Snow is an important climate regulator because it greatly
increases the surface albedo of middle and high latitudes of the Earth.
Earth system models (ESMs) often adopt two-stream approximations with
different radiative transfer techniques, the same snow therefore has
different solar radiative properties depending whether it is on land or on
sea ice. Here we intercompare three two-stream algorithms widely used in
snow models, improve their predictions at large zenith angles, and introduce
a hybrid model suitable for all cryospheric surfaces in ESMs. The algorithms
are those employed by the SNow ICe and Aerosol Radiative (SNICAR) module
used in land models, dEdd–AD used in Icepack, the column physics used
in the Los Alamos sea ice model CICE and MPAS-Seaice, and a two-stream
discrete-ordinate (2SD) model. Compared with a 16-stream benchmark model,
the errors in snow visible albedo for a direct-incident beam from all three
two-stream models are small (

Snow cover on land, land ice, and sea ice, modulates the surface energy balance of middle and high latitudes of the Earth, principally because even a thin layer of snow can greatly increase the surface albedo. Integrated over the solar spectrum, the broadband albedo of opaque snow ranges from 0.7 to 0.9 (e.g., Wiscombe and Warren, 1980; Dang et al., 2015). In contrast, the albedo of other natural surfaces is smaller: 0.2, 0.25, and 0.5–0.7 for damp soil, grassland, and bare multi-year sea ice, respectively (Perovich, 1996; Liang et al., 2002; Brandt et al., 2005; Bøggild et al., 2010). The accumulation, evolution, and depletion of snow cover thus modify the seasonal cycle of surface albedo globally. In particular, snow over sea ice absorbs more solar energy and begins to melt in the spring, which forms melt ponds that bring the sea ice albedo to as low as 0.15 to further accelerate ice melt (Light et al., 2008, 2015). An accurate simulation of the shortwave radiative properties of snowpack is therefore crucial for spectrally partitioning solar energy and representing snow–albedo feedbacks across the Earth system. Unfortunately, computational demands and coupling architectures often constrain representation of snowpack radiative processes in Earth system models (ESMs; please refer to Table 1 for all abbreviations used in this work) to relatively crude approximations such as two-stream methods (Wiscombe and Warren, 1980; Toon et al., 1989). In this work, we intercompare two-stream methods widely used in snow models and then introduce a new parameterization that significantly reduces their snowpack reflectance and heating biases at large zenith angles, to produce more realistic behavior in polar regions.

Abbreviations used in this paper and their references. Last access date for all cited URLs in this table is 22 July 2019.

Snow albedo is determined by many factors including the snow grain radius,
the solar zenith angle, cloud transmittance, light-absorbing particles, and
the albedo of underlying ground if snow is optically thin (Wiscombe and
Warren, 1980; Warren and Wiscombe, 1980); it also varies strongly with
wavelength since the ice absorption coefficient varies by 7 orders of
magnitudes across the solar spectrum (Warren and Brandt, 2008). At visible
wavelengths (0.2–0.7

Several parameterizations have been developed to compute the snow solar
properties without solving the radiative transfer equations and some are
incorporated into ESMs or regional models. Marshall and Warren (1987) and Marshall (1989) parameterized snow albedo in both visible and near-IR bands
as functions of snow grain size, solar zenith angle, cloud transmittance,
snow depth, underlying surface albedo, and black carbon content. Marshall
and Oglesby (1994) used this in an ESM. Gardner and Sharp (2010) computed
the all-wave snow albedo with similar inputs. This was incorporated into the
regional climate model RACMO
(

More complex models that explicitly solve the multiple-scattering radiative
transfer equations have also been developed to compute snow solar
properties. Flanner and Zender (2005) developed the SNow Ice and Aerosol
Radiation model (SNICAR) that utilizes two-stream approximations (Wiscombe
and Warren, 1980; Toon et al., 1989) to predict heating and reflectance for a
multilayer snowpack. They implemented SNICAR in the Community Land Model
(CLM) to predict snow albedo and vertically resolved solar absorption for
snow-covered surfaces. Before SNICAR, CLM prescribed snow albedo and
confined all solar absorption to the top snow layer (Flanner and Zender,
2005). Over the past decades, updates and new features have been added to
SNICAR to consider more processes such as black carbon–ice mixing states
(Flanner et al., 2012) and snow grain shape (He et al., 2018b). Concurrent
with the development of SNICAR, Briegleb and Light (2007) improved the
treatment of sea ice solar radiative calculations in the Community Climate
System Model (CCSM). They implemented a different two-stream scheme with
delta-Eddington approximation and the adding–doubling technique (hereafter,
dEdd–AD) that allows CCSM to compute bare, ponded, and snow-covered sea ice albedo
and solar absorption profiles of multilayer sea ice. Before these
improvements, the sea ice albedo was computed based on surface temperature,
snow thickness, and sea ice thickness using averaged sea ice and snow
albedo. dEdd–AD has been adopted by the sea ice physics library Icepack
(

