Estimation of the direct-to-diffuse solar irradiance ratio
The ratio of diffuse to direct irradiance is required to perform accurate
correction of the measured spectrum (e.g. ). The
calculation of such ratio requires the knowledge of the atmospheric profiles
and of the local cloud optical thickness, τ, at 550 nm.
We thus estimate this local cloud optical thickness using both the
atmospheric profiles from ALADIN-Climate and local meteorological
observations at Col de Porte to overcome the coarse
resolution of ALADIN-Climate. The first step consists in estimating a
relationship between τ and the broadband direct (SWdir) to
total (SWtot) ratio. For this purpose, the SBDART
detailed radiative model was used to calculate
SWdir over SWtot from varying cloud optical
thicknesses and mean atmospheric conditions at Col de Porte (aerosols optical
thickness, total ozone column and total water vapour column). The mean
atmospheric conditions were derived from ALADIN-Climate outputs and the
measured 2 m air temperature average over the measurements period. A
regression equation (Eq. ) was derived from those results.
τ=μslog(1+0.8μs-1.95μs2+1.67μs3-0.771-SWdir/SWtot),
where μs=cosθs is the cosine of the solar
zenith angle.
As a second step, Eq. () was used to estimate τ from
SWdir over SWtot ratio measured at Col de Porte.
Finally, the outputs of the ALADIN-Climate (namely hourly total aerosols
optical thickness, total water vapour column and total ozone column) together
with measured air temperature and estimated τ were used as inputs for
the SBDART model in order to compute the hourly diffuse
and direct spectral irradiance. The ratio between these variables is used in
our analysis. Note that the broadband SWdir and
SWtot estimated from Col de Porte measurements and simulated
with SBDART agree within ±10 W m-2. The accuracy of the simulated
spectral direct to diffuse solar irradiance ratio has not been evaluated due
to the absence of measurements. The difference in elevation between the
ALADIN-Climate grid cell and Col de Porte site might lead to an
overestimation of the diffuse fraction in the visible wavelengths.
Spectral albedo dependence on snow SSA and impurity content
In order to relate albedo to snow properties, we use the theoretical
formalism of . Several assumptions are made: (i) the
snowpack is horizontally and vertically homogeneous, which means only one
bulk SSA and impurity content value are used to explain albedo variations;
(ii) the surface is flat; (iii) snow phase function and single scattering
albedo are implicitly described by the asymmetry factor, g, the absorption
enhancement parameter, B, and SSA; (iv) the surface and the sensor are
perfectly horizontal. The effect of surface slope is investigated in
Sect. .
Schematic drawing of the tilted snow surface (blue plane) and
associated angles.
The grey plane corresponds to the horizontal plane. The black reference frame is the one attached to the tilted surface.
The grey arrow represents the vertical with respect to the horizontal plane. H(ϕ) is the horizon line in the tilted reference
frame. θs̃ and ϕs̃ are the zenith and azimuth sun angles in the tilted reference frame. θn is the slope elevation and ϕn the aspect.
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Raw measured albedo for a clear-sky day (blue lines,
20 March 2014) and for a cloudy day (red lines,
1 February 2014) at 10:00 (dashed lines), 12:00
(solid lines) and 14:00 (dash-dotted lines).
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(a) Spectral albedo simulated using Eq. ()
for an horizontal surface and Eq. () for tilted surface
(ϕn=300∘, θn=5∘) using estimated diffuse-to-total-irradiance ratio and ϕs and θs for
8 March 2014 at noon. The black lines correspond to
SSA = 40 m2 kg-1 and the blue lines to
SSA = 5 m2 kg-1. (b) Example of measured, predicted
albedo and theoretical albedo on an horizontal surface for three dates in the
season. For the red lines (and blue and black), optimal SSA is
36 m2 kg-1 (6.6, 2.6) and optimal cimp is
0.5 ng g-1 (15.1, 328). A is set to 0.943.
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Under these conditions, following and , snow albedo can be written as
αth(λ,θ)=rdiff(λ,θ)αdiff(λ)+(1-rdiff(λ,θ))αdir(λ,θ),αdiff(λ)=exp(-σ(λ,SSA,cimp)),αdir(λ,θ)=exp(-37(1+2cosθ)σ(λ,SSA,cimp)),σ(λ,SSA,cimp)=64π3λρiceSSA(1-g)2ni(λ)B+3ρicecimpρimpImmimp2-1mimp2+2,
where rdiff(λ,θ) is the diffuse-to-total-irradiance ratio, ρ=917 kg m-3 is the ice density at
0 ∘C and ni(λ) is the imaginary part of the ice
refractive index taken either from (default) or from
. B=1.6 and g=0.85 are constant and taken from
. mimp is the complex refractive index of BC
taken from . ρimp is BC density and is
set to 1270 kg m-3 according to .
cimp is the effective impurity mass per unit of snow mass
(kg kg-1) where effective stands for BC optically equivalent content.
