Frozen ground can be important to flood production and is often
heterogeneous within a watershed due to spatial variations in the available
energy, insulation by snowpack and ground cover, and the thermal and moisture
properties of the soil. The widely used continuous frozen ground index (CFGI)
model is a degree-day approach and identifies frozen ground using a simple
frost index, which varies mainly with elevation through an elevation–temperature relationship. Similarly, snow depth and its insulating
effect are also estimated based on elevation. The objective of this paper is
to develop a model for frozen ground that (1) captures the spatial
variations of frozen ground within a watershed, (2) allows the frozen
ground model to be incorporated into a variety of watershed models, and
(3) allows application in data sparse environments. To do this, we
modify the existing CFGI method within the gridded surface subsurface
hydrologic analysis watershed model. Among the modifications, the snowpack
and frost indices are simulated by replacing air temperature (a surrogate for
the available energy) with a radiation-derived temperature that aims to
better represent spatial variations in available energy. Ground cover is also
included as an additional insulator of the soil. Furthermore, the modified
Berggren equation, which accounts for soil thermal conductivity and soil
moisture, is used to convert the frost index into frost depth. The modified
CFGI model is tested by application at six test sites within the Sleepers
River experimental watershed in Vermont. Compared to the CFGI model, the
modified CFGI model more accurately captures the variations in frozen ground
between the sites, inter-annual variations in frozen ground depths at a given
site, and the occurrence of frozen ground.
Introduction
Frozen ground (also known as frozen soil or soil frost) is important to
predicting stormflows produced by certain watersheds (Shanley and Chalmers,
1999; McNamara et al., 1997; Prèvost et al., 1990; Woo, 1986). Several
plot-scale studies have shown that frozen ground can impede infiltration and
thus enhance runoff (Bayard et al., 2005; Dunne and Black, 1971; Stähli
et al., 1999). Several of these studies have also shown that frozen ground
is highly variable temporally and spatially (Campbell et al., 2010; Shanley
and Chalmers, 1999; Stähli, 2017), which affects the amount and type of
runoff (Wilcox et al., 1997). The presence, spatial pattern, and depth of
frozen ground are driven by mass (water) and energy balances. The energy
available from the atmosphere to thaw the soil is subject to the insulation
of the snowpack (Pearson, 1920; Willis et al., 1961) and ground cover, including any vegetation, woody debris, and leaf litter (Brown, 1966;
Diebold, 1938; Fahey and Lang, 1975; Sartz, 1973; Stähli, 2017).
MacKinney (1929) found that ground cover reduced the depth of frost
penetration by 40 % at a test site in Connecticut. Additionally, the
presence and depth of frozen ground is affected by soil moisture (Fox, 1992;
Willis et al., 1961) and the thermal conductivity of the soil (Farouki,
1981; Johansen, 1977).
Frozen ground has proven difficult to simulate within hydrologic models due
to complex interactions of energy and water between the atmosphere,
snowpack, and soil (Dun et al., 2010; Kennedy and Sharratt, 1998; Lin and
McCool, 2006). Physically based models of frozen ground, such as the
simultaneous heat and water (SHAW) model (Flerchinger and Saxton, 1989), the
coupled heat and mass transfer model for soil–plant–atmosphere systems
(COUP) (Jansson 2001; Jansson and Karlburg, 2010), and the distributed
water–heat coupled (DWHC) model (Chen et al., 2007) have large parameter and
forcing data requirements – such as wind speed, relative humidity, and
short- and long-wave radiation – which restricts their applicability in
many watershed. Additionally, these types of models either include, or are
tightly coupled to soil moisture models, which can limit their applicability
in models that do not explicitly simulate soil moisture content. To reduce
data and parameter requirements and increase applicability, simple
temperature-index or degree-day methods (Molnau and Bissell, 1983;
Rekolainen and Posch, 1993) remain widely used within watershed models,
including LISFLOOD (De Roo et al., 2001; Van Der Knijff et al., 2010),
CREAMS (Rekolainen and Posch, 1993), and the gridded surface subsurface
hydrologic analysis (GSSHA) model (Downer and Ogden, 2004). Degree-day
approaches typically accumulate the daily average temperature as a frost
index (∘C-days). When the frost index exceeds a threshold, the
soil is considered frozen and impermeable to infiltration. The sudden
restriction on infiltration can be an incorrect assumption, especially in
forested environments where frozen soils often still experience infiltration
(Lindstrom et al., 2002; Nyberg et al., 2001; Shanley and Chalmers, 1999). A
limitation of degree-day approaches is that they are often untested against
observed frost data because the frost index is not a physical property that
can be compared to measurements. However, degree-day methods have been
successful in capturing increased runoff from frozen ground events (Molnau
and Bissell, 1983), and higher frost index values have been shown to
correlate to deeper frost depths (Vermette and Christopher, 2008; Vermette
and Kanack, 2012). Spatial variations of frozen ground within degree-day
methods are typically based on variations in temperature (which are
estimated from an elevation–temperature relationship) and variations in
snowpack insulation (which are also typically inferred from an
elevation–temperature relationship). Such reliance on elevation may lead to
errors because Stähli (2017) found no clear connection between elevation
and presence of frozen ground at test sites in the Swiss pre-Alpine zone.
