The TI snow model was implemented into GSSHA by Follum et al. (2014), who provides additional information about the model. Although GSSHA allows a variable time step for multiple processes, it always uses an hourly time step (Δt) for snow calculations. GSSHA utilizes a structured grid in which each cell can have a different air temperature Ta (∘C) and precipitation P (m h-1). Air temperature is the primary driver of snowpack dynamics in the TI model and is estimated as follows: 1Ta=Tg+∅Eg-Ec, where Tg (∘C) is the air temperature at a gage, ∅ is a linear lapse rate (∘C km-1), and Eg and Ec (m) are the elevations of the temperature gage and the grid cell where Ta is being calculated, respectively. Precipitation accumulates as SWE (m) when Ta≤Tpx, where Tpx is the freezing point (0∘C by default). The precipitation P is multiplied by a uniform multiplication factor (Scf), which crudely represents snowpack sublimation and redistribution of snow due to wind (Anderson, 2006). The resultant effective precipitation (Peff) is added to the SWE.

Before the snowpack begins to melt, its heat deficit (or cold content) must be overcome. The change in heat deficit ΔDt (mm of SWE), due to a temperature difference between the snow surface and air, is calculated using the following equation: 2ΔDt=Nmf,maxΔt/6Mf/Mf,maxATI-Tsur, where Tsur is the snow surface temperature, and ATI is the antecedent temperature index (∘C), which is calculated using Ta and the antecedent snow temperature index parameter ATIPM (see Anderson, 2006, for details regarding Tsur and ATI). Nmf,max is the maximum negative melt factor (mm ∘C-1 (6 h)-1), which is a parameter. Mf is the melt factor (mm ∘C-1Δt-1), which is calculated as follows: 3Mf=Δt/6SvAvMf,max-Mf,min+Mf,min, where Sv and Av are seasonal melt adjustments that change by Julian day, and Mf,max and Mf,min are the maximum and minimum melt factors (mm ∘C-1 (6 h)-1), which are parameters.

Once the heat deficit is overcome, SWE decreases as melt occurs. During normal conditions, the melt M (mm of SWE) is 4M=MfTa-Tmbase+0.0125PefffrTrΔt, where Tmbase is the temperature at which melt begins (0 ∘C by default), fr is the fraction of any precipitation that is rain (assumed equal to 1 when Ta>0∘C, otherwise set to 0), and Tr is the precipitation temperature (assumed equal to Ta or 0∘C, whichever is greater). During rain-on-snow events (more than 1.5 mm of rainfall in the previous 6 h), M is calculated from a simple energy balance: M=σTa+2734-2734Δt+0.0125PefffrTr5+8.5fuΔt/6rhesat-6.11+0.00057PaTa, where σ is the Stefan–Boltzmann constant, fu is the average wind function (mm mb-1 (6 h)-1) (see Anderson, 2006, for details), rh is the relative humidity (assumed to be 0.9 during rain-on-snow events) (Anderson, 1973, 2006), Pa is atmospheric pressure (mb) (either measured or calculated from elevation) (Anderson, 2006), and esat is the saturation vapor pressure (mb) (calculated based on Smith, 1993). The ripeness of the snowpack affects the amount of melt that is released and is controlled by the liquid holding capacity Lhc, which is a specified percentage of the ice in the snowpack (Anderson, 2006).

For frozen ground calculations, the snow depth is needed from the snow model. The snow depth Ds (cm) is found from the SWE and the snowpack density. GSSHA uses the single-layer snow density functions from SNOW-17 (Anderson, 1976, 2006). The density of newly fallen snow ρn (gm cm-3) varies between 0.05 (Ta≤-15∘C) and 0.15 (Ta=0∘C) according to the following equation: 6ρn=0.05+0.0017Ta+151.5. Increases in snowpack density ρx from compaction, destructive metamorphism, and melt metamorphism due to the presence of liquid water are calculated as (Koren et al., 1999): 7ρx,t=ρx,t-1eB2B2eB1 where, 8B1=c3c5dtec4Ts-cxβρx,t-1-ρd,and9B2=Wt-1c1dte0.08Ts-c2ρx,t-1. The variable t is an index for time, W is the ice portion of the snow pack (cm, W=100Sswe,t-1) where Sswe is the snow water equivalent on the ground in m, Ts is the average snow pack temperature (∘C, calculated based on Anderson, 2006), and ρd is the threshold density above which destructive metamorphism decreases (ρd is set to 0.15 gm cm-1 based on Anderson, 2006). Finally, β=0 if ρx,t-1≤ρd, and β=1 if ρx,t-1>ρd, c1=0.026 cm-1 h-1, c2=21 cm3 gm-1, c3=0.005 h-1, c4=0.10 ∘C-1, and c5=2.0 if there is liquid water in the snowpack and c5=1.0 if there is not (see Anderson, 1976, 2006, for details).

