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.
Introduction
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.
DataTest 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.
Test area in Sodankylä in northern Finland. The premises of
the Arctic Space Centre of FMI are situated in Tähtelä (T). The
operational albedo measurements are located in the upper part of the rectangle surrounding T and the NorSEN mast at the lower part. The aapa mire test site
Mantovaaranaapa is marked with M and the forest clearing site Hirviäkuru
with H. The corner coordinates of the area given in the WGS84 system. CC BY
4.0 National Land Survey of Finland (04/2020).
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 variation range and median values of the air temperature and
snow surface temperature during the SNORTEX campaign in Sodankylä in
11–19 March and 20–27 April 2009. The corresponding variation of the
snow density and snow water equivalent are given as well. The measurements
were carried out during 09:00 and 17:00 h local time (Manninen and Roujean,
2014). The values in brackets are those measured at the reference site
NorSEN mast (67.3621∘ N, 26.63445∘ E) in
Tähtelä. The total number of individual measurements was altogether
118 (17) for the reference site. The elemental carbon and organic carbon
concentrations measured near the NorSEN mast (67.364011∘ N,
26.635891∘ E) in March and April are shown as well (Meinander et
al., 2020).
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.
The variation range and median values of snow grain size (defined
as the maximum diameter of the smallest snow grain in each sample) and
density of the topmost layer and the snowpack during the SNORTEX campaign in
Sodankylä in 11–19 March and 20–27 April 2009.
Parameter Grain diameter ofGrain diameterDensity of topDensity oftop layer [mm]snowpack [mm]layer [kg m-3]snowpack [kg m-3]MarchMin0.250.2511059Median0.51.5143173Max1.53.25317345AprilMin0.250.256511Median22272259Max3.54433433Surface 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.
Examples of surface roughness of snow at Mantovaaranaapa on 22 April 2009. The black background of the plate is 1 m wide.
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:
1logσ=a+blogx,2L=k0+kx,
where a, b, k0 and k are constants, and k0=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
(β=arctan(Δz/Δx)), 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).
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.
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).
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.
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 m2) 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.
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 m2, 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.
MethodsSmooth 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.
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 αw (Lucht et al., 2000,
Schaepman-Strub et al., 2006) is related to the average number of
facet-to-facet scattering events <n>:
αw=αw0<n>+1,
where αw0 is the white-sky albedo of the bulk volume.
Second, for direct illumination, the black-sky albedo αb(θi) (Lucht et al., 2000, Schaepman-Strub et al., 2006)
is approximately related to the αw0, <n> and the average number <m(θi)> of
facet-to-facet scattering events in direct illumination conditions by
αb=αb0αw0<n>1-αw0<m>+11-αw0<n>+1,
where αb0(θi) 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 αw0
are produced using the TARTES model (Sect. 3.1).
Facet structure of a randomly rough surface of spherical
scatterers. The arrows indicate an example of a possible ray path involving
facet-to-facet scattering.
Surface characteristics calculated from the plate measurements in
March and April. (a) The relative height distributions and (b) the
distributions of the slope angles (in degrees) at the average measured
spatial resolution of 0.26 mm.
The albedo α in mixed illumination conditions is typically estimated
using the weighted mean approximation of the two extreme values αw and αb (Lucht et al., 2000; Schaepman-Strub et al.,
2006; Román et al., 2010):
α=fαw+1-fαb,
where f is the fraction of diffuse irradiance. According to Eqs. (3)–(5) the
blue-sky albedo is then estimated from
α=fαw0<n>+1+1-fαb0αw0<n>1-αw0<m>+11-αw0<n>+1=αw0<n>fαw0+1-fαb01-αw0<m>+11-αw0<n>+1.
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.
The average number of reflections <ns> of an individual ray hitting the surface as a function of the rms slope angle
β (in radians) for two incident solar zenith angle values θi
for the snow profiles measured in the SNORTEX campaign (Manninen and Roujean,
2014; Anttila et al., 2014) in March and April 2009.
The average zenith angle θo of the reflected escaping
individual ray as a function of the rms slope angle β (in radians)
for two irradiance solar zenith angle values θi for the snow
profiles measured in the SNORTEX campaign (Manninen and Roujean, 2014;
Anttila et al., 2014) in March and April 2009. The sign of the zenith angle
is positive for forward reflection and negative for backward reflection.
