Author Contributions
Conceptualization, A.D. and Z.J.; Data curation, A.D., J.I.P., J.G., S.Y. and L.C.; Formal analysis, A.D., Z.J., Y.D. and S.Y.; Funding acquisition, Z.J.; Investigation, X.Z., J.G. and R.X.; Methodology, Y.C.; Validation, L.C.; Writing—original draft, A.D.; Writing—review & editing, A.D., Z.J., Y.D., X.Z., J.I.P. and L.M.
Figure 1.
The solar zenith angle (SZA) distribution of the Polarization and Directionality of the Earth’s Reflectances (POLDER) and field-measured data.
Figure 1.
The solar zenith angle (SZA) distribution of the Polarization and Directionality of the Earth’s Reflectances (POLDER) and field-measured data.
Figure 2.
The simulated data of the bicontinuous photon tracking (bic-PT) model (red dots) and the reconstructed bidirectional reflectance distribution function (BRDF) shapes by the RossThick-Snow (RTS) model (green) and RossThick-Roujean (RTR) model (black) in the principal plane (PP) of the red band (670 nm), where (a), (b) and (c) represent the SZAs equal to 0°, 40° and 70°, respectively. The equivalent grain radius is 0.1 mm, the b parameter is 1, and the snow density is 0.1 g/cm3.
Figure 2.
The simulated data of the bicontinuous photon tracking (bic-PT) model (red dots) and the reconstructed bidirectional reflectance distribution function (BRDF) shapes by the RossThick-Snow (RTS) model (green) and RossThick-Roujean (RTR) model (black) in the principal plane (PP) of the red band (670 nm), where (a), (b) and (c) represent the SZAs equal to 0°, 40° and 70°, respectively. The equivalent grain radius is 0.1 mm, the b parameter is 1, and the snow density is 0.1 g/cm3.
Figure 3.
The density plots comparing the simulated data with the RossThick-Roujean (RTR)model (blue dots) and RossThick-Snow (RTS)model (red dots) in the 670 nm (a) and 865 nm (b) bands.
Figure 3.
The density plots comparing the simulated data with the RossThick-Roujean (RTR)model (blue dots) and RossThick-Snow (RTS)model (red dots) in the 670 nm (a) and 865 nm (b) bands.
Figure 4.
The spectral albedo simulated by the bic-PT model (black) and the spectral albedo retrieved by the RTS (blue) and RTR models (red), where (a), (b) and (c) represent SZAs of 0°, 40° and 70°, respectively. The equivalent grain radius is 0.1 mm, the b parameter is 1, and the snow density is 0.1 g/cm3.
Figure 4.
The spectral albedo simulated by the bic-PT model (black) and the spectral albedo retrieved by the RTS (blue) and RTR models (red), where (a), (b) and (c) represent SZAs of 0°, 40° and 70°, respectively. The equivalent grain radius is 0.1 mm, the b parameter is 1, and the snow density is 0.1 g/cm3.
Figure 5.
The comparison of albedos retrieved by these two kernel-driven models with the albedo simulated by the bic-PT model, and the two wavelengths are 670 nm (a) and 865 nm (b). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 5.
The comparison of albedos retrieved by these two kernel-driven models with the albedo simulated by the bic-PT model, and the two wavelengths are 670 nm (a) and 865 nm (b). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 6.
The comparison of the albedos retrieved by the two kernel-driven models with the “validation” albedo, and the two wavelengths are 670 nm (a) and 865 nm (b). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 6.
The comparison of the albedos retrieved by the two kernel-driven models with the “validation” albedo, and the two wavelengths are 670 nm (a) and 865 nm (b). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 7.
The angle distribution of the “validation” data (a) and three typical azimuthal samplings (b), (c) and (d) (i.e., ID = 0, 4 and 9), where the red dots represent the sun’s direction, and the black dots represent the view directions.
Figure 7.
The angle distribution of the “validation” data (a) and three typical azimuthal samplings (b), (c) and (d) (i.e., ID = 0, 4 and 9), where the red dots represent the sun’s direction, and the black dots represent the view directions.
Figure 8.