The CFGI model was originally developed as a lumped model for flood forecasting in the Pacific Northwest, but it has been used in distributed models as well (De Roo et al., 2001; Van Der Knijff et al., 2010). The rationale of the CFGI method is that air temperature ultimately controls the ground temperature, but its impact is moderated by the insulating effects of any snowpack. The presence of frozen ground is determined by the frozen ground index F (∘C-days), which is calculated by 10Ft=Ft-1A-Ta,de-0.4KsDs, where Ta,d is the average daily air temperature (∘C), A is a daily decay coefficient, and Ks is the snow reduction coefficient (cm-1). The daily decay coefficient (A) controls the persistence of the F values, and Ks controls the insulation from the snowpack. Molnau and Bissell (1983) recommended changing Ks depending on whether Ta,d is above or below freezing (denoted as Ks,Ta>0∘C and Ks,Ta<0∘C, respectively).

Higher values of F indicate a higher likelihood that the ground is frozen. Once F exceeds a specified threshold (Fthreshold), the ground is considered frozen and infiltration is restricted. Molnau and Bissell (1983) found the ground to be frozen when F>83 ∘C-days and thawed when F<56 ∘C-days. When F is between these values, the ground could be either frozen or thawed. It is worth noting that F does not depend on soil moisture, which is known to affect the initialization and depth of frozen ground (Kurganova et al., 2007; Willis et al., 1961).

The RTI model makes two modifications to the TI model: (1) it uses a radiation-derived temperature Trad(∘C) to better describe the available energy, and (2) it estimates spatially varying snowpack sublimation based on solar radiation approximations.

The RTI model replaces Ta in Eqs. (4) and (5) with a radiation-derived proxy temperature Trad (∘C). In those equations, Ta is used to conceptually represent the energy available to the snowpack. Trad has a similar purpose but is intended to improve the estimation of available energy. Trad is calculated by assuming that the radiation terms dominate the energy balance at the snow surface so that outgoing longwave radiation balances the net incoming shortwave and longwave radiation (Follum et al., 2015). Thus 11RLW↑=RSW,net+RLW↓, where RLW↑ is outgoing longwave radiation, RSW,net is the net incoming shortwave radiation, and RLW↓ is the downwelling longwave radiation. The right side of Eq. (11) represents the energy that is supplied to the snowpack via the atmosphere. RLW↑ (W m-2) is the radiative response of the snowpack to that energy. Using the Stefan–Boltzmann law, RLW↑ can be written in terms of temperature Trad: 12Trad=RSW,net+RLW↓εsnowσ1/4-273.15, where εsnow is the emissivity of snow (assumed to be 0.97) and σ is the Stefan–Boltzmann constant.

RSW,net is calculated as follows: 13RSW,net=(1-αs)RSW↓, where αs is the albedo of the snowpack, which is calculated based on the time elapsed since the most recent snowfall and whether melt is occurring (Henneman and Stefan, 1999). RSW↓ is the incident shortwave radiation, which is calculated using the following: 14RSW↓=RSW,0φrφatmφcφvφsφt, where RSW,0 is the solar constant (Liou, 2002), φr accounts for distance from the Earth to the sun (based on Julian day, TVA, 1972), φatm accounts for atmospheric scattering (based on elevation, Allen et al., 2005), φc accounts for absorption by clouds (based on fractional cloud cover, TVA, 1972), φv accounts for vegetation (set equal to the vegetation transmission coefficient Kv (Bras, 1990), a vegetation-specific parameter ranging from 1.0 for no canopy coverage to 0.0 for complete canopy coverage), φs accounts for the slope/aspect of the terrain (based on latitude, slope, and azimuth angle, Duffie and Beckman, 1980), and φt accounts for topographic shading (based on elevation, azimuth angle, and solar elevation angle).

RLW↓ is calculated from the contributions of the atmosphere (including clouds) and the canopy: RLW↓=σεaTa+273.1541.0+0.17N21-Fc15+FcσεcTcanopy+273.154, where εa is the air emissivity (0.757 when snow is present based on Bras, 1990), N is the fractional cloud cover, Fc is the fractional canopy cover (estimated from leaf area index LAI following, Liston and Elder, 2006; Pomeroy et al., 2002), εc is the canopy emissivity (assumed equal to 1 following Sicart et al., 2004), and Tcanopy is the canopy temperature (∘C) which is assumed equal to Ta following DeWalle and Rango (2008).