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 ns 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.
ResultsSurface 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 (R2=0.97)
to April (R2=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 ns per ray on the surface has
an increasing trend during the melting season (Fig. 5). It correlates well (R2=0.83) with the rms slope angle
(β, in radians) of the profile. A good general fit to all measured
plate profile data was
ns=1+0.355332cosθi4β-1.082751-exp1.75cosθi1/4β4,
where θi is the solar zenith angle. The mean zenith angle
θo to which the radiation escapes from the surface (when mirror
reflection from the facet is assumed) is even more strongly correlated
(R2=0.93) with β (Fig. 6):
θo=-0.9252391-11+0.25θi32-1.299821-11+0.75θi32log(β).
All angles (β, θi, θo) are given in radians
in the above equations, and θo>0(θo<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).
Relationship between rms slope angle and the horizontal distance
between the points used for its determination according to the laser
scanning data of 18 March 2010. For comparison the mean value and variation
range of the rms slope angle values derived from the 10 plate profiles for
horizontal resolution 0.25 mm in the same area in the same day are shown in
red. The dashed black curve is the regression to the 36 laser-scanning-based
points (black polyline), and the grey shaded area covers the 80 % variation
range of rms slope angle at the distance in question.
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,
ns 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.
The relationship between the albedo values corresponding
to the solar zenith angle value of 73∘ measured operationally at
Sodankylä and the regression (albedo = 0.84–0.29
b-0.008k0) based on measured surface
roughness parameters b (Eq. 1) and
k0 (Eq. 2) at the NorSEN
mast in March and April 2009. The darkness of the markers is related to the
fraction of diffuse irradiance.
Variation range (grey) of the individual snow reflectance spectra
measured using the ASD spectrometer in Hirviäkuru (67.38∘ N,
26.85∘ E) on 13 March and in Mantovaaranaapa (67.4∘ N,
26.72∘ E) on 22 April. The albedo simulations using the TARTES
model are shown for fractal grains (blue) and spheres (red). The solid lines
indicate the mean value and the dotted lines the minimum and maximum curves
of the day in question. The ASD measurements were carried out in the same
area as the grain size and density measurements, but the impurity
measurements were daily values measured at Tähtelä (67.37∘ N, 26.63∘ E).
Evolution of measured surface albedo in Tähtelä
(67.37∘ N, 26.63∘ E) on 12–19 March and 20–28 April 2009. The corresponding variation range of simulated albedo values based
on simultaneous grain size and density measurements and temporally
interpolated impurity content in the same day are shown in grey. The
minimum, mean and maximum values are indicated with darker grey curves. The
vertical error bars marked for some of the measured albedo values are
based on the variation range of the measured ASD-spectra-based broadband
reflectance values during the same day in the larger test area in
Sodankylä.
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.
Ratio of simulated total and bulk white-sky albedo values for all
1381 profiles and snow pit and impurity data in Sodankylä in 11–19 March
and 20–28 April 2009. The daily mean, minimum and maximum values are
indicated by the darker curve, and the variation range is shaded.
Simulated albedo values at the NorSEN mast based on density,
grain size and surface roughness measurements of snow in 12–19 March 2009
and in 20–28 April 2009 and temporally interpolated impurity content
data. The mean values are shown separately for the TARTES model containing
only the volume scattering contribution (grey curve) and the TARTES model
combined with the surface scattering model of this study (blue or red). The
daily variation range based on diverse profiles, grain size and density
measured at the site and temporally interpolated impurity content are shown
as the area shaded by light blue or light red colour. The darkness of the
empirical points indicates the fraction of diffuse irradiance during the
measurement.
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 σ/L (determined for 0.60 m distance) correlated relatively well with β, with the R2 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 σ/b (Eq. 1),
with its R2 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
σ/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 R2=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 R2=0.90,
when the parameters b and k0 (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.
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.
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 αw0 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 ns. Namely, m=ns-1 for the local solar
zenith (θil) angle range derived using the ray tracing method.