The field-measured data (red dots) and reconstructed BRDF shapes using the RTS model (green) and RTR model (black) in the PP and red band (670 nm), where (a) and (b) represent the cases, and the SZAs equal 50° and 70°, respectively.
Figure 8.
The field-measured data (red dots) and reconstructed BRDF shapes using the RTS model (green) and RTR model (black) in the PP and red band (670 nm), where (a) and (b) represent the cases, and the SZAs equal 50° and 70°, respectively.
Figure 9.
The density plots comparing simulated data with the RTR model (blue dots) and RTS model (red dots) in the 670 nm (a) and 865 nm (b) bands.
Figure 9.
The density plots comparing simulated data with the RTR model (blue dots) and RTS model (red dots) in the 670 nm (a) and 865 nm (b) bands.
Figure 10.
The comparison of black-sky albedo (BSA) (red dots) and white-sky-albedo (WSA) (blue dots) retrieved by the two kernel-driven bands in the red (a) and near-infrared (NIR) bands (b) (i.e., 670 nm and 865 nm). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 10.
The comparison of black-sky albedo (BSA) (red dots) and white-sky-albedo (WSA) (blue dots) retrieved by the two kernel-driven bands in the red (a) and near-infrared (NIR) bands (b) (i.e., 670 nm and 865 nm). The central dotted lines are 1:1 lines, the outer dotted lines are 0.02 offset lines, and the outermost dotted lines are 0.05 offset lines.
Figure 11.
The field-measured data (red dots) and reconstructed BRDF shapes using the RTS model (green) and RTR model (black) in the PP and red band (670 nm), where (a) and (b) represent the cases, and the average SZAs equal 68° and 69°, respectively.
Figure 11.
The field-measured data (red dots) and reconstructed BRDF shapes using the RTS model (green) and RTR model (black) in the PP and red band (670 nm), where (a) and (b) represent the cases, and the average SZAs equal 68° and 69°, respectively.
Figure 12.
The comparison of BSA (blue) and WSA (red) retrieved by these two kernel-driven models in 6 spectral channels (i.e., 490, 565, 670, 765, 865 and 1020 nm). Where (
a) and (
b) represent the bias and relative error (RE) of snow albedo retrieved by these two models as a function of the wavelength. We further compare the difference between these two models for the POLDER shortwave albedos derived by using Equation (1). The results are shown in
Figure 13 and are generally consistent with the results of the narrowband albedo. The shortwave albedo retrieved by the two models also has a high correlation (R
2 = ~0.95), especially in the BSA. However, the shortwave albedo retrieved by the RTR model shows a significant difference relative to the RTS model (P < 0.05), with their differences, which are 1.43% and 1.54%, respectively. Some points are beyond the range of ±0.02, accounting for 17.01% and 15.10% in the BSA and WSA, respectively. This difference is most probably attributed to their ability in fitting POLDER multiangle observations because the potential uncertainties resulting from various factors are completely consistent between the two models.
Figure 12.
The comparison of BSA (blue) and WSA (red) retrieved by these two kernel-driven models in 6 spectral channels (i.e., 490, 565, 670, 765, 865 and 1020 nm). Where (
a) and (
b) represent the bias and relative error (RE) of snow albedo retrieved by these two models as a function of the wavelength. We further compare the difference between these two models for the POLDER shortwave albedos derived by using Equation (1). The results are shown in
Figure 13 and are generally consistent with the results of the narrowband albedo. The shortwave albedo retrieved by the two models also has a high correlation (R
2 = ~0.95), especially in the BSA. However, the shortwave albedo retrieved by the RTR model shows a significant difference relative to the RTS model (P < 0.05), with their differences, which are 1.43% and 1.54%, respectively. Some points are beyond the range of ±0.02, accounting for 17.01% and 15.10% in the BSA and WSA, respectively. This difference is most probably attributed to their ability in fitting POLDER multiangle observations because the potential uncertainties resulting from various factors are completely consistent between the two models.
Figure 13.
The comparison of the broadband BSA (a) and WSA (b) retrieved by these two kernel-driven models. The thick dashed lines are 1:1 lines, the thin dashed lines are 0.02 and 0.05 offset lines deviating from the 1:1 lines, respectively.