Because the TI model uses Ta to drive snowpack dynamics, those dynamics are only directly associated with the downwelling longwave radiation from the air, which is a component of RLW↓. Furthermore, the spatial variations in the available energy only depend on the variations of Ta, which are inferred from elevation. Trad in the RTI model considers both RSW,net and RLW↓ and thus accounts for heterogeneity in topographic orientation and shading as well as canopy cover. The TI model partially accounts for seasonal variation in solar radiation and snow albedo by empirically adjusting Mf as shown in Eq. (3). In the RTI model, seasonal variations in solar radiation and snow albedo are included in Trad, so a constant melt factor Mf is used (Follum et al., 2015).

The TI model uses a uniform multiplication factor (Ssf) that is applied to the precipitation to account for sublimation, but sublimation is known to vary spatially (Musselman et al., 2008; Rinehart et al., 2008; Veatch et al., 2009). Most sublimation methods depend on relative humidity and wind speed (e.g., Pomeroy, 1988; Liston and Elder, 2006), which are often unavailable in data sparse environments. However, Gustafson et al. (2010) linked differences in sublimation rates to the amount of solar radiation a location receives. In the RTI model a simple approach is used to estimate hourly sublimation rates Ssub (cm h-1) as follows: 16Ssub=Ssub,dRSW↓RSW↓,flat, where Ssub,d (cm d-1) is the watershed-average daily maximum sublimation amount (a parameter), and RSW↓,flat is the daily shortwave radiation for a flat cell within the watershed on a cloud-free day. Thus, locations with higher RSW↓ (e.g., open areas and south-facing slopes in the Northern Hemisphere) will have higher values of Ssub. The method neglects wind speed and relative humidity, but does vary sublimation rates based on spatial patterns of solar radiation.

The CFGI model is modified in three ways to create the modCFGI model. First, the average daily proxy temperature Trad,d is used in place of Ta,d to represent available energy. Second, ground cover (leaf litter, woody debris, etc.) is included as an insulator in the frozen ground index. And third, an option is included to estimate frost depth based on the frozen ground index. The frost depth calculation is optional because it requires soil moisture estimates and may not be needed in many hydrologic models that only require the occurrence (not depth) of frozen ground.

The CFGI uses Ta,d in Eq. (10) to represent the energy that is available to heat the ground surface. In the modCFGI model, Ta,d is replaced with Trad,d. Trad,d is calculated using αs (see Eqs. 12 and 13) when snow is present, and the albedo of the land cover when snow is not present. By using Trad,d, the modCFGI model is expected to better represent the spatial heterogeneity of energy supply due to variations in the topography and canopy cover within a watershed.

The insulation by the ground cover is included by modifying Eq. (10) which becomes 17Ft=Ft-1A-Trade-0.4KsDs+KgcDgc, where Kgc is the ground cover reduction coefficient (cm-1) and Dgc is the depth of ground cover (cm). This formulation retains the original form of the CFGI model but includes insulation from both snowpack and ground cover. F can still be used to identify the occurrence of frozen ground, which may be sufficient for many hydrologic models. However, because F is not a measurable quantity, an option to extend modCFGI to calculate frost depth is also needed.

Frost depth is calculated using F and the modified Berggren equation. As originally proposed (and described by DeWalle and Rango, 2008), the Berggren equation relates the number of degree days in the freezing/thawing period U (∘C-days) to the maximum frost depth Zmax (m) as follows: 18Zmax=λ48Uδ-1Ωm1/2, where λ is a dimensionless coefficient that accounts for changes in sensible heat of the soil, δ (J m-3) is the latent heat of fusion of the soil, and Ωm (J m-1 h-1 ∘C-1) is the mean thermal conductivity of the frozen and unfrozen soil layers. The derivation and corresponding assumptions (i.e., linear soil temperature gradients, Aldrich Jr., 1956) do not reveal any major impediments to adapting this equation for a shorter time step. In addition, Fox (1992), Woo et al. (2004), and Carey and Woo (2005) have used a layered version of the Stefan equation, which is similar to Eq. (18) to simulate daily frost depths with daily input data. Thus, the modified Berggren equation is applied at a daily timescale and revised to become: 19Zd=λ[48F-Fthresholdδ-1Ωm]1/2, where Zd is the depth of frozen ground (m). By using the difference between F and Fthreshold, the degree-days of the current freezing/thawing period is utilized, which is similar to the use of U in the original equation. Zd is only calculated once the ground begins to freeze (when F>Fthreshold). Zd deepens as F becomes increasingly larger than Fthreshold. When F decreases (due to increasing Trad), so does the thickness of frost depth. No frost remains when F falls below Fthreshold.