And n is the weighted mean of ns-1, where the weights are cos(θil)⋅sin(θil). The results are shown in
Figs. 11 and 12.
First, the ratio αw/αw0 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 αw/αw0
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 αw/αw0 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).
Measured and modelled blue-sky albedo values at Mantovaaranaapa
on 22 April 2009. Each modelled point is an average corresponding to three
individual plate profiles taken from the same surface. The empirical albedo
values are likewise averages of the three individual points recorded using the
Kipp & Zonen albedometer CM14 at the time of taking the profile photos.
The shaded area shows the variation range of the broadband reflectance
values measured using the ASD spectrometer.
Measured principal plane BRFs in Mantovaaranaapa on 22 April 2009
of one smooth snow and three rough snow cases, with a few individual
profiles each.
Mantovaaranaapa on 22 April 2009.
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.
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 (R2=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.
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.
Calculated average scattering angle distributions for surface
scattering (i.e. zenith angles of reflected radiation, positive for forward
directions and negative for backward directions) for profiles measured
during the SNORTEX campaign in 2009 (blue) and in Mantovaaranaapa on 22 April 2009 (red) for incidence angle values 20 and 60∘.
The mean values and variation range of the ratio rfb of
backward to forward scattering for diverse solar zenith angle values for all
profiles measured in March and April 2009. These values represent the
surface scattering contribution only.
Solar zenithMean80 % variationanglerfbrange of rfb20∘0.350.21–0.4940∘0.660.36–1.0260∘1.270.50–2.2780∘4.051.23–8.43
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.
Surface roughness parameter a as a function of surface roughness
parameter b (Eq. 1) of the dominantly backscattering and dominantly forward
scattering profiles for incidence angles 20, 40 and
60∘.
Snow surface structure evolution during 16–18 March 2009. The centimetre scale is shown in the right images.
Microscale snow surface structure evolution during 16–18 March 2009. In the left images the scale of the grid is 1 mm. In the right images
the grids correspond to 1, 2 and 3 mm from left to right.
(a) The snow surface rms height and correlation length measured
with the plate method during 16–18 March 2009. The values correspond to
the distance 0.6 m. (b) The rms slope angle β (in radians) of
individual plate snow profiles during 16–18 March 2009. The linear
regression for β vs. time is shown for 16 March, with the R2 value
included.
Discussion
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 αw0 and
αb0 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 R2=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.
Conclusions
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 (R2=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.
Deriving the formulas for surface roughness effect on scatteringScattering 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 i0=i0(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
a0 and s0, respectively, are (Smolander and Stenberg, 2005;
Rautiainen and Stenberg, 2005; Stenberg and Manninen, 2015)
A1a0=1-ω1-pωi0,A2s0=ω-pω1-pωi0.
For simplicity the dependence of ω, i0, a0 and s0 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 ps. 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 ps are not identical.
The radiation escaping the snowpack altogether without hitting another
facet, r0, is
r0=1-psqs0=1-psqω-pω1-pωi0,
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
i1=psqs0=psqω-pω1-pωi0.
The absorbed (a1) and scattered (s1) amounts of radiation by the
second volume scattering sequence are
A5a1=1-ω1-pωi1=psq1-ωω-pω1-pω2i0,A6s1=ω-pω1-pωi1=psqω-pω1-pω2i0.
The amounts of radiation escaping (r1) and entering the following
scattering sequence (i2) of another facet are
A7r1=1-psqs1=1-pspsq2ω-pω1-pω2i0,A8i2=psqs1=ps2q2ω-pω1-pω2i0.
Formulas for the corresponding radiation components in the following
facet-to-facet scattering round are
A9a2=1-ω1-pωi2=ps2q21-ωω-pω21-pω3i0,A10s2=ω-pω1-pωi2=ps2q2ω-pω1-pω3i0,A11r2=1-psqs2=1-psps2q3ω-pω1-pω3i0,A12i3=psqs2=ps3q3ω-pω1-pω3i0.
Further on, the components corresponding to the jth round are
A13aj=1-ω1-pωij=psjqj1-ωω-pωj1-pωj+1i0,A14sj=ω-pω1-pωij=psjqjω-pω1-pωj+1i0,A15rj=1-psqsj=1-pspsjqj+1ω-pω1-pωj+1i0,A16ij+1=psqsj=psj+1qj+1ω-pω1-pωj+1i0.
