Improvement of Snow Albedo Simulation Considering Water Content

Highlights

What are the main findings?

• Developed a snow albedo model that explicitly accounts for liquid water content (LWC) by integrating the Maxwell–Garnett mixing rule, Mie scattering theory, and a four-stream discrete ordinates adding method.
• The LWC on the surface of snow has a stronger impact on albedo, and snow with smaller particle sizes is more sensitive to changes in LWC.

What are the implications of the main findings?

• Achieved improved accuracy in albedo simulations under certain conditions when compared with observations.
• Demonstrated a strong ability to express physical mechanisms and maintain stable performance in complex environments, making it applicable to wet snow containing impurities.

Abstract

By combining the Maxwell–Garnett mixing rule, Mie scattering, and the four-stream discrete ordinates adding method, a snow albedo model with explicit consideration of water content was constructed, and the influence of snow water content on snow albedo simulation was systematically analyzed. The results indicate that liquid water content is the key factor contributing to significant changes in albedo in the near-infrared band. The albedo of snow with small particle sizes is more sensitive to water content. The water content in the surface layer of snow has a more pronounced effect on reducing albedo. The actual measurement cases at the stations on the Tibetan Plateau, Xinjiang, and Northeast China show that the model established here provides a good simulation of albedo accuracy, with a bias of −0.0069 and a Root Mean Square Error (RMSE) of 0.0583 compared to the observations. This indicates that the model has a strong ability to express physical mechanisms and performs stably in complex environments, thereby demonstrating good regional applicability. This model can also be applied to wet snow containing impurities in the future.

1. Introduction

Snow, a key component of the cryosphere, plays a crucial role in the climate system, and its variability has far-reaching impacts on the surface energy balance, hydrological cycle, and regional climate [1,2]. Snow albedo is the primary parameter governing the absorption and scattering of solar radiation by the snowpack. It controls the surface radiative budget and directly modulates land-atmosphere energy exchange, thereby influencing atmospheric circulation and local climatic conditions [3,4,5]. The fidelity of snow albedo simulations depends strongly on the physical properties of the snowpack, such as snow depth (SD), density, microstructure morphology, and liquid water content (LWC), which directly shape its scattering, absorption, and reflection characteristics and thus its optical behavior [6,7,8,9,10,11].

Snow albedo modeling evolved from empirical approximations toward physically based schemes. Early models typically expressed albedo as a function of observable or measurable snow properties (e.g., effective grain radius, snow age, and SD). For example, Wiscombe and Warren [6,12] employed empirical functions describing the exponential dependence of albedo on grain size and snow age; this approach performs reasonably well for clean, dry snow conditions but is limited for wet or impurity-laden snow. Gardner and Sharp [10] further incorporated the effects of impurities, especially black carbon and moisture, combined with grain size variations to quantify albedo changes. Although such models are simple to apply and computationally efficient, they are largely empirical and lack a complete physical description, and their accuracy and applicability depend heavily on the availability and quality of observations [13,14,15]. Common observations include ground-based measurements and satellite retrievals. Ground sites can provide high-precision parameters, such as SD, particle size, density, and LWC [4,6,11,16,17,18], although their spatial representativeness is limited, particularly across sparsely instrumented, high-elevation regions such as the Tibetan Plateau [1]. Satellite remote sensing (e.g., Moderate-Resolution Imaging Spectroradiometer (MODIS) and passive microwave sensors) offers broad spatial coverage and continuous monitoring [19], yet its accuracy can be degraded by clouds, atmospheric conditions, and retrieval algorithms, leading to substantial errors in complex terrains. These limitations constrain the performance and general applicability of empirical models.