For the original modified Berggren equation, λ can be estimated annually from Aldrich Jr. (1956) using U, the mean annual air temperature, and the soil water content ω (% of dry weight). Here, λ is calculated using daily differences between F and Fthreshold, the mean annual air temperature, and daily ω values. Thus, soil moisture is included in the calculation of Zd even though it is not included in the calculation of F. Furthermore, δ is estimated daily as 20δ=δfρω/100, where δf is the latent heat of fusion of water (0.334 MJ kg-1 at 0∘C) and ρ is the dry soil density. Ωm is estimated as follows (Farouki, 1981; Johansen, 1977): 21Ωm=Ωsat-Ωdryω+Ωdry, where Ωdry and Ωsat are the thermal conductivity of dry and saturated soil, respectively. Ωsat is calculated as the geometric mean of the conductivities of the materials within the soil profile (Farouki, 1981; Johansen, 1977): 22Ωsat=Ωs(1-ntotal)Ωice(nice)Ωwater(ntotal-nice), where Ωs, Ωice, and Ωwater are the thermal conductivity of solids, ice, and water, respectively (Farouki, 1981). ntotal is the porosity, and nice is 23nice=ntotalZd/H, where H (m) is the soil thickness.

The TI/CFGI and RTI/modCFGI models are tested at the W-3 sub-basin (Fig. 1) of the SREW. The study period is 1 October 2005 through 30 September 2010, which is water year (WY) 2006 through 2010. The SREW was founded in 1958 primarily for studies of snow accumulation, melt, and runoff (Anderson, 1973, 1976; Dunne and Black, 1970a, b; Dunne and Black, 1971; Shanley, 2000; Shanley and Chalmers, 1999). The W-3 sub-basin is located at 44∘29′ N and 72∘09′ W. Elevations range between 348 and 697 m, and the area is approximately 8.5 km2 (based on the National Elevation Dataset, Gesch et al., 2002). The basin is primarily forested with deciduous (57.7 %), evergreen (7.8 %), and mixed (15.3 %) trees (based on the 2006 National Land Cover Database (NLCD), Fry et al., 2011). Approximately 14.6 % of the land cover is pasture/hay and cultivated crops. These open areas are typically below an elevation of 525 m, which is the approximate limit for viable agriculture (Shanley and Chalmers, 1999). The W-3 sub-basin is extensively gaged for both hydrometeorology and hydrology by the US Geological Survey (USGS) and collaborators from federal agencies and universities. Additional basin information and data are provided by Shanley et al. (1995), Shanley and Chalmers (1999), and the USGS website (https://nh.water.usgs.gov/project/sleepers/index.htm, last access: 7 November 2016).

Two snow sites and 35 frost sites within W-3 were monitored by the Vermont Field Office of the USGS. At the snow sites, SWE and snow depth were measured approximately weekly, and both sites are used in the present study. At the frost sites, snow depth and frost depth were measured periodically (between 0 and 14 measurements in a given winter). Frost depth was measured using CRREL-Gandahl frost tubes (Ricard et al., 1976), which are filled with a methylene blue solution. The frost depth is identified by a change in color within the tube (blue indicates thawed, clear indicates frozen). Vermette and Kanack (2012) provide images and descriptions of similar frost tubes, and Shanley and Chalmers (1999) provides detailed descriptions of the frost tubes at SREW. The frost sites (labeled FS in Fig. 1) are clustered in six parts of the watershed. For this paper, one site from each cluster (FS4, FS11, FS21, FS24, FS30, and FS40) was selected for analysis. The selected sites are far enough apart to be relatively independent but still capture the variations in elevation and land cover classification within the watershed.

The TI and CFGI models require hourly precipitation and temperature data, which were obtained from the USGS. Precipitation was measured at the W9 weir and R3 snow site (Fig. 1). The USGS then creates a single spatially averaged precipitation time series by weighting the measurements using the distribution of elevation (based on personal communication with James Shanley of the Vermont Field Office of the USGS on 14 November 2016). The W9 gage receives more weight because the watershed includes elevations both above and below this site. Hourly temperature was measured at the W9 site, which has an elevation of 520 m.

