A versatile method for computing optimized snow albedo from spectrally ﬁxed radiative variables : VALHALLA v1.0

. In climate models, the snow albedo scheme generally calculates only a narrowband or broadband albedo, which leads to signiﬁcant uncertainties. Here, we present the Versatile ALbedo calculation metHod based on spectrALLy ﬁxed radiative vAriables (VALHALLA, version 1.0), to optimize spectral snow albedo calculation. For this optimization, the energy absorbed by the snowpack is calculated by the spectral albedo model Two-streAm Radiative TransfEr in Snow (TARTES) and the spectral irradiance model Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). This calculation takes into 5 account the spectral characteristics of the incident radiation and the optical properties of the snow, based on an analytical approximation of the radiative transfer of snow. For this method, 30 wavelengths, called tie points ( tps ), and 16 reference irradiance proﬁles are calculated to incorporate the absorbed energy and the reference irradiance. The absorbed energy is then interpolated for each wavelength between two tps with adequate kernel functions derived from radiative transfer theory for snow and the atmosphere. We show that the accuracy of the absorbed energy calculation primarily depends on the adaptation of 10 the irradiance of the reference proﬁle to that of the simulation (absolute difference < 1 Wm − 2 for broadband absorbed energy and absolute difference < 0.005 for broadband albedo). In addition to the performance in accuracy and calculation time, the method is adaptable to any atmospheric input (broadband, narrowband), and is easily adaptable for integration into a novel for calculating large spectral bands. The low spectral resolution of these ﬂuxes therefore leads to uncertainties in the determination of radiative variables such as snow albedo that are key for energy exchanges at the surface. This study presents a new method VALHALLA for calculating the spectral albedo of snow based on the determination of key atmospheric and snow variables explaining variations in absorbed energy using spectrally ﬁxed variables. For this method, tie points ( tps ) and reference irradiance proﬁles are calculated to incorporate the absorbed

In Tuzet et al. (2017) and later studies, TARTES was used for calculations of radiative transfer with a spectral resolution of 20 nm. This resolution is the best compromise between the accuracy of radiation and calculation time, which is still very important, and makes this model configuration computationally expensive. 115 2.2 SBDART 2.1.1 :::::::: Radiative :::::::: transfer :: in ::: the ::::::::::: atmosphere, :::::::: SBDART The model Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART, Ricchiazzi et al., 1998b) is used for radiative transfer calculation in clear-sky and cloudy conditions in the atmosphere. SBDART uses Discrete Ordinate Radiative Transfer (DISORT, Stamnes et al., 1988) to solve the radiative transfer equation in the atmosphere vertically homogeneous. This model 120 is organized to permit up to 65 atmospheric layers and 40 radiation streams. The main input parameters used in this study are the aerosol optical depth (AOD), the cloud-layer optical depth (τ ), the boundary layer aerosol type selector (IAER) and SZA.
Between two tie-points tp n and tp n+1 , we assume that the absorbed energy can be approximated by : To determine these variables, which take into account all snow and illumination properties, an optimization by the leastsquare method is used. Indeed, D and J are mutually dependent.

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In the context of optimization, variable D is written in : with : and J : ::::::::::::::::: The optimization is realised on the variable G tpn+1 tpn and uses absorbed energy E abs and total irradiance E ref for tp n+1 .