The objective of this paper is to develop a model for frozen ground that (1) captures the spatial
variations of frozen ground within a watershed, (2) allows the frozen ground model to be incorporated into a variety of
watershed models, and (3) allows application in data sparse environments
where limited forcing data may prohibit use of energy balance methods. In
this paper, we use the GSSHA watershed model and develop the frozen ground
model by modifying the commonly used conceptual frozen ground index (CFGI)
(Molnau and Bissell, 1983) method in four ways. First, the CFGI method is
coupled to an improved snowpack model that more accurately captures the
spatial heterogeneity of the snowpack. In past applications of GSSHA, the
CFGI method has been coupled with a temperature-index (TI) snowpack model
based on SNOW-17 (Anderson, 1973, 2006). However, Follum et al. (2015)
proposed a radiation-derived temperature index (RTI) snow model that
uses a proxy temperature instead of air temperature to represent the energy
available to the snowpack. Compared to the TI model, the RTI model more
directly includes the effects of shortwave radiation and canopy cover and
was shown to better represent the spatial variations of snow cover and snow
water equivalent (SWE) in the Senator Beck Basin in Colorado. The RTI model
is adopted to simulate the snowpack in the present study. Second, the
effects of shortwave radiation and canopy cover are included in the CFGI
model when calculating the energy available at the snow or ground surface.
These effects are included by using a similar radiation-derived proxy
temperature when calculating the frost index. Third, the insulation effects
of ground cover are included by modifying the frost index equation. Fourth,
an option is included to compute frost depth as a function of the frost
index value. The modified Berggren equation and similar Stefan equation have
been previously used to estimate frost depth from degree-days (Carey and
Woo, 2005; DeWalle and Rango, 2008; Fox, 1992; Woo et al., 2004); a similar
approach is used here to convert the frost index to frost depth.
The following sections first describe the existing TI and CFGI models within
GSSHA. The combination of these two models serves as the baseline or control
case for the experiments. Then, the RTI snow model and the modified CFGI
frozen ground model (referred to as modCFGI) are described. Finally, the
results of the TI/CFGI model and RTI/modCFGI models are compared to each
other and to observations of snow depth, SWE, and frost depth at the
Sleepers River experimental watershed (SREW) in Vermont.
MethodologyTI snowpack model
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:
Ta=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:
ΔDt=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:
Mf=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
M=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:
ρn=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):
ρx,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).
CFGI frozen ground model
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
Ft=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).
RTI snowpack model
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
RLW↑=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:
Trad=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:
RSW,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:
RSW↓=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:
Ssub=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.
modCFGI frozen ground model
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
Ft=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:
Zmax=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:
Zd=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
δ=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):
Ωm=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):
Ωsat=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
nice=ntotalZd/H,
where H (m) is the soil thickness.
W-3 sub-basin in the Sleepers River experimental watershed. Sites
used in this study are identified with red triangles and blue snowflakes.
Basin delineation and elevation contours (m) are based on the 1/3 arcsec
National Elevation Dataset, land cover classification is based on the 2006
National Land Cover Database, and sources of the background imagery include
ESRI, DigitalGlobe, Earthstar Geographics, CNES/Airbus DS, GeoEye, USDA FSA,
USGS, Getmapping, Aerogrid, IGN, IGP, and the GIS User Community.
Model applicationStudy area
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.
Model inputs
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.
Allowable ranges and calibrated values for the TI and RTI model
parameters using a model-independent parameter estimation and uncertainly analysis (PEST). Dashes indicate parameters that are not required in
the associated model. The sensitivity ranking for each parameter is shown in
parentheses.
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.
Parameter estimation and calibration
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).