To reduce reliance on observations while incorporating additional snow physics, Kokhanovsky and Zege [20] developed a dry-snow albedo model grounded in optical scattering and radiative transfer theory accounting for the granular structure of snow and multiple scattering, using geometric optics or Mie theory combined with corrected two-stream radiative transfer approximations; this framework was widely applied in quantitative studies of snow optical properties. Flanner and Zender [21] further parameterized the modulation of albedo by impurity concentration and grain size through empirical relationships and simplified radiative transfer treatments, enabling rapid estimates of black-carbon effects. However, neither approach explicitly treats snow LWC as an independent variable or parameter despite the fact that increasing water content substantially alters snow scattering and can induce a nonlinear decline in albedo [5,22]. In recent years, the SNICAR (Snow, Ice, and Aerosol Radiative) model [23,24] has advanced snow albedo simulations by supporting multiwavelength, multi-layer snowpack with embedded impurities. SNICAR explicitly represents liquid water by linearly weighting the complex refractive indices of ice and water and employs a two-stream radiative transfer approximation. In this scheme, the relative contributions of ice and water are weighted according to their respective volume fractions in the mixed-phase snow. Although Gardner and Sharp [10] explored a higher-order (16-stream) discrete-ordinate radiative transfer (DISORT) model to improve the treatment of multiple scattering in snow, its computational expense limits its routine application. By contrast, a fast and accurate four-stream discrete-ordinate adding radiative transfer scheme successfully used in atmospheric applications [25] remains underutilized for snowpack radiative transfer.

To better account for LWC as a key control in snow optics, while retaining computational efficiency and improving the accuracy of multiple-scattering calculations, the complex refractive index of ice–water mixtures is computed using the Maxwell–Garnett mixing rule, layer-resolved snow optical properties are derived using a Mie scattering scheme, and radiative transfer is simulated using a four-stream discrete ordinate adding method. Based on these components, a snow albedo model that explicitly incorporates the effects of LWC was developed. Section 2 describes the data and methods used in the study. Section 3 presents the results of the idealized single-layer and two-layer experiments and provides an evaluation of our model simulations. Section 4 and Section 5 present the discussion and conclusions, respectively.

2. Data and Methods

2.1. Data

In the current study, the following datasets were utilized.

2.1.1. Complex Refractive Indices of Ice and Liquid Water

The temperature-dependent complex refractive index (CRI) data for ice and liquid water were used (download from https://refractiveindex.info) (accessed on 1 June 2025), specifically the CRI of ice at −7 °C [26] and the CRI of water at 0 °C and −10 °C [27]. Figure 1 illustrates the wavelength dependence on the real Re(m) and imaginary Im(m) parts of the CRI for ice and water. The real part of ice is generally lower than that of water and varies more smoothly with the wavelength. The CRI of liquid water is sensitive to temperature; as the temperature decreases, the scattering is slightly enhanced [27].

Figure 1. Wavelength dependence of the complex refractive index for ice and liquid water, including real Re (m) and imaginary Im (m) parts (download from https://refractiveindex.info, accessed on 1 June 2025).

In the ultraviolet-visible range, the scattering of ice and water is extremely strong (the imaginary part of the complex refractive index is about 10⁻⁹–10⁻⁸), and the increases in water content have little effect on the properties of snow cover. In the near-infrared (NIR) range, the absorption by ice and water is enhanced (the imaginary part increases by 3 or 4 orders of magnitude), and there are significant phase differences in the absorption bands of ice and water. At absorption peaks near ~1.4 µm and ~1.9 µm, water absorbs more strongly than ice, whereas near ~1.6 µm, it is up to an order of magnitude less absorptive. These contrasts indicate that liquid water has a non-negligible influence on the optical properties of snow. An increase in water content may lead to a notable enhancement in snow absorption in certain bands [26].

2.1.2. Ground-Based Observations

The in situ measurements used in this study were obtained from the National Cryosphere Desert Data Center (NCDC, China; http://www.ncdc.ac.cn), specifically the “Snow Observation Dataset along Typical Transects in China (2017–2021)” for Transect 4 over the Tibetan Plateau [28]. This dataset provides variables including snow LWC, grain size, bulk density, and surface albedo, covering multiple snowpack stages (accumulation, quasi-stability, and ablation). Measurements were acquired using a combination of manual observations and instrument-based sampling, including layer-resolved surveys with snow forks, shovels, and snow tubes, and were subjected to rigorous quality control to ensure data completeness and reliability. These observations provide critical ground truth for evaluating snow optical properties and albedo simulations.

