by Manninen
(2003), and its application to the snow profiles is presented by Anttila et al. (2014). The snow surface roughness is close to a Brownian fractal
surface (Anttila et al., 2014) so that the logarithm of the rms height
*σ* depends linearly on the logarithm of the length *x* of the analysed
profile used for its calculation, and the corresponding correlation length
*L* is linearly related to *x*:

where *a*, *b*, *k*_{0} and *k* are constants, and *k*_{0}=0 for an ideal Brownian
surface (Russ, 1994). For each profile the values of the constants were
calculated by linear regression using varying sliding window sizes, i.e.
varying values of *x* (Anttila et al., 2014).

In addition, the rms slope angles *β*, i.e. arcus tangent of the slopes
$(\mathit{\beta}=\mathrm{arctan}(\mathrm{\Delta}z/\mathrm{\Delta}x\left)\right)$, were calculated for the measured
spatial resolution, which was on average 0.26 mm. The vertical precision
was about 0.1 mm and the horizontal precision 0.04 mm (Manninen et al.,
2012).

## 2.4 Surface roughness from laser scanning

In addition to the plate measurements, laser scanning data for snow roughness were utilized. The laser scanning data used in this study have been acquired using the FGI ROAMER system (Kukko et al., 2007). The system, including a FARO Photon 120 laser scanner, a NovAtel SPAN GPS–IMU system, and data synchronizing and recording devices, was mounted on a sledge, which was towed by a snowmobile. The data acquisition covers a 2.5 km zone at each side of an official snowmobile track (see Kukko et al., 2013, for examples of profiles, the snowmobile track and other details). The landscape covered sparse pine forests and open bogs. The absolute precision of these measurements was analysed by Kaasalainen et al. (2011) to be better than 5 cm, while the relative accuracy (which is more relevant for observing the snow roughness) was found to be 0.7–2 mm for a static system and better than 10 mm when the snowmobile was moving. The best repeatability was achieved at ranges closer to the scanning system, i.e. below 5 m. The data quality and precision were controlled using control points measured with a virtual reference station (VRS) precision GPS (Leica SR530 receiver + AT502 antenna).

The laser profiles (about 16 profiles per 1 m at 3 m s^{−1}
snowmobile velocity)
measured on 18 March 2010 were used to analyse the variation of the slope
angles in a larger area than was possible using the plate profiles. The
profiles covered an area that was 2.4 km long, and the width extended into
3.2 m at both sides of the snowmobile. The slope angles for successive
points were determined for each scan of the whole data set. The slope angles
were then binned according to the horizontal distance between the successive
points, with a bin width of 10^{−5} m. Then the root-mean-square value of
the slope angles was determined for each horizontal distance bin, and a regression function for the dependence of slope angles on distance between
successive points was derived.

## 2.5 Snow impurity content

The snow impurity was measured by filtering a melted sample of snow. The quartz filters were analysed using the NIOSH 5040 protocol. The increase in the median amount of impurities from March to April is obvious from Table 1 (Meinander et al., 2013; Meinander et al., 2014; Meinander et al., 2020). The detection limit of the thermal–optical OCEC method is 0.2 µgC, and the uncertainty of the OCEC is estimated to be ± 0.2 µgC (±5 % relative error for higher loaded samples). The relative portion (±5 %) is composed of the instrument variation and slight variations due to sample deposit inhomogeneity and sample handling, as we recently discussed more in detail in Meinander et al. (2020).

## 2.6 Surface albedo

The surface albedo was operationally measured at Sodankylä
(67.36664^{∘} N, 26.628253^{∘} E downward; 67.36695^{∘} N, 26.62973^{∘} E reflected) with a 1 min interval using Kipp
& Zonen CM11 pyranometers. The site is surrounded by trees and houses, so
that shadowing takes place in certain azimuth directions, when the solar
elevation is very low. Hence, the measured white-sky albedo values are
considered more reliable than the blue-sky values. The least shadowed
azimuth direction in early March corresponded to the solar zenith angle
value of 73^{∘} in the afternoon. Thus, the blue-sky albedo values
used in the analysis were all taken from the afternoon, when the solar
zenith angle equalled 73^{∘}. This means that the azimuth direction
used increased a bit during the spring, but it did not cause any additional
shadowing problem. Yet, the clear-sky albedo of 12 March was replaced with
the diffuse albedo dominating that day, because the clear-sky albedo value
of a narrow time window seemed unrealistically small. Albedo values measured
at the NorSEN mast on 21 April 2009 using a portable Kipp & Zonen CM14
albedometer were used to calibrate the operationally measured albedo data in
order to correct for the slight difference in location of the upward- and
downward-looking pyranometer used for operational measurements.

The portable albedometer was used in April 2009 to measure the snow surface albedo in the same areas where the snow pits were made (Sect. 2.2). The instrument was installed on a short boom affixed on a lightweight camera tripod for easy transport. The tripod legs affect somewhat the reflected radiation measurements, and therefore a first-order correction, multiplication by 1.055, was applied. It was based on estimation of the solid angle blocked by the tripod legs from a fisheye lens photograph with a camera mounted onto the albedometer position on the tripod, assuming a constant albedo of 0.1 for the dark carbon fibre legs. The albedometer was calibrated against a reference pyranometer at FMI in Helsinki prior to each campaign. The albedometer was carefully levelled on the tripod before measurements at each location and the stability of levelling monitored regularly, as melting snow may become unstable during the day.

## 2.7 Spectral reflectance

The spectral reflectance of snow was measured using the ASD FieldSpec Pro JR
spectrometer on several days, specifically in the perfectly overcast
conditions on 13 March and on the perfectly clear-sky day of 22 April, during
the campaign in 2009. The irradiance spectra were measured as well. Every
spectrum is an average of 30 individual spectra. The spectrometer was
calibrated by the manufacturer prior to the campaigns. The instrument was
powered on at least 15–20 min before each measurement to ensure an even
operating temperature. A Spectralon panel (0.125 × 0.125 m^{2}) was used
as white reference for the reflectance measurements. The Spectralon panel was
housed in a container with two orthogonal spirit levels, placed on the snow
and levelled. Narrow-view foreoptics were used to ensure that the field-of-view (FOV) fits
fully onto the Spectralon panel. This was further visually confirmed by looking
through the foreoptic before inserting the fibre optic cable. The white
reference was measured, and then the tripod was carefully rotated so that the
foreoptic pointed into pristine snow. Tripod leg shadowing on the measured
area was carefully avoided for both white reference and snow measurements.
Most measurements took place from a height of 0.5–0.6 m with an 8^{∘}
foreoptic. The spectrometer was optimized before each measurement.