SNICAR and dEdd–AD solve the multiple-scattering radiative transfer
equations and provide much improved solar radiative representations for the
cryosphere, though their separate development and implementation created an
artificial divide for snow simulation. In ESMs that utilize both SNICAR and
dEdd–AD, such as the Community Earth System Model (CESM,

In this paper, we evaluate the accuracy and biases of three two-stream
models listed in Table 2, including the algorithms used in SNICAR and dEdd–AD, for representing reflectance and heating. In Sects. 2–4, we
describe the radiative transfer algorithms and calculations performed in
this work. The results and model intercomparisons are discussed in Sect. 5. In Sect. 6, we introduce a parameterization to reduce the simulated
albedo and heating bias for solar zenith angles larger than 75

Two-stream radiative transfer algorithms evaluated in this work, including algorithms that are currently implemented in Earth system models CESM and E3SM.

In this section, we summarize the three two-stream models and the benchmark DISORT model with 16 streams. These algorithms are well documented in papers by Toon et al. (1989), Briegleb and Light (2007), Jin and Stamnes (1994), and Stamnes et al. (1988). Readers interested in detailed mathematical derivations should refer to those papers. We only include their key equations to illustrate the difference among two-stream models for discussion purposes.

SNICAR is implemented as the default snow shortwave radiative transfer
scheme in CLM and the E3SM land model (ELM). It adopts the two-stream algorithms
and the rapid solver developed by Toon et al. (1989) to compute the solar
properties of multilayer snowpacks. These two-stream algorithms are derived
from the general equation of radiative transfer in a plane-parallel media:

SNICAR itself implements all three two-stream algorithms in Toon et al. (1989): Eddington, quadrature, and hemispheric mean. In practical
simulations, it utilizes the Eddington and hemispheric-mean approximations
to compute the visible and near-IR snow properties, respectively (Flanner et
al., 2007). In addition to its algorithms, SNICAR implements the
delta transform of the fundamental input variable asymmetry factor (

Icepack, CICE, and MPAS-Seaice use the same shortwave radiative scheme
dEdd–AD developed and documented by Briegleb and Light (2007). Sea ice is
divided into multiple layers to first compute the single-layer reflectance
and transmittance using two-stream delta-Eddington solutions to account for
the multiple scattering of light within each layer (Equation set 50,
Briegleb and Light, 2007), where the name “delta” implies dEdd–AD
implements the delta transform to account for the strong forward scattering
of snow and sea ice (Eqs. 2a–2c, Wiscombe and Warren, 1980). The
single-layer direct albedo and transmittance are computed by equations

The computed single-layer reflectance and transmittance of direct and diffuse components are then combined to account for the interlayer scattering of light to compute the reflectance and transmission at every interface (Equation set 51, Briegleb and Light, 2007), and eventually the upward and downward fluxes (Equation set 52, Briegleb and Light, 2007). These upward and downward fluxes at each optical depth are then used to compute the column reflectance and transmittance, and the absorption profiles for any multilayered media, such as snowpacks on land and sea ice.

In nature, a large fraction of sea ice is covered by snow during winter. As
snow melts away in late spring and summer, it exposes bare ice, and melt
ponds form on the ice surface. Such variation in sea ice surface types
requires the shortwave radiative transfer model to be flexible and capable
of capturing the light refraction and reflection. Refractive boundaries
exist where air (refractive index

In this paper, we apply dEdd–AD to snowpacks that can be treated as uniform refractive media such as the land snow columns assumed in SNICAR for model evaluation. An ideal radiative treatment for snow should, however, keep the potential to include refraction for further applications to snow on sea ice or ice sheets. Therefore, in addition to these two widely used algorithms in Icepack and SNICAR, we evaluate a third algorithm (Sect. 2.3) that can be applied to layered media with either uniform or nonuniform refractive indexes.

