Parameterizations of Snow Cover, Snow Albedo and Snow Density in Land Surface Models: A Comparative Review

Abstract

Snow plays a vital role in the interaction between land and atmosphere in the state-of-the-art land surface models (LSMs) and the real world. While snow plays a crucial role as a boundary condition in meteorological applications and serves as a vital water resource in certain regions, the acquisition of its observational data poses significant challenges. An effective alternative lies in utilizing simulation data generated by Land Surface Models (LSMs), which accurately calculate the snow-related physical processes. The LSMs show significant differences in the complexities of the snow parameterizations in terms of variables and processes considered. In this regard, the synthetic intercomparisons of the snow physics in the LSMs can give insight for further improvement of each LSM. This study revealed and discussed the differences in the parameterizations among LSMs related to snow cover fraction, albedo, and snow density. We selected the most popular and well-documented LSMs embedded in the earth system models or operational forecasting systems. We examined single-layer schemes, including the Unified Noah Land Surface Model (Noah LSM), the Hydrology Tiled ECMWF Scheme of Surface Exchanges over Land (HTESSEL), the Biosphere-Atmosphere Transfer Scheme (BATS), the Canadian Land Surface Scheme (CLASS), the University of Torino land surface Process Interaction model in Atmosphere (UTOPIA), and multilayer schemes of intermediate complexity including the Community Noah Land Surface Model with Multi-Parameterization Options (Noah-MP), the Community Land Model version 5 (CLM5), the Joint UK Land Environment Simulator (JULES), and the Interaction Soil-Biosphere-Atmosphere (ISBA). Through the comparison analysis, we emphasized that inclusion of geomorphic and vegetation-related variables such as elevation, slope, time-varying roughness length, and vegetation indexes as well as optimized parameters for specific regions, in the snow-related physical processes, are crucial for further improvement of the LSMs.

1 Introduction

Physical processes related to snow play an essential role in interacting with the heat and moisture flux between the atmosphere and the land surface, as well as between the land surface and soil. In addition, snow is a significant water resource for the high mountainous area (Deng et al. , 2019) or semi-arid region (Sankey et al. , 2015). For these reasons, accurate observation or modeling of snow amount or snow cover distribution is required to predict snow-related hazards or snow-affected meteorological conditions or to manage water resources effectively. However, the snow’s spatial and temporal heterogeneity causes difficulties in observation, understanding, and modeling (Webster et al. , 2018).

Measuring changes in snow cover and snow albedo poses significant challenges directly from ground observation systems. Moreover, the spatial measurement of snow depth is a non-trivial task. Therefore, there are efforts to enhance the observation of snow data. For instance, Hedrick et al. (2015) used regional-scale lidar-derived measurements, Fernandes et al. (2018) measured the change of snowpack using lightweight unoccupied vehicle videos, and López-Moreno et al. (2011) discussed sampling strategies and average depth measurements related to snow depth data acquisition. Land surface models (LSMs), in particular, offer a distinct advantage in capturing changes in snow elements by incorporating modeling variables related to snow, as ground observation are unable to directly measure these specific snow elements.

Snow cover plays a vital role in surface albedo, which leads to the exchange of energy and water flux between land and atmosphere. If the snow cover fractions are underestimated, the land surface albedo will be underestimated too (Roesch et al. , 2001), and vice versa. In the spring and in the first part of summer, when solar radiation is at its most intense, snow cover significantly affects the global radiation balance, and snow cover change significantly impacts nature and human systems. Snow is a crucial water source in rivers and groundwater, particularly in semi-arid regions (Armstrong and Brun , 2008). Additionally, given the value of snow cover as a recreational and travel resource in mid-latitude mountainous regions, it becomes crucial to monitor snow cover changes accurately.

Besides, snow albedo is an important physical property that causes energy interactions between the land surface and the atmosphere, along with the thermal barrier properties of snow and the possibility of phase changes due to latent heat (Armstrong and Brun , 2008). In particular, albedo — the reflectivity of the ground during incident solar radiation — is vital for the land surface’s energy balance (Warren and Wiscombe , 1981). Wiscombe and Warren (1980) and Warren and Wiscombe (1980) created early models of snow albedo, and many researchers have improved the snow albedo simulation for attaining the continuous snow data (Zhong et al. , 2017; He et al. , 2021; Usha et al. , 2022). Even though the remotely sensed snow albedo data is more accurate than BATS aged-based model (Molotch and Bales , 2006; Bair et al. , 2019), snow albedo simulation in LSM remains a valuable approach, offering a straightforward means of interpolation between periods of in situ or remotely sensed measurements. Even if we attain the snow albedo simulation data from the LSM, its performance strongly relies on the quality of the input data. For instance, input data can be under/over-estimating precipitation. Second, the simulated grid point may have an elevation different from the station point, and these differences can influence the simulation result of LSM.

Since snow depth serves as a direct indicator of snowfall amount, it is imperative to predict or observe accurately to prevent accidents caused by snow. For example, snow can lead to delays for both ground and air traffic and heavy snow disasters for crops and livestock. In addition, it causes accidents, injuries, and fatalities resulting from snowfall or snow avalanches (Armstrong and Brun , 2008). Meanwhile, snow density is closely related to the snow depth in terms of timely changing variables in the LSM modeling. Therefore, parameterization method employed by LSMs to estimate snow density and depth using other atmospheric variables can aid in preventing snow-related hazards in areas with lack or scarcity of observations.

As a result of analyzing previous studies, there were studies related to model development or improvement for one or two LSM models. For example, there were snow cover studies (Brun et al. , 1992; Yang et al. , 1997; Roesch et al. , 2001; Brown et al. , 2006; Roesch and Roeckner , 2006; Brun et al. , 2008; Dutra et al. , 2009; Ma et al. , 2019), and snow albedo studies (Roesch and Roeckner , 2006; Mölders et al. , 2007; Dutra et al. , 2009; Wang and Zeng , 2010; Ma et al. , 2014; Bartlett and Verseghy , 2015; Zhong et al. , 2017; Wang et al. , 2020), and studies related to snow density (Gottlib , 1980; Longley , 1960; Dutra et al. , 2009; Dawson et al. , 2017). As a study that directly compared the snow model, Pedersen and Winther (2005) compared and verified the snow albedo parameterization method, and Voordendag et al. (2021) mainly made a simulation comparison of the snow depth change and sublimation process. Mott et al. (2018) reviewed seasonal changes in wind-induced snow cover at three different scales. He et al. (2020) conducted a review on the possibility of improvement by comparing models, including the land surface model for soil thermal conductivity parameterization schemes.

Large-scale projects that conducted model comparison experiments on snow include the Program for Intercomparison of Land-Surface Parameterization Schemes (PILPS) Phase 2d (Pitman and Henderson-Sellers , 1998; Slater et al. , 2001), Phase 2e (Bowling et al. , 2003), Phase 1 of the Snow Model Intercomparison Project (SnowMIP1; Etchevers et al. , 2004), and Phase 2 (SnowMIP2; Essery et al. , 2009; Rutter et al. , 2009). These projects have shown that large errors occur in warmer winters or warmer regions concerning the time of snow melting in the simulation of snow cover changes (Krinner et al. , 2018). In addition, they suggested that more temporal and spatial data could improve their understanding of the snow model and reduce uncertainty about the climate feedback process (Menard et al. , 2021). The Earth System Model-Snow Model Intercomparison Project (ESM-SnowMIP; Krinner et al. , 2018) is an extension of the Land Surface, Snow and Soil Moisture Model Intercomparison Project (LS3MIP; van den Hurk et al. , 2016) at a local scale. The aim was to evaluate snow models and quantify snow-related climate feedback (Krinner et al. , 2018). Menard et al. (2021) compared 27 models of ESM-SnowMIP and compared the model ranking through the Normalized Root Mean Square Error (NRMSE) of Snow Water Equivalent (SWE) and surface temperature and the bias of SWE, surface temperature, albedo, and soil temperature.

Although the above projects have performed model verification and performance evaluation, systematic analysis of the types and complexity of variables used in the parameterization of the snow physics process remains insufficient. Menard et al. (2021) gave a summarized comparison of snow-related characteristics and their references briefly for twenty-seven models from 22 modeling teams. Even if their work was valuable and insightful, we needed to focus on the listing and comparing the considered variables for main snow related parameterizations. Since each model’s documentation in ESM-SnowMIP is not perfect or even not provided, we decided to review the following LSMs that have comprehensive and informative documentation for analyzing the snow-related parameterization: 1) the Community Land Model version 5 (CLM5; Lawrence et al. , 2018) — the land model for the Community Earth System Model (CESM; Danabasoglu et al. , 2020); 2) the Hydrology Tiled ECMWF Scheme of Surface Exchanges over Land (HTESSEL; Dutra et al. , 2009) — the operational land model for the European Centre for Medium-Range Weather Forecasts (ECMWF); 3) the Canadian Land Surface Scheme (CLASS; Verseghy , 2012) — the land surface schemes in the Canadian Earth system model (CanESM; Swart et al. , 2019) developed by the Environment and Climate Change Canada (ECCC); 4) the Joint UK Land Environment Simulator (JULES; Best et al. , 2011) — the land surface schemes in both the physical climate model HadGEM3-GC3 (Menary et al. , 2018) and the United Kingdom’s Earth System Model (UKESM; Sellar et al. , 2019); and 5) the Interaction Soil-Biosphere-Atmosphere (ISBA; Decharme et al. , 2016), which constitutes the ‘nature’ tile of the platform SURFEX (SURFace EXternalisée; Masson et al. , 2013) as the externalized land surface scheme in the Centre National de Recherches Météorologique Coupled global climate Model (CNRM-CM; Voldoire et al. , 2012).