2.1.3. Satellite Retrievals

To validate the model simulations, we employed the Moderate Resolution Imaging Spectroradiometer (MODIS) global albedo product MCD43C3 Collection 6. The product has a spatial resolution of 0.05° and a nominal daily temporal resolution and provides white-sky (WSA; diffuse-hemispheric) and black-sky (BSA; directional-hemispheric) albedos. To enhance comparability with the simulations, we combined MODIS WSA and BSA using ERA5-derived partitioning between diffuse and direct shortwave radiation to obtain an effective all-sky surface albedo for model evaluation. MCD43C3 was widely used in land surface and snow albedo studies, and additional information is available on the MODIS project website: https://modis.gsfc.nasa.gov/, accessed on 1 June 2025.

2.1.4. ERA5 Reanalysis

We used the ERA5 dataset from the European Center for Medium-Range Weather Forecasts [29]. This dataset offers hourly atmospheric variables from 1940 to the present with a high horizontal resolution of 0.25° × 0.25°. Here, we selected key times from 17 to 21 January 2021 (14:00 h on January 17; 12:00 h, 13:00 h, and 15:00 h on January 19; 12:00 h on January 20; and 15:00 h on January 21) and extracted surface solar radiation downwards (SSRD) and total-sky direct solar radiation at the surface (FDIR). These fields were used to weight the MODIS WSA and BSA to derive the all-sky surface albedo inputs for model validation.

2.2. Methods

In the following subsections, we provide a concise overview of the methods used in this study. Figure 2 presents the calculation flowchart of the model.

Figure 2. Flowchart illustrating the model construction process.

2.2.1. Maxwell–Garnett Mixing Rule

To explicitly incorporate LWC and provide a more robust physical basis for simulating the optical behavior of wet or melted snow, we computed the effective complex refractive index (CRI) of ice–water mixtures using the Maxwell–Garnett (MG) mixing rule [30,31]. In our formulation, ice is treated as the host (matrix), and liquid water forms a coating around ice grains, yielding a representative “ice-core water-shell” microstructure. As liquid water exhibits a much stronger near-infrared (NIR) absorption than that of ice (Figure 1), such coatings enhance the bulk absorption of composite particles, thereby modifying the optical properties of snow.

The Maxwell–Garnett mixing rule can effectively characterize the dielectric content of such asymmetric composite structures, making it suitable for describing the modification of ice optical properties by water in wet or melted snow. The expression of the Maxwell–Garnett mixing rule for a two-component mixture is expressed as:

$${\displaystyle \frac{\mathsf{\epsilon }-{\mathsf{\epsilon }}_1}{\mathsf{\epsilon }+2{\mathsf{\epsilon }}_1}} = {\mathrm{V}}_2{\displaystyle \frac{{\mathsf{\epsilon }}_2-{\mathsf{\epsilon }}_1}{{\mathsf{\epsilon }}_2+{2\mathsf{\epsilon }}_1}}$$

(1)

where $\mathsf{\epsilon }$, ${\mathsf{\epsilon }}_1$, ${\mathrm{V}}_2$, and ${\mathsf{\epsilon }}_2$ are the effective dielectric constant of the mixture, the dielectric constant of the host material (ice in this study), the volume fraction, and the dielectric constant of the composite material (water), respectively. A solution to the Maxwell–Garnett mixing rule was reported by Wu et al. [31]. As shown in Equation (2), by adding more terms to the right-hand side of Equation (1), the complex refraction index of wet snow containing impurities can be calculated. Considering wet snow containing one impurity as an example, the formula is shown in Equation (2). If more impurities are included, Equation (2) can be similarly expanded:

$${\displaystyle \frac{\mathsf{\epsilon }-{\mathsf{\epsilon }}_1}{\mathsf{\epsilon }+2{\mathsf{\epsilon }}_1}} = {\mathrm{V}}_2{\displaystyle \frac{{\mathsf{\epsilon }}_2-{\mathsf{\epsilon }}_1}{{\mathsf{\epsilon }}_2+{2\mathsf{\epsilon }}_1}}+{\mathrm{V}}_3{\displaystyle \frac{{\mathsf{\epsilon }}_3-{\mathsf{\epsilon }}_1}{{\mathsf{\epsilon }}_3+2{\mathsf{\epsilon }}_1}}$$

(2)

where ${\mathrm{V}}_3 \mathrm{and} {\mathsf{\epsilon }}_3$ are the volume fraction and effective dielectric constant of the impurity, respectively.