A refractive boundary also exists between the atmosphere and the ocean, and
models have been developed to solve the radiative transfer problems in the
atmosphere–ocean system using the discrete-ordinate technique (e.g., Jin and
Stamnes, 1994; Lee and Liou, 2007). Similar to the two-stream algorithms of
Toon et al. (1989) used in SNICAR, Jin and Stamnes (1994) also developed
their algorithm from the general equation

For two-stream approximations of this method, analytical solutions of upward
and downward fluxes are coupled at each layer interface to generate

In addition to the mathematical technique, the accuracy and speed of radiative
transfer algorithms depend on the number of angles used for flux estimation
in the upward and downward hemispheres. SNICAR, dEdd–AD, and 2SD use one
angle to represent upward flux and one angle to represent downward flux;
hence they are named the two-stream algorithm. Lee and Liou (2007) use two
upward and two downward streams. Jin and Stamnes (1994) documented the
solutions for any even number of streams. The computational efficiency of
these models is lower than that of two-stream models while their accuracy is
better. To quantify the accuracy of the three two-stream algorithms for snow
shortwave simulations, we use the 16-stream DIScrete-Ordinate Radiative
Transfer model (DISORT) as the benchmark model (

In this work, we focus on the performance of two-stream algorithms for pure
snow simulations. The inputs for these three models are the same:
single-scattering properties (SSPs, i.e., single-scattering albedo

In snow, photon scattering occurs at the air–ice interface, and the
absorption of photons occurs within the ice crystal. The most important
factor that determines snow shortwave properties is the ratio of total
surface area to total mass of snow grains, also known as “the specific surface area”
(e.g., Matzl and Schneebeli, 2006, 2010). The specific surface area (

The input SSPs of snow grains are computed using Mie theory at a fine
spectral resolution for a wide range of ice effective radius

In climate modeling, snow albedo computation at a fine spectral resolution
is expensive and unnecessary. Instead of computing spectrally resolved snow
albedo, wider-band solar properties are more practical. For example, CESM
and E3SM aggregate the narrow RRTMG bands used for the atmospheric radiative
transfer simulation into visible (0.2–0.7

The band albedo

Spectral and total down-welling solar flux at surface computed
using SWNB2 for

The spectral reflectance of pure deep snow computed using two-stream models
and 16-stream DISORT is shown in Fig. 2. The snow grain radius is 100

The spectral albedo of pure snow computed using 16-stream DISORT,
SNICAR, dEdd–AD, and 2SD models, for clear-sky (direct beam at solar zenith
angle 60

In both sky conditions, the errors of snow albedo are larger at near-IR
wavelengths ranging from 1.0 to 1.7

Integrated over the visible and near-IR wavelengths, the error in band albedos computed using two-stream models for different cases is shown in Figs. 3–6.

The difference in direct snow albedo (

Figure 3 shows the error in direct band albedo for fixed snow grain radius
of 100

The difference in direct snow albedo (

For a fixed solar zenith angle of 60

The difference in direct snow albedo (

Figure 6 is similar to Fig. 5, but shows the diffuse snow albedo. In the visible band, SNICAR and dEdd–AD generate similar errors in that they both underestimate the albedo as snow grain size increases and snow depth decreases. 2SD overestimates the albedo with a maximum error of around 0.015. In the near-IR, two-stream models tend to overestimate snow albedo, while the magnitude of biases produced by SNICAR and 2SD is 1 order larger than that of dEdd–AD with the maximum error of 0.035 generated by SNICAR. As a result, the all-wave diffuse albedos computed using dEdd–AD are more accurate than those computed using SNICAR and 2SD.

The difference in diffuse snow albedo (

Figures 7, 8, and 9 show the errors in reflected shortwave flux caused by
snow albedo errors seen in Figs. 3, 4, and 6. In general, two-stream
models produce larger errors in reflected direct near-IR flux (Figs. 7 and 8), especially with the 2SD model: the maximum overestimate of reflected
near-IR flux is 6–8 W m

Error in reflected direct solar flux given albedo errors shown in Fig. 3.

Error in reflected direct solar flux given albedo errors shown in Fig. 4.

Error in reflected diffuse solar flux given albedo errors shown in Fig. 6.

In general, dEdd–AD produces the most accurate albedo and thus reflected
flux for both direct and diffuse components. SNICAR is similar to dEdd–AD
for its accuracy of direct albedo and flux, yet generates large error for
the diffuse component. 2SD tends to overestimate snow albedo and reflected
flux in both direct and diffuse components and shows the largest errors
among three two-stream models. Although the differences between algorithms
are small, they can have a notable impact on snowpack melt. For example,
compared to dEdd–AD, SNICAR and 2SD overestimate the diffuse albedo by

Figure 10 shows absorption profiles of shortwave flux computed using the
16-stream DISORT model, with errors in absorbed fractional solar flux
computed using two-stream models. The snowpack is 10 cm deep and is divided
into five layers, each 2 cm thick. The snow grain radii are set to 100