## 2.8 BRF

The bidirectional reflectance factor BRF of snow was measured using the
Finnish Geodetic Institute's Field Goniospectrometer (FIGIFIGO; Peltoniemi et
al., 2005, 2015, 2014). FIGIFIGO consists of a motorized arm of length of 2 m, moving the optics head ±90^{∘} around nadir, and an ASD Field Spec
PRO FR spectrometer recording the spectrum in the range of 350–2400 nm.
The azimuth is turned manually, and all angles and coordinates are recorded
automatically, based on inclination, direction and position sensors. The
footprint is around 0.10 m in diameter. FIGIFIGO gives spectrally resolved
BRF data, relative to Spectralon reference standard (of the size of 0.25 × 0.25 m^{2}, connected to a screw-adjustable mount and levelled with a
bubble level), from which also spectral albedo can be evaluated by fitting a
polynomial function and integrating over the hemisphere. However, as the
system is not absolutely calibrated in the field setup, external solar
spectrum is needed for deriving real broadband albedos and BRF. In the
results shown, a mean solar spectrum is used that may differ several percent
from the real-time one.

In Mantovaaranaapa, three sets of rough snow were measured, as well as one set of smoother snow formed by a thin layer of windblown grains. Another set of thin and rough snow was measured in Korppiaapa, but this was not used in the present study. The sunlight measurements were complemented by set of artificial light measurements of smoother snow near the NorSEN mast.

## 3.1 Smooth snowpack albedo modelling

The TARTES model (available at: https://snowtartes.pythonanywhere.com/, last access: 8 March 2020) was used to estimate the snowpack white-sky and black-sky albedo values (Warren, 1984; Warren and Brandt, 2008; Kokhanovsky and Zege, 2004; Baldridge et al., 2009; Libois et al., 2013, 2014; Picard et al., 2016). It is a fast and easy-to-use optical radiative transfer model and represents the snowpack as a stack of horizontal homogeneous layers. Each layer is characterized by the snow grain size, snow density, impurities amount and type, and two parameters for the geometric grain shape: the asymmetry factor and the absorption enhancement parameter. The albedo of the bottom interface can be prescribed (here 0.13), although the bottom interface only markedly impacts thin snowpacks (< 5 cm depth). The model is based on the Kokhanovsky and Zege (2004) formalism. The required input values for the model (density and grain size profile) were provided by the snow pit measurements of the SNORTEX campaign (Manninen and Roujean, 2014; Sect. 2.2). The amount of impurities was temporally interpolated from the values of measured days. The black-sky albedo values of the bulk snowpack were derived by weighting the spectral albedo with the standard top-of-atmosphere (TOA) spectrum ASTMG173. The white-sky albedo values of the bulk snowpack were derived by weighting the spectral albedo with measured diffuse irradiance spectra of the cloudy day 13 March 2009.

The black-sky albedo values were calculated for three local incidence angle values of each plate surface roughness profile: the mean, the mean minus 1 standard deviation, and the mean plus 1 standard deviation of the individual local incidence angle values determined for each slope of the surface roughness profiles. The nominal incidence angle was set to the solar zenith angle value at the time of the measurements of the plate surface roughness profiles and the density and grain size values of the snowpack layers. The blue-sky albedo values were obtained from the black-sky and white-sky albedo values using the fraction of diffuse irradiance operationally measured at Tähtelä.

## 3.2 Rough snowpack albedo modelling

From the theoretical point of view there is a difference in scattering from a snowpack having an ideally planar surface and a rough surface, because the rough surface may have an incidence angle distribution that markedly differs from the Gaussian distribution of incidence angles produced by a random volume of spherical scatterers partly shading each other. In addition, the roughness may cause a markedly higher amount of multiple scattering, thus reducing the amount of radiation escaping the target.

Scattering from randomly rough continuous surfaces is related to the characteristic size of the surface roughness with respect to the wavelength used (Beckmann and Spizzicchino, 1963; Ulaby et al., 1982; Tsang et al., 1985; Fung, 1994). When the surface roughness of a randomly rough continuous surface is large compared to the wavelength of the electromagnetic wave, the scattering of the wave from the surface can be approximated by scattering from random facets (i.e. using the Kirchhoff approximation), whose slopes determine the scattering directions. As the shortwave illumination covers the wavelength range of about 300–2500 nm, all structures in the millimetre scale (or above) are large compared to the wavelength, so that a facet-based surface scattering calculation is reasonable. Each facet is taken to represent a volume of random scatterers, and the local incidence angle of the incoming radiation is the angle between the normal of the facet and the solar zenith angle (Fig. 3). The surface of a snowpack is not a continuous solid surface, but when the snowpack surface is rough, the incidence angle distributions of the scattering elements may deviate from that of a planar surface with randomly oriented scatterers. In addition, it is possible that a photon escaping one facet hits another facet. The snowpack scattering can then be thought to have elements both of bulk volume scattering and surface scattering. The following 2D analysis demonstrates this idea.

Multiple scattering between facets can be taken into account using the
photon recollision probability theory (Knyazikhin et al., 1998; Panferov et
al., 2001; Smolander and Stenberg, 2005; Rautiainen and Stenberg, 2005;
Stenberg et al., 2008; Stenberg and Manninen, 2015; Stenberg et al., 2016).
The formulation is shown in Appendix A separately for diffuse and direct
irradiance. The essential equations are repeated here. Firstly, for the
diffuse case, the white-sky albedo *α*_{w} (Lucht et al., 2000,
Schaepman-Strub et al., 2006) is related to the average number of
facet-to-facet scattering events $<n>$:

where *α*_{w0} is the white-sky albedo of the bulk volume.

Second, for direct illumination, the black-sky albedo *α*_{b}(*θ*_{i}) (Lucht et al., 2000, Schaepman-Strub et al., 2006)
is approximately related to the *α*_{w0}, $<n>$ and the average number < *m*(*θ*_{i})> of
facet-to-facet scattering events in direct illumination conditions by

where *α*_{b0}(*θ*_{i}) is the black-sky albedo of the
bulk part of the snowpack (i.e. the albedo without the surface roughness
contribution). In this study the bulk albedo values *α*_{w0}
are produced using the TARTES model (Sect. 3.1).