2.2.2. Mie Scattering

In this study, the Mie scattering theory was employed to calculate the scattering and absorption properties of particles with respect to solar radiation. This theory, derived as an exact solution to Maxwell’s equations, is applicable to the radiative interactions of spherical particles whose sizes are comparable to the incident wavelength. By specifying the complex refractive index and particle radius, key optical parameters such as the extinction efficiency (Q_ext), single-scattering albedo (ω), and asymmetry factor (g) were obtained [32].

2.2.3. Four-Stream Radiative Transfer Approximation Scheme

To accurately simulate snow albedo under varying liquid water content and vertical stratification, we employed the four-stream discrete ordinates adding scheme [25]. By simultaneously considering radiative fluxes in four distinct directions, this model provides a more detailed representation of scattering anisotropy than that by the two-stream approximation. It is more suitable for conditions involving high solar zenith angles, complex layered structures, and multiple scattering, and it has been verified in the atmospheric applications [25]. In this study, the four discrete zenith-direction cosine values (μ) and their associated directional weights (a) are used as follows: μ₁ = 0.2113248654, μ₂ = 0.7886751346, μ_-1= −μ₁, μ_-2= −μ₂, a₁ = a_-1 = 0.5, a₂ = a_-2 = 0.5. where μ_i < 0 represents upward fluxes and μ_i > 0 represents downward fluxes [25,33]

2.2.4. Evaluation Metrics

To quantitatively evaluate the model performance, two statistical indicators were employed: Bias and Root Mean Square Error (RMSE). Bias represents the mean systematic deviation between the simulated and observed values, indicating whether the model tends to overestimate (positive Bias) or underestimate (negative Bias) the observations. RMSE reflects the overall magnitude of the prediction error and is sensitive to large deviations. Both metrics were computed as follows:

$$\mathrm{B}\mathrm{ias} = {\displaystyle \frac{1}{n}}\sum _{i=1}^n(S_i-T_i)$$

(3)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(S_i-T_i)}^2}$$

(4)

where S_i and T_i are the simulated and observed values, respectively, and n is the sample size. Bias can take either positive or negative values, with values closer to 0 indicating smaller systematic error. RMSE is always non-negative, and a smaller RMSE corresponds to higher model accuracy.

3. Results

The results obtained from both the idealized experiments and the case study on the Tibetan Plateau simulations, using the data and approaches introduced earlier, are presented in this section.

3.1. Results from Idealized Experiments

3.1.1. Snow Optical Properties

The optical properties of the layered snow were calculated based on Mie scattering theory [32]. We computed the optical characteristics of pure ice (dry snow) across the 0.2–2.6 µm spectrum for different grain sizes, including the Q_ext, ω, and g. The results (Figure 3) reveal distinct wavelength- and size-dependent behaviors. The Q_ext increases in an oscillational manner with wavelength, with smaller grains exhibiting stronger extinction, particularly pronounced in the near-infrared region (Figure 3a). The ω approaches unity in the visible band, indicating scattering dominance, while it decreases rapidly in the near-infrared with significant fluctuations. Larger grains showed reduced scattering efficiency (Figure 3b), the g generally increased with the wavelength, and larger snow grains demonstrated more pronounced forward scattering (Figure 3c).

Figure 3. Calculated Q_ext, ω, and g for pure ice (dry snow) with grain sizes of 50, 100, 200, 500, and 1000 µm.

These findings are consistent with those of earlier studies [9,34,35,36], confirming that snow optical behavior in the visible range is dominated by scattering, whereas absorption becomes significantly stronger in the near-infrared region.

3.1.2. Single-Layer Idealized Experiment

To systematically investigate the influence of LWC on the optical properties of snow, we first conducted an idealized single-layer experiment. The snow layer was assumed to be a homogeneous medium with three representative grain radii of 50, 200, and 500 µm. The LWC varied from 0% to 100% by volume with specific increments of 0%, 0.1%, 1%, 3%, 5%, 10%, 20%, and up to 100%. The spectral range covers 0.2–2.6 μm, encompassing major solar shortwave radiation energy bands.

Figure 4 shows the computed optical properties of snow with varying LWC and grain sizes. For the Q_ext (Figure 4a–c), minimal variation is observed within the 0.2–1.0 μm band, with increasing water content showing negligible effect. In the near-infrared region (1.0–2.6 μm), however, a significant enhancement in the absorption occurs. As the grain size increases from 50 to 500 μm, the Q_ext values gradually decrease, consistent with that of the saturation of scattering and absorption in larger particles under the geometric optics approximation.