The RTI and modCFGI models also require cloud cover data, which were obtained from the National Centers for Environmental Information (NCEI, https://www.ncdc.noaa.gov/, last access: 7 November 2016). The hourly cloud cover classification data (clear, few clouds, broken sky, etc.) were collected at the Edward F. Knapp State Airport (44 km southwest of the basin) and the Morrisville-Stowe State Airport (36 km west of the basin). The classification data were converted to cloud cover percentages using the method from Follum et al. (2015). Cloud cover data are routinely measured at most airports in the US (data archived at NCEI) as well as many meteorological stations. For simulation of frost depth (and comparison to frost depth observations), soil moisture and evapotranspiration were also simulated. These two components additionally require hourly relative humidity, wind speed, and atmospheric pressure data, all of which were obtained from a meteorological station at the Fairbanks Museum in Saint Johnsbury, VT (11 km southeast of the basin) with missing values replaced with hourly data from the two airports.

All the models require elevation data to determine the spatial patterns of snow and frozen ground. W-3 was delineated using the 1/3 arcsec (∼ 9 m) National Elevation Dataset (Gesch et al., 2002). The RTI and modCFGI models additionally require land cover classifications, which were obtained from the 2006 NLCD (Fry et al., 2011) and have a 30 m resolution. The classifications of some grid cells were changed to match the land covers observed in the field. In particular, the grid cell containing R3 was changed from deciduous forest to pasture/hay, FS11 was changed from mixed forest to evergreen forest, and FS21 was changed from developed to mixed forest. Both FS24 and FS30 are classified as pasture/hay, where FS24 is a managed pasture and FS30 is an unmanaged pasture (Ann Chalmers, Vermont Field Office of the USGS, personal communication, 15 November 2016). For example, during field observations in November 2016, FS24 had manure spread throughout the field, while FS30 was not fertilized.

Soil classification data are also required for calculating frost depth, and were obtained from the Digital General Soil Map of the United States (Soil Survey Staff, Natural Resources Conservation Service, United States Department of Agriculture, Web Soil Survey, available online at http://websoilsurvey.nrcs.usda.gov/, last access: 10 August 2016). Almost the entire W-3 basin is classified as fine sandy loam. The Watershed Modeling System (Aquaveo, 2013) was used to develop the GSSHA model with a 30 m structured grid. This resolution is adequate to capture the spatial heterogeneity of the basin while remaining computationally efficient.

The model-independent parameter estimation and uncertainly analysis (PEST) method (Doherty et al., 1994) was used to calibrate seven parameters in the TI model and eight parameters in the RTI model. PEST is a nonlinear local search parameter estimator that calibrates numerous parameters simultaneously to produce the best fit between simulated results and observations. WY 2006 and 2007 were used as the calibration period. The TI and RTI snow models were calibrated first to minimize the sum of the squared residuals between simulated and observed snow depths at the eight sites (six frost sites and two snow sites).

Table 1 displays the allowable range, calibrated value, and sensitivity ranking for the calibrated snow parameters. Goodness of fit statistics as well as description of the affects each parameter has on the snow simulations are described in the results and discussion section of this paper. The allowable ranges for ATIPM, fu, Lhc, Nmf,max, Mf , Mf,max, and Mf,min are based on physical limitations and typical ranges in the literature (Follum et al., 2015). LAI can be estimated from seasonal and annual relationships to remotely sensed normalized difference vegetation index (NDVI) values (Wang et al., 2005). However, snowpack affects the measurement of greenness in high latitude regions (Beck et al., 2006). Thus, LAI and Kv values were calibrated based on land cover classifications with forested land covers being categorized as deciduous forest (including deciduous forest, woody wetlands, and mixed forest) or evergreen forest. LAI and Kv values for non-forested land cover classifications were set to 0.0 and 1.0, respectively. Tpx and Tmbase were not calibrated (both set to 0 ∘C) because the temperature data were post-processed by the Vermont USGS and are expected to be accurate. By comparing the temperature measurements at W9 and the Fairbanks Museum (elevation of ∼ 212.4 m), ∅ was estimated at 6.6 ∘C km-1. All snow density parameters are set based on Anderson (1973, 2006).

The PEST results indicate that the TI model's snow depths are most sensitive to Scf, Mf,max, ATIPM, and Mf,min. For the RTI model, snow depths are most sensitive to Kv (deciduous), ATIPM, LAI(evergreen), and Mf (Table 1). The calibrated deciduous Kv is near the top of the allowable range (1.0) and LAI is near the bottom (0.103), indicating that the snow in the deciduous forest behaves similarly to the open pasture areas where Kv= 1 and LAI= 0.