As shown in the first column of Fig. 10, for new snow with a radius of 100

Comparison of light-absorption profiles derived from two-stream
models and 16-stream DISORT. The left-most column shows fractional band
absorptions computed using 16-stream DISORT. The right three panels show the
errors of all-wave, visible, and near-IR fractional absorptions calculated
using two-stream models. The top and bottom panels are for clear-sky and
cloudy-sky conditions (solar zenith angle of 60

Comparing to 16-stream DISORT, two-stream models underestimate the column solar absorptions for new snow, and they overestimate them for old snow, especially for the surface snow layer and the underground. Overall, dEdd–AD gives the most accurate absorption profiles among the three two-stream models, especially for new snow.

It has been pointed out in previous studies that the two-stream approximations become poor as solar zenith angle approaches 90

Figure 11 shows the direct near-IR albedo and fractional absorption of 2 m thick snowpacks consisting of grains with radii of 100 and
1000

We define and compute

Error in semi-infinite snow albedo computed using dEdd–AD before

When the solar zenith angle exceeds 75

Although the errors of direct near-IR albedos are large for large solar zenith angles, the absolute error in reflected shortwave flux is small (Figs. 7 and 8) as the down-welling solar flux reaches snowpack and decreases as solar zenith angle increases (Fig. 1b). However, such small biases in flux can be important for high latitudes where the solar zenith angle is large for many days in late winter and early spring.

ESMs often use band-averaged SSPs of snow and aerosols for computational efficiency, rather than using brute-force integration of spectral solar properties across each band (per Eq. 11). In addition to using different radiative transfer approximations, SNICAR and dEdd–AD also adopt different methods to derive the band-averaged SSPs of snow for different band schemes.

In SNICAR, snow solar properties are computed for five bands: one visible band
(0.3–0.7

The band-averaged SSPs of snow grains are computed following the
Chandrasekhar mean approach (Thomas and Stamnes, 1999, their Eq. 9.27;
Flanner et al., 2007). Specifically, spectral SSPs of snow grains are
weighted into bands according to surface incident solar flux typical of
midlatitude winter for clear- and cloudy-sky conditions. In addition, the
single-scattering albedo

In dEdd–AD, the snow-covered sea ice properties are computed for three bands:
one visible band (0.3–07

SNICAR and dEdd–AD also use different approaches to avoid numerical
singularities. In SNICAR, singularities occur when the denominator of term

Based on the intercomparison of three two-stream algorithms and their implementations in ESMs, we formulated the following surface shortwave radiative transfer recommendations for an accurate, fast, and consistent treatment for snow on land, land ice, and sea ice in ESMs.

First, the two-stream delta-Eddington adding–doubling algorithm by Briegleb
and Light (2007) is unsurpassed as a radiative transfer core. The evaluation
in Sect. 5 shows that this algorithm produces the least error for snow
albedo and solar absorption within snowpack, especially under overcast
skies. This algorithm applies well to both uniformly refractive media such
as snow on land, and to nonuniformly refractive media, such as
bare, snow-covered, and ponded sea ice and bare and snow-covered land ice. Numerical
singularities occur only rarely (when

Second, any two-stream cryospheric radiative transfer model can incorporate
the parameterization described in Sect. 6 to adjust the low bias of direct
near-IR snow albedo and high bias of direct near-IR solar absorption in
snow, for solar zenith angles larger than 75

Third, in a cryospheric radiative transfer model, one should prefer physically based parameterizations that are extensible and convergent (e.g., with increasing spectral resolution) for the band-averaged SSPs and size distribution of snow. Although the treatments used in SNICAR and dEdd–AD are both practical since they both reproduce the narrowband solar properties with carefully derived band-averaged inputs as discussed in Sect. 7, the snow treatment used in SNICAR is more physically based and reproducible since it does not rely on subjective adjustment and empirical coefficients as used in dEdd–AD. Specifically, the empirical adjustment to snow grain radius implemented in dEdd–AD may not always produce compensating errors. For example, in snow containing light-absorbing impurities such adjustment may also lead to biases in aerosol absorption since the albedo reduction caused by light-absorbing particles does not linearly depend on snow grain radius (Dang et al., 2015). For further model development incorporating nonspherical snow grain shapes (Dang et al., 2016; He et al., 2018a, b), such adjustment on grain radius may fail as well. Moreover, SNICAR computes the snow properties for four near-IR bands, which helps capture the spectral variation in albedo (Fig. 2) and therefore better represents near-IR solar properties. It is also worth noting that unlike the radiative core of dEdd–AD, SNICAR is actively maintained, with numerous modifications and updates in the past decade (e.g., Flanner et al., 2012; He et al., 2018b). Snow radiative treatments that follow SNICAR conventions for SSPs may take advantage of these updates. Note that any radiative core that follows SNICAR SSP conventions must be called twice to compute diffuse and direct solar properties.