Figure 4. Q_ext, ω, and g of ice–water mixtures calculated using the Maxwell–Garnett mixing rule for grain sizes of 50 μm, 200 μm, and 500 μm (In this experiment, the solar incident zenith angle is 0°, and the snow depth is 5 cm.).

Regarding the ω (Figure 4d–f), high values (close to 1) are maintained across the UV-visible spectrum, with little influence from water content. In the near-infrared band (1.0–2.6 μm), ω shows a fluctuating decrease with increasing LWC, indicating that liquid water has a significant influence. Larger grains exhibit greater sensitivity to water content in this spectral region, showing a more pronounced reduction in ω.

The variations in g are shown in Figure 4g–i. Similarly, g remained largely unaffected by water content in the UV-visible range. In the near-infrared region, as water content increases, g increases at some wavelengths where water vapor absorption is stronger than that of ice (Figure 1), such as near 1.4 μm. However, at wavelengths where water vapor absorption is weaker than that of ice (Figure 1), g decreases with increasing water content, such as near 1.5 μm. Overall, larger snow grains demonstrate stronger forward scattering, an effect more pronounced under wet conditions.

Figure 5 shows the spectral albedo and transmittance of dry snow (0% LWC) for different grain sizes (50 μm, 200 μm, and 500 μm) and snow densities (0.1 g cm⁻³, 0.2 g cm⁻³, and 0.3 g cm⁻³). The results indicate that snow albedo decreases only gradually in the visible band, whereas it exhibits a sharp oscillatory decrease in the near-infrared region with increasing wavelength. Additionally, albedo prominently decreased with increasing grain size.

Figure 5. Albedo of dry snow (0% LWC) for grain sizes (r) of 50 μm, 200 μm, and 500 μm, and densities (ρ) of 0.1 g cm⁻³, 0.2 g cm⁻³, and 0.3 g cm⁻³.

Figure 6 shows the relative difference in albedo between wet (containing liquid water) and dry snow. In the visible region (0.2–0.8 μm), the difference in albedo is minimal due to the weak absorption by both ice and liquid water. In contrast, the near-infrared region shows a substantial reduction in albedo for wet snow, exceeding 80% at certain wavelengths driven by the strong liquid water absorption features near 1.4 μm, 1.9 μm, and 2.6 μm (see Figure 2). The peaks in albedo reduction closely align with those of these absorption bands. The influence of liquid water on albedo changes with increasing grain size. The albedo of snow with small particle size increases and decreases more rapidly with the increasing of LWC. It indicates that snow with smaller grains is more sensitive to changes in LWC.

Figure 6. Relative difference in albedo ((wet snow albedo − dry snow albedo)/dry snow albedo × 100%) between wet snow and dry snow (0% LWC), for snowpacks with grain sizes of 50 μm, 200 μm, and 500 μm and densities of 0.1 g cm⁻³, 0.2 g cm⁻³, and 0.3 g cm⁻³.

3.2. Two-Layer Idealized Experiment

To further investigate the influence of snow stratigraphy and the vertical distribution of LWC on the optical properties of snow, a two-layer idealized experiment was designed. The snowpack was assumed to consist of upper and lower layers, with the spectral range remaining 0.2–2.6 μm. Four typical scenarios were defined (Table 1). In Case 1, both layers exhibited low LWC. In Case 2, the upper layer was dry, whereas the bottom layer had a higher LWC due to melting. Both layers showed high LWC as a result of extensive melting in Case 3. The upper layer had a high LWC due to melting, whereas the bottom layer remained relatively dry in Case 4.

Table 1. Experimental design for the two-layer idealized simulations.