Figure 13.
The comparison of the broadband BSA (a) and WSA (b) retrieved by these two kernel-driven models. The thick dashed lines are 1:1 lines, the thin dashed lines are 0.02 and 0.05 offset lines deviating from the 1:1 lines, respectively.
Table 1.
The input parameters of the bicontinuous photon tracking (bic-PT) model.
Table 1.
The input parameters of the bicontinuous photon tracking (bic-PT) model.
Parameters | Value | Step | Unit |
---|
Monte Carlo superposition (N) | 1000 | - | - |
photon total | 50,000 | - | - |
equivalent grain radius | 0.05–0.50 | 0.05 | mm |
structure parameter (b) | 1–16 | 5 | - |
snow density | 0.1–0.5 | 0.1 | g/cm3 |
wavelength number | 2 | - | - |
wavelength | 0.67–0.865 | 0.195 | μm |
solar zenith angle | 0–70 | 10 | degrees (°) |
view zenith angle | 5–70 | 5 | degrees (°) |
relative azimuth angle | 1–359 | 2 | degrees (°) |
snow depth | 1.5 | - | m |
soil reflectance | 0 | - | - |
streams number | 32 | - | - |
order number | 4 | - | - |
Table 2.
The location, targets and measurement dates of the field-measured bidirectional reflectance factors (BRFs).
Table 2.
The location, targets and measurement dates of the field-measured bidirectional reflectance factors (BRFs).
Date | Site | Latitude | Longitude | Sample |
---|
Apr. 2005 | Sodankylä, Etupiha | 67.0021° | 27.2430° | natural snow |
Apr. 2007 | Tahtela, Sodankylä | 67.3622° | 26.6344° | natural snow |
Apr. 2008 | Sodankylä | 67.3628° | 26.6355° | new snow; old snow |
Mar. 2009 | Masala | 60.1719° | 24.5542° | natural snow |
Apr. 2009 | Kommattivaara, Sodankylä | 67.4211° | 26.7923° | natural snow |
Jun.–Jul. 2010 | Summit | 72.5961° | -38.4219° | natural snow |
Mar. 2010 | Sodankylä | 67.3627° | 26.6356° | natural snow |
Mar. 2013 | Luoman Asema | 60.1721° | 24.5486° | natural snow; snow + dust |
Apr. 2013 | Sodankylä | 67.3958° | 26.6141° | natural snow; snow + volcanic sand, soot, and silt |
Table 3.
The reconstructed bidirectional reflectance distribution function (BRDF) parameters and statistics of these two kernel-driven models for three typical solar zenith angles (SZAs) (i.e., SZA = 0°, 40° and 70°), and the BRDF parameter fgeo for the RossThick-Snow (RTS)model is fsnw. The fiso(λ), fvol(λ), fgeo(λ) and fsnw(λ) are the weight components of the isotropic scattering, volume scattering, geometric-optical scattering and snow kernels, respectively.
Table 3.
The reconstructed bidirectional reflectance distribution function (BRDF) parameters and statistics of these two kernel-driven models for three typical solar zenith angles (SZAs) (i.e., SZA = 0°, 40° and 70°), and the BRDF parameter fgeo for the RossThick-Snow (RTS)model is fsnw. The fiso(λ), fvol(λ), fgeo(λ) and fsnw(λ) are the weight components of the isotropic scattering, volume scattering, geometric-optical scattering and snow kernels, respectively.
Model | SZA (°) | fiso | fvol | fgeo | R2 | RMSE | Bias | α |
---|
RTR | 0 | 1.026 | 0.020 | 0.151 | 0.991 | 0.007 | 0.000 | -- |
40 | 0.961 | 0.000 | 0.048 | 0.333 | 0.031 | 0.000 | -- |
70 | 0.786 | 0.301 | 0.000 | 0.495 | 0.105 | 0.000 | -- |
RTS | 0 | 0.868 | 0.000 | 1.158 | 0.999 | 0.003 | 0.000 | 0.00 |
40 | 0.869 | 0.411 | 1.960 | 0.936 | 0.009 | 0.000 | 0.05 |
70 | 0.845 | 0.167 | 0.538 | 0.965 | 0.028 | 0.000 | 0.30 |
Table 4.