Allowable ranges and calibrated values for the CFGI and modCFGI
model parameters using PEST. Dashes indicate parameters that are not
required in the associated model. The sensitivity ranking for the modCFGI
parameters are shown in parentheses.
Values of soil parameters used to calculate soil moisture in the
single-layer Green and Ampt infiltration model.
ParameterUnitsValueSaturated hydraulic conductivitycm h-12.040Effective porositycm3 cm-30.407Residual water contentcm3 cm-30.038Field capacitycm3 cm-30.166Wilting pointcm3 cm-30.075Capillary headcm8.570Pore distribution arithmetic meancm cm-10.466
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.
Results and discussionSnow depth and SWE (TI vs. RTI)
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.
Simulated maps of snow depth (TI and RTI models) within the W-3
watershed for 23 February 2007. No observed maps of snow depth are
available, but the map shows the differences between the temperature-based
(TI) model and the modified (RTI) model.
TI and RTI simulated snow depth at all eight test sites within the
W-3 watershed.
Statistics for TI and RTI snow depth values at all eight test sites,
and statistics for TI and RTI SWE values at the R3 and R25 snow test sites.
Values are shown for calibration period (WY 2006–2007), validation period
(WY 2008–2010), and overall (WY 2006–2010). RMSE values closer to zero and
NSE values closer to one indicate better fit.
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.
Ta and Trad values
at all eight test sites within the W-3 watershed between 1 and 15 March 2005.
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.
TI and RTI simulated SWE at R3 and R25 snow sites within the W-3
watershed.
Frost depth (CFGI vs. modCFGI)
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.
Simulated maps of frost depth (CFGI and modCFGI models) within the
W-3 watershed for 23 February 2007. No observed maps of frost depth are
available, but the map shows the differences between the temperature-based
(CFGI) model and the modified (modCFGI) model.
Observed frost depth compared against simulated (CFGI and modCFGI)
frost depth at all six selected frozen ground test sites within the W-3
watershed.
Statistics for CFGI and modCFGI frost depth at all six frost sites.
Values are shown for calibration period (WY 2006–2007), validation period
(WY 2008–2010), and overall (WY 2006–2010). RMSE values closer to zero and
NSE values closer to one indicate better fit. No frost was present at FS4
and FS21 during the validation period, resulting in an inability to
calculate NSE. Statistics for a recalibrated modCFGI model without ground
cover (labeled as ”modCFGI no gc”) are also shown.
Land coverFrost depth modelCalibration Validation Overall SiteRMSE (cm)NSERMSE (cm)NSERMSE (cm)NSEFS4Deciduous forestCFGI8.2-5.05.7NA7.2-3.8modCFGI2.50.40.2NA1.90.7modCFGI no gc26.5-62.916.0NA22.6-45.9FS11Evergreen forestCFGI15.5-1.29.9-1.613.1-0.6modCFGI12.6-0.58.2-0.810.7-0.1modCFGI no gc17.5-1.810.9-2.114.6-1.0FS21Deciduous forestCFGI12.5-24.28.4NA11.8-10.9modCFGI3.9-1.50.0NA3.5-0.1modCFGI no gc26.2-109.814.3NA24.3-49.4FS24PastureCFGI17.5-188.57.3-12.312.4-49.1modCFGI1.4-0.32.3-0.32.0-0.3modCFGI no gc28.2-490.013.5-44.320.6-137.4FS30PastureCFGI27.5-5.86.2-2.417.70.2modCFGI11.7-0.224.9-55.020.9-0.1modCFGI no gc18.1-2.011.4-10.814.40.5FS40Deciduous forestCFGI14.2-22.610.1-164212.6-20.9modCFGI3.4-0.31.8-52.12.8-0.1modCFGI no gc36.9-157.921.8-763131.2-133.1
Number of true positive (both simulated and observed data show
frost depth), true negative (both simulated and observed data show no frost
depth), false positive (simulated data shows frost depth but observed data
does not), and false negative (simulated data shows no frost depth but
observed data shows frost depth) occurrences during the entire test period.
The Accuracy is the sum of the true positive and true negative divided by
the total number of observations.
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 %).
Observed frost depth compared against simulated (modCFGI with and
without ground cover included) frost depth at all six selected frozen ground
test sites within the W-3 watershed. The modCFGI model without ground cover
is labeled as ”modCFGI no gc”.
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.