Sample	Layer	Moisture Content (%)	Density (g cm−3)
Case1	Layer1	1	0.1
	Layer2	1	0.2
Case2	Layer1	1	0.1
	Layer2	60	0.2
Case3	Layer1	60	0.1
	Layer2	60	0.2
Case4	Layer1	60	0.1
	Layer2	1	0.2

The simulated spectral albedo for these four cases is shown in Figure 7. Cases 1 and 2, where the upper layer contained less liquid water and exhibited significantly higher albedo than that in Cases 3 and 4, particularly in the near-infrared region. These results demonstrate that snow albedo depends not only on the total LWC but also on its distribution. The liquid water present in the surface layer had a stronger reducing effect on the albedo than that by the water located deeper in the snowpack. The greatest reduction in reflectance was observed when liquid water was present in the surface layer of the snowpack.

Figure 7. Albedo spectra obtained from the two-layer idealized experiments.

3.3. Snow Albedo Model Validation in the Real Case

To evaluate the performance of the snow albedo model developed here under actual snow conditions, a validation experiment was conducted using observational data from the Tibetan Plateau (TP), Xinjiang (XJ), and Northeast China (NE). The selected sites (Figure 8), distributed across different elevations and covering various underlying surface types, including high mountains, river valleys, and plains, were selected to ensure regional representativeness. The geographic coordinates, observation period, and physical snow parameters (grain size, SD, density, LWC, and layered structure) for each site are summarized in Table 2. Data were obtained from the snow observation dataset (2017–2021) of the National Cryosphere Desert Data Center (NCDDC). In addition, the snow grain radius was derived using a correction factor based on the International Classification for Seasonal Snow on the Ground [37]. The selection of these stations takes into account that the grid distance from the satellite observation data does not exceed 500 m and that no significant terrain differences exist, enabling a more accurate assessment of model performance.

Figure 8. Observation sites in Xinjiang, the Tibetan Plateau, and Northeast China. Color shading indicates elevation.

Table 2. Geographic coordinates and snow physical variables for the observational sites named by XJ-Xinjiang, TP-Tibetan Plateau, and NE-Northeast China.

Station	Site	(Longitude, Latitude)	Time	Snow Depth (cm)	Layer (cm)	Moisture Content (%)	Density (g cm−3)	Particle Size (μm)
XJ-1	Altay	(87.932°N,47.787°E)	2018/8/9 16:40	13	0–5	0.526	0.1291	180
					5–10	1.468	0.1383	150
					10–13	8.9723	0.3443	165
XJ-2	Gumul County	(88.842°N,44.737°E)	2018/12/20 16:00	10	0–5	0.2233	0.1493	169
					5–10	0.8647	0.0999	158
XJ-3	Nanshan Astronomical Observatory	(87.235°N,43.435°E)	2018/1/23 13:40	9.5	0–5	0	0.1127	145
					5–9.5	0.037	0.1229	128
XJ-4	Baiyanggou	(87.235°N,43.435°E)	2019/1/14 13:10	10	0–5	2.58	0.1407	186
					5–10	0	0.259	213
TP-1	Gangba Village, Dari County, Golog Prefecture	(97.236°N,33.485°E)	2018/2/4 9:30	10	0–5	1.6193	0.1249	190
					5–10	0.035	0.0936	190
TP-2	Zhenqin Town, Chengduo County, Yushu Prefecture	(97.235°N,33.486°E)	2018/2/25 13:00	10	0–5	0.354	0.1951	200
					5–10	0	0.2398	121
TP-3	Horba Township, Zhongba County, Shigatse Prefecture, TAR	(30.450°N, 82.673°E)	2021/01/17 14:00	8	0–8	0.0300	0.1281	175
TP-4	Pulan Town, Pulan County, Ngari Prefecture, TAR	(30.186°N, 81.192°E)	2021/01/19 12:00	10	0–5	0.2673	0.1661	193
					5–10	0.0377	0.1558	174
TP-5	Pulan Town, Pulan County, Ngari Prefecture, TAR	(30.183°N, 81.176°E)	2021/01/20 12:00	15	0–5	0.3080	0.1646	191
					5–10	0.0377	0.1592	194
					10–15	0.0380	0.1724	135
TP-6	Dingsong, Pulan County, Ngari Prefecture, TAR	(30.256°N, 81.097°E)	2021/01/19 12:00	12	0–5	0.1130	0.1518	173
					5–12	0.0000	0.3965	138
TP-7	Mentuo Township, Gar County, Ngari Prefecture, TAR	(31.335°N, 80.620°E)	2021/01/19 15:00	10	0–5	2.4230	0.1566	174
					5–10	0.4597	0.1645	177
TP-8	Dongru Township, Ritu Count, Ngari Prefecture, TAR	(34.084°N, 80.359°E)	2021/01/20 12:00	7	0–7	0.0820	0.1282	137
NE-1	Genhenan	(121.137°N,50.59°E)	2018/3/9 16:50	15	0–5	0	0.1156	263
					5–10	0.1417	0.1075	189
					10–15	2.0777	0.0366	121
NE-2	Mishan North	(131.84°N,45.586°E)	2018/1/2 12:05	16	0–14	2.106	0.0867	185
					14–16	0.9173	0.0949	525