Following , we introduce a scaling factor A to account for
several artefacts and shortcomings in the measurement technique. This scaling
factor relates the measured albedo, αmeas, and the
theoretical albedo obtained from Eq. (),
αth.
αmeas(λ,θ)=Aαth(λ,θ)
Effect of the slope on measured albedo
The slope of the snow surface introduces a change in the solar irradiance
with respect to the solar radiation incoming on a perfectly horizontal
surface. This change is of crucial importance for our application since it is
wavelength dependent. In addition to the change in the effective sun
incident angles, the surface slope modifies (i) the solid angle under which
the sky is viewed from the surface and thus the incoming amount of diffuse
solar radiation and (ii) the total amount of radiation received by the snow
surface since it receives some of the radiation reflected by the adjacent
slopes. The upward radiation measured by the horizontal sensor is also
modified with respect to what would happen if the sensor were parallel to the
surface since part of the field of view sees the atmosphere and not the snow
surface. In the following, we assume that (i) both diffuse solar radiation
and reflected radiation are isotropic and (ii) the surface slope is small and
local enough not to modify significantly the solid angles under which the
incoming and reflected radiations are measured with respect to the solid
angles that would apply in the case of an horizontal surface. With these
assumptions, the slope of the surface only affects the effective sun zenith
and azimuth angles and thus the direct solar irradiance (see details in
Appendices and ) . Note that for fully cloudy
days, under these assumptions, the slope has consequently no influence on the
measured albedo.
Let θs and ϕs be the direction of sun with
respect to a perfectly horizontal surface, θn and ϕn the slope and aspect of the surface, respectively,, and
θs̃ and ϕs̃ the effective
direction of the sun with respect to the tilted surface
(Fig. ). Then
cosθs̃=Kcosθs (e.g.
), where K is the relative change in the cosine of the
sun effective incident angle to the local tilt of the snow surface:
K=cosθn+tanθssinθncos(ϕs-ϕn).
This leads to the following relationship between the measured and the
theoretical albedo on a tilted surface (see details in
Appendices and ):
αmeas(λ,θs)=A(rdiff(λ,θs)αdiff(λ)+(1-rdiff(λ,θs))Kαdir(λ,θs̃)).
The variations of measured spectral albedo with SSA, cimp and
surface slope are illustrated in Fig. a. The effect of the
anisotropy of diffuse solar radiation is discussed in
while the effect of the anisotropy of the reflected radiation is discussed in
and . The anisotropy of the sky
diffuse component and of the reflected radiation is second order as long as
the cosine response correction is small
().
Figure illustrates the raw measured albedo diurnal
cycles for a cloudy day (red lines) and for a clear-sky day (blue lines). As
expected from Eq. (), the diurnal cycle is more pronounced for
the clear-sky day, the albedo evolution being non-symmetric with respect to
solar noon probably due to both slope and changes in snow properties effects.
Albedo variations analysis method
In order to relate variations of spectral albedo to variations of surface
snow properties, we apply the following methodology to the measured albedo.
The main idea of the methodology is to fit Eq. (8) using optimal
parameters for each spectrum. Equation (8) indeed contains four unknowns, namely
A, SSA, cimp and K. For moderate to high amounts of
impurities in snow, several (A, cimp, SSA) triplets can lead
to approximately the same modelled spectrum. For this reason, we chose to
first set the scaling factor A to a constant value for the whole season
(step 1). Optimal SSA, cimp and K are then estimated
for each spectrum (step 2). The diurnal cycles of parameter K are
then used to estimate a daily value for surface slope and aspect
(step 3). Although this step is not required for the estimation of
optimal SSA and cimp, it provides a further verification of the
physical consistency of the methodology. Finally, the measured spectra are
analysed with respect to the presence of liquid water (step 4).
Distribution of the scaling factor, where A is for overcast days. Spectra
used for this analysis corresponds to θs smaller than
65∘ and RMSD between predicted and measured spectrum lower than 0.02
over the 400–1050 nm range.