The albedo *α* in mixed illumination conditions is typically estimated
using the weighted mean approximation of the two extreme values *α*_{w} and *α*_{b} (Lucht et al., 2000; Schaepman-Strub et al.,
2006; Román et al., 2010):

where *f* is the fraction of diffuse irradiance. According to Eqs. (3)–(5) the
blue-sky albedo is then estimated from

Obviously, surface roughness decreases the white-sky albedo and typically also the black-sky and blue-sky albedo. Only when $<m>$ is larger than $<n>$, surface roughness can increase the black-sky albedo of bright targets. The effect of surface roughness is non-negligible even when the roughness is not large. On the other hand, the larger the roughness is (i.e. the larger $<n>$ and $<m>$ are), the larger the effect is for darker targets. Hence, for snow the effect is larger in the near-infrared than in the visible wavelengths and in midwinter roughness alters the broadband albedo only slightly, whereas during the melting season, when the snow is darker, the effect of the roughness may be much larger. Thus, the effect of surface roughness would be larger for old snow than for new snow, which may explain part of the quick darkening of snow during the melting season.

## 3.3 Ray tracing analysis of surface roughness

Scattering from the snow profiles measured using the plate was analysed by a
ray tracing method using 1000 equally spaced rays per profile per direction.
The number of hits *n*_{s} on the surface (unity for single reflection and
larger for multiple surface scattering) and the direction of the escaping
reflected ray were calculated as a function of the zenith angle of the
incoming ray with an interval of 2^{∘} from 0 to
80^{∘}. Scattering from the surfaces was calculated by assuming
mirror reflection from smooth facets of continuous material, which has often
been also assumed in the computation of ice crystal single-scattering
properties in the solar spectral region (Nousiainen and McFarquhar, 2004;
Zhang et al., 2004). This approximation neglects the impact on scattering
due to snow structures smaller than the measurement resolution
(∼ 0.1 mm for plate measurements). For each angle the case was
calculated separately for rays coming from the left and from the right, and
the two results were unified to improve statistics. This choice was
motivated by the known fact that, even when the snow is highly forward
scattering, it is with respect to the direction of the incoming solar
radiation, not the wind direction. Thus, the dominant scattering direction
moves with the sun during the day. In some cases, the ray was trapped to
infinite reflection from facet to facet. But these quite rare (< 1 %) cases were excluded from the statistical analysis. Since the surface
roughness profiles produce only 2D information and scattering angles differ
markedly in 2D and 3D, the calculated ray-tracing-based 2D BRFs were
converted to 3D versions assuming that each facet has besides the measured
vertical angle also an azimuth angle obeying a random uniform distribution
between 0 and 180^{∘}. In fact, calculations were made for the range
0–90^{∘}, assuming the case to be symmetrical with respect to
azimuth angle, like in constructing the FIGIFIGO-based BRFs. The 3D
conversions make the peaks of the 2D scattering angle distributions slightly
less distinct. No atmospheric contribution was included in either data set.

Late in spring the scattering from a snowpack often also contains a component that is something between volume and surface scattering, namely deep narrow pits generated by impurities that have sunk downwards in the snowpack due to melting caused by absorption of solar radiation. This kind of an effect is not easy to take into account either in volume scattering or surface scattering, because they do not affect the roughness or density in a random way. Their contribution to albedo is beyond the scope of this study.

## 4.1 Surface roughness and inputs for albedo modelling

To start with, we consider the snow surface roughness from the plate
measurements. The rms slope angle calculated from the plate roughness
measurements had an increasing trend from March to April
(Fig. 4). The surface height distributions
developed towards a more Gaussian distribution from March (*R*^{2}=0.97)
to April (*R*^{2}=0.99). However, it should be noted that individual
profiles could deviate markedly from Gaussianity, as evidenced by the ratio
of skewness to standard deviation. The 90 % quantile of this ratio for
individual profiles was 0.36 in March and 0.13 in April. The ratio was not
negligible for the monthly average distributions either (0.17 in March and
−0.04 in April). Furthermore, the autocorrelation functions were in most
cases not Gaussian (Anttila et al., 2014). The most common (41 %)
autocorrelation function (ACF) type was multiscale exponential, and 66 %
of the profiles had multiscale ACF (Anttila et al., 2014).

The mean number of individual reflections *n*_{s} per ray on the surface has
an increasing trend during the melting season (Fig. 5). It correlates well (${R}^{\mathrm{2}}=\mathrm{0}.$83) with the rms slope angle
(*β*, in radians) of the profile. A good general fit to all measured
plate profile data was

where *θ*_{i} is the solar zenith angle. The mean zenith angle
*θ*_{o} to which the radiation escapes from the surface (when mirror
reflection from the facet is assumed) is even more strongly correlated
(*R*^{2}=0.93) with *β* (Fig. 6):

All angles (*β*, *θ*_{i}, *θ*_{o}) are given in radians
in the above equations, and ${\mathit{\theta}}_{o}>\mathrm{0}({\mathit{\theta}}_{o}<\mathrm{0}$) for forward-scattering (backward scattering). Obviously, the
probability for backward scattering increases with increasing incidence
angle and increasing *β*. Since *β* increases with time, also the
probability for stronger backscattering from the snow cover increases with
time. Indeed, it is well known that older snow cover is less strongly
forward scattering than new midwinter snow (Peltoniemi et al., 2010).

Unfortunately, the *β* values depend strongly on the scale of the
measurements. The laser scanning data are well suited to demonstrate this,
because the horizontal distance between successive data points increases
from the beginning to the end of the scan line. On 18 March 2010 plate
profile measurements and laser scanning were carried out in the same
relatively flat wetland area of Mantovaaranaapa (67.4^{∘} N,
26.7^{∘} E). The laser scanning data covered a 2.4 km long and about
3.2 m wide area on each side of the snowmobile route. Altogether 10 plate
profile measurements were taken directly after the scanning, at about
100–200 m intervals starting from the western edge of the scan route (Kukko
et al., 2013). The average rms slope angle of the 10 plate profiles was
30.7^{∘} (*β*=0.54) with an 80 % variation range of
24.7– 34.3^{∘} (*β*=0.43–0.60). Consequently,
*n*_{s} would then vary in the range 1–1.5 (Fig. 5). The rms slope angles were calculated also from the laser scanning
profiles as a function of horizontal increments, which were within the range
5–100 mm. The number of points per distance varied between 4 thousand
and 2.2 million. Nonlinear regression to the 36 points in the range 5–100 mm produced an exponential curve that approaches the mean rms slope
angle value obtained from the 10 plate profiles of the same area and
Fig. 7. Shorter increments of the laser data
could not be reliably used in the analysis.