Additionally, we examined the LSMs that are not included in the ESM-SnowMIP: 1) the Unified Noah Land Surface Model (Noah LSM; Koren et al. , 1999) — the operational land surface model of the National Centers for Environmental Prediction (NCEP); 2) the Community Noah Land Surface Model with Multi-Parameterization Options (Noah-MP; Niu et al. , 2011); 3) the Biosphere-Atmosphere Transfer Scheme (BATS; Dickinson et al. , 1993); and 4) the University of Torino land surface Process Interaction model in Atmosphere (UTOPIA; Cassardo , 2015). The Noah LSM and Noah-MP are the current and future generations of Korean Integrated Model (KIM; Hong et al. , 2018) operational forecasting systems of the Republic of Korea, respectively. In addition, we analyzed BATS model because albedo parameterization of the Common Land Model (CoLM; Li et al. , 2017), JSBACH (Reick et al. , 2021), and JSBACH3-PF (Ekici et al. , 2014) as well as Noah-MP is adapted from BATS method (Dickinson et al. , 1986). The snow-related model performance of UTOPIA is as reliable as HTESSEL from the sensitivity test from Terzago et al. (2020). Besides, UTOPIA has an even stronger performance, lowering the temporal resolution of input data (Terzago et al. , 2020). We decided to choose a selection of representative models to derive ideas for future model improvement and concretely compare and analyze the snow cover, snow albedo, and snow density parameterization.

Therefore, this study attempts to understand the variables and parameters considered in each land surface model’s current situation by looking at the parameters of three significant snow-related variables—snow cover, snow albedo, and snow density—through nine LSMs: Noah LSM, HTESSEL, BATS, UTOPIA, CLASS, Noah-MP, CLM5, JULES, and ISBA. By comparing parameterization methods, we intend to provide insights into variables that may be considered in different models. Sections 2, 3, and 4 compare and analyze the parameterization of snow cover, snow albedo, and snow density on the nine LSMs, respectively. In the snow density part, we focused on the snow density of new snow because snow compaction due to destructive metamorphism or overburden, melting, and drifting snow is too complicated to cover in this review. Section 5 provides insights for LSM improvement by discussing the comparison results and significant variables in other previous studies. Finally, we present summaries in Section 6.

2 Snow Cover Parameterization in LSMs

Model studies show that snow cover changes significantly influence the land surface energy balance and ground temperature (Thomas and Rowntree , 1992; Viterbo and Betts , 1999). Since snow acts as a sort of ‘lid’, keeping soil surface temperature almost constant (or damping the air temperature fluctuations), and reflecting most of the incident solar radiation. The snow-covered ground albedo is much higher than the snow-free ground albedo, accurate prediction of the presence or absence of snow is crucial for understanding near-surface energy balance. Snow cover can change grassland albedo by approximately 3-4 times and mountain albedo by 2-3 times (Thomas and Rowntree , 1992; Betts and Ball , 1997; Jin et al. , 2002; Barlage et al. , 2010). Armstrong and Brun (2008) presented snow depth, snow density, and SWE as fundamental properties when describing snow cover.

Table 1 Variables or parameters for parameterization of snow cover in nine LSMs

Table 1 expressed the variables considered in the nine LSMs, including the main variables such as snow depth, snow density, roughness length and SWE. All LSMs except BATS, UTOPIA, JULES, ISBA used the SWE to parameterize snow cover. The snow cover parameterization of Noah LSM and CLM5 considered the threshold of SWE. Snow cover formula in both Noah LSM and CLM5 considers the land surface type through the SWE threshold parameter. Especially, CLM5 takes into account sub-grid geomorphic variation, the main difference among LSMs. HTESSEL is the simplest of the nine LSMs and considers the SWE and snow density which is a proxy for snow age (denser snow leads to a lower snow cover fraction). The CLASS model used the snow depth and the snow depth threshold instead of the SWE and the SWE threshold, and the snow depth is calculated from SWE and snow density. The BATS, UTOPIA, JULES, ISBA and Noah-MP model are different from the others in that it utilizes snow depth and roughness length to parameterize snow cover. Besides, they used different values of roughness length according to the land surface type.

Table 2 Snow cover parameterization in nine different LSMs

Table 2 presents the different formulations for snow cover fraction in the nine LSMs. The SWE, the SWE threshold value ($W_{\max }$) when the snow cover is 1, and the distribution shape parameter ($P_s$) determine the snow cover parameterization of the Noah LSM (Eq. (a) in Table 2) (Koren et al. , 1999). If the SWE exceeds $W_{\max }$, the snow cover is assumed to be 1. Anderson (1973) considered the range of $P_s$ as a value between 2 and 4 in the empirical equation for snow cover fraction, and the Noah LSM applied a fixed value of 2.6. The primary empirical values — the $W_{\max }$ and the $P_s$ — affect the land surface albedo because the snow cover fraction variation controls the snow albedo. The Noah LSM provides $W_{\max }$ values according to the land surface types based on the modified International Geosphere-Biosphere Program (IGBP) MODIS Noah classification. The Noah LSM designate $W_{\max }$ values as 0.08 m, 0.04 m, 0.025 m, 0.02 m, and 0.01 m for forests, short vegetation, tundra, low vegetation areas such as snow, glacial areas, and water bodies, respectively. Wang and Zeng (2010) showed that the $W_{\max }$ values for grassland and forest tend to have similar simulation results to the observed values when adjusted to 0.01 m and 0.2 m instead of 0.04 m and 0.08 m for each vegetation type. Livneh et al. (2010) improved the $W_{\max }$ value to 0.02 m for non-forest and 0.04 m for the forest, reflecting the study area’s vegetation properties. Therefore, we need to find an appropriate $W_{\max }$ value that can guide the snow cover fraction following its observation value depending on the land surface type.

HTESSEL parameterized the snow cover fraction using a formulation of SWE and low and high densities of snow, which showed a different route for the snow cover parameterization. The existing HTESSEL snow cover was calculated only as an SWE function (Eq. (b)). However, the new scheme introduced snow density in addition to the SWE (Eq. (c)). The existing parameterization method was expressed as extreme values of both snow densities, while the modified parameterization method was improved to reflect the changing snow density. The new scheme is straightforward, but it reflects the variability of snow cover between the beginning of a cold season (low snow density) and the end of an ablation season (high snow density) (Dutra et al. , 2009). At the beginning of the cold season, it is easy to cover the entire grid even if there is a small SWE. On the other hand, since snow melting that creates non-snow patches at the snow depletion period, more SWE is required for snow cover to be 1 in a single grid. HTESSEL introduced snow density to the simulation of seasonal snow cover variation, thus devising a snow cover parameterization scheme that could reflect the physical process of snow.

In BATS, the snow cover fraction is measured in terms of the snow depth, so it changes over time by subtracting the sublimation rate and snow melting from the snowfall rate. BATS parameterized the snow cover fraction, divided into areas with and without vegetation (Dickinson et al. , 1993). For land surfaces without vegetation, it was parameterized with a roughness length of 0.1 m and a snow depth as in Eq. (d), while for land surface with vegetation it was parameterized as a function of the snow depth and the roughness length multiplied by 10 (Eq. (e)). Yang et al. (1997) described the total fraction of the grid square covered by snow ($f_{s}$) as

$$\begin{aligned} f_{s}=f_{s,g}(1-\sigma _{f})+f_{s,v}\sigma _{f}, \end{aligned}$$

(1)

where $\sigma _{f}$ is green vegetation fraction, $f_{s,g}$ and $f_{s,v}$ are fraction of ground covered by snow and fraction of vegetation covered by snow, respectively. Eq. (d) from Table 2 clarified as

$$\begin{aligned} f_{s,g}=D_{sn}/(10z_{0g}+D_{sn}), \end{aligned}$$

(2)

where $D_{sn}$ is snow depth and $z_{0g}$ is roughness length for bare soil whose value is designated as 0.01 m (Yang et al. , 1997). The formation of Eq. 2 is same as Eq. (e) in Table 2.

Fig. 1

Original (Eq. (d) in Table 2: ground fraction covered by snow, Eq. (e): vegetation fraction covered by snow) and new formulation (Eq. (f): snow cover fraction depending on the surface type) for snow cover fraction in BATS model (Yang et al. , 1997) when land cover type is short grass ($z_0 = 0.02$ m)

Yang et al. (1997) upgraded the snow cover fraction scheme using different roughness length according to the land surface type and snow depth (Eq. (f)). This new scheme reflects the perspectives from Jordan et al. (2008) that snow cover and albedo are related to the vegetation and roughness length. Yang et al. (1997) showed that the simulated snow cover fraction and surface albedo have close agreement with the observed ones, especially over short vegetation. Figure 1 demonstrates the comparison of BATS formulation (Eqs. (d), (e), (f)) for snow cover fraction in the case of short grassland cover. Eq. (f) may be appropriate for grasses, and agricultural lands (Yang et al. , 1997) because it illustrates that the higher the snow depth, the faster cover the short grass. BATS focused on the snow-ground process and did not distinguish between the soil temperature and the lowest layer of snow. Geothermal heat can reach the upper surface of the snow by conduction. Nevertheless, it was assumed to ignore snow melting at the bottom of the snowpack (Dickinson et al. , 1993).

UTOPIA also considered the surface type using the vegetation cover and roughness length. At first, it is identical to BATS, which defined the total fraction for the snow-covered area with and without vegetation cover as Eq. 1. However, existing Eq. (d) and Eq. (e) from the BATS could not reach the whole coverage of snow in that upgraded version of BATS took into consideration the roughness length and snow depth as Eq. (f) (Fig. 1). UTOPIA also formulated a snow cover scheme as Eq. (g) considering both the roughness length, which can be varied with vegetation types, and snow depth when the snow depth is shallower than the roughness length value divided by 0.13 (Cassardo , 2015).

