3.1. Snow Surface Albedo
 Albedo was estimated throughout the snowmelt season (i.e., 6 April to 27 July 1997) using three approaches: (1) a pixel-specific albedo decay function based on 3 AVIRIS scenes (5 May, 21 May, and 18 June 1997) [Molotch et al., 2004], (2) a decay function developed from first principles for the BATS model, and based on solar zenith angle, snow temperature, and snow age [Dickinson et al., 1993], and (3) an empirically derived standard albedo decay function (SADF) based solely on snow surface age [U.S. Army Corps of Engineers, 1956].
 Snow surface albedo was obtained from the AVIRIS scenes using a spectral unmixing model that has a measurement error of 0.02 [Painter et al., 2003]. The 17-m resolution AVIRIS data were degraded to 30 m to match that of the digital elevation model. During time periods without snowfall between two AVIRIS acquisitions, daily pixel-specific albedo estimates were obtained by linearly interpolating albedo as a function of time. When snowfall occurred between two acquisitions and after the final acquisition, albedo was estimated using a linear AVIRIS albedo decay function (AADF) that we derived using the AVIRIS-observed pixel-specific albedo and the snow age at the time of observation. Snow age was determined from precipitation and temperature data from the meteorological stations in the catchment (Figure 1); snow age was set to 0 when precipitation occurred at temperatures below 0°C. Pixel-specific albedo estimates were obtained from the AADF as follows:
where αAh is the estimated albedo for snow age xh, αi and αk are AVIRIS-observed albedo values for acquisitions corresponding to snow ages xi and xk respectively. After a snowfall event the AADF was reset to the maximum AVIRIS-observed albedo value of the 3 acquisitions (i.e., 0.84); snow albedo tends to be fairly uniform immediately after snowfall, becoming spatially heterogeneous as metamorphic processes occur at different rates across a basin. Here we applied a single decay function for each pixel and thus we assumed that the spatial variability in albedo decay rates were relatively consistent after each snowfall event.
 The SADF estimates daily albedo using:
where, αsh is the SADF snow surface albedo estimate for snow age xh.
 Hourly pixel-specific BATS albedo estimates, αb were obtained using [Dickinson et al., 1993; Yang et al., 1997]:
where αVIS is snow albedo for wavelengths <0.7 μm, αIR is snow albedo for wavelengths >0.7 μm, and fage is dimensionless snow age. We assume that snowfall events deposited at least 10 mm of water equivalent and therefore set ts and fage = 0 at the time of snowfall. After snowfall ts is calculated as:
where N indicates the current time step; and r1 represents the impact of vapor diffusion on snow surface grain growth:
where Tsn (K) is snow surface temperature. The effect of meltwater refreeze is represented by r2:
The effect of dust and soot on snow surface albedo is represented by r3 and is set to 0.3. The factor of solar illumination angle, fz was calculated as
where b is adjustable based on observations but was set to 2 (as per BATS) and θz is the pixel-specific solar illumination angle determined using the method of Dozier and Frew .
 We mapped the spatial differences between the two ground-based parameterizations and AVIRIS-observed albedo on 5 and 21 May and 18 June 1997. Because the BATS parameterization includes terms for solar illumination angle and snow temperature we compared BATS albedo estimates corresponding to the specific hour of each AVIRIS acquisition as snow temperature and illumination angles vary during the day.
 Multivariate linear regressions were used to relate differences between AVIRIS-observed albedo and modeled albedo to elevation, slope, aspect, average incident solar radiation, and maximum upwind slope (i.e., variables that potentially influence spatial variability in snow surface metamorphic processes and therefore albedo). Elevation, slope and aspect were derived from the level 1 standard U.S. Geological Survey digital elevation model (DEM). Average incident solar radiation throughout the snowmelt season was calculated from the hourly solar radiation surfaces described below. Maximum upwind slope, a terrain-based parameter designed to account for the effect of wind redistribution on snow accumulation [Winstral et al., 2002], is included because wind can significantly alter snow surface roughness and grain size. Molotch et al.  provide a more detailed explanation of the derivation of these variables.
 Meteorological influences on albedo differences were also estimated using multivariate linear regressions relating albedo differences in each pixel to corresponding pixel values of accumulated degree days and accumulated incident solar radiation. Accumulated degree days were calculated as the summation of the average daily air temperature above 0°C for each day between the last snowfall event and the AVIRIS acquisition date. Accumulated incident solar radiation was calculated as the summation of average daily incident solar radiation over the same period. These two variables were used as solar radiative and sensible heat fluxes account for the majority of energy exchange between the snowpack and the atmosphere [Cline, 1997; Marks and Dozier, 1992] and therefore snowpack metamorphism and snow surface albedo. The techniques used to distribute air temperature and solar radiation observations over the basin are described in the next section.