The satellite-derived albedo data used in this study here were obtained from the MODIS Global Albedo Product. The WSA and BSA values corresponding to Band 5 (1.23–1.25 μm) of MODIS for each of the six sites are provided in Table 3. To facilitate the comparison between simulated and observed albedos, the satellite-derived albedo was calculated using a weighting method based on the ratio of direct to diffuse radiative flux densities from the ERA5 reanalysis dataset for the corresponding time period. The formula used was as follows:

$$\mathsf{\alpha } = {\mathsf{\alpha }}_{\mathrm{black}}·{\mathrm{F}}_{\mathrm{dir}} + {\mathsf{\alpha }}_{\mathrm{white}}·{\mathrm{F}}_{\mathrm{diff}}$$

(5)

where $\mathsf{\alpha }$, ${\mathsf{\alpha }}_{\mathrm{black}}$, ${\mathsf{\alpha }}_{\mathrm{white}}$, ${\mathrm{F}}_{\mathrm{dir}}$, and ${\mathrm{F}}_{\mathrm{diff}}$ represent the actual albedo, BSA, WSA, the proportion of direct radiation to total downwelling radiation (a ratio between 0 and 1), and the proportion of diffuse radiation, respectively, with F_dir + F_diff = 1.

Table 3. Retrieved WSA and BSA values and the corresponding albedo derived from them for these sites.

Station	Time	Longitude	Latitude	White Sky Albedo	Black Sky Albedo	Albedo
XJ-1	2018/08/09 16:40	87.9318	47.7868	0.3835	0.3545	0.3676
XJ-2	2018/12/20 16:00	88.8425	44.7371	0.3365	0.3510	0.3383
XJ-3	2018/01/23 13:40	87.2350	43.4345	0.2508	0.2594	0.2547
XJ-4	2019/01/14 13:10	87.2352	43.4346	0.2340	0.2436	0.2399
TP-1	2018/02/04 09:30	97.2362	33.4851	0.3225	0.3157	0.3189
TP-2	2018/02/25 13:00	97.2355	33.4862	0.3636	0.3505	0.357
TP-3	2021/01/17 14:10	82.6732	30.4504	0.3985	0.3947	0.3973
TP-4	2021/01/19 12:10	81.1923	30.1856	0.2932	0.2876	0.2925
TP-5	2021/01/19 13:30	81.1764	30.1834	0.2869	0.2812	0.2847
TP-6	2021/01/19 15:50	81.0966	30.2562	0.2950	0.2894	0.2937
TP-7	2021/01/20 12:40	80.6204	31.3346	0.3138	0.3095	0.3131
TP-8	2021/01/21 13:10	80.3590	34.0839	0.2339	0.2340	0.2339
NE-1	2018/03/09 16:50	121.1370	50.5901	0.2769	0.2787	0.2784
NE-2	2018/01/02 12:05	131.8397	45.5864	0.3275	0.3577	0.3362

In addition, this study also calculated the results of a comparative model to better evaluate the accuracy of the new model. This comparative model explicitly represents the mixture by linearly weighting the complex refractive indices of ice and water and employs a two-stream radiative transfer approximation, which is the same as SNICAR. It is referred to as the “Liner + 2S” model.

Figure 9 presents the spectral albedo (0.2–2.6 μm) at all sites for two model simulation results and their differences. All sites exhibit the typical snow signature: high albedo in the visible (<1.0 μm) and a sharp decrease into the near-infrared. This spectral behavior is consistent with the results from the preceding idealized experiments. Compared with the “Liner + 2S” model, the new model (“Maxwell + 4S”) reveals a smaller albedo in most near-infrared spectra, with a localized maxima difference approaching 0.15 near 1.0, 1.2, 1.4, and 1.8 μm (Figure 9c).