CLASS parameterized the snow cover fraction using the snow depth ($D_{sn}$) and the threshold value of snow depth ($D_{sn,lim}$) (Eq. (h)) (Verseghy , 2012). The threshold of the snow depth ($D_{sn,lim}$) applied to 0.1 m. If the snow depth ($D_{sn}$) is greater than 0.1 m, the snow cover is assumed to be 1. When the snow depth is smaller than 0.1 m, parameterization for snow cover fraction uses Eq. (h) in Table 2. In this formula, CLASS calculates the snow depth from the SWE divided by the snow density (Eq. (i)). Melton et al. (2019) recently suggested another parameterization for snow cover with the exponential form (Eq. (j)) proposed by Brown et al. (2003). In CLASS, the snow depth threshold assumed the same value regardless of the land surface type, so there could be a limitation in that it cannot follow the variability of the snow cover fraction according to the land cover type.

Noah-MP parameterizes the snow cover fraction using roughness length, snow depth, and snow density. At first, Yang et al. (1997) proposed the Hyperbolic Tangent(tanh) formulation (Eq. (f)) because they need to reflect two stages of snow depth and snow albedo relationship: sharply increasing albedo stage and slowly increasing albedo stage per changing snow depth. This formulation has a limitation of too fast covering snow, so Eq. (f) does not elaborately consider the seasonal variation of snow cover patterns. Niu and Yang (2007) perceived this limitation of Eq. (f) which is fit for shorter vegetation, and revised formulation as Eq. (k) using a melting factor for snow cover fraction ($F_{melt}$). $F_{melt}$ (Eq. (l)) calculates using fresh snow density, 100 kg m$^{-3}$, and current snow density (Eq. (m)) with determining melting curves parameter M which can be adjustable.

Fig. 2

Depletion curves of Snow Cover Fraction (SCF) varied with the ratio between snow water equivalent ($W_s$) and threshold SWE above which $f_s$ is 100% ($W_{max}$) depending on the shape parameter $N_{melt}$ (Swenson and Lawrence , 2012). $N_{melt}$ is calculated by the standard deviation of the elevation within a grid cell ($\sigma _{topo}$) as Eq. (p) in Table 2. For this Fig. 2, $N_{melt}$ is shortened to N for the abbreviation. The line colors and styles differentiate snow cover fraction according to N and corresponding $\sigma _{topo}$

CLM5 parameterizes the accumulation and depletion of snow cover fraction separately (Lawrence et al. , 2018). The parameterization of the snow cover fraction during the snow accumulation phase (Eq. (n)) include the new snow amount ($q_{snow}\Delta t$), a constant ($k_{accum}$) whose default value is 0.1, and the snow cover fraction ($f_s^n$) from the previous time step. The snow cover fraction parameterization during the snow ablation phase (Eq. (o)) shows similarity with Noah LSM in that SWE and SWE thresholds influence the depletion curve. Swenson and Lawrence (2012) developed the empirically derived expression Eqs. (n) and (o). Significantly, CLM5 newly incorporates a parameter $N_{melt}$ that changes depending on the geomorphic variability within the grid cell as shown in Eq. (p) in Table 2 (Swenson and Lawrence , 2012; Lawrence et al. , 2018). $N_{melt}$ controls the shape of the snow-covered area (Fig. 2). If $N_{melt}$ values are low, the topographic variability of regions are high. CLM5 calculates $N_{melt}$ from the standard deviation of the 1 km-resolution DEM elevation within a grid cell. However, in the glacier region, $N_{melt}$ implemented a direct value of 10 because the terrain there could be very flat. This new parameterization scheme roughly demonstrates the results of Jordan et al. (2008) in that the topographic variation of the ground causes changes in the snow cover depth.

Snow cover fraction parameterization in JULES (Eq. (q)) is identical to BATS snow cover fraction for vegetation (Eq. (e)). JULES adopted only the snow cover fraction using snow density and roughness length for bare soil (Best et al. , 2011) and ignored the vegetation fraction for calculating the snow cover fraction. Snow cover fraction parameterization in ISBA was similar to BATS in that ISBA separates the snow cover fraction formula for bare soil (Eq. (r)) and vegetation (Eq. (s)) in Table 2. ISBA formulation of snow cover fraction for bare soil is similar to CLASS using snow depth and snow depth thresholds. Besides, the ISBA formulation structure of snow cover fraction for vegetation is the same as BATS; the coefficient is different between ISBA and BATS as 2 and 10, respectively. The vegetation roughness length in ISBA is designated as a specific value depending on each vegetation type (Decharme et al. , 2019). In particular, ISBA used Eq. 1 with green vegetation fraction for the average between the snow cover fraction in the vegetated ($f_{s,v}$) and non-vegetated ($f_{s,g}$) as a part of a grid cell concept similar to BATS (Decharme et al. , 2016). Among the nine LSMs, only BATS, UTOPIA, and ISBA consider green vegetation fraction directly for snow cover parameterization, and BATS, UTOPIA, Noah-MP, JULES, and ISBA account for roughness length.

3 Snow Albedo Parameterization in LSMs

Snow albedo is the rate of reflection of incident radiation on snowpack, which plays an essential role in the energy balance of the land surface over snow during the melting season (Warren and Wiscombe , 1980; Wiscombe and Warren , 1980). One of the land surface model’s main objectives is the accurate distribution of incident radiation into radiation, sensible heat, and latent heat fluxes. Net radiation is a concept that includes both upward & downward shortwave radiation and upward & downward longwave radiation in the surface energy balance. The surface albedo — the ratio of reflected solar radiation to the incident solar radiation — determines the energy transfer between the atmosphere and land surface. Typical albedo values are as high as 80-90 % on a non-melting snow surface (Armstrong and Brun , 2008) and are expressed as various values according to the state of snow surface (fresh, dry snow; old, dry snow; wet snow; melting ice/snow) or the land surface type (snow-covered forest, snow-free vegetation/soil, water) (Table 3).

Table 3 Typical ranges for surface albedo (Armstrong and Brun , 2008)

Table 4 Variables considered for snow albedo parameterization in nine LSMs

Table 5 Snow albedo parameterization in nine different LSMs

One of the most rapid changes in land surface albedo occurs by the shift between presence and absence of snow cover. Noah LSM, UTOPIA, CLM5, and JULES use the snow cover fraction as a significant variable in the albedo parameterization (Table 4). Thus, they calculate snow albedo by summing the snow-free albedo and snow-covered albedo according to the snow cover ratio. In Jordan et al. (2008), the interception of radiation by vegetation strongly changes the albedo in the snow-covered area. The vegetation type, vegetation density, snow accumulation on the canopy, and incident radiation can affect snow albedo alteration (Jordan et al. , 2008). For instance, Noah LSM, HTESSEL, BATS, UTOPIA, and CLM5 consider the vegetation effects on snow albedo parameterization indirectly by using the maximum snow albedo for each vegetation type or applying the green vegetation fraction value. The albedo for each grid point can be calculated by linearly weighting the snow-free albedo and snow and vegetation albedo in the snow-covered area. In particular, BATS, CLM5, and JULES were relatively complex land surface models because they identify direct & diffuse albedo, visible & near-infrared albedo separately, taking into account all the effects of the solar zenith angle, and ice grain size. Additionally, BATS and CLM5 considers dust and carbon.

Table 5 presents snow albedo parameterization in nine different LSMs. In the Noah LSM, the albedo is parameterized by a weighted average of the snow-free albedo and the snow-covered albedo (Eq. (a) in Table 5). The snow albedo is closely related to the snow cover fraction (Brun et al. , 2008). The snow-covered patch can be distributed differently due to the slope or altitude, the spatial direction of a snowdrift, and the snow melting period. Moreover, the snow albedo on the surface can increase noticeably as the vegetation rate decreases. When there is short or tall vegetation, the snow-covered area may exist in the form of a patch. The surface albedo is parameterized to account for this vegetation effect, as shown in Eq. (b), using the green vegetation fraction ($\sigma _f$).

Noah LSM treated the maximum albedo for fresh snow by calculating the satellite-based snow albedo ($\alpha _{\max ,sat}$), the maximum snow albedo coefficient ($\alpha _{\max ,CofE}$), and the proportionality coefficient (C) as shown in Eq. (c). The proportionality coefficient is fixed at 0.5, and Livneh et al. (2010) adjusted it to 1. Thus, only the maximum snow albedo coefficient ($\alpha _{\max ,CofE} = 0.85$) determines the maximum albedo value of newly fallen snow. The maximum albedo of new snow equal to 0.85 in Noah LSM was the same value used in the Distributed Hydrology Soil Vegetation Model (DHSVM; Wigmosta et al. , 1994) and Variable Infiltration Capacity Macroscale Hydrologic Model (VIC; Andreadis et al. , 2009). In SnowModel default (Liston and Elder , 2006), and CLASS model (Verseghy , 1991), this value was set to 0.8, while SnowModel (Sproles et al. , 2013) used the maximum albedo value after new snowfall as 0.8 in unforested and 0.6 in forested areas. According to the snow melting condition, the new maximum snow albedo used in the SNOWPACK (Bartelt and Lehning , 2002) was 0.95 in the non-melting condition and 0.7 in the melting condition. In the case of BATS (Dickinson et al. , 1993), it was set to 0.85/0.65 (visible/near-infrared area) in the non-melting condition and 0.75/0.15 in the melting condition.