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Step 1: estimate the scaling factor <inline-formula><mml:math id="M88" display="inline"><mml:mi>A</mml:mi></mml:math></inline-formula>
A seasonal value of A is estimated using Eq. () with
rdiff=1 (selected fully cloudy days from visual inspection of
the photographs and θs smaller than 65∘)
and a non-linear least squares method (provided by Python
scipy.optimize.leastsq function). To avoid avoid multiple solutions (A
and cimp) in case of moderate to high amounts of impurities in
snow, only the spectra from the beginning of season were used for this
estimation.
(a) Illustration of the determination of slope and aspect
(step 3) for clear-sky days. Green diamonds represent the diurnal
cycle of parameter K retrieved after step 2 and blue crosses show
the simulated K using the estimated θn and ϕn from
step 3. The scaling factor was set to A=0.943.
(b) Optimal daily values of θn (in black) and ϕn (in
blue) from step 3. Vertical error bars are obtained using quantiles
25 and 75 of the distribution of A. Daily values with range of aspect
uncertainty larger than 80∘ are discarded. Spectra are also discarded
when the RMSD between predicted and measured spectrum is lower than 0.022 over
the 400–1050 nm range. Measured snow depth is shown in red.
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Effect of liquid water presence on measured spectra. (a)
Example of two measured spectra on 9 March at 09:00 in blue and 13:12 in
green and the associated wavelength of minimum reflectance represented by the
vertical dotted lines. The vertical black line corresponds to
λwater. (b) Distribution of wavelength of
minimum albedo in Eq. (). Spectra are discarded when the RMSD
between predicted and measured spectra is lower than 0.022 over the
400–1050 nm range. λwater is indicated by the black
vertical line.
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Quantiles 25 and 75 of A distribution are used to propagate uncertainties on
the near-surface and surface properties predicted from the spectra.
Step 2: estimate optimal SSA and <inline-formula><mml:math id="M107" display="inline"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mi mathvariant="normal">imp</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula>
Once the value of A is set, optimal SSA and log10(cimp)
values are estimated from the spectrum within 400–1050 nm along with K
using Eq. () and the non-linear least square optimization
method. For cloudy days, the optimization is performed with K=1. Clear-sky
days are selected as days with τ<0.01, τ being estimated using
Eq. (). Illustration of step 2 is provided for three
spectra in Fig. b.
After these steps, spectra are filtered based on the root mean square
deviation (RMSD) between the measured spectrum and the optimal spectrum
calculated with Eq. () with a threshold of 0.022. This
filtering ensures that the measurement artefacts are reasonably accounted for
in the optimal snow surface properties estimation. The threshold value is
higher than the one used in and was set to account for the
discrepancies between the modelled and measured albedo spectra due, among
others, to the impurity type assumption in the model. Indeed, the model used
in this study only includes BC. Alpine snowpacks are frequently
affected by deposition of Saharan dust (e.g. Di Mauro et al., 2015) that has
a reddish spectral signature. The presence of red dust in snow can induce
discrepancies between the modelled spectrum with BC only and the
measured spectrum (Fig. 4 in ). This “dusty” pattern is
clearly visible on the black measured spectrum of Fig. b in
the visible wavelengths (400–500 nm).
Note that by using a seasonal A value for every spectra, we assume that the
measurement artefacts are the same under cloudy and clear-sky conditions
and for varying solar zenith angles.
Step 3: estimate daily optimal surface slope and aspect
Section explains how the measured albedo varies with the
surface slope and aspect. The slope and aspect below the sensor evolves in
time with respect to precipitation, melt and snow transportation by the wind.
Unfortunately no measurement of surface slope and aspect is available during
this winter season. Consequently, using K diurnal cycles and
Eq. (), we estimate daily optimal values of surface slope angle
and aspect for fully clear-sky days.
This step is not required for the estimation of optimal SSA and
cimp but it indirectly validates that K optimization has
not compensated for other artefacts than slope. In other words, estimating
physically consistent values of daily slope and aspect from K diurnal
cycles further ensures the consistency of the spectra correction.
Step 4: detect if the snow surface is wet or dry
Taking benefit of the spectral shift of the absorption feature at 1030 nm in
presence of liquid water , we apply the following method to
distinguish wet from dry snow. The spectra are first averaged using a 20 nm
moving window average in order to reduce noise before minimum calculation. We
then compute the wavelength of minimum albedo in the 1000–1050 nm range and
apply the following criteria to detect liquid water presence:
argminλ∈[1000,1050]nm(α(λ))<λwater.
λwater threshold has been set after studying the
distribution of argmin(α) on the whole spectra dataset.