Figure 9. Spectral albedo (0.2–2.6 μm) simulated for the observation sites using the Linear + 2S and Maxwell + 4S models, together with the corresponding difference colormap (“Linear + 2S”-“Maxwell + 4S”).

The results indicate that the presence of liquid water markedly modifies the scattering characteristics of snow grains. At high-elevation, low-temperature sites, the influence of LWC on the optical properties was minimal and remained similar to that of dry snow. In contrast, sites at lower elevations or those more susceptible to precipitation exhibited more pronounced liquid water effects. In real environments, the optical properties of snow are modulated by both LWC and its vertical distribution, which is consistent with the conclusions from the multi-layer idealized experiments (Figure 7).

To evaluate the simulation accuracy of the two schemes under real observation conditions, the bias and root-mean-square error (RMSE) between the modeled albedo and the satellite-retrieved albedo were calculated. Figure 10 presents the point-by-point comparison for the 1.23–1.25 μm band. The “Liner + 2S” model (Figure 10a) yields a positive bias of 0.0206 and an RMSE of 0.0637, indicating a systematic overestimation relative to the satellite product. In contrast, the “Maxwell + 4S” model (Figure 10b) reduces the bias to −0.0069 and the RMSE to 0.0583, with the data points lying closer to the 1:1 reference line, thereby demonstrating a substantially improved agreement with the satellite retrievals. This improvement reflects the potential of the new model in enhancing the simulation of albedo on wet snow surfaces.

Figure 10. One-to-one comparison between the modeled albedo and the satellite-derived albedo integrated over the 1.23–1.25 μm band.

4. Discussion

This study established a wet snow albedo model. The influence of water content on near-infrared albedo was analyzed through single-layer and two-layer idealized experiments, as well as real observation experiments, and the accuracy of the new albedo model was also evaluated. Although this study verified the accuracy and generalization ability of the new model to a certain extent, a few limitations remain. First, the Maxwell–Garnett mixing rule and Mie theory assume an idealized “ice-core water-shell” configuration and spherical particles, neglecting complex microphysical processes such as non-spherical snow grains [9] and interconnected liquid water structures. These simplifications may affect the generalizability of the model. Second, spatiotemporal matches among ground observations, satellite data, and reanalysis datasets can introduce uncertainties. The limited ground-based observation data also make it difficult to carry out large-scale case comparison experiments in this study, which may lead to incomplete conclusions.

In the future, we will conduct further research to continuously improve the model, including incorporating non-spherical scattering and examining the applicability and accuracy of the model under conditions of wet snow containing various impurities [35] with multiple layers.

5. Conclusions

A snow albedo model explicitly accounting for LWC was developed by integrating the Maxwell–Garnett mixing rule, Mie scattering theory, and a four-stream discrete ordinate adding radiative transfer method. The influence of liquid water on snow albedo was systematically investigated through single-layer and multi-layer idealized experiments as well as real-case evaluations.

The single-layer idealized experiments demonstrated that LWC is a key factor affecting albedo in the near-infrared region. Snow with smaller grain sizes exhibits higher sensitivity to liquid water, resulting in a more substantial decrease in albedo. Two-layer experiments further revealed the important role of surface snow water content. Albedo reduction was stronger when liquid water was concentrated near the surface, and the decrease was most pronounced when the entire snow layer was wet.

Validation based on in situ observations from multiple sites across Xinjiang, the Tibetan Plateau, and Northeast China further supports these findings. After explicitly incorporating the layered liquid water content, the albedo simulated using the Maxwell–Garnett mixing rule and four-stream radiative transfer scheme achieved a bias of only 0.004 and a reduced RMSE, outperforming the commonly used linear mixing rule with the two-stream model (e.g., the SNICAR model). This indicates that the proposed model maintains high accuracy and physical consistency under both wet-snow conditions and vertically stratified snowpacks. The point-to-point comparison with satellite-retrieved albedo also shows that the new model effectively suppresses the systematic near-infrared overestimation that occurs in the “linear + 2S” model, resulting in a distribution of points that lies much closer to the 1:1 reference line. These results further confirm the robustness and regional applicability of the model at larger spatial scales.