Livneh et al. (2010) adapted the albedo-decay parameterization proposed by Wigmosta et al. (1994) for the DHSVM. The albedo-decay scheme can change the coefficient over time depending on whether the snow is accumulated (slow decay) or ablated (fast decay) (Eq. (d)). The albedo decay scheme in Noah LSM reflects the snow accumulation season as a default setting. At the snow accumulation, the snow temperature is $<273.16$ K, and A and B are set to 0.94 and 0.58 in Eq. (d), respectively. When the snow decays, the snow temperature is $\ge 273.16$ K, A and B are set to 0.82 and 0.46 in Eq. (d), respectively. Storck (2000) calculated A’s values in the albedo-decay scheme as 0.92 and 0.7 for the accumulation and ablation seasons, respectively, based on observations of the shelterwood site. Sun et al. (2019) showed that the A value at the time of snow accumulation could be in the possible range 0.87-0.99 in the same formula (Eq. (d)) as the Noah LSM. They derived 0.9-0.96 as the optimal value ranges for snow accumulation phase for different areas. While the possible range for the A value at the time of snowfall ablation season was set to 0.82-0.91, the optimal value was almost the same range at 0.83-0.91 (Sun et al. , 2019). In addition, Ma et al. (2014) suggested improved values only for the time of snow accumulation, and A was in the range 0.9-0.94, and B was in the range 0.5-0.66 for different areas.

The original scheme of the snow albedo for HTESSEL follows the formulation of Baker et al. (1990), Verseghy (1991), Douville et al. (1995) and separates into melting and non-melting cases. Its formulation structure is the same as the past version of ISBA parameterization (Mölders et al. , 2007). If the snow does not melt, the albedo uses a linearly decaying coefficient ($\tau _{a}=0.008$). When the snow melting exceeds 0 or when the snow surface temperature is $>2$ K lower than the freezing point temperature, HTESSEL parameterized the albedo as Eq. (e) in Table 5 with the exponential decay factor ($\tau _{f}=0.24$) and the minimum albedo ($\alpha _{\min ,H}=0.5$). In other words, the higher the temperature, the more quickly the snow albedo decreases. When the amount of new snowfall exceeds 1 kg m$^{-2}$ hr$^{-1}$, the maximum snow albedo is set to 0.85. However, Pedersen and Winther (2005) and Mölders et al. (2007) pointed out that the original scheme has a disadvantage of too low threshold value when used for numerical forecasting in snow albedo parameterization. The parameterization was improved as shown in Eq. (f), so the snow albedo was continuously reset. HTESSEL assumed that 10 kg m$^{-2}$ of new snowfall is required to reset the maximum snow albedo to 0.85 consistently.

The original scheme uses a fixed value of 0.15 for albedo under a canopy regarding the vegetation’s influence. However, this is insufficient to describe the variability of snow albedos according to the vegetation characteristics. As a result of observations in a forest area with snow intercepted by vegetation, the albedo appeared to be 0.3 immediately after heavy snowfall and changed to 0.2 after a few days according to the forest type (Betts and Ball , 1997; Viterbo and Betts , 1999). The snow intercepted by vegetation disappears due to wind blowing in cold temperatures or snow melting in warm temperatures. Therefore, HTESSEL set the albedo for each vegetation condition differently according to the 16 types of IGBP vegetation. HTESSEL used the white-sky albedo climate values (2000-2004) in the 0.3-5 $\mu$m band of the northern hemisphere with snow on the ground suggested by Moody et al. (2007).

BATS calculates the snow albedo by classifying the diffuse and direct solar radiation and parameterizes each albedo by separating it into visible and near-infrared radiative regions. Concerning the diffuse radiation, when the solar zenith angle is less than 60°, the new snow albedo for visible radiation incident is 0.95, and the new snow albedo for near-infrared solar radiation is 0.65. The albedo decay parameterization method applies over time (Dickinson et al. , 1993) (Eqs. (g) and (h) in Table 5). As a factor associated with snow decay, BATS calculates the decreasing term $F_{age}$ through Eq. 3-11 (Wang et al. , 2020): $F_{age}$ give the fractional reduction of snow albedo due to growth of snow grain size and accumulation of dirt and soot by

$$\begin{aligned} F_{age}=\frac{tauss^{t}}{\left( {tauss}^{t}+1\right) }, \end{aligned}$$

(3)

where ${tauss}^{t}$ is a nondimensional age of snow, defined as

$$\begin{aligned} {tauss}^{t}= {\left\{ \begin{array}{ll} 0, &{} \text{ for } \hspace{0.05in} {swe}^{t}=0 \hspace{0.05in} \text{ or } \hspace{0.05in} {swe}^{t}>800~\text {mm}; \\ \max {\left( 0,{sge}^{t}\right) }, &{} \text{ for } \hspace{0.05in} {0< {swe}^{t} < 800~\text {mm},} \end{array}\right. } \end{aligned}$$

(4)

with ${swe}^{t}$ representing the snow water equivalents of the current time step. The nondimensional age of snow ${sge}^{t}$ depends on a model prognostic variable $\delta _{a}$ as follows.

$$\begin{aligned} {sge}^{t}=\left( {tauss}^{t-1}+\delta _{a}\right) \times \left( 1-\delta _{s}\right) , \end{aligned}$$

(5)

where $\delta _{s}$ is expressed as

$$\begin{aligned} \qquad \delta _{s}=\frac{\max {\left( 0, {swe}^{t}-{swe}^{t-1}\right) }}{{swe}_{c}}. \end{aligned}$$

(6)

Here, ${swe}^{t}-{swe}^{t-1}$ is the change of snow water equivalent (in mm or kg m$^{-2}$) in one time step $\Delta t$, and ${swe_{c}}$ indicates the critical value of new snow water equivalent to fully cover the old snow, assumed to be 1 mm; and $\delta _{a}$ is parameterized as

$$\begin{aligned} \qquad \delta _{a}=10^{-6} \times \Delta t \times \left( {Age}_{1}+{Age}_{2}+{Age}_{3}\right) . \end{aligned}$$

(7)

In (7), $Age_1$ represents the effect of growing snow particle size by vapor diffusion which is formulated as

$$\begin{aligned} {Age}_{1}=\exp {({arg})} \end{aligned}$$

(8)

where

$$\begin{aligned} \qquad {arg} =5000\left( \frac{1}{T_{f}}-\frac{1}{T_{g}}\right) , \end{aligned}$$

(9)

with $T_{f}$ the freezing temperature (273.16 K) and $T_{g}$ the ground temperature (K); $Age_2$ reflects the effect of recrystallization of melting moisture or close to it and is represented as

$$\begin{aligned} {Age}_{2}=\min {\left[ 1, \exp {(10 \times {arg})}\right] }; \end{aligned}$$

(10)

and $Age_3$ represents the effects of dust and soot as follows:

$$\begin{aligned} {Age}_{3}= {\left\{ \begin{array}{ll} 0.01 &{} \text{ over } \text{ Antarctica }; \\ 0.3 &{} \text {elsewhere.} \end{array}\right. } \end{aligned}$$

(11)

BATS designated $Age_3$ as 0.01 for Antarctica and 0.3 for other areas (Dickinson et al. , 1993).

Concerning the direct radiation, BATS parameterizes the new snow albedo using factors such as the diffuse snow albedo and the solar zenith angle (Eqs. (i) and (j) in Table 5). $F_{zenith}$ is a factor to account for solar zenith angle impact, calculated as

$$\begin{aligned} F_{zenith}= & {} \frac{1}{b}\left\{ \frac{b+1}{[1+2 b \cos (zen)]}-1\right\} , F_{zenith}\nonumber \\ {}= & {} 0 \text{ if } \cos (zen)>0.5, \end{aligned}$$

(12)

where b can be adjustable to best available data, and it is designated $b = 2.0$ in BATS (Dickinson et al. , 1993). Equation 12 has the property for all b that it vanishes at $cos(zen) = 0.5$, so $F_{zenith}$ values equal to 0. The value of b is unity at $cos(zen) = 0$ which is the same as the solar zenith angle in the horizontal plane, so $F_{zenith}$ is equal to 1. In Wang et al. (2020), the method of parameterizing $F_{zenith}$ was further simplified as follows:

$$\begin{aligned} F_{zenith}=\max \left\{ \frac{1.5}{[1+\cos (zen)]}-0.5,0\right\} , \end{aligned}$$

(13)

where $F_{zenith}$ is a value between 0 and 1; for example, when the solar zenith angle exceeds 60 degrees, the snow albedo for visibility area increases.

Table 6 Typical albedo values for fresh snow, old dry snow, and melting snow in CLASS model (Verseghy et al. , 2020)

In UTOPIA, snow albedo parameterization followed by Verseghy (1991), Douville et al. (1995), and Sun et al. (1999). In particular, the formulation structure is the same as in HTESSEL. However, the criteria for applying the formula differ between deeper snow and shallower snow with the factor of snow depth ($D_{sn}$). For deeper snow ($D_{sn} > 0.2$ m), the magnitude of $\alpha _{sn}$ decreases exponentially with time from a fresh snow value of 0.85 (Robinson and Kukla , 1984) and the exponential coefficient for decrease of melting snow albedo ($\tau _f$) in Eq. (k). In the case of shallower snow ($D_{sn} \le 0.2$ m), a linear coefficient for the decrease of albedo used for shallow melting snow ($\tau _a = 0.071$) and shallow dry snow ($\tau _a = 0.006$)(Sun et al. , 1999) in Eq. (k).

UTOPIA considered the contribution of solar elevation (McCumber , 1980) using a formulation:

$$\begin{aligned} \alpha _{\text {sn}} = \alpha _{\text {sn0}} + \alpha _{\text {z}} - 0.03, \end{aligned}$$

(14)

where $\alpha _{\text {z}}$ is the all albedoes during daytime by function of the solar angle $\gamma$:

$$\begin{aligned} \alpha _z=\left\{ \begin{array}{ll} 0 &{} \text{ if } \gamma \le 3^{\circ } \\ \frac{\exp \left[ 0.003286\left( 90^{\circ }-\gamma \right) ^{1.5}-1\right] }{100} &{} \text{ if } \gamma >3^{\circ }. \end{array}\right\} \end{aligned}$$

(15)

UTOPIA calculates snow albedo in case of fresh snowfall above pre-existing snow cover. When the quantity of fresh snowfall surpasses a predefined empirical threshold, the snow albedo is reset to its maximum value ($\alpha _{\max ,H} = 0.85$). Otherwise, the snow albedo is determined as a weighted average, taking into account both the albedo of the newly fallen snow and the pre-existing snow:

$$\begin{aligned} \alpha _{sn0}\!=\!\left\{ \begin{array}{ll} \alpha _{\max ,H} &{} \text{ if } \Delta t P_{sn} \ge P_{sn,white} \\ \alpha _{\max ,H} P_{sn} \Delta t+\alpha _{sn}^{t} \frac{P_{sn,white}-P_{sn} \Delta t}{P_{sn,white}} &{} \text{ if } \Delta t P_{sn}<P_{sn,white}. \end{array}\right\} \end{aligned}$$

(16)

where $\Delta tP_{sn}$ is the fresh snowfall and $P_{sn,white}$ is the empirical threshold ($10^{-5}~\text {m}$).