As the laser scanner covered a much larger area than the plate profiles,
that data give an estimate of the rms slope angle variation in a larger area and are based on a larger number of individual slope angle values. For
the laser data set, the median difference between the 90 % quantile curve
of the slope angle values and the rms slope angle value curve was
6.1^{∘}. The corresponding median difference for the 10 %
quantile curve from the rms slope angle curve was −5.9^{∘}. For the
plate profiles, 90 % and 10 % quantile values of the rms slope angle
differed from the mean rms slope angle value by 3.6 and
−6.0^{∘}. Thus, the larger area covered by the laser scanner shows a
larger variation of the rms slope angle, as one could expect. Obviously, the
laser scanner data can be extrapolated to estimate *β* at higher
horizontal resolution than the measurements directly enable, but then
*β* has to be analysed as a function of the horizontal distance
increment (Fig. 7). However, the strong variation
of *β* with spatial resolution suggests that using less scale-dependent
surface roughness descriptors would be desirable, if they just can provide
the information needed.

The relationship between other surface roughness parameters (such as rms
height *σ* and correlation length *L*) and *β* is in general not
strong, even for a Gaussian surface height distribution (Beckmann and
Spizzicchino, 1963). For the whole period (3 March–28 April 2009) the
ratio of $\mathit{\sigma}/L$ (determined for 0.60 m distance) correlated relatively well with *β*, with the *R*^{2} values being 0.70, 0.62 and 0.67
for the whole data range, March and April, respectively, but the best
descriptor of *β* was found to be $\mathit{\sigma}/b$ (Eq. 1),
with its *R*^{2} values for the linear correlation being 0.78, 0.68 and 0.82 for
the whole data range, March and April, respectively (data shown in
the Supplement). *β* tends to increase with the progress of the
melting season (0.002 rad per day). Likewise, its correlation with
$\mathit{\sigma}/b$ increases during the melting season. It was therefore examined
whether the measured surface albedo correlates well with the measured
surface roughness parameters. Using just the rms height (derived for a 0.60 m horizontal scale) as an explanatory variable of the albedo, the coefficient
of determination was *R*^{2}=0.81 (data shown in Supplement).
The relationship between the albedo and surface roughness parameters that
are scale-independent in a large range (Manninen, 2003) was then evaluated.
Indeed, a simple linear regression for the data of March and April 2009
produced a coefficient of determination value as high as *R*^{2}=0.90,
when the parameters *b* and *k*_{0} (see Eqs. 1 and 2) were used as explanatory
variables (Fig. 8). While correlation is not a
proof of causality, this result supports the view that surface roughness
affects the albedo.

## 4.2 Snow albedo spectra: measured vs. modelled

Two examples of snow nadir reflectance spectra measured with the ASD spectroradiometer are shown in Fig. 9. On 13 March, the sky was completely overcast, whereas on 22 April it was perfectly clear. For comparison, corresponding albedo spectra modelled using TARTES are shown. The ASD reflectance spectra were scaled so that the derived broadband reflectance value matched the calibrated operationally measured broadband albedo value. The scaling factor was 0.994 for the diffuse case of 13 March and 0.937 for the clear-sky case of 22 April. No BRF was available for the clear-sky case for the location of ASD, but for old rough snow in the area it was relatively flat (see Sect. 4.5), so that the comparison of the spectral reflectance and albedo values seems reasonable enough to enable the choice of the grain shape to be used in the TARTES model calculations. The spectra modelled with TARTES accounted for the empirical grain size and density values, as well as for black carbon but not for organic carbon (Table 1), which included needles and various tree trash deposited on snow. In March, the impurity content in surface snow was very low, while in April it was roughly 3 times higher (Table 1). In March a better fit is obtained using fractal grains, the result of which is also supported by photos taken of snow grains and the fact that the snow was fresh. In April the modelled albedo favoured the use of spherical grains rather than fractals in the calculations. The photos taken about the snow grains also supported the use of spherical grains in April. Thus, the TARTES results seem to represent the snowpack in March well, but in April when the melting has been going on for a longer time the modelled albedo values are higher than the empirical reflectance values both in the visible and near-infrared wavelengths (less than ∼ 1 µm), which dominate the value of the broadband albedo of snow. The grain size estimation uncertainty was not significant in April, because the grains were already very rounded, but in March the definition and estimation of the grain size was challenging. However, even if the actual grain size for fractal grains was as much as 1 mm larger than estimated, this would not in all cases provide the measured albedo value using the TARTES model without contribution of surface roughness. As the median grain size of the top layer was in March 0.5 mm and the corresponding maximum value was 1. 5 mm (Table 2), it is in practice highly improbable to make an error of 1 mm or more using a graded plate with 1 mm scale.

## 4.3 Broadband albedo: measured vs. modelled

The evolution of the operationally measured broadband albedo in Tähtelä is shown for the periods 12–19 March and 21–28 April in Fig. 10 together with the corresponding albedo values simulated using the TARTES model assuming a flat surface and the grain size and density measurements of the day in the Sodankylä area. Following the justifications outlined above and on the basis of photos taken in every test site, fractal grain shapes were used in March and spheres in April. The simulated values tend to exceed the measured ones, especially in April. In March the grain size estimation was difficult, because of the small dimensions of the very complex grain shapes. Hence, some but not all of the difference between the measured and modelled results may be explained by that. On the contrary, in April the grains were already very large and rounded, so that the difference between the modelled and measured values should not come from grain size uncertainty. Besides, the variation range of simulated albedo values is rather small compared to that of the empirical broadband reflectance values. However, it will be demonstrated next that taking into account the surface roughness decreases the difference between simulated and empirical albedo estimates.

The albedo model taking into account both volume and surface scattering
(Eqs. 3–5) was applied so that *α*_{w0} was the value
provided by the TARTES model based on the measured density and grain size
profile and impurity content. The values for *n* and *m* were derived from the
empirical values of *n*_{s}. Namely, $m={n}_{\mathrm{s}}-\mathrm{1}$ for the local solar
zenith (*θ*_{il}) angle range derived using the ray tracing method.
And *n* is the weighted mean of *n*_{s}−1, where the weights are cos(*θ*_{il})⋅sin(*θ*_{il}). The results are shown in
Figs. 11 and 12.