3.2. Snowmelt and Snowpack Mass Balance
 Precipitation inputs to the snowpack during the ablation season were small and therefore we calculate snow water equivalent as:
where Mj is the melt flux at time step j, SWEn is the SWE of each pixel at time step n, and SWE0 is the initial SWE or SWE at the time step corresponding to the field campaigns. By knowing any two of the three terms in the above expression the mass balance can be closed. We manipulated this expression in two independent simulations to estimate snow water equivalent (case 1) and snow extent (case 2). Field measurements of SWE were held out of case 1 simulations and then used as an independent check on model performance. In the case 2 simulations field measurements of SWE were used to initialize the model and remotely sensed snow covered area (SCA) data, held out of the simulation, were used to evaluate model performance. This approach allows us to evaluate model performance with respect to snowpack mass balance using two measurements that are independent of streamflow. Previous work has primarily used streamflow observations to evaluate snowmelt model performance, potentially misinterpreting results since streamflow incorporates the combined processes of snowmelt, sublimation, infiltration and evapotranspiration integrated over a range of conditions throughout a watershed. Here, we used streamflow only to evaluate model performance with respect to the timing of snowmelt.
 SWE was measured during three intensive field campaigns (6–12 April, 8–15 May, and 16–18 June, 1997) involving over 300 snow depth measurements and over 50 snow density measurements on average. Binary regression trees were used to interpolate point observations across the basin at 30-m resolution [Molotch et al., 2005].
 Case 1 simulations, aimed at reconstructing SWE at the time of each snow survey, were performed by solving equation (4) for initial SWE:
Here the daily vertical melt flux, Mvj was calculated following Brubaker et al. :
where Sj (W m−2) is incoming solar radiation, αj is the snow surface albedo, Lj (W m−2) is incoming thermal radiation, Lj (W m−2) is outgoing thermal radiation, Mq (0.026 cm day−1 W−1 m2) is an energy to water depth conversion, Tdj (°C) is the average daily air temperature above 0°C, ar is the restricted degree day coefficient (0.09 cm °C−1) [Brubaker et al., 1996]. Hereafter we refer to the product of Tdj and ar as “turbulent transfer” although it is actually a parameterization rather than an explicit calculation. SCAj (values range from 0 to 1) was obtained using pixel-specific snow cover depletion curves:
where SCAj is the estimated fractional SCA at time step j, SCAi and SCAk are remotely sensed fractional SCA values derived from Landsat TM data (described below) preceding and subsequent to time step j, respectively, and Mvi and Mvk are the vertical melt flux summation values corresponding to the remote sensing acquisition time steps i and k, respectively.
 The initial SWE estimates were then compared with the interpolated observations of SWE from Molotch et al. . Pixel-specific absolute errors for simulations using the different albedo parameterizations were determined and multivariate regression models, as previously described for albedo differences, were used to assess physiographic influences on model performance.
 In case 2, snow extent was simulated by subtracting Mvj from the interpolated observations of SWE from Molotch et al. ; when Mvj equals SWE0, SWEj and SCAj are equal to zero. Errors of omission were determined as the percentage of observed snow covered pixels (i.e., % SCA > 0) simulated to be snow-free. Errors of commission were determined as the percentage of observed snow-free pixels simulated to be snow covered. Map accuracies (percent) were then determined by subtracting from 100 the sum of the omission and commission errors.
 Model forcings were distributed over the terrain, using a combination of geometric models and interpolation between point measurements. Air temperature surfaces were created by deriving an environmental lapse rate in the basin for each hour of the model run using air temperature data from three meteorological stations. We then applied the lapse rate to every pixel in the basin using the DEM [Colee et al., 2000; Daly et al., 2000; Raskin et al., 1997; Thornton et al., 1997; Willmott and Matsuura, 1995].
 TOPORAD [Dozier, 1980; Dozier and Frew, 1990] was used to model incoming solar radiation at the Emerald and Topaz Lake meteorological stations (Figure 1). Using LOWTRAN7 [Dubayah, 1991; Kneizys et al., 1988], we calculated the atmospheric transmission parameters that caused TOPORAD to match the observed incoming solar radiation values at the meteorological stations. This was done for 5 different atmospheric conditions, ranging from clear sky to cloud cover. Using these atmospheric parameters, TOPORAD was then used to model the incoming solar radiation across the basin.
 We modeled incoming thermal radiation using a method similar to that of Colee et al.  and Cline et al. . Using the interpolated temperature surfaces (as described above) and relative humidity surfaces (interpolated using the same method as the temperature surfaces) we applied the equations of Idso  to the modeled relative humidity and temperature of each pixel to compute the incoming longwave radiation. The RMSE over the modeling period was 34 W m−2 at the Emerald Lake meteorological station and 40 W m−2 at the Topaz Lake station. A detailed evaluation of the techniques used to distribute model forcings is presented by Colee et al. .
 Outgoing thermal radiation was estimated as
where ɛs is snow emissivity and is set to 0.99, and δ is the Stefan-Boltzmann constant (5.67 × 10−8 W K−4 m−2). Snow temperature, Tsn at each pixel was set to the air temperature value two hours before the current time step (i.e., a 2-hour lag function) constrained to a maximum of 273.15 K.
 Seven satellite images from the Landsat Thematic Mapper (TM) acquired on 6 and 22 April, 8 May, 9 and 25 June, and 11 and 27 July 1997 were used to construct fractional SCA maps across the basin using the spectral mixture analysis algorithm of Rosenthal and Dozier . AVIRIS scenes on 5 and 21 May were also used to obtain fractional SCA using a spectral mixture analysis model for subpixel SCA, grain size, and albedo [Painter et al., 2003]. AVIRIS SCA surfaces were also degraded to 30-m resolution from their native resolution of 17-m.