CLASS assumes a single snow albedo for direct and diffusive radiation. If the snow depth exceeds 1 cm, the new snow albedo is supposed to be 0.84 in the original CLASS. CLASS calculated the new snow albedo as the weighted average of 0.84 and the ground albedo before snowfall. Fresh snowfall albedo in CLASS (Eq. (l)) has the same structure as HTESSEL’s new snowfall albedo parameterization (Eq. (f)). However, there is a difference in that CLASS assigned a value of 1 mm for the SWE threshold of 100% snow cover. In addition, CLASS assumed that the snow albedo in the current time step is over 0.55 (Eq. (m)).

Recently, Melton et al. (2020) described the Canadian Land Surface Scheme including Biogeochemical Cycles (CLASSIC; v.1.0) as the coupled model framework of the CLASS (Verseghy , 2017) and the Canadian Terrestrial Ecosystem Model (CTEM; Melton and Arora , 2016). CLASSIC model parameterized snow albedo decay based on the data of Aguado (1985), Robinson and Kukla (1984), and Dirmhirn and Eaton (1975). Melton et al. (2019) showed that it is the same as empirical exponential decay functions applied to CLASS-CTEM by Verseghy (2017). The background old snow value was 0.55 in the original CLASS (Eq. (m)). However, the background old snow value ($\alpha _{sn,total,old}$) applied as 0.5 if the snow melting ratio was $>0$ (melting snow condition) unless it applied as 0.7 in the case of dry snow in the updated CLASS in CLASSIC (Eq. (n)). The CLASSIC classified three categories: total albedo, visible albedo, and near-infrared albedo that have typical values for fresh snow, old dry snow, and melting snow as shown in Table 6 (Aguado , 1985; Robinson and Kukla , 1984; Dirmhirn and Eaton , 1975). CLASSIC calculated the albedos of visible and near-infrared as Eqs. (o) and (p) for dry snow and Eqs. (q) and (r) for melting snow, respectively.

Noah-MP implemented two options for parameterizing snow albedo from BATS and CLASS schemes. The BATS scheme of snow albedo accounts for fresh snow albedo, variation in snow age, solar zenith angle, snow grain size growth, and dirt or soot on snow (Niu et al. , 2011). The CLASS scheme of snow albedo considers fresh snow albedo and snow age (Niu et al. , 2011). Noah-MP uses an old version of the CLASS scheme as Eqs. (l) and (m), so it does not distinguish the wet and dry snow conditions as well as snow albedo of the visible and near-infrared area. Niu et al. (2011) describes that the CLASS scheme tend to decrease snow surface albedo easily compared to the BATS scheme due to the strong aging factor.

CLM5 calculates the ground albedo by a weighted average of the overall direct beam albedo and diffuse albedo. The albedo value in CLM5 adapted MODIS MCD43B3 v5 data for 2001-2015 (Lawrence et al. , 2019). It parameterizes the direct and diffuse albedos using the fraction of the ground covered with snow as in Eqs. (s) and (t) in Table 5, respectively (Lawrence et al. , 2018). In addition, the direct beam albedo and diffuse albedo are various according to the land surface type. The representative types are glaciers, unfrozen lakes, frozen lakes, and soil surfaces. Glacier albedos have a prescribed value depending on the range of visible and near-infrared (Cuffey , 2010), so the visible albedo is 0.6 and near-infrared albedo is 0.4 regardless of direct or diffuse albedo (Eqs. (u) and (v) in Table 5). Frozen lake albedos have the same fixed value as glacier albedo (Bonan , 1996), direct and diffusive albedos of an unfrozen lake are parameterized following the cosine of the solar zenith angle (Eq. (w)) (Lawrence et al. , 2018).

Soil albedo depends on the soil color and varies with the moisture content of the first soil layer (Bonan , 1996). Soil albedo is calculated differently according to the change in soil moisture content ($\Delta = 0.11-0.40 \text { }\theta _1 > 0$) based on the saturated albedo and dry albedo determined according to the soil color (Eq. (x)). The soil color of the CLM5 is pre-specified for the CLM grid using the method of Lawrence and Chase (2007). The soil color is adjusted to one of 20 averaging the fitted monthly soil colors over all snow-free months and compared with the MODIS monthly local solar noon all-sky surface albedo described in Strahler et al. (1999) and Schaaf et al. (2002).

CLM5 simulates the snow albedo and solar absorption within the snow layer with the Snow, Ice, and Aerosol Radiative Model (SNICAR), which was implemented in CLM version 3 (Flanner , 2005; Flanner et al. , 2007). SNICAR includes the two-stream radiative transfer solution of Toon et al. (1989) and uses the theory of Wiscombe and Warren (1980). The albedo and the vertical absorption profile depend on the solar zenith angle, the albedo of the surface under the snow, the mass concentration of atmospheric-deposited aerosols (black carbon, mineral dust, and organic carbon), and the ice effective grain size (Lawrence et al. , 2018).

JULES has two types of scheme: the diagnostic albedo scheme and the prognostic albedo scheme. The diagnostic albedo calculates snow albedo as Eq. (y1) in Table 5 using surface temperature ($T_g$), snow aging parameter ($k_1$), snow albedo threshold temperature ($T_c$), and cold deep snow albedo depending on the vegetation type ($\alpha _{cds}$) (Best et al. , 2011). JULES derives the weighted average of surface albedo as Eq. (y2) considering snow depth ($D_{sn}$) and surface masking snow depth ($D_m$). The prognostic albedo in JULES calculates total surface albedo by using snow cover fraction, snow albedo, and snow-free albedo like Eq. (a) of Noah LSM. The prognostic albedo scheme calculates both diffuse visible and near-infrared areas as Eqs. (y3) and (y4), respectively (Best et al. , 2011). The diffuse scheme includes timely changing grain size (r) and fresh snow grain size ($r_0$). For the direct radiation albedo, r in Eqs. (y3) and (y4) need to be replaced by effective grain size ($r_e$) as

$$\begin{aligned} r_e=[1+0.77(\mu -0.65)]^2r, \end{aligned}$$

(17)

calculated using the cosine of the zenith angle ($\mu$) and grain size (r). The current state of grain size is calculated every time step with snowfall rate $S_f$ by

$$\begin{aligned} r(t+\delta t)=[r(t)^2+\frac{G_r}{\pi }\delta t]^{1/2}-[r(t)-r_0]\frac{S_f\delta t}{d_0}, \end{aligned}$$

(18)

where $d_0$ is fresh snow depth and $r_0$ is 50 $\mu$m for fresh snow (2.5 kg m$^{-2}$ fresh snow mass required to refresh the albedo) and $G_r$ is the empirical growth rate of snow grain area expressed as

$$\begin{aligned} G_r= {\left\{ \begin{array}{ll} 0.6 &{} T_g=T_m \text { (melting snow);}\\ 0.06 &{} T_g<T_m, \text { }r<150 \text { }\mu \text {m} \text { (cold fresh snow);}\\ A_s \exp {(-4550/T_g)} &{} T_g<T_m, \text { }r>150 \text { } \mu \text {m} \text { (cold aged snow).} \end{array}\right. } \end{aligned}$$

(19)

In Eq. 19, $A_s$ is set to $0.23 * 10^6 \mu$ m$^2$ s$^{-1}$. The prognostic albedo scheme in JULES consider direct and diffuse albedo from visible and near-infrared area same as BATS and CLM5. This scheme incorporates ice grain size affected by surface temperature and snowfall rate as the significant factor to calculate snow albedo.

ISBA calculates snow albedo from 3 distinct bands (Eqs. (z1), (z3), and (z5) in Table 5), the first band with the ultraviolet and visible range (0.3-0.8 $\mu$m), the second and the third band with near-infrared ranges (0.8-1.5 and 1.5-2.8 $\mu$m) following Brun et al. (1992). In the ISBA model, the total net shortwave radiation ($Q_{\textrm{sn}}$) is calculated as the sum of the absorption in each spectral band, k,

$$\begin{aligned} Q_{\textrm{sn}}(i)=&R_{\textrm{sw}} \sum _{k=1}^3\left[ \omega (k)\left( 1-\alpha _{\textrm{sn}}(k)\right) \right. \nonumber \\&\left. \exp \left( -\sum _{j=1}^i\left[ \beta _{\textrm{sn}}(k, j) \Delta z(j)\right] \right) \right] , \end{aligned}$$

(20)

where $R_{\textrm{sw}}$ (W $\text {m}^{-2}$) represents the absorption of incident shortwave solar radiation, and $\omega$ is the empirical weight of each spectral band equal to 0.71, 0.21, and 0.08 for first, second, and third band ranges, respectively. In Eq. 20, the extinction coefficient of snow, denoted as $\beta _{\textrm{sn}}$, depends on snow density and optical diameter of snow. The snow albedo is determined as a function of the snow’s optical diameter and the age of the first layer of the snowpack. For the first snow layer, snow albedo account for snow optical diameter, the near surface atmospheric pressure, and the age of the first snow layer in days (Eq. (z1)). Only first snow layer considers atmospheric pressure and time of snow albedo decay ratio by impurities. The albedo scheme for the second and the third snow layer only take into account snow grain size. The optical diameter of snow ($d_{\textrm{opt}}(i)$) is parameterized as

$$\begin{aligned} d_{\textrm{opt}}(i)=\min [d_{max},g_1 + g_2 \times \rho _{\textrm{s}}(i)^4 + g_3 \times \min (15, A_{sn}(i))], \end{aligned}$$

(21)

where $d_{max}$ is the maximum snow diameter value, and $g_1$ ($1.6\times 10^{-4}$ m), $g_2$ ($1.1\times 10^{-13}$ m$^{13}$ kg$^{-4}$) are coefficients calculated from Anderson et al. (1976). $g_3$ ($0.5\times 10^{-4}$ m day$^{-1}$) is the increase rate of the optical diameter of snow with snow age in Eq. 21. $A_{sn}$ is age of the first snow layer expressed in days. $\beta _{sn}$ is the extinction coefficient of snow affected by snow density and optical diameter of snow (Eqs. (z2), (z4), and (z6) in Table 5). Bohren and Barkstrom (1974) used a constant $3.8\times 10^{-3}$ m$^{2.5}$ kg$^{-1}$ for the snow extinction coefficient, but it changed to 0.00192 and 0.01098 for the first and the second snow layers (Eqs. (z2) and (z4)).