First, the ratio ${\mathit{\alpha}}_{\mathrm{w}}/{\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}$ of the total white-sky
albedo and the bulk white-sky albedo provided by the TARTES model is
considered in Fig. 11. In March, this ratio varies
mainly between 0.97 and 0.99, indicating that small-scale surface roughness
decreases the snow albedo typically by 1 %–3 %. With the progress of snowmelt, the effect of surface roughness increases markedly. On 26–27 April
(Julian days 116–117), the median of ${\mathit{\alpha}}_{\mathrm{w}}/{\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}$
falls below 0.9, indicating an over 10 % decrease in snow albedo. The
relative difference between the total and bulk albedo values is about the
same for the black-sky case as for the white-sky case, but the solar zenith
angle naturally slightly complicates that relationship (see Eqs. 3 and 4).
The larger variation of the ${\mathit{\alpha}}_{\mathrm{w}}/{\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}$ values in
the latter part of April (after Julian day 112) is related to vigorous
melting of the snowpack, since then the measured temperatures were about
0 ^{∘}C throughout the snowpack (Manninen and Roujean, 2014).

The modelled albedo values are further compared to observations in Fig. 12. The variation range of the simulations shown with the background shading is based on the variation of the grain size, density and local incidence angle of the measured profiles. Overall, the inclusion of surface roughness significantly improves the agreement of modelled albedos with the observed ones. A notable overestimation remains, however, on Julian days 77–78 and will be discussed in Sect. 5. All in all, the model is robust enough to be reasonably applied to empirical data of grain size, density and surface roughness.

Obviously, taking into account the surface roughness contribution improves the match of empirical and modelled results, but still it is very clear that the grain size, grain shape and SWE (or density) are dominant parameters, because the amount of volume scattering also affects directly the amount of surface scattering. The variation range of the modelled albedo is much larger for clear-sky cases than diffuse cases, which is understandable as some of the variation in clear-sky cases comes from the solar zenith angle variation during the day.

## 4.4 Albedo during snow metamorphosis on 22 April

The modelled albedo results were compared with the Kipp & Zonen CM14
albedometer measurements carried out in Mantovaaranaapa on 22 April 2009,
which was a perfectly clear day (Fig. 13). One
snow pit was measured during 09:00–10:00 UTC at 67.40735^{∘} N,
26.72357^{∘} E. The albedometer was positioned in its vicinity and it
recorded the metamorphosis process shown by the linear (*R*^{2}=0.998)
decrease in the albedo with time. Surface roughness of 54 individual profiles was retrieved with the plate method with typically 10 m
incremental distance in perpendicular directions covering an area of about
100 m × 100 m (Fig. 2). Each position of the plate
was photographed three times, so that 18 separate profiles were
characterized. Hence, the modelled albedo results were averaged to get one
value per surface sample. The light grey band in
Fig. 13 shows the variation range of the
broadband-converted reflectance spectra measured with the ASD spectrometer
in the same area at the same time. Obviously, the modelled profile albedos
fit in that range. The mean of the empirical albedo values is 0.67 and the
modelled mean values are 0.72 and 0.68 for volume scattering only and for
both volume and surface scattering, respectively. Taking into account the
snow surface roughness thus improved the average modelled albedo estimate.

## 4.5 BRF

Since the contribution of surface roughness to the total albedo is markedly smaller than the contribution of the bulk volume scattering (i.e. scattering of a snowpack with an ideal plane surface), it is clear that the volume scattering dominates also the BRF. However, the contribution of surface roughness is not negligible, and the BRF of the surface scattering component may differ markedly from the bulk volume component, resulting in a complex total BRF. The ray-tracing-based surface BRFs (without any volume scattering contribution) were compared with empirical BRFs measured using FIGIFIGO (Fig. 14) in Mantovaaranaapa on 22 April 2009. The area is an aapa mire, which late in spring affects the snow properties markedly (Fig. 15). Sporadically the snow had melted and refrozen.

The comparison of the ray-tracing- and surface-roughness-based BRFs with the
empirical FIGIFIGO-based BRFs was made in the principal plane, since the
azimuth information of the former ones was just a statistical assumption to
convert the 2D principal plane BRF to the 3D principal plane BRF. The surface
scattering BRFs are typically peaked to the direction *θ* of forward
scattering matching mirror reflection of the surface plane of the snowpack.
In addition, a backward peak in the direction *θ*-90^{∘} is
also strong in most cases (Fig. 16). The balance
between forward and backscattered intensity varies with incidence angle so
that large incidence angles favour surface backscattering due to roughness
(Fig. 17, Table 3).
Although mere surface scattering would lead to dominantly backward
scattering in Mantovaaranaapa due to the large incidence angle values
(55–64^{∘} for FIGIFIGO BRFs), the empirical BRFs
measured using FIGIFIGO dominated by the volume scattering are still
dominantly forward scattering. The balance between forward and backward volume
scattering is related to the grain shape (Peltoniemi et al., 2010). But
indeed, the smoothest snow sample produced the least backscattering, and the ratio
of the backward to the forward-scattered amount of radiation was 0.58,
whereas the corresponding ratio was on average 0.73 for the rough BRFs.
This result is quite in line with the general ray tracing analysis results
that surface roughness increases the fraction scattered backwards
(Table 3). Also, in a previous theoretical study of
Gaussian surfaces it was shown that roughness affects the maximum
direction of backward scattering (Jämsä et al., 1993). For the
profiles measured in Mantovaaranaapa on 22 April the ratio of the backward-
and forward-scattered radiation amounts in the incidence angle range of the
FIGIFIGO measurements varied from 1.0 to 1.46. One has to take into account
that the plate profiles register roughness in 1 m scale, whereas FIGIFIGO
measures samples of 0.10 m diameter. Hence the largest spatial roughness may
not necessarily show up as strongly in the FIGIFIGO results.

In this study, the equations combining the volume scattering and surface
scattering were derived using the photon recollision theory (Appendix A),
because this theory could be extended to include the surface scattering
effect. However, to describe properly the volume scattering of real
snowpacks, it is essential to pay attention also to layer structure, grain
shapes and various types of impurities, so that a more complex
description is typically needed for realistic volume scattering estimation.
In principle, the photon recollision theory could be extended to take the
layer structure of the snowpack into account by just letting the photon
recollision probability *p* to be a function of the depth, i.e. *p*=*p*(*z*), where *z*
would be the distance from the bottom or top surface. However, that is
beyond the scope of this study. Hence, the surface scattering part is
developed so that in principle it can be combined with any volume scattering
method. One just applies the estimates of *α*_{w0} and
*α*_{b0} derived with the chosen volume scattering model in
Eqs. (3)–(5). Hence, to obtain more realistic volume scattering estimates for
the snowpack, we used the TARTES model for volume scattering in the
simulations.

