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 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 the irradiance of the reference profile to that of the simulation (absolute difference

Solar irradiance is an essential source of energy to snow and ice surfaces

In addition to the above-mentioned requirements, accuracy in the estimation of the energy absorbed at the snow surface can be achieved through spectral calculation of the albedo but remains numerically expensive. This also requires spectral calculations of the solar irradiance that are not available most of the time in climate models. This is usually overcome in most global and regional climate models by computing broadband or narrowband albedo to estimate the energy budget at the snow and ice surfaces

To overcome these uncertainties while maintaining an adequate calculation time to remain competitive, new methods are developed. One of them, recently developed by

Here, we describe a novel method for accurately calculating the solar energy absorbed by the snowpack based on the determination of spectrally fixed radiative variables. The method is named VALHALLA for Versatile ALbedo calculation metHod based on spectrALLy fixed radiative vAriables (version 1.0). This method maintains adequate accuracy of absorbed energy values while reducing calculation time irrespective of the radiative transfer scheme used for the atmosphere. While VALHALLA like SNOWBAL is a coupling scheme, VALHALLA fulfils a different niche than SNOWBAL since it allows accurate calculation when only broadband atmospheric inputs are available and accounts for snow property variations. SNOWBAL requires accurate snow radiative transfer calculations for a limited number of wavelengths and an adequate representation of the atmosphere, i.e. cloud content, water vapour, SZA, direct-to-diffuse irradiance ratio. VALHALLA requires accurate radiative transfer calculations for both snow and the atmosphere for a limited number of wavelengths. The proposed method takes advantage of the spectral characteristics of incident radiation and optical snow properties based on the analytical approximation of the radiative transfer within the snowpack provided by

The VALHALLA method relies on accurate calculations of the solar radiation absorbed by the snowpack for a small number of selected wavelengths, named tie points (TPs) in the following. The number of tie points is kept as small as possible to limit the computing resources. Between these tie points, the VALHALLA method interpolates the absorbed radiation based on kernel functions that reflect the main spectral variations of the absorbed radiation across the solar spectrum. The general reasoning of the method consists of assuming that the spectral variation between tie points can be approximated using the refractive index of ice. The calculation of the absorbed radiation at the tie points can be performed with any radiative transfer model. In the following, we selected the Two-streAm Radiative TransfEr in Snow (TARTES;

TARTES calculates spectral albedo in a multilayer snowpack when the physical properties of each layer and the angular and spectral characteristics of the radiation are known

We used two shape parameters that are relevant for the optical properties of snow: the asymmetry parameter

In

The model Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART;

The spectral direct albedo, also called directional–hemispherical reflectance,

When neglecting the spectral variations due to the presence of LAPs in Eq. (

The fraction of absorbed energy in the snowpack with respect to the incoming energy,

For the atmosphere, we use the Beer–Lambert law to express the first-order spectral variations of the incoming solar radiation. The Beer–Lambert law establishes a relationship between the radiation transmitted through a given medium

As a consequence, we assume that the absorbed energy by a snowpack for a given wavelength

The VALHALLA method is based on precise calculation of the absorbed energy at the tie points (TPs) and on the interpolation between these wavelengths based on the general shape of the spectrum given in the equation above (Eq.

The method uses a reference irradiance profile with a spectral resolution of 1 nm,

For each TP, the absorbed energy and irradiance are calculated using TARTES–SBDART and used for determining the values of variables

Between two tie points

To determine these variables, which take into account all snow and illumination properties, an optimization by the least-square method is used. Indeed,

In the context of optimization, variable

The optimization is realized on the variable

Namely, an optimization method is used to solve Eq. (

Spectral positions of TPs on an example of an absorbed energy profile for a snowpack without LAPs.

The tie points, TPs, are the reference wavelengths for absorbed energy and total irradiance. For all types of snow and cloud cover, a total of 30 TPs are selected as a compromise between accuracy and computational time (Fig.