Table 7 Variables considered for snow density and snow depth parameterization

4 Snow Density Parameterization in LSMs

Snowfall amounts are measured with the snow depth and SWE when the snow melts. Typical snow rates are 1 cm h$^{-1}$ or 0.8 mm h$^{-1}$. The LSMs normally calculate snow depth from the SWE divided by the snow density. The density of newly fallen snow is approximately 20-300 kg m$^{-3}$, and the dry snow density is about 60-120 kg m$^{-3}$ with the condition of low-to-moderate winds, while it increases with the condition of wet snow, sleet, and wind-packed snow. When precipitation mixes snow and rain, the new snow density exceeds 300 kg m$^{-3}$. When snow falls by strong winds, the snow particles become fragmented and small-grained. As the wind speed of 1 m s$^{-1}$ increases, the density gradually increases to 20 kg m$^{-3}$ (Jordan et al. , 1999). Therefore, snow characteristics (dry/wet) and wind conditions in addition to temperature affect the new snow density.

Table 8 Snow density parameterization in nine different LSMs

The LSMs considered different variables for calculating the snow density and snow depth (Table 7). Noah LSM, HTESSEL, UTOPIA, CLM5, ISBA, and Noah-MP used air temperature to calculate the new snow density. Among these LSMs, HTESSEL, CLM5, and ISBA had in common that they considered both the temperature and wind effects when calculating the new snow density. Jordan et al. (2008) also recognized the effect of wind speed, and the variation of wind speed caused by air pressure affects the snow depth on the slope, leading to snow erosion on the slope’s upwind side and snow deposition on its lee side. BATS includes complex snow physics such as SWE, snow grain size, and the effects of dust and soot for calculating snow density variation. CLASS parameterizes the snow density updated over time with the maximum snow density which is calculated by the snow depth and the empirical constant according to the snow temperature. The time tendency of snow density in ISBA is affected by the snow density, vertical stress, snow viscosity, and wind-driven compaction rate in each layer. The snow density over a time step in JULES is updated by compactive viscosity, ice and liquid water contents for the mass of snow above the layer, snow density, and the temperature of the k-th snow layer.

The LSMs parameterize the new snow density differently to estimate the snow depth as in Table 8. The new snow depth in Noah LSM is calculated from the new SWE divided by the new snow density (Eq. (a)). If the sum of the existing and the new snow depth is $<0.001$, the LSM took the higher value of the snow density between the new and the existing snow density. If the sum of the existing and the new snow depth is $\ge 0.001$, the snow density is parameterized from the existing and new SWE divided by the sum of the existing and the new snow depth. Noah LSM uses the equation of Gottlib (1980) for calculating the new snow density (Koren et al. , 1999). The density of new snow is a function of 2 m temperature above the ground when the 2 m temperature exceeds –15$\,^{\circ }\textrm{C}$. The density of fresh snow is 0.05 g cm$^{-3}$ when the 2 m air temperature is $\le -15\,^{\circ }\textrm{C}$ (Eq. (b) in Table 8). According to the parameterization method, Fig. 3 illustrates the relationship between temperature and new snow density. The minimum value of density of new snowfall is 0.05 g cm$^{-3}$, and the density of new snowfall tends to increase as the temperature increases. Wang and Zeng (2010) incorporate the minimum value of the liquid water portion stored in the snowpack during snowmelt for the improvement of the existing snow density simulation in Noah LSM because snow density changes abruptly at the surface condition of near freezing temperature. More importantly, they considered the shaded forest canopy to reduce the overestimation of potential evapotranspiration, snow sublimation and melt due to the overestimated net solar radiation. Considering the vegetation shading effects improves the SWE and snow depth simulation.

Fig. 3

The relationship between the 2 m air temperature ($\,^{\circ }\textrm{C}$) and the new snow density (g cm$^{-3}$) in Noah LSM

In HTESSEL, the snow depth value is inversely proportional to the snow density and snow cover fraction and is proportional to SWE (Dutra et al. , 2009) (Eq. (c)). When calculating the snow depth, using the snow density and snow mass (similar to the SWE) is similar to those of Noah LSM, and the consideration of the snow cover fraction is similar to that in CLM5. Eq. (d) shows the equation for obtaining the new snow density in HTESSEL. This equation is converted from a constant value to an equation using the equation of CROCUS (Brun et al. , 1989; Brun et al. , 1992) and is a parameterization method that expresses the new snow density as a function of the temperature and wind speed (Dutra et al. , 2009). In Eq. (d), $a_{sn}$ is 109 kg m$^{-3}$, $b_{sn}$ is 6 kg m$^{-3}$, and $c_{sn}$ is 26 kg m$^{-3.5}s^{0.5}$. At this time, the snow density is limited to the range 50-450 kg m$^{-3}$.

The BATS model calculates the snow depth as shown in Eq. (e) in Table 8 by dividing the snow mass by the snow density (Dickinson et al. , 1993; Yang et al. , 1997). The snow density is parameterized as (Eq. (f)) with a snow aging factor according to the snow model and data of Anderson et al. (1976). The change of the snow aging factor is affected by the growth of snow particle size and accumulation of dust and soot (Dickinson et al. , 1993; Yang et al. , 1997). The snow aging factor explained in Section 3 with Eq. 3 to 11.

UTOPIA calculated snow density continuously by using current snow density ($\rho _{sn}^n$), snow water equivalent ($W_s$), new snow density ($\rho _{sn}$), and new snowfall rate ($P_{sn}$).

$$\begin{aligned} \rho _{sn}^{n+1}=\frac{\rho _{sn}^n W_s+\rho _{sn} P_{sn} \Delta t}{W_s+P_{sn} \Delta t}, \quad \rho _{\text{ sn,min } } \le \rho _{sn}^{n+1} \le \rho _{\text{ sn,max } }, \end{aligned}$$

(22)

it assumed that snow density varies during snowfall and snow melt expressed as this Eq. 21. In UTOPIA, snow depth is closely related to snow density. Significantly, Eq. (g) in Table 8 can parameterize new snow density. Its formulation is the same as Noah LSM, but its range is designated by minimum and maximum snow density, respectively ($\rho _{\text {sn,min}}$,$\rho _{\text {sn,max}}$).

CLASS also derives snow depth dividing SWE by snow density (Eq. (h) in Table 8). In the case of the maximum snow density, CLASS 2.7 used 300 kg m $^{-3}$ as a fixed value, whereas CLASS 3.6 parameterized as a function of the snow depth (Brown et al. , 2006; Verseghy , 2012). If snow temperature below 0$\,^{\circ }\textrm{C}$, snow density decreases exponentially with time from the maximum mean density which is the calculated background snow density (Eq. (i)). The equation is derived from field measurements of Longley (1960) and Gold (1958) and express similarly to the albedo. Bartlett et al. (2006) found that the modified formula produced good results in the Boreal Ecosystem Research and Monitoring Sites (BERMS) (Old Aspen, Old Jack Pine, and Old Black Spruce). In the equation of the maximum snow density, the empirical constant ($A_s$) is 450 for cold snow case ($T_{snow}<0\,^{\circ }\textrm{C}$), and 700 for snow close to the melting point ($T_{snow}=0\,^{\circ }\textrm{C}$) according to Tabler et al. (1990) (Eq. (j)). This empirical constant is consistent with the results of Wakahama (1968) and Colbeck (1973), which both showed a tendency to increase the compression rate of snow density with wet snow (Brown et al. , 2006).

Noah-MP calculates snow density or snow depth considering destructive metamorphism by temperature change, compaction by the weight of the overlying snow layers, and snow melting metamorphism according to Anderson et al. (1976). Noah-MP’s snow layer determines by the total snow depth (Yang and Niu , 2003). If the snow depth is < 0.045 m, snow layer does not exists. If the snow depth is $\ge 0.045$ m, the first snow layer has a layer thickness as the total snow depth. When snow depth is $\ge 0.05$ m, two layers produces, and when snow depth is $\ge 0.15$ m, a third layer is created. The third snow layer is thick due to the compaction following Sun et al. (1999) (Niu et al. , 2011). Snow depth is dividing snow on the ground by bulk density snowfall which is set to 120 when fresh snow density is smaller than 120 and parameterized as the following fresh snow density (Eqs. (k) and (l)). Fresh snow density in Noah-MP is calculated according to the Hedstrom and Pomeroy (1998) by Eq. (m) in Table 8. Fresh snow density mainly account for the surface air temperature and freezing temperature.