The CFGI and modCFGI frozen ground models were calibrated to minimize the sum of squared residuals between the simulated and observed frost depths at the six frost sites. For purposes of comparison the modified Berggren equation was also added to the CFGI model to calculate frost depth. Table 2 displays the allowable range, calibrated value, and sensitivity ranking of each calibrated frozen ground parameter. Goodness of fit statistics as well as description of affects each parameter has on the frost depth simulations are described in the results and discussion section. Fthreshold was calibrated for both the CFGI and modCFGI models with the upper range based on Molnau and Bissell (1983). Three Kgc values were calibrated for the modCFGI frozen ground model: one for the managed pasture site FS24 (Kgc,FS24), one for the unmanaged pasture site FS30 (Kgc,FS30), and one for all other frozen ground sites (Kgc).

Following Molnau and Bissell (1983), multiple combinations of A (0.8 and 0.97), and Ks,Ta<0∘C and Ks,Ta>0∘C (0.08, 0.2, and 0.5) values were tested with A= 0.97, Ks,Ta<0∘C= 0.08, and Ks,Ta>0∘C= 0.5 producing frost indices that best replicate the rise and fall of the frost depth as well as the timing of the peak frost depth. Depth of ground cover for each land cover type was obtained from field observations in November 2016. Specifically, Dgc=6 cm for deciduous forest (fallen leaves), Dgc=2 cm for evergreen forest (fallen leaves), Dgc=4 cm for pasture (grass), and Dgc=0 cm for all other land cover types.

The modified Berggren equation requires soil moisture, which can be simulated using several methods in GSSHA (Downer and Ogden, 2006). To facilitate extension of these results to other hydrologic models, the commonly used single-layer Green and Ampt infiltration model (Green and Ampt, 1911) with soil moisture redistribution between rainfall events (Ogden and Saghafian, 1997) is utilized to calculate infiltration. Soil moisture is tracked using a simple bucket approach, accounting for infiltration, evapotranspiration, and groundwater recharge as described in Downer (2007). The soil layer thickness (H) is set to 0.5 m for both the soil moisture calculations and frost depth equations. Soil infiltration parameters are set based on published values for the W-3 soil type (Downer and Ogden, 2006; Rawls et al., 1982, 1983; Rawls and Brakensiek, 1985) and are shown in Table 3. Evapotranspiration, which can reduce the soil moisture, is simulated using a Penman–Monteith approach (Monteith, 1965, 1981) with parameters estimated based on land cover (Downer and Ogden, 2006). The dry soil density (ρ= 1137 kg m-3) and dry soil thermal conductivity (Ωdry= 792 J m-1 h-1 ∘C-1) are set based on measurements of fine sandy loam by Nikolaev et al. (2013).

For the CFGI model, the calibrated Fthreshold value (Table 2) is relatively close to the lower bound value of 56 ∘C-days found in Molnau and Bissell (1983). For the modCFGI model, the calibrated Fthreshold value is at the lower bound. The Fthreshold value is expected to be lower for the modCFGI model than the CFGI model. The modCFGI model incorporates the insulation by ground cover directly using Kgc and Dgc, whereas the CFGI model can only account for those effects by adjusting the Fthreshold value. It is also worth noting that Kgc,FS30 has a very low value (minimum of allowable range), which suggests that insulation from grass in an unmanaged pasture is very small. This could be the result of snow falling within the grass of the unmanaged pasture, thus making any insulating contribution from the grass very small.

Figure 2 shows maps of simulated snow depth on 23 February 2007 from the TI and RTI snow models. The spatial variability in the TI snowpack is entirely based on elevation (due to the inference of local air temperature from elevation). Higher elevations have deeper snowpack due to lower air temperatures. The RTI snowpack also varies with elevation but shows variation due to land cover as well. In particular, pasture areas have slightly shallower snowpack than surrounding areas due to higher sublimation rates and higher absorbed shortwave radiation. North-facing slopes also have more snow than south-facing slopes due to lower absorbed shortwave radiation. Although no maps of observed snow depth are available for comparison, large-scale distributions of snowpack are known to be controlled by elevation, land cover, and slope/aspect (Fassnacht et al., 2017; Jost et al., 2007), which is more consistent with the RTI model.

Figure 3 shows the snow depths from the TI and RTI models at all eight test locations and compares them to the observations. Root mean squared error (RMSE) and Nash–Sutcliffe efficiency (NSE) are shown in Table 4 for the calibration period (WY 2006–2007), validation period (WY 2008–2010), and complete period (WY 2006–2010). The TI and RTI models track closely together at the eight test locations despite differences in the snow depth shown in Fig. 2. Differences between the TI and RTI snowpack at the test sites are small (Fig. 3 and Table 4). The RTI model performs slightly better than the TI model in overall average RMSE (15.69 vs. 15.71 cm), while the TI model performs slightly better in overall average NSE (0.58 vs. 0.53). The observed snow depth is relatively low in WY2008 and 2009 at two of the pasture sites (FS24 and FS30) compared to the other sites. Specifically in WY2008 the small snow depth observations are not captured within either model. The R3 site is also classified as pasture yet has a higher snowpack in WY2008 and 2009. The higher snowpack at this pasture site may be explained by the proximity of R3 to forested areas, which may reduce the wind and help preserve the snowpack. Neither model considers wind effects.