Fourth, a surface cryospheric radiative transfer model should flexibly
accommodate coupled simulations with distinct atmospheric and surface
spectral grids. Both the five-band scheme used in SNICAR and the three-band scheme
used in dEdd–AD separate the visible from near-IR spectrum at 0.7

Last, it is important to note that, although we only examine the performance of the dEdd–AD for pure snow in this work, this algorithm can be applied to the surface solar calculation of all cryospheric components with or without light-absorbing particles present. First, Briegleb and Light (2007) proved its accuracy for simulating ponded and bare sea ice solar properties against observations and a Monte Carlo radiation model. Second, In CESM and E3SM, the radiative transfer simulation of snow on land ice is carried out by SNICAR with prescribed land ice albedo. Adopting the dEdd–AD radiative core in SNICAR will permit these ESMs to couple the snow and land ice as a nonuniformly refractive column for more accurate solar computations since bare, snow-covered, and ponded land ice is physically similar to bare, snow-covered, and ponded sea ice, and the latter is already treated well by the dEdd–AD radiative transfer core. Third, adding light-absorbing particles in snow will not change our results qualitatively. Both dEdd–AD and SNICAR simulate the impact of light-absorbing particles (black carbon and dust) on snow and/or sea ice using self-consistent particle SSPs that follow the SNICAR convention (e.g., Flanner et al., 2007; Holland et al., 2012). These particles are assumed to be either internally or externally mixed with snow crystals; the combined SSPs of mixtures (e.g., Appendix A of Dang et al., 2015) are then used as the inputs for radiative transfer calculation. The adoption of the dEdd–AD radiative transfer algorithm in SNICAR, and the implementation of SNICAR snow SSPs in dEdd–AD enables a consistent simulation of the radiative effects of light-absorbing particles in the cryosphere across ESM components.

In summary, this intercomparison and evaluation has shown multiple ways
that the solar properties of cryospheric surfaces can be improved in the
current generation of ESMs. We have merged these findings into a hybrid
model SNICAR-AD, which is primarily composed of the radiative transfer
scheme of dEdd–AD, five-band snow–aerosol SSPs of SNICAR, and the
parameterization to correct for snow albedo biases when solar zenith angle
exceeds 75

SNICAR-AD is now implemented in both the sea ice (MPAS-Seaice) and land (ELM) components of E3SM. More simulations and analyses are underway to examine its impact on E3SM model performance and simulated climate. The results are however beyond the scope of this work and will be thoroughly discussed in a future paper.

In this work, we aim to improve and unify the solar radiative transfer
calculations for snow on land and snow on sea ice in ESMs by evaluating the
following two-stream radiative transfer algorithms: the two-stream
delta-Eddington adding–doubling algorithm dEdd–AD implemented in sea ice
models Icepack, CICE, and MPAS-Seaice, the two-stream delta-Eddington and
two-stream delta-Hemispheric-Mean algorithms implemented in snow model
SNICAR, and a two-stream delta-discrete-ordinate algorithm. Among these
three models, dEdd–AD produces the most accurate snow albedo and solar
absorption (Sect. 5). All two-stream models underestimate near-IR snow
albedo and overestimate near-IR absorption when solar zenith angles are
larger than 75

The data and models are available upon request to Cheng Dang (cdang5@uci.edu). SNICAR and dEdd–AD radiative transfer core can be found at

CD and CZ designed the study. CD coded the offline dEdd-AD and 2SD schemes, performed two-stream and 16-stream model simulations, and wrote the paper with input from CZ and MF. CZ performed the SWNB2 simulations. MF provided the base SNICAR code and snow optical inputs.

The authors declare that they have no conflict of interest.

The authors thank Stephen G. Warren and Qiang Fu for insightful discussions on radiative transfer algorithms. The authors thank Adrian Turner for instructions on installing and running MPAS-Seaice. The authors thank David Bailey and the one anonymous reviewer for their constructive comments that improved our paper. This research is supported as part of the Energy Exascale Earth System Model (E3SM) project, funded by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research.

This research has been supported by the U.S. Department of Energy (grant no. DE-SC0012998).

This paper was edited by Dirk Notz and reviewed by David Bailey and one anonymous referee.