CLM5 calculates the snow depth next time, adding new snow depth to the snow depth at the current time (Lawrence et al. , 2018). CLM5 parameterizes the new snow depth as the rate of change ($\Delta z_{sno}$, m), which is a function of the bulk density of newly fallen snow ($\rho _{sn}$, kg m$^{-3}$), the snow cover fraction ($f_{s}$), and the rate of solid precipitation reaching the ground ($q_{grnd,ice}$, kg m$^{-2}$ s$^{-1}$) (Eq. (n)). In particular, CLM5 considers the snow cover for estimating snow depth, like HTESSEL. The density of newly fallen snow parameterization incorporates summing the fresh snow density according to the temperature and wind (Eq. (o)). Eq. (p) shows the fresh snow density according to the temperature ($\rho _{T}$) in CLM5 (Anderson et al. , 1976; van Kampenhout et al. , 2017), and the equation has a similar structure to that of the Noah LSM. However, CLM5 has a difference in that it added the critical temperature, the freezing temperature of water (in K) + 2 K. Besides, the lowest air temperature parameterized in CLM5 does not use a constant value like -0.05. CLM5 utilized wind dependent term for the new snowfall density ($\rho _{w}$) as Eq. (p) from van Kampenhout et al. (2017) who considers both wind and temperature.

JULES has a multi-layer snow model, whose snow density on k-th layer over a time step of length $\delta t$ can be parameterized as Eq. (q) in Table 8. Snow density scheme in JULES account for the reference snow temperature ($k_s = 4000$ K), the snow melting point ($T_m = 273.15$ K), the temperature of the k-th snow layer ($T_k$, K), reference snow density ($\rho _{0} = 50$ kg m$^{-3}$), snow density of the k-th snow layer ($\rho _{k}$, kg m$^{-3}$), and compactive viscosity ($\eta _{0} = 10^7$ Pa s), and the mass of snow above the middle layer ($M_{k} = 0.5\left( I_{k}+W_{k}\right) +\sum _{i=1}^{k-1}\left( I_{i}+W_{i}\right)$), where $I_{k}$ (kg m$^{-2}$) and $W_{k}$ (kg m$^{-2}$) are an ice content and a liquid water content, respectively (Best et al. , 2011). The layer thickness ($d_{k}$, m) which have the relationship with density and mass expressed by $\rho _{k}d_{k}=I_k+W_k$. This formulation is identical meaning with Noah LSM, BATS, CLASS, and Noah-MP.

The newly fallen snow density in ISBA is identical with fresh snow density scheme in HTESSEL (Eq. (d) in Table 8). ISBA expresses the time tendency of snow density as Eq. (s) like JULES, but the specific terms are different. The snow density changes by snow compaction due to vertical stress, the snow viscosity change (Brun et al. , 1989), and wind-induced densification of snow layers (Brun et al. , 1997). Especially, compaction rate of wind-driven snow densification parameterized as $\tau _{\textrm{W}}(i)$ which describes precisely in Appendix B (Wind-induced densification of near-surface snow layers) from Decharme et al. (2016), and it is unique consideration of ISBA. The vertical stress in each layer can be parameterized as Eqs. (t) and (u) and vertical stress is also considered in JULES. It accounts for layer snow depth changes per time step ($\Delta z$), layer snow density ($\rho _{sn}$) and the acceleration of gravity (g) in every snow layer, but only the first snow layer used half mass of the uppermost layer as vertical stress effect. The snow viscosity in Eq. (v) is calculated by reference snow viscosity ($\eta _{0} = 7622370$ Pa s), reference snow density ($\rho _{0} = 250$ kg m$^{-3}$), snow density in i-th layer ($\rho _{sn}(i)$), snow temperature in i-th layer ($T_{sn}(i)$), and the constants ($a_{\eta } = 0.1$ K$^{-1}$, $b_{\eta } = 0.023$ m$^3$ kg$^{-1}$, and $\Delta T_{\eta } = 5$ K). The decrease of viscosity with the presence of liquid water ($f_{\textrm{W}}(i)$) is calculated by Eq. (w), where $W_{l\max }(i)$ (kg m$^{-2}$) and $W_l(i)$ (kg m$^{-2}$) refer to the maximum liquid-water-holding capacity and liquid water content in i-th layer.

5 Future Outlook

We visualized the current state of the art of snow cover fraction parameterization as Fig. 4 by checking all significant variables of snow models in the supplementary table from Menard et al. (2021). Figure 4 shows that only a few LSMs used specific variables to discern the snow cover fraction, so snow model developers can easily compare differences in considered variables among LSMs. The nine land surface models examined in this study calculate the snow cover fraction through “the threshold of the SWE and the SWE,” for Noah LSM, and HTESSEL, “the critical value of the snow depth and snow depth,” for CLASS. In addition, BATS, UTOPIA, ISBA, JULES and Noah MP considers “the snow depth and roughness length” for deriving snow cover fraction (Fig. 4). Noah MP additionally concerned snow density as the melting factor, and BATS, UTOPIA, and ISBA used the vegetation fraction separating the bare soil ground and vegetation covered ground for applying different roughness length according to the vegetation type and snow depth. Therefore, it is crucial to properly set the parameter values for “the threshold of snow depth or SWE” and the roughness length depending on the land surface type. In this regard, Lim et al. (2022) suggested the micro-genetic algorithm for attaining the optimal values of snow related parameter.

Fig. 4

Conceptual diagram key variables of snow cover fraction by snow model intercomparison of this study and the ESM-SnowMIP. BATS, UTOPIA, and ISBA incorporate snow depth, roughness length, and vegetation fraction as key parameters in their approaches for snow cover fraction parameterization. Meanwhile, Noah LSM, CLM5, and HTESSEL account for snow water equivalent (SWE) as a significant factor. Notably, CLM5 incorporates topographic variables, while HTESSEL integrates snow density into its framework. CLASS, Noah-MP, and JULES, on the other hand, all incorporate snow depth as a foundational parameter. JULES extends its consideration to encompass roughness length, while Noah-MP additionally includes snow density and roughness length in its formulation. ORCHIDEE I and ORCHIDEE E are employed in their implicit and explicit snow versions, respectively, in CMIP5 (Coupled Model Intercomparison Project Phase 5) and CMIP6 (Coupled Model Intercomparison Project Phase 6)

It is significant to derive an optimal parameter value that can reflect the local characteristics based on the same land type from the observation data. For instance, accurate snow cover estimation is crucial because snow cover fraction affects snow sublimation, soil heat flux, surface emissivity, and albedo in Noah LSM. That is why parameter improvements were attempted in Livneh et al. (2010) and Wang and Zeng (2010). The parameter values according to the land cover type provided in Noah LSM were set differently according to the height of the vegetation. The current version of the Noah LSM reflects the SWE for snow cover parameterization but does not consider the snow depth, snow density, roughness length, and green vegetation fraction. Therefore, considering snow depth (CLASS, RUC, ORCHIDEE-I, SPONSOR in Fig. 4) or snow density (HTESSEL, EC-EARTH, Noah-MP, CoLM in Fig. 4) for snow cover fraction parameterization could be better because these variables have timely varying characteristics due to the temperature. In addition, current state-of-the-art LSMs considers fixed roughness length value according to the vegetation cover (JULES, BATS, ISBA, UTOPIA, and Noah MP). Therefore, a seasonal variation of roughness length needs to calculate snow cover fraction for reflecting vegetation effect.

Point-specific or 100% snow cover means that LSMs assume they do not express subgrid heterogeneity. They inform ground conditions with or without snow (Fig. 4). Since most LSMs have been developed for the atmospheric model, the representation of snow-covered and snow-free areas is significant for energy flux as it directly influences the lower boundary condition of the model. Land surface schemes (CABLE-SLI, ESCIMO, JSBACH, JSBACH3-PF, MATSIRO, SWAP, VEG3D, SURFEX-ISBA, JULES-GL7, JULES-UKESM, and JULES-I), hydrological model (CRHM), snow physics model (Crocus, SMAP, and SNOWPACK) impose complete snow cover or switch off their subgrid parameterizations, according to Menard et al. (2021). Among them, however, we identified the parameterization of snow cover fraction in SURFEX-ISBA (land surface scheme in CNRM-CM with SURFEX version 8.0), JULES-GL7, JULES-UKESM (land surface schemes in HadGEM3-GC3 and UKESM with JULES version 5.3), and JULES-I (land surface scheme in HadCM3 with JULES version 4.8), which concerns specific variables including roughness length as illustrated in the diagram in Fig. 4.

Table 9 Spatial variability of snow depth and related processes and predictors (slightly updated from Table 2; Copyright 2011 by the American Geophysical Union. Clark et al. , 2011)

The land surface model’s newly added snow cover parameterization method incorporates the sub-grid topography variation in CLM5 (Swenson and Lawrence , 2012). Ma et al. (2019) used this method which considers the sub-grid unit’s terrain characteristics to calculate the snow cover fraction. Initially, Douville et al. (1995) proposed a formula with the irregular snow cover distribution for mountainous areas, including snow depth, roughness length, and the standard deviation of the subgrid orography (m). Roesch and Roeckner (2006) also measured snow cover fraction in the ECHAM climate model using the standard deviation of the sub-grid elevation and SWE. Recently, Miao et al. (2022) accounted for the topographic effects and added a modified topographic factor (Li et al. , 2019) to the snow cover fraction parameterization. They emphasized that including topographic effects for the snow cover simulation can reduce the “cold bias” in the winter season due to the overestimation of snow cover and snow albedo. Even though this method shows the importance of the topographic factor, it has a limitation due to only taking a variable called “elevation” into account as the standard deviation for the feature of the geomorphic setting.