The findings that surface roughness in general decreases albedo and that the effect is larger for larger solar zenith angles are quite in line with the recent results obtained by applying a new rough-surface ray-tracing (RSRT) model to artificially generated surface roughness of snow (Larue et al., 2020). This study, however, extends these findings to smaller-scale roughness down to the sub-millimetre scale. The discussion about the effect of the varying local incidence angle on a rough surface and shadowing effects has been going on for decades, but until recently the emphasis has been on large-scale features (Warren et al., 1998; Kokhanovsky and Zege, 2004; Lhermitte et al., 2014). The essential advantage in studying the roughness from the theoretical point of view by generating artificial roughness with known dimensions and orientation is that one can then study the effect of each parameter involved separately (Larue et al., 2020). However, it is not trivial to generalize those results to natural snow, because the deterministic periodic structures may generate scattering features that will not be present for scattering from randomly rough surfaces. The advantage of the statistical approach presented in this study is that it does not make assumptions about the surface roughness characteristics but deals with the surfaces provided by nature. In addition, the derived formulas of the rough surface albedo are mathematically very simple and depend only on very few parameters, which makes their use very easy.

The ray tracing analysis of this study showed that the backward scattering increases with increasing surface roughness and increasing incidence angle of the illumination. However, that analysis concentrates only on surface scattering without any volume scattering contribution. Combining the surface and volume scattering contributions to BRF, perhaps using an adding procedure for radiative transfer, would be an interesting topic for future work.

Although the surface scattering model used here includes multiple scattering from the surface, it may be that a surface layer containing very deep cavities (in the scale of a few centimetres) would benefit from some special attention, like in the case of large-scale penitentes (Lhermitte et al., 2014). Namely, surface roughness measurements methods are usually designed for typical roughness of about the same variation range horizontally and vertically and not for extremely deep pits. To some extent the pit structure will be taken into account by the volume scattering models, since they affect the density of the surface layer of the snowpack. However, their very anisotropic (vertical) orientation is not well described by random scattering of a layer with reduced density. The pits act like illumination traps so that a larger part of illumination reaches lower layers of the snowpack before it is absorbed or scattered upwards. Therefore, the bulk albedo is smaller than for a completely random volume of the same density.

An example case is offered to illustrate this effect. Slight snow
precipitation took place on 16 March so that a very fluffy surface structure
of large dendritic snow crystals was formed on the snowpack
(Figs. 18 and 19).
The rms slope angle based on the plate method showed an increase with time
from about 0.38 to about 0.63 with *R*^{2}=0.68. The rms height and
correlation length also manifested clear evolution during those days
(Fig. 20). A related change is obvious also using
the roughness parameters *a* and *b* (Anttila et al., 2014). Yet, the surface
roughness measurements based on the plate method or laser scanning are not
able to catch the deep pit structure of the surface, because of shadowing
effects. Therefore, the simulated albedo is higher than the empirical one
(Fig. 12, Julian days 76–78). However, even if the
2D-surface roughness were characterized properly, the surface scattering
model based on a statistical approach of random scatterers would not be
ideal for a case with a distinct periodic surface structure. For example,
one could analyse separately the percentage of illumination that will be
completely trapped by the deep cavities and reduce the simulated total
albedo with that fraction.

The plate profile method has the advantage of high spatial resolution. Its main drawback is that it can be used only when the snow is relatively soft. It has been successfully used in Finnish Lapland (Anttila et al., 2014) and at Greenland Summit Station (Manninen et al., 2016), but Antarctic snow is typically so hard that it is not possible to immerse the plate in it. In addition, the icy and crusty snow surface of Finnish Lapland in 2018 and 2019 turned out to be too hard for the plate. For laser scanning, however, the hardness of the snowpack does not cause any problems. Indeed, laser scanning shows great potential for measuring snow surface roughness as it can cover large areas with high point precision accuracy. It is a particularly good method for measuring larger-scale roughness from 0.05–0.1 m upwards. The limiting factor in finer-scale roughness measurements is the data resolution and footprint size of the laser beam. So far, the scanners with the highest point density, accuracy and smallest spot size are meant for indoor use, but as the technology improves, smaller and smaller features become measurable also outdoors. In addition, the fractal nature of snow surfaces enables extrapolation of surface roughness from the centimetre scale to the millimetre scale (Kukko et al., 2013). Another benefit of using laser scanning for surface roughness measurements is that it leaves the surface intact. This enables repeatable measurements of the same surface, giving a means to study the evolution of surfaces in time. The backscattering intensity of the laser beam is typically stored for each point measured by laser, and in the most modern scanners also the range deviation is stored. These features have so far not been widely used, but they could potentially be used in the future for surface scattering property measurements and snow surface classifications.

Finally, considering satellite retrievals, it is expected that for an ideally flat surface the impact of roughness would be essentially the same at the satellite resolution as that in the scale of in situ measurements. However, the larger the satellite pixel is, the larger the spatial scale roughness that has to be taken into account is. The derived model is applicable to take into account roughness of all relevant scales, but the problem is how to estimate the average multiscale number of facet-to-facet scattering events. It is anticipated that due to the fractal nature of snow the small-scale estimate of the average number of facet-to-facet scattering events is a reasonable first-order estimate for the corresponding multiscale value, but a related detailed analysis is beyond the scope of this study.

A method was developed to model the effect of surface roughness on albedo
besides the volume scattering. It can be combined with any volume scattering
model. Applying measured surface roughness values to the model produced
results closer to measured values than only volume scattering simulations
made with the TARTES model. The surface roughness is described by the
average number of surface scattering events per ray, which is currently
estimated from the rms slope angle values of the measured surface roughness
profiles. High empirical correlation (*R*^{2}=0.9) of albedo with just
two surface-roughness-related parameters supports the importance of surface
roughness to albedo.

The albedo modelling results also taking into account the surface roughness indicate that it may decrease the albedo by about 1 %–3 % in midwinter and even more than 10 % during late melting season. The effect is largest for low solar zenith angle values and lower bulk snow albedo values. Hence, the effect is larger during early and late times of day everywhere, and it increases during the melting season especially at high latitudes, where the sun elevation is lower. Increasing surface roughness also favours more backward scattering.