The amounts absorbed and scattered by the surface and volume, considering up
to n facet-to-facet scattering events, are then
A17a=∑j=0naj=∑j=0npsjqj1-ωω-pωj1-pωj+1i0=1-ω1-pωi0∑j=0npsqω-pω1-pωj,A18s=∑j=0nsj=∑j=0npsjqjω-pω1-pωj+1i0=ω-pω1-pωi0∑j=0npsqω-pω1-pωj.
Correspondingly, the upward-escaping radiation is
r=∑j=0nrj=∑j=0n1-pspsjqj+1ω-pω1-pωj+1i0=1-psqω-pω1-pωi0∑j=0npsqω-pω1-pωj.
Note that when the number n of additional facet scattering sequences the photon
has before it escapes altogether is zero and ps=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
A20a=1-ω1-pω11-psqω-pω1-pωi0,A21r=1-psqω-pω1-pω11-psqω-pω1-pωi0.
The total white-sky (diffuse) albedo αw of the snowpack is then
αw=ri0=1-psqω-pω1-pω11-psqω-pω1-pω.
The white-sky (diffuse) albedo of the bulk part of the snowpack (without
facet-to-facet scattering) αw0 is
αw0=qs0i0=qω-pω1-pω.
Hence, the total white-sky albedo is simply
αw=1-psαw011-psαw0.
Estimation of the parameter ps 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. ps 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 nmax, the following
relation is obtained:
∑n=0n=nmaxαw0i0q1-αw0n+11-αw0f(n)=αw0i0q11-psαw0,
which reduces to the following equation:
∑n=0n=nmaxαw0n+1f(n)=1-1-αw01-psαw0.
The value for ps can be solved from the above equation and is
ps=1-∑n=0n=nmaxαw0nf(n)1-∑n=0n=nmaxαw0n+1f(n).
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 αw0n+1 represented by
the left part of Eq. (A26) can be approximated with a high precision with
αw0<n>+1. This was indeed the case in the
measured profiles. In March the normalized distribution was approximately
f(n)=2⋅exp(-2n) and in April f(n)=1.5⋅exp(-1.5n). 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 ps can be estimated from
ps=1-αw0<n>1-αw0<n>+1.
The relationship between the total albedo and the bulk albedo is then
αw=1-psαw011-psαw0=αw0<n>+1.
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 αw 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
<n>=∑j=0∞pssji0=psω-pω1-pω∑j=0∞psqω-pω1-pωj=psω-pω1-pω11-psqω-pω1-pω=psαw0q1-psαw0.
Combining Eqs. (A28) and (A30), it is possible to estimate q for values <n> larger than 0, and it is
q=psαw0<n>1-psαw0=αw01-αw0<n><n>1-αw0.
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.
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 θi, i.e. i0=i0(λ,θi), in addition to the wavelength.
The photon recollision probability in the bulk snowpack will be denoted for
the first scattering sequence p1=p1(θil) 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 θil. Then, the absorbed radiation of
the volume will be (Stenberg and Manninen, 2015)
a0=1-ω1-ωp-p11-pωi0,
and the radiation scattered by the volume will be
s0=ω1-p1-ωp-p11-pωi0.
When p1=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 ps1=ps1(θil) for the first facet-to-facet scattering
sequence and by psr for the latter facet-to-facet scattering sequences.
One should notice that psr is not necessarily equal to ps 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, r0, is
r0=1-ps1q0s0=1-ps1q0ω1-p1-ωp-p11-pωi0,
where q0=q0(λ,θi) denotes the fraction of
the scattered radiation escaping upwards from the snowpack during the first
volume scattering sequence. The radiation hitting another facet is
i1=ps1q0s0=ps1q0ω1-p1-ωp-p11-pωi0.
The further scattering sequences are assumed to be independent of the solar
zenith angle of the original incident radiation. The absorbed (a1) and
scattered (s1) amounts of radiation by the second volume scattering
sequence are then
A36a1=1-ω1-pωi1=ps1q0ω1-ω1-p1-ωp-p11-pω2i0,A37s1=ω-pω1-pωi1=ps1q0ωω-pω1-p1-ωp-p11-pω2i0.