The statistical results of the RossThick-Roujean (RTR) model for the different angular samplings in the red (670 nm) band.
Table 4.
The statistical results of the RossThick-Roujean (RTR) model for the different angular samplings in the red (670 nm) band.
ID | R2 | RMSE | Bias | RE(%) | T-test | P value |
---|
0 | 0.929 | 0.018 | 0.014 | 1.462 | 2.177 | 0.000 |
1 | 0.937 | 0.017 | 0.013 | 1.387 | 2.080 | 0.000 |
2 | 0.956 | 0.015 | 0.011 | 1.184 | 1.798 | 0.000 |
3 | 0.976 | 0.010 | 0.008 | 0.836 | 1.283 | 0.000 |
4 | 0.982 | 0.006 | 0.003 | 0.406 | 0.546 | 0.001 |
5 | 0.968 | 0.005 | -0.003 | 0.439 | 0.432 | 0.008 |
6 | 0.896 | 0.013 | -0.009 | 1.052 | 1.690 | 0.000 |
7 | 0.673 | 0.023 | -0.016 | 1.847 | 3.042 | 0.000 |
8 | 0.377 | 0.031 | -0.022 | 2.493 | 4.076 | 0.000 |
9 | 0.495 | 0.026 | -0.018 | 2.039 | 3.257 | 0.000 |
Table 5.
The statistical results of the RossThick-Snow (RTS) model for the different angular samplings in the red (670 nm) band.
Table 5.
The statistical results of the RossThick-Snow (RTS) model for the different angular samplings in the red (670 nm) band.
ID | R2 | RMSE | Bias | RE(%) | T-test | P value |
---|
0 | 0.877 | 0.008 | 0.003 | 0.602 | 0.493 | 0.001 |
1 | 0.894 | 0.008 | 0.003 | 0.551 | 0.495 | 0.001 |
2 | 0.925 | 0.007 | 0.003 | 0.451 | 0.507 | 0.001 |
3 | 0.952 | 0.005 | 0.003 | 0.364 | 0.478 | 0.001 |
4 | 0.976 | 0.004 | 0.002 | 0.293 | 0.337 | 0.021 |
5 | 0.991 | 0.002 | 0.000 | 0.175 | 0.005 | 0.971 |
6 | 0.952 | 0.007 | -0.003 | 0.410 | 0.576 | 0.000 |
7 | 0.838 | 0.015 | -0.008 | 0.972 | 1.397 | 0.000 |
8 | 0.633 | 0.025 | -0.016 | 1.904 | 2.824 | 0.000 |
9 | 0.599 | 0.030 | -0.020 | 2.391 | 3.429 | 0.000 |
Table 6.
The BRDF parameters and statistics of these two models for the field-measured data, where the BRDF parameter fgeo for the RTS model is fsnw. The fiso(λ), fvol(λ), fgeo(λ) and fsnw(λ) are the weight components of the isotropic scattering, volume scattering, geometric-optical scattering and snow kernels, respectively.
Table 6.
The BRDF parameters and statistics of these two models for the field-measured data, where the BRDF parameter fgeo for the RTS model is fsnw. The fiso(λ), fvol(λ), fgeo(λ) and fsnw(λ) are the weight components of the isotropic scattering, volume scattering, geometric-optical scattering and snow kernels, respectively.
Model | SZA (°) | fiso | fvol | fgeo | R2 | RMSE | Bias | α |
---|
RTR | 50 | 0.954 | 0.000 | 0.000 | 0.000 | 0.050 | 0.000 | -- |
70 | 0.964 | 0.119 | 0.000 | 0.056 | 0.161 | 0.000 | -- |
RTS | 50 | 0.973 | 0.000 | 0.838 | 0.950 | 0.011 | 0.000 | 0.19 |
70 | 1.006 | 0.000 | 0.721 | 0.961 | 0.033 | 0.000 | 0.30 |