## A1 Scattering of diffuse radiation

The scattering of light in canopies has for several years successfully been
described with spectral invariants and the so-called photon recollision
theory, *p* theory (Knyazikhin et al., 1998; Panferov et al., 2001; Smolander
and Stenberg, 2005; Rautiainen and Stenberg, 2005; Stenberg et al., 2008;
Stenberg and Manninen, 2015; Stenberg et al., 2016). The central parameter,
the photon recollision probability *p*, is spectrally invariant and depends on
the amount of scattering surface in the volume. Canopies do not have distinct
upper surfaces; hence the *p* theory is developed so far only for a scattering
volume, but it has already been successfully combined with forest floor
scattering also for snow-covered cases (Manninen and Stenberg, 2009). Here
the *p* theory is applied to snowpack scattering taking into account that the
snowpack has a distinct surface, which may be rough.

A simple way to take into account the surface roughness effect on scattering
is to consider every facet of the snowpack as a separate volume of
scatterers. When the irradiance *i*_{0}=*i*_{0}(*λ*) first arrives
at the facet, it enters a volume scattering sequence, which can be described
with the spectrally invariant photon recollision probability *p* of the bulk
part of the snowpack and the single-scattering albedo of the snow grains
*ω*=*ω*(*λ*) (Knyazikhin et al., 1998; Panferov et
al., 2001). The radiation absorbed and radiation scattered by the volume of the facet
*a*_{0} and *s*_{0}, respectively, are (Smolander and Stenberg, 2005;
Rautiainen and Stenberg, 2005; Stenberg and Manninen, 2015)

For simplicity the dependence of *ω*, *i*_{0}, *a*_{0} and *s*_{0} on the
wavelength *λ* is not shown explicitly in the equations. The
radiation escaping the volume of the facet either escapes altogether or hits
another facet and experiences another volume scattering sequence. The
probability of the latter case is defined to be the surface photon
recollision probability *p*_{s}. Because the snow grains in the bulk part are
completely surrounded by other snow grains while in the surface only close to half of the surrounding volume may contain snow grains, it is essential to
assume that *p* and *p*_{s} are not identical.

The radiation escaping the snowpack altogether without hitting another
facet, *r*_{0}, is

where *q*=*q*(*λ*) is the fraction of the volume scattering escaping
upwards. Essentially it corresponds to *Q* defined by Stenberg et al. (2016,
Eq. 24), but as it does not contain the fraction scattered upwards by the
surface, it is not the total upward-scattered fraction of light. Hence *q* is
used here instead of *Q*. Theoretically the values for *q* are in the range 0–1,
being larger the thicker and denser the scattering layer is. The
radiation hitting another facet is

The absorbed (*a*_{1}) and scattered (*s*_{1}) amounts of radiation by the
second volume scattering sequence are

The amounts of radiation escaping (*r*_{1}) and entering the following
scattering sequence (*i*_{2}) of another facet are

Formulas for the corresponding radiation components in the following facet-to-facet scattering round are

Further on, the components corresponding to the *j*th round are

The amounts absorbed and scattered by the surface and volume, considering up
to *n* facet-to-facet scattering events, are then

Correspondingly, the upward-escaping radiation is

Note that when the number *n* of additional facet scattering sequences the photon
has before it escapes altogether is zero and *p*_{s}=0, there is no
facet-to-facet scattering; i.e. the case is the normal volume scattering
case. The total absorbed radiation and upward-escaping radiation of the snowpack are
derived as infinite geometrical sums and are

The total white-sky (diffuse) albedo *α*_{w} of the snowpack is then

The white-sky (diffuse) albedo of the bulk part of the snowpack (without
facet-to-facet scattering) *α*_{w0} is

Hence, the total white-sky albedo is simply

Estimation of the parameter *p*_{s} is not trivial, but when the distribution
of *n* or its average value denoted by $<n>$ is known from
measurements, then a reasonable estimate can be obtained from the total
scattered energy *s* (Eq. A18) as follows. First, *s* is derived using the finite
sum formulation of Eq. (A18) with a surface recollision probability of unity;
i.e. *p*_{s} set to 1. Second, *s* is derived using the probabilistic infinite
sum formulation of Eq. (A18); i.e. *n* is set to infinity. These two estimates
must equal. Taking into account Eq. (A23) and the normalized distribution
*f*(*n*) of the values of *n* from zero to its maximum value *n*_{max}, the following
relation is obtained:

which reduces to the following equation:

The value for *p*_{s} can be solved from the above equation and is

In practice, the number of facet-to-facet scattering sequences *n* is often
small, so that *f*(*n*) is dominated by values *n*=0 and *n*=1. When this is the
case, the weighted mean of ${\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}^{n+\mathrm{1}}$ represented by
the left part of Eq. (A26) can be approximated with a high precision with
${\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}^{<n>+\mathrm{1}}$. This was indeed the case in the
measured profiles. In March the normalized distribution was approximately
$f\left(n\right)=\mathrm{2}\cdot \mathrm{exp}(-\mathrm{2}n)$ and in April $f\left(n\right)=\mathrm{1.5}\cdot \mathrm{exp}(-\mathrm{1.5}n)$. Hence,
the albedo estimation using $<n>$ instead of the
distribution of *n* caused in March at most an underestimation of the total
albedo by 0.004 and in April by 0.014. As reliable estimation of *f*(*n*) for single
profiles is challenging, it is recommended to simply use the mean value
$<n>$. Then, the value of *p*_{s} can be estimated from

The relationship between the total albedo and the bulk albedo is then

When the surface does not cause additional scattering, <*n**>=*0 and the total albedo equals the bulk albedo. The larger the $<n>$ is, the smaller *α*_{w} is.

For a single photon the number of facet-to-facet volume scattering rounds is naturally an integer number. For the ensemble of the photons, $<n>$ can be estimated to be

Combining Eqs. (A28) and (A30), it is possible to estimate *q* for values $<n>$ larger than 0, and it is

For <*n**>=*1, *q* equals the bulk white-sky albedo, and for larger
(smaller) values of $<n>$ it is slightly smaller (larger) for medium
albedo values.

## A2 Scattering of direct radiation

For the direct component of solar illumination one has to take into account
that the irradiance depends on the solar zenith
angle *θ*_{i}, i.e. ${i}_{\mathrm{0}}={i}_{\mathrm{0}}(\mathit{\lambda},{\mathit{\theta}}_{i})$, in addition to the wavelength.
The photon recollision probability in the bulk snowpack will be denoted for
the first scattering sequence *p*_{1}=*p*_{1}(*θ*_{il}) and for
latter sequences *p*, assuming that the incidence angle dependence is lost
after the first volume scattering sequence. Here the local incidence angle
of the facet is denoted by *θ*_{il}. Then, the absorbed radiation of
the volume will be (Stenberg and Manninen, 2015)

and the radiation scattered by the volume will be

When *p*_{1}=*p* the above formulas reduce to Eqs. (A1) and (A2) as they should.