Clark et al. (2011) presented a list of predictors related to the cause of spatial variation in snow depth according to spatial scale, as shown in Table 9. The list included both the elevation and slope among the significant predictors for snow temporal and spatial distribution. Since the spatial variation of snow depth is highly relevant to snow cover, we can apply essential predictors to the snow cover fraction parameterization. Anderson et al. (2014) also provided insights into the snow physics process that regulates the relationship between snow distribution and topographic characteristics and argued that snow cover correlates with altitude, aspect, canopy density, and snow accumulation/ablation. Younas et al. (2017) applied elevation, slope, and aspect combination for a subgrid-scale snow parameterization on the CLASS. In particular, Newman et al. (2014) showed that existing land surface models have vertical complexity rather than horizontal complexity in that the LSMs can be improved by using multivariate mosaic subgrid approach with similarity analysis (e.g., k-means clustering) within a catchment or model grid box. Aas et al. (2016) also commented the problem of fixed reference SWE, and used a tiling approach based on a terrain parameterization accounting for wind effects (Winstral et al. , 2002) in a coupled WRF-Noah.

Besides, Mott et al. (2018) suggested that the characteristics of snow cover change according to the mountain range scale, the ridge, and the slope scale may differ. This idea provides worth variables considering when applying scales to interesting areas in LSMs. First, snow accumulation depends on the climate, elevation, and vegetation at the mountain-range scale (López-Moreno et al. , 2008; Clark et al. , 2011; Anderson et al. , 2014). Especially, elevation is the most significant variable for precipitation pattern at the mountain range scale (Mott et al. , 2018). Second, snow deposition patterns at the ridge-scale reveal more considerable spatial variability of snow or enhanced snow deposition over leeward slopes by preferential deposition of snowfall (Mott and Lehning , 2010; Gerber et al. , 2017). Third, at the slope scale, the highest model resolutions show the snow redistribution processes such as saltation and turbulent suspension (Mott et al. , 2018).

Noah LSM, CLM5, and JULES calculates the land surface albedo by summing the ground albedo without snow and the snow albedo using the snow cover fraction. Other LSM models using the Noah LSM’s snow albedo decay scheme were VIC and DHSVM. Livneh et al. (2010) introduced the parameter value of albedo decay parameterization. Storck (2000), Sun et al. (2019), and Ma et al. (2014) also enhanced this parameter to fit the characteristics of the study area. Furthermore, maximum snow albedo was approximately 0.6-0.95 in other LSMs and journal articles (Sproles et al. , 2013; Verseghy , 1991; Wigmosta et al. , 1994; Andreadis et al. , 2009; Yang et al. , 1997; Bartelt and Lehning , 2002). In Noah LSM, snow albedo is affected by snow cover; it affects the latent heat of the freezing rain, the heat flux of the snow surface, and the phase change heat flux for snow melting. Therefore, the maximum snow albedo is the main parameter of snow albedo parameterization, so setting the maximum snow albedo’s optimal value is essential.

In addition to the HTESSEL, BATS, and CLASS models, ISBA includes a prognostic albedo scheme in which the previous snow albedo value affects the current snow albedo value. At first, CLASS did not consider the snow albedo changes according to the spectral range, similar to HTESSEL (Eqs. (l) and (m) in Table 5). Noah-MP also used this scheme (Wang et al. , 2020). After that, CLASS applied the difference between visible and near-infrared snow albedo according to spectral range like CLM5, BATS, JULES, and ISBA; it used an empirical formulation that has the difference between dry snow (Eqs. (o) and (p) in Table 5) and melting snow (Eqs. (q) and (r) in Table 5). Melton et al. (2019) used the spectral method instead of the empirical exponential decay function in order to apply the results of Wiscombe and Warren (1980) that the growth of snow particles, soot/dirt precipitation affects snow albedo. BATS’s snow albedo parameterization method was actively accepted in CLASS, and there was a slight difference in applying to freshly fallen snow with a near-infrared value of 0.73, unlike BATS’s 0.65. In addition, the structure of the equation reflecting the effect of the solar zenith angle was slightly different. The CLASS model in CLASSIC shows the most significant difference from other land surface models in that they reflected canopy albedo (Bartlett and Verseghy , 2015; Melton et al. , 2020).

In particular, CLASS has the lowest albedo bias range among the JULES-I, JULES-GL7, JULES-UKESM, Surfex-ISBA, CLM5, CLASS, and HTESSEL from part of Fig. 2 in Menard et al. (2021). Furthermore, Park and Park (2016) successfully implemented vegetation effects into the Noah MP parameterization scheme reagarding snow albedo, specifically by incorporating leaf area index (LAI) and stem area index (SAI) with optimized values based on different land use types. Their study demonstrated that this inclusion led to improved performance in winter surface albedo. Hence, it is imperative for researchers in future studies to consider the effect of vegetation for the improvement of snow albedo simulation. Thackeray et al. (2014) demonstrated that the presence of a canopy, which leads to sublimation and wind-driven unloading, can influence the consistency of snow during the mid-winter season. This is why albedo parameterization may present a bias towards higher values, as it does not account for the vegetation effect combined with the wind factor. For instance, when temperatures rise above freezing, snow on the canopy can melt instantaneously in the LSM model; however, realistically canopy snow is less sensitive to temperature.

Since many LSMs calculate the snow depth as the SWE ratio to the snow density, it is vital to parameterize the new snow density affected by fresh snowfall. The new snow density’s minimum value differed slightly depending on the type of LSM or the study area, so it was necessary to adjust the parameter value. In the snow density parameterization, while Noah LSM and Noah-MP solely consider temperature, CLM5, HTESSEL, and ISBA took wind and temperature into consideration. However, Föhn (1976, 1985) raised the question of the representativeness of the measurement of the new snow depth in precipitation gauges in the mountains because they ignored the topographical characteristics such as the mountain peak, the wind-slope, and the leeward slope or depressions. The records from rainfall gauges surrounded by valley plains help infer the depth of new snow depths over large areas (Föhn , 1976; Föhn , 1985). Therefore, it is imperative to reflect the influence of topographic conditions’ variation (elevation and slope) for mountainous areas’ snow depth. Clark et al. (2011) proposed to derive snow depth by considering topographic variations such as wind slope vs. leeward slope, depression, crests, and the variation of critical elevation. Spatial variation of snow depth and temporal consistency of snow cover diversified due to snow freezing temperature, slope, aspect, and radiation. Hojatimalekshah et al. (2020) examined the relationship between tree canopy and snow depth and Kantzas et al. (2014) investigated the change of the snow regime according to vegetation dynamics through a ground model; these studies show that consideration of the vegetation effect is necessary for the snow parameterization method. Therefore, it is imperative to integrate factors such as vegetation density, leaf area index (LAI), stem area index (SAI), vegetation type, and snow behavior in sheltered areas into the snow density parameterization with respect to snow depth. In breif, the inclusion of geomorphic and vegetation-related variables can influence the simulation of snow depth with spatial and temporal variations.

Essery et al. (2013) emphasized that there is no clear link between model complexity and performance. Nevertheless, they highlighted the importance of considering the storage and refreezing of liquid water within the snow for snow density and albedo parameterization. Additionally, Sun et al. (2023) recognized that no single model can perform optimally across different regions or time periods. As a solution, they suggested that integrating multiple models for simulating snow can be an effective approach to obtain high-quality snow simulation results.

6 Conclusions

In this study, we have identified significant variables for calculating snow cover, snow-covered surface albedo and snow density, by examining the snow physics parameterizations in nine LSMs. These findings provide insights for further enhancement of snow-related parameterizations. Each model has different uncertainties in simulating the snow cover, snow albedo, and snow density. For this reason, finding better ways to simulate the snow-related parameterizations can help manage primary water resources related to snow cover and develop Earth system models considering the land surface energy budget. We presented a comparison table so that readers can identify the variables considered in various LSMs with information on snow-related schemes. This study confirmed that snow cover, snow albedo, and snow density parameterizations differ for each land surface model in terms of complexity and incorporated variables or parameters. The comparative analysis for each element is summarized as follows.

For the snow cover parameterization, the snow decay scheme in CLM5 considers the subgrid’s topographical variation. Besides JULES, BATS, UTOPIA, ISBA, and Noah MP used snow depth and roughness length for calculating the snow cover fraction. BATS, UTOPIA, and ISBA consider vegetation fraction, while Noah MP accounts for the melting factor using snow density.

For the snow albedo parameterization, Noah LSM, UTOPIA, CLM5, and JULES use the snow cover fraction. Noah LSM, UTOPIA, CLM5, BATS, and HTESSEL used the maximum snow albedo for each vegetation type or applied the green vegetation fraction. Especially BATS, CLM5, and JULES considered solar zenith angle and ice grain size to calculate snow albedo using direct & diffuse albedo and visible & near-infrared albedo. BATS and CLM5 take into account the effect of impurities like dust or carbon on snow. CLASS does not consider solar zenith angle and was recently upgraded by calculating visible and near-infrared albedo for dry and melting snow, respectively.

In terms of snow density, Noah LSM and Noah-MP consider only temperature, whereas HTESSEL, CLM5, and ISBA reflect temperature and wind to parameterize the density of new snowfall. ISBA and JULES include internal snow processes such as vertical stress, snow temperature, and snow viscosity. In particular, ISBA accounts for wind-driven snow compaction precisely. BATS calculates snow density using the growth of snow particle size and the accumulation of dust and soot. CLASS calculates the maximum snow density as a function of snow depth, increasing exponentially with wet snow over time.

We suggested what variables need to be considered to reflect the snow’s physical properties properly in the future outlook section. In particular, at a high-resolution spatio-temporal scale, we highlight the importance of geomorphic and vegetation conditions in snow parameterization. First, topographic features such as slope aspect, slope gradient, and altitude have great importance in snow ground physics. Second, we found that vegetation-related factors such as vegetation density, vegetation type, vegetation cover fraction, LAI, and SAI are crucial for snow parameterization at the mountain range or ridge scale. In future studies aimed at improving LSM, we recommend a parameterization that incorporates the topography and vegetation characteristics as well as optimized snow-related parameters considering local geomorphic and land cover conditions.