[9] MODIS data have enabled the analysis of snow properties beyond snow covered area (SCA) because its radiometric range in the VIS does not saturate at the large radiances from snow. Other multi-spectral sensors such as the pre-Landsat-8 Thematic Mappers and the NASA Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) saturate at reflectances usually less than 50% in the VIS, leaving the impact of LAI unknowable. Fortunately, the dynamic range in the VIS bands of the Operational Line Imager (OLI) on Landsat-8 should not saturate over snow, thereby enabling frequent rigorous validation of the MODDRFS product. The location and dynamic range of MODIS band passes enables the detection of changes in absorption in the VIS and changes in grain size expressed in the NIR/SWIR (Figure 1d).

#### 3.1. MODDRFS

[10] The MODDRFS algorithm infers per-pixel radiative forcing by LAI in snow using MODIS surface reflectance data (Terra MODIS MOD09GA, Aqua MODIS MYD09GA; we speak to MOD09GA only for brevity) and a coupled radiative transfer model for snow. MODDRFS determines the spectral reflectance differences between the measured MODIS spectrum and the modeled clean snow spectrum of the same OGR. Integration of the band-wise multiplication of this spectral difference with local spectral irradiance that accounts for terrain variations gives the instantaneous at-surface radiative forcing (W m^{−2}).

[11] MODDRFS first determines those pixels that can provide more robust retrievals (relatively free of mixing) from the MODIS Snow Covered Area and Grainsize (MODSCAG) fractional snow and vegetation products [*Painter et al.*, 2009]. MODSCAG finds maximal snow cover in winter/spring from a time series approach, from which we determine the per-pixel potential for complete cover. Among those pixels, we then use each acquisition's retrievals of fractional vegetation to remove those pixels that have a mixed cover of vegetation that has been exposed since maximum cover. We do not take this step for the mixed cover of rock because it would be confused with and remove pixels impacted by LAI (which introduces error -section 5.2 below).

[12] As with the spectral albedo of snow, the spectral hemispherical-directional reflectance factor (HDRF [*Schaepman-Strub et al.*, 2006] – the MOD09GA retrieval) of snow also varies with grain size (Figure 1). We estimate the OGR and identify the clean snow HDRF through the normalized difference grain size index (NDGSI):

where *MODIS*_{2} is the MOD09GA surface reflectance in MODIS band 2 (band center ∼0.858 *μ*m) and *MODIS*_{5} is the MOD09GA surface reflectance in MODIS band 5 (band center ∼1.240 *μ*m) (Figure 1d). NDGSI has a logarithmic relationship with OGR due to the decreasing changes in HDRF with increases in OGR, and is sensitive to solar zenith angle (SZA) (Figure 1b). From the OGR, we determine the clean snow spectrum for the same OGR and SZA with the discrete ordinates solution to the radiative transfer equation. The description of these clean snow spectra is given in the auxiliary material.

[13] We use the Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART) model in conjunction with the 3 arc second Shuttle Radar Topography Mission (SRTM) digital elevation model (DEM) to estimate per-pixel clear sky incident spectral irradiance. The University of Wisconsin's MODIS Terra overpass predictor (http://eosweb.ssec.wisc.edu/) was used to determine the time of acquisition and solar ephemeris. Direct and diffuse spectral irradiance are first modeled in SBDART for a range of SZA and elevation bands. Incident spectral irradiances are then determined at 1/5th MODIS pixel spatial resolution and up-scaled to obtain the mean per-MODIS pixel irradiance spectrum. Per-pixel terrain slope and aspect give the plane to which we correct the direct component of level-surface solar irradiance according to the following relationship:

where *β* is the local solar zenith angle, *θ*_{s} is the solar zenith angle for a level surface, *ϕ*_{s} is the solar azimuth angle, *θ*_{n} is the surface slope, and *ϕ*_{n}is the surface aspect. We then determine corrected per-pixel solar spectral irradiances,*E*_{corrected,λ}, according to:

where *E*_{direct} and *E*_{diffuse}are the direct and diffuse spectral irradiances. This calculation assumes that the diffuse and terrain-scattered irradiances are identical. It is valid to first order and ameliorates the need for a complex, intensive topographic treatment for this global product.

[14] Because MODIS does not measure the entire hemisphere-reflected flux, we must use the modeled understanding of the relationship between the MODIS-derived spectral HDRF and hemispherical spectral albedos [*Schaepman-Strub et al.*, 2006]. Here we use scalars, *c*, between the directional reflectance spectrum at the observation geometry, *R*, and the spectral albedo for the same irradiance geometry:

where *λ* denotes the wavelength, *θ*_{0} and *θ*_{r} the solar and viewing zenith angles, *ϕ*_{r} the relative azimuth and *ζ*_{atm} the atmospheric properties.

[15] Because of the lack of spectral continuity in the MODIS spectrum we do not know the wavelength at which the measured and modeled clean spectra diverge (Figure 1c). To account for this, the measured spectrum is fit to the clean spectrum at *MODIS*_{2} (band center wavelength ∼0.858 *μ*m and upper end ∼0.876 *μ*m) and we determine the radiative forcing for the wavelength range down to 0.35 *μ*m. Moreover, because of the discrete bands, we model the irradiances and spline albedos to a continuous spectrum of 0.01 *μ*m bands across the range 0.350–0.876 *μ*m. We then retrieve the radiative forcing estimate, *F*, from the following:

where *E*_{corrected, λ} is the corrected irradiance, *α*_{clean, λ} is the clean snow spectral albedo of the MODIS OGR, *α*_{MODIS, λ} is the spectral albedo of the MODIS pixel, and Δ*λ* is 0.01 *μ*m. The retrieved radiative forcings are then instantaneous values and not daily averages convolved with the diurnal cycle of irradiances.