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In this section, we compare the simulated broadband absorbed energy resulting from VALHALLA for 30 tps with that obtained with TARTES-SBDART for the same spectral range between 320 and 4000 nm. We first analyse the impact of incident solar radiation, cloud cover conditions and snow properties on the errors in the estimated absorbed energy and albedo. The efficiency of the method is then compared to the TARTES-SBDART calculation for different spectral resolutions ranging from 1 nm (reference simulations) to 100 nm. erties, the median error on absorbed energy exhibits a stronger sensitivity to τ than to AOD. The median errors are small for the light absorbing particles concentration (for soot and dust) are provided for the first two layers of the snowpack. For layer 3(not true for snow without LAPs) and layer 4, the values of all input parameters, :::::: besides ::: soot :::: and ::: dust ::::::: contents, : are fixed :::::: constant. :: For :::: layer :: 4, ::: all :::: input :::::::: parameters ::: are :::::: constant. Soot concentration (ng g :: ng ::: g −1 ) 0, 100, 200 Dust concentration(ng g :: ng ::: g −1 ) 0, 25000, 50000 kg :::: m −2 ) 600 Soot concentration (ng g :: ng ::: g −1 ) 0 Dust concentration(ng g :: ng ::: g −1 ) 0 values of τ equal to 0.1, 0.5 and 10 (absolute difference < 1 W m −2 ) but remain larger for values equal to 0.0, 0.9 and 5.0 (e.g. median error = 3.6 W m −2 for τ = 5 and SZA = 10°). Errors are lower when using an adequate reference irradiance profile (τ of simulation (τ simu ) equal to τ of reference (τ ref ), τ = 0.5 and 10) and the calculated absorbed energy is therefore very sensitive to τ (median errors between 1.5 and -3.6 W m −2 ). Regarding AOD, the median errors are small (absolute difference 255 < 0.5 W m −2 ) and show little changes with τ . This demonstrates that AOD exerts a very small influence on the median error and thus on the calculation of the energy absorbed by the method. Concerning the properties of the snow cover, the SSA value of the first layer has little impact on the error on : of : the absorbed energy. For the different SSA values, the median errors are small (absolute difference < 0.5 W m −2 ) and vary little depending on the value studied. The presence of LAPs in the snowpack leads to an increase in the median error (absolute difference < 1 W m −2 ) compared to pure snow (absolute difference < 0.1 260 W m −2 ). Overall the method slightly overestimates the energy absorbed by the snowpack (mostly positive errors). The error is not very sensitive to the physical properties of the snowpack and to the AOD. However, the error is very sensitive to τ of the simulations and thus to the τ chosen for the reference profile. The sensitivity to cloud conditions is investigated in more details in the next section.
We presented a calculation method VALHALLA :: the :::::::::::: VALHALLA :::::: method : for calculating absorbed energy and albedo based on a calculation of the main variables explaining the variations in absorbed energy using spectrally fixed radiative variables.
We determined 30 tps, corresponding to the local minima and maxima of the absorbed energy at which the exact calculation of the absorbed energy is performed. In addition, we used 16 different reference irradiance profiles to interpolate between these tps. We evaluated the accuracy of the method for several atmospheric and snow properties that influence the amount of 345 energy reaching the ground and snow albedo, such as τ , AOD, SSA and LAPs content. We have shown that absorbed energy and albedo errors due to the use of this method are small (absolute difference < 1 W m −2 for broadband absorbed energy and absolute difference < 0.005 for broadband albedo) and correspond to a factor 6 in terms of computation times compared to calculations made at 20 nm resolution.

Conclusions
In climate models, energy fluxes are most often given for narrow and large spectral bands. The low spectral resolution of these fluxes therefore leads to uncertainties in the determination of radiative variables such as snow albedo that are key for energy exchanges at the surface. This study presents a new method VALHALLA for calculating the spectral albedo of snow based on the determination of key atmospheric and snow variables explaining variations in absorbed energy using spectrally 435 fixed variables. For this method, tie points (tps) and reference irradiance profiles are calculated to incorporate the absorbed energy and the reference irradiance. The absorbed energy is then interpolated for each wavelength present between two tps with adequate kernel functions derived from radiative transfer theory for snow and the atmosphere.
For the different properties of the atmosphere and snow studied, the cloud-layer optical depth (τ ) and the LAP content of the snow cover are the main variables influencing the calculation of the absorbed energy by the method. Indeed, when the value 440 of τ of the simulation is equal to that of the reference irradiance profile, the method converges towards a value of absorbed energy close to that calculated as a reference. On the other hand, when this value is not equal to that of the reference profile, differences in absorbed energy are noticeable at certain wavelengths. For snowpacks containing LAPs, the method encounters difficulties in representing the variation in absorbed energy at the beginning of the spectrum and therefore generates significant differences in energy. For the other properties studied, the variables influencing the amount of energy absorbed by the snow, 445 such as SZA and SSA, influence the calculation by the methodwhen the reference irradiance profile is inadequate ::: The ::: use ::: of ::::::: reference ::::::: profiles :::: with :: an :::::::: adequate ::::: value :: of :::: SZA :: is :::::::: necessary :: to ::: the ::::: good ::::::: accuracy ::: of ::: the :::::: method.
The VALHALLA method, therefore, determines the absorbed energy for all wavelengths between 320 and 4000 nm using 30 tps. This number of tps is necessary for a good representation of the absorbed energy when the snow contains LAPs.
Despite an overestimation of the energy absorbed by the method, the results obtained with 30 tps are similar to the results 450 of a TARTES-SBDART at 20 nm. This results in a reduction of the calculation time by a factor of 6 (30 tps versus 180 wavelengths). In addition to the performance in calculation time, the method is versatile and adaptable to any atmospheric input (broadband, narrowband).
In conclusion, the development of the method VALHALLA presented here allows a considerable reduction in calculation time while maintaining a good representation of the spectral albedo. One of the perspectives would be to integrate this method 455 in a radiative scheme of a global or regional climate model in order to drastically reduce the calculation time and to largely improve the albedo calculation compared to more common broadband and/or narrowband calculations.
Code and data availability. The VALHALLA v1.0 development and data presented and described in this article (Veillon et al., 2020) is available for download at https://doi.org/10.5281/zenodo.4570565.
TARTES is freely avaibable on the website: http://pp.ige-grenoble.fr/pageperso/picardgh/tartes/. SBDART is freely available on the web- Competing interests. The authors declare that they have no conflict of interests.