The snow depths from the two models are similar at each location (Fig. 3) because, on average, the available energy to melt snow (Ta in the TI model and Trad in the RTI model) is similar (Fig. 4). However, the diurnal variation of Trad is typically greater than that of Ta. Trad is derived from a simple radiation balance (i.e., neglecting other terms in the thermal energy balance). Thus, Trad is higher than Ta during the day due to high RSW↓ values, and it is typically lower than Ta at night because RSW↓ reduces to 0 and εa (set to 0.757) in Eq. (15) limits the affect Ta has on RLW↓ and therefore Trad. As shown in Fig. 4, the available energy is also similar between these locations. The elevation difference between the highest and lowest elevation site is approximately 300 m, corresponding to a maximum temperature difference of approximately 2 ∘C between the sites. Also, the test sites are typically located on shallow slopes so topographic aspect has little influence on the energy available to melt the snowpack (i.e., Trad). All land cover classifications except evergreen forest (FS11) have Kv values at or near 1 and LAI values at or near 0, which reduces any variations due to land cover. Trad at FS11 (evergreen forest) is different from the other seven sites because its low Kv value (0.308) reduces RSW,net during the day, and a high LAI value (1.0) increases RLW↓ during day and night.

Figure 5 shows the simulated (both TI and RTI models) and observed SWE values, and Table 4 shows the associated performance metrics at the R3 and R25 snow sites. The TI and RTI models are only calibrated to snow depth, but SWE is calculated first and then combined with snow density to determine snow depth. Both models use the same method to calculate snow density. Both models also exhibit similar behavior and performance at the two sites, which is consistent with their similar snow depths discussed earlier (Fig. 3 and Table 4). Overall, this suggests that the snow density equations used within GSSHA are relatively accurate at the W-3 watershed. Thus, accurate estimates of snow depth typically correspond to accurate estimates of SWE as well.

Figure 6 shows simulated frost depth maps for 23 February 2007 using the CFGI and modCFGI models (no maps of observed frost depths are available for comparison). In the CFGI model, the frost depths mainly depend on elevation. Colder temperatures at higher elevations generally result in greater snowpack, which insulates the ground and produces smaller frost depths. However, at the beginning of the snow season when the snowpack is shallow, low temperatures at high elevations create deep frost in the higher elevations of the watershed. Later, deeper snowpack at high elevations insulate the ground, while the frost depth increases at lower elevations. This reversal in the elevation dependence can produce an inversion (localized minima in frost depth), as seen between the 500 and 650 m contour lines in Fig. 6. The modCFGI frost depth also has some elevation dependence, but the spatial variation mainly follows land cover classification, which is similar to observations of frozen ground in the Swiss pre-Alpine zone (Stähli, 2017). This variation is partly due to the use of Trad and the increased heterogeneity in the snow depth. The effect of snowpack can be seen by comparing hillslopes with the same land cover but different orientations, such as along the 500 m contour south of FS11. Lower Trad values on northeast-facing slopes result in deeper snowpack than the southwest-facing slopes (Fig. 2). This deeper snowpack produces shallower frost depths on the northeast-facing slopes due to insulation by the snow. However, the spatial pattern of frost depth is more heavily affected by the land cover. Land cover's impact largely occurs through the associated ground cover. This effect can be seen by comparing the deep frost at the unmanaged pasture (near FS30) with the shallower frost depth at the deciduous forest areas near FS4, FS21, and FS40. The low ground cover reduction coefficient at the unmanaged pasture (Kgc,FS30) reduces the insulation from the ground cover, creating deeper frost compared to the deciduous forest areas. The larger than expected role of ground cover in the modCFGI model may occur because ground cover is present during the initiation, deepening, and decrease of frost depth, while the snowpack is much more variable throughout the season.