The amounts of radiation escaping (r1) and entering the following
scattering sequence (i2) are
A38r1=1-psrqs1=1-psrps1qq0ωω-pω1-p1-ωp-p11-pω2i0,A39i2=psrqs1=psrps1qq0ωω-pω1-p1-ωp-p11-pω2i0.
Formulas for corresponding radiation components in the following round are
A40a2=1-ω1-pωi2=psrps1qq0ω1-ωω-pω1-p1-ωp-p11-pω3i0,A41s2=ω-pω1-pωi2=psrps1qq0ωω-pω21-p1-ωp-p11-pω3i0,A42r2=1-psrqs2=1-psrpsrps1q2q0ωω-pω21-p1-ωp-p11-pω3i0,A43i3=psrqs2=psr2ps1q2q0ωω-pω21-p1-ωp-p11-pω3i0.
The components corresponding to the jth round are
A44aj=1-ω1-pωij=psrj-1ps1qj-1q0ω1-ωω-pωj-11-p1-ωp-p11-pωj+1i0,A45sj=ω-pω1-pωij=psrj-1ps1qj-1q0ωω-pωj1-p1-ωp-p11-pωj+1i0,A46rj=1-psrqsj=1-psrpsrj-1ps1qjq0ωω-pωj1-p1-ωp-p11-pωj+1i0,A47ij+1=psrqsj=psrjps1qjq0ωω-pωj1-p1-ωp-p11-pωj+1i0.
The amounts absorbed and scattered by the surface and volume, considering up
to m facet-to-facet scattering events, are then
A48a=∑j=0maj=ps1q0psrq1-ω1-p1-ωp-p11-pω1-pi0∑j=0mpsrqω-pω1-pωj,A49s=∑j=0msj=ps1q0psrqω1-p1-ωp-p11-pωi0∑j=0mpsrqω-pω1-pωj.
Correspondingly, the upward-escaping radiation is
r=∑j=0mrj=1-psrps1q0psrω1-p1-ωp-p11-pωi0∑j=0mpsrqω-pω1-pωj,
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
r=1-psrps1q0psrω1-p1-ωp-p11-pω11-psrqω-pω1-pωi0.
The total black-sky (direct) albedo αb of the snowpack is then
αb=ri0=1-psrps1q0psrω1-p1-ωp-p11-pω11-psrqω-pω1-pω.
The black-sky (directional) albedo of the bulk snowpack αb0 is
αb0=q0s0i0=q0ω1-p1-ωp-p11-pω.
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
αb=1-psrps1αb0psr11-psrαw0.
Estimating ps1 or psr (whether equal to ps 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.
ps1psr11-psrαw0=1-αw0<m>+11-αw0.
Unfortunately, there are two variables, ps1 and psr, to solve
but only one equation. Hence, the theory does not provide an exact solution
for both ps1 and psr, but only psr remains explicitly in the
equation of the black-sky albedo:
αb=1-psrαb01-αw0<m>+11-αw0.
When the assumption that psr=ps is valid, the following formula
is obtained for the black-sky albedo using Eq. (A28):
αb=αb0αw0n1-αw0<m>+11-αw0<n>+1.
When <m>=<n>, the black-sky albedo reduces to αb0⋅αw0<n>. Further on, when
αb0=αw0, 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 psr=ps is good. Hence, further studies are needed
for proper estimation of psr and ps1.
The average number of facet-to
-facet scattering rounds <m> is estimated to be
<m>=ps1s0i0+∑j=1∞pssji0=ps1αb0q0-psαb0q0+∑j=0∞pssji0=ps1-psαb0q0+ps1αb0q1-psαw0.
When ps1→ps, q0→q and αb0→αw0, <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 αb0 and αw0 and q and q0. The
estimate for q0 can now be derived from Eq. (A58).
Data availability
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.
Author contributions
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.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
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.
Financial support
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).
Review statement
This paper was edited by Marco Tedesco and reviewed by Alexander Kokhanovsky and one anonymous referee.
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