The first surface scattering sequence shall be treated separately, because
the volume scattering depends on the incident solar zenith angle. The
surface photon recollision probability is denoted by *p*_{s1}=*p*_{s1}(*θ*_{il}) for the first facet-to-facet scattering
sequence and by *p*_{sr} for the latter facet-to-facet scattering sequences.
One should notice that *p*_{sr} is not necessarily equal to *p*_{s} of the
diffuse case, although they certainly approach each other asymptotically.
The reason for not taking them to be identical immediately is the typically
small number of surface scattering events per ray of relatively smooth
midwinter snow surfaces. The radiation escaping the facet altogether after
just volume scattering, *r*_{0}, is

where ${q}_{\mathrm{0}}={q}_{\mathrm{0}}(\mathit{\lambda},{\mathit{\theta}}_{i})$ denotes the fraction of the scattered radiation escaping upwards from the snowpack during the first volume scattering sequence. The radiation hitting another facet is

The further scattering sequences are assumed to be independent of the solar
zenith angle of the original incident radiation. The absorbed (*a*_{1}) and
scattered (*s*_{1}) amounts of radiation by the second volume scattering
sequence are then

The amounts of radiation escaping (*r*_{1}) and entering the following
scattering sequence (*i*_{2}) are

Formulas for corresponding radiation components in the following round are

The components corresponding to the *j*th round are

The amounts absorbed and scattered by the surface and volume, considering up
to *m* facet-to-facet scattering events, are then

Correspondingly, the upward-escaping radiation is

where *m* is the number of facet-to-facet scattering events. The total
radiation escaping the snowpack upwards is derived again as an infinite
geometrical sum

The total black-sky (direct) albedo *α*_{b} of the snowpack is then

The black-sky (directional) albedo of the bulk snowpack *α*_{b0} is

Also taking into account Eq. (A23), the relationship between the total black-sky albedo and the bulk snowpack black-sky and white-sky albedo is

Estimating *p*_{s1} or *p*_{sr} (whether equal to *p*_{s} or not) is not
trivial, but like in the case of diffuse irradiance one can benefit from the measured average value of *m* denoted by $<m>$ by requiring
that the total scattered energy (Eq. A49) is the same, whether it is
calculated from the probabilistic infinite sum or a deterministic sum with
surface recollision probability of unity, i.e.

Unfortunately, there are two variables, *p*_{s1} and *p*_{sr}, to solve
but only one equation. Hence, the theory does not provide an exact solution
for both *p*_{s1} and *p*_{sr}, but only *p*_{sr} remains explicitly in the
equation of the black-sky albedo:

When the assumption that *p*_{sr}=*p*_{s} is valid, the following formula
is obtained for the black-sky albedo using Eq. (A28):

When $<m\mathit{>=}<n>$, the black-sky albedo reduces to ${\mathit{\alpha}}_{\mathrm{b}\mathrm{0}}\cdot {\mathit{\alpha}}_{\mathrm{w}\mathrm{0}}^{<n>}$. Further on, when
*α*_{b0}=*α*_{w0}, the total black-sky and
white-sky albedo values are equal as well, if $<m>=<n>$. The above approximation of the black-sky albedo is reasonable only when
the assumption *p*_{sr}=*p*_{s} is good. Hence, further studies are needed
for proper estimation of *p*_{sr} and *p*_{s1}.

The average number of facet-to -facet scattering rounds $<m>$ is estimated to be

When *p*_{s1}→*p*_{s}, *q*_{0}→*q* and *α*_{b0}
→*α*_{w0}, $<m>$ approaches $<n>$, as it should. It
should be noted that the number of facet-to-facet scattering rounds in
direct illumination ($<m>$) and in diffuse illumination ($<n>$) are not necessarily equal, with the difference being related to the difference
of *α*_{b0} and *α*_{w0} and *q* and *q*_{0}. The
estimate for *q*_{0} can now be derived from Eq. (A58).

Some data of the SNORTEX campaign is directly listed in the report (Manninen and Roujean, 2014). The plate snow profile data and the ASD data are available upon request from Finnish Meteorological Institute. The FIGIFIGO measurements and the laser scanning data are available upon request from the Finnish Geospatial Research Institute, National Land Survey.

The supplement related to this article is available online at: https://doi.org/10.5194/tc-15-793-2021-supplement.

Albedo data were measured/retrieved and analyzed by TM and AR. Albedo modelling was carried out by TM, EJ, AR, JP, RP, J-LR, OH and PR. Plate surface roughness measurements and analysis were carried out by KA and TM. Laser scanning surface roughness measurements and analysis were carried out by SK, HK, AKu, KA and TM. FIGIFIGO BRF measurements and analysis were carried out by JP, JS and TH. Snow density and grain size profile measurements were carried out by PL, NS, LT and AKo. Snow impurity measurements and analysis were carried out by AKo, HS and OM. ASD snow reflectance spectra were measured and analyzed by AR, AKo, HS and TM. Ray tracing analysis was carried out by TM. Synthesis of analyses was carried out by TM, JP, PR and J-LR. Writing of the manuscript was carried out by everybody.

The authors declare that they have no conflict of interest.

The co-operation with the operational units of FMI in Helsinki and in Sodankylä and all SNORTEX campaign participants is gratefully acknowledged. The authors are indebted to Tuure Karjalainen for taking the plate-surface-roughness-related photos in the field in 2009. Panu Lahtinen, Niilo Siljamo and Laura Thölix took the other photos. The authors are also grateful to Markku Ahponen and Veikko Mylläri for the snow depth, density and SWE measurements of the remote points outside the intensive test area and to Markku Ahponen, Veikko Mylläri and Anita Sassali for participating in snow impurity sampling. The authors would like to thank Alexander Kokhanovsky, Ghislain Picard and an anonymous reviewer for beneficial comments.

The work was financially supported by the Academy of Finland in the projects SnowAPP (315497), OPTICA (295874), Reflectance of Boreal forests (120949), Scattering (260027), Arctic Absorbing Aerosols and Albedo of Snow (A4), (254195), and NABCEA (296302) and by EUMETSAT in the projects Satellite Application Facility on Climate Monitoring (CM SAF), Satellite Application Facility on Support to Operational Hydrology and Water Management (H SAF) and Satellite Application Facility on Land Surface Analysis (LSA).

This paper was edited by Marco Tedesco and reviewed by Alexander Kokhanovsky and one anonymous referee.

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