Figure 7 shows the frost depths from the CFGI and modCFGI models along with the frost depth observations. The RMSE and NSE values during the calibration, validation, and overall periods are shown in Table 5. The simulated frost depth remains more constant amongst the sites when using the CFGI model, which produces similar maximum frost depths for a given year independent of the land cover. The modCFGI results deviate considerably from the CFGI results, producing greater frost depths at the unmanaged pasture (FS30) and evergreen (FS11) sites and smaller frost depths at the deciduous (FS4, FS21, and FS40) and managed pasture (FS24) sites. These simulated differences between the sites are consistent with the observations. The decreased frost depth in the deciduous forest and managed pasture result from their high measured litter depth (Dgc=6 cm) and high reduction coefficient (Kgc,FS24=1.887 cm-1), respectively. The two pasture sites (FS24 and FS30) differ considerably in the observed frost depth with FS30 consistently having deeper frost. This difference likely occurs because FS24 is managed and FS30 is not. With the exception of the validation period at FS30, the modCFGI model performs better (lower RMSE and higher NSE values) than the CFGI model. The difference in performance is most pronounced at the deciduous sites (FS4, FS21, and FS40) where the average overall NSE value is -11.9 for the CFGI model and 0.20 for the modCFGI model.

In hydrologic models, capturing the presence of frozen ground is important because even shallow frost with high moisture content (concrete frost) has the potential to impede infiltration (Dunne and Black, 1971). Therefore, the ability of the CFGI and modCFGI models to accurately capture the presence of frozen ground is evaluated. Whenever frost observations are available, the simulated frost depths are categorized as follows: true positive (both simulated and observed data show frost), true negative (both simulated and observed data show no frost), false positive (simulated data shows frost but observed data shows no frost), or false negative (simulated data shows no frost but observed data shows frost). Table 6 shows the number of observations in each category for each test site. The table also shows the model accuracy, which is calculated as the percent of the observations that are correctly classified (true positive or true negative). The CFGI and modCFGI models perform similarly in capturing true positives at FS4, FS21, FS24, and FS40, while modCFGI has more true positives at FS11 and FS30. The lower true positives and higher false negatives indicate that the CFGI model tends to underestimate the presence of frozen ground at FS11 and FS30. Overall, both the CFGI and modCFGI models capture most of the frozen ground events, with the modCFGI model performing better than the CFGI model at five sites and worse at one site (FS21). The average accuracy of the modCFGI model is 15.2 % higher than the CFGI model, with the largest increase in accuracy at FS11 (29.8 %).

A simple test is employed to explore the modification that contributes most to the increased accuracy of the modCFGI model. This test removes ground cover from the modCFGI model, recalibrates, and then compares the results to observations. When ground cover is removed, the calibrated Fthreshold value is 83 ∘C-days, which is at the top of the calibration range. This change indicates that ground cover has a large impact on the appropriate value of this threshold. Figure 8 shows the simulated frost depths using the modCFGI model with and without ground cover for each test site. Performance metrics for the modCFGI model with and without ground cover are shown in Table 5. Variability in frost depth between the sites is diminished when ground cover is removed, leading to large errors between simulated and observed frost depth. When ground cover is removed, the frost depth results decrease in accuracy (higher RMSE values and lower NSE values) compared to the complete modCFGI model. The only exception is the overall period at FS30, which is also the only site where the CFGI model outperforms the full modCFGI model. These results suggest that inclusion of ground cover is an important reason why the modCFGI model outperforms the CFGI model.

The sensitivity of the modCFGI results to soil moisture is also examined. Soil moisture does not affect the calculation of F, but it is included within the modified Berggren equation (Eqs. 18 and 19) in the calculation of δ (Eq. 20) and Ωm (Eq. 21). Soil moisture was simulated using a single-layer Green and Ampt approach. However, no soil moisture measurements are available at any of the test sites to evaluate the accuracy of the simulated values. Sensitivity of the modCFGI model to volumetric soil moisture is tested by artificially setting the soil moisture to either the residual water content (θlow) or the effective porosity (θhigh), which are the lower and upper bounds for soil moisture values within the model. Figure 9 shows the modeled frost depths from the modCFGI model using θlow, θhigh, and the soil moisture from the Green and Ampt approach (θsim, which is identical to modCFGI in Figs. 7 and 8). Also shown are the observed frost depths for reference only. The frost depth from the θsim case is similar to the frost depth from the θhigh because the simulated soil moisture is usually close to the effective porosity. Frost depth increases when θlow is used, which coincides with other studies (Fox, 1992; Willis et al., 1961). The timing of the frozen ground (when it begins and ends) is identical in all three of the simulations. The consistent timing occurs because soil moisture is not used to calculate F and the same Fthreshold (which controls when frozen ground begins) was used for all three simulations. This result highlights a deficiency in the modeling framework. Specifically, soil moisture should be considered for determining the initiation of frozen ground because wet soils have a higher specific heat capacity and require more energy loss to cool and freeze the soil (Kurganova et al., 2007).