To account for a representative set of atmospheric conditions, different reference irradiance profiles depending on SZA and cloud cover are chosen. These profiles are calculated by SBDART simulations with a spectral resolution of 1 nm for two cloud cover types. For simulations of clear-sky and partly cloudy conditions, reference irradiance profiles with values of

Atmospheric parameters of reference irradiance profiles. Each irradiance profile is calculated for eight values of SZA and two values of

The main SBDART input parameters used in this study are the aerosol optical depth (AOD), cloud layer optical depth (

Atmospheric parameters of simulations. Each irradiance profile is calculated for eight values of the SZA, five values of IAER, three values of the AOD, and five values of

The main TARTES input parameters used in this study are the surface specific area (SSA) of the first layer of the snowpack, the snow water equivalent (SWE) for each layer of the snowpack, and the LAP concentration. We consider a snowpack with three layers of varying thickness and density (Table

Snow properties of simulations. The spectral albedo is calculated for eight values of SZA and five values of SSA for the snowpack first layer. For snow without LAPs, the SWE values are provided for the first three layers of the snowpack. For snow with LAPs, the SWE and the light-absorbing particle concentration (for soot and dust) are provided for the first two layers of the snowpack. For layer 3, the values of all input parameters, besides soot and dust contents, are constant. For layer 4, all input parameters are constant.

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.

Figure

Median error of the broadband absorbed energy for varying SZA, the

Figure

Figure

Bias on the broadband absorbed energy

Figure

Figure

Bias on the broadband absorbed energy

Figure

Figure

Bias on the broadband absorbed energy

In Fig.

Errors on the broadband albedo for different constant spectral resolutions (left) and comparison with the errors of our method (right). For these resolutions, the broadband albedo is computed by TARTES–SBDART and is compared to the one computed at 1 nm resolution. The red lines indicate the median (same as in Fig.

We presented 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 energy reaching the ground and snow albedo, such as

We have shown that the absorbed energy calculated by VALHALLA is very sensitive to

The accuracy of the method is sensitive to the locations and to the number of TPs. An increase or a decrease in the number of TPs could improve or alter the representation of the absorbed energy. Using an overly large number of TPs leads to a decrease in the calculated error but increases the calculation time, especially if the TP number is increased at the beginning of the spectrum to compensate for the oscillations of the absorbed energy when the snowpack contains LAPs. For a lower TP number, the oscillations at the beginning of the spectrum due to LAPs are not well represented by the method, and this leads to a significant increase in the error. With the use of 15 TPs, the error on the broadband albedo increases globally by a factor of 10 to 15 for snow containing LAPs. With 10 TPs, the error increases by a factor of 25 and 50 for the same type of snow. The effect of LAPs on the absorbed energy is therefore poorly represented when the number of TPs is too low. The use of 30 TPs is therefore a good compromise between precision for snowpacks containing LAPs and calculation time.

The SNOWBAL coupling scheme from

VALHALLA and SNOWBAL fulfil two different niches. SNOWBAL indeed required accurate snow radiative transfer calculation and accurate atmospheric conditions (cloud water content, direct-to-diffuse irradiance radiation, etc.). VALHALLA requires both snow and atmosphere radiative transfer calculations for the TPs. This difference together with the need for more than 15 TPs implies that the computational cost of VALHALLA is higher than the computational cost of SNOWBAL. However, the accuracy of the SNOWBAL methods depends on the number and range of the narrowband solar radiation available. SNOWBAL accuracy increases when the sub-band spectral variability is reduced. Here, we used the VALHALLA method with broadband solar irradiance inputs, i.e. the worst case. The method was also tested with narrowband solar radiation inputs (from AROME, not shown;

The VALHALLA method has been developed to provide accurate calculation of the solar energy absorbed by the snowpack at low computational cost compared to full spectral calculation. The VALHALLA method requires accurate calculation of the spectral absorbed energy for the TPs. In the study, this is based on the TARTES and SBDART models, but any other radiative model could be used (e.g. SNICAR for snow,

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 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 (

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 of 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 terms of 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 into 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.

The VALHALLA v1.0 development and data presented and described in this article are available for download at

TARTES is available as a python module at

FV, MD, and MF started this project and developed the method. FV ran the simulations and wrote the first draft of the paper. FV, MD, and CA performed the analysis. All authors discussed and revised the paper.

The authors declare that they have no conflict of interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The authors are grateful to Ghislain Picard and Quentin Libois for discussion on the VALHALLA method. The authors are also grateful to the two referees, Joseph Cook and Christiaan Van Dalum, for very helpful and relevant comments on the paper.

CNRM/CEN is part of Labex OSUG@2020 (ANR-10-LABX-0056). This work was partly funded by CNES APR MIOSOTIS and by ANR grant EBONI (ANR-16-CE01-0006). Marie Dumont has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (IVORI, grant agreement no. 949516).

This paper was edited by Fabien Maussion and reviewed by Christiaan van Dalum and Joseph Cook.