Simulated frost depths from the modCFGI model using simulated soil
moisture (θsim), a constant high soil moisture
(θhigh), and a constant low soil moisture
(θlow) at all six selected frozen ground test
sites within the W-3 watershed.
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).
Conclusions
The main purpose of this paper was to better estimate the spatial pattern of
frozen ground for distributed watershed modeling by modifying an existing
degree-day frozen ground model (CFGI), which uses a frost index value to
determine whether the ground is frozen or not. The modifications to the CFGI
model include (1) use of a radiation-derived temperature index (RTI) snow
model instead of a standard temperature-index (TI) snow model, (2) use of a
radiation-derived proxy temperature (Trad) instead of air
temperature (Ta) in the calculation of the frost index,
(3) inclusion of ground cover (litter, debris, grass, etc.) as an insulator
of the ground from air temperatures, and (4) an option to use a version of
the modified Berggren equation to calculate frost depths based on the frost
index values. The CFGI and modCFGI models were tested using the GSSHA
hydrologic model over a five-year period within the W-3 watershed, which is
part of Sleepers River experimental watershed in Vermont. The model results
were compared against snow depth at eight sites, snow water equivalent at two
sites, and frost depth at six sites. The primary conclusions of the paper are
as follows:
The RTI snow model produces much more complex spatial patterns of snow depth
than the TI snow model for the W-3 watershed. The TI model, which is based
on SNOW-17 (Anderson, 2006), only produces spatial variation using
elevation. The RTI model accounts for elevation, hillslope orientation,
canopy shading, and longwave radiation from the canopy through the use of
the radiation-derived proxy temperature. It also includes a simple
sublimation method based on solar radiation. Thus, its snow depths exhibit
spatial heterogeneity based on elevation, slope/aspect, and land cover, all
of which are known to affect the large-scale distribution of observed snow
depths (Fassnacht et al., 2017; Jost et al., 2007).
Both the RTI model and TI model produce accurate results for the eight snow
depth sites at W-3. Two of the eight sites also measure snow water
equivalent, where the RTI and TI model also show similarly accurate results.
The eight test sites have similar topographic attributes and primarily
differ in their land covers, which include pasture, deciduous forest, and
evergreen forest. Because the leaves have typically fallen prior to snow
accumulation, all but the evergreen site behave similarly in snow
accumulation and ablation.
The modCFGI frost model produces more complex spatial patterns of frost
depth than the CFGI frost model for the W-3 watershed. The CFGI model uses
elevation to infer the spatial variation of air temperature. It additionally uses
the TI model for snow depth, which also depends on elevation. Thus, the
simulated frost depths at W-3 primarily reflect the watershed elevations. In
contrast, the modCFGI model uses the radiation-derived proxy temperature to
infer the energy available to heat the ground and the RTI model to simulate
snow depth. Furthermore, it accounts for the insulating effects of ground
cover (in addition to snowpack), which also depends on the land cover. Thus,
the frost depths simulated by the modCFGI model at W-3 depend on the local
elevation, hillslope orientation, and land cover, all of which are known to
affect the distribution of frozen ground (Fox, 1992; MacKinney, 1929; Wilcox
et al., 1997; Willis et al., 1961).
The modCFGI model produces more accurate frost depths than the CFGI for all
but one of the six test sites in the W-3 watershed. Overall, the modCFGI
model more accurately captures the inter-annual variability in frost depth
at a given site and variability of frost depth between sites. Although both
the CFGI and modCFGI capture the majority of frozen ground events observed,
the modCFGI model has 15.2 % better accuracy in capturing the presence of
frozen ground, which is expected to be important for capturing runoff that
is produced by frozen ground.
A key reason for the difference in performance between the two frost models
is that the modCFGI model includes the insulation of the ground by ground
cover while the CFGI model does not. When ground cover is removed from the
modCFGI model its results for W-3 are less accurate and the variability in
simulated frost depth between the sites is limited. Ground cover is likely
important in this watershed because it is relatively thick and is also
present at all stages of the winter while snowpack is not.
Overall, the modCFGI model provides improved spatial representation of
frozen ground while requiring only cloud cover estimates as additional
forcing data (more forcing data may be required if soil moisture is
simulated to obtain frost depth). Limited data requirements should make
modCFGI well suited for data sparse environments. Hydrologic models often
need to account for the presence of frozen ground, which in data-sparse
environments often means using simple degree-day approaches that typically
vary frozen ground with elevation only (as was shown with the CFGI model).
To calculate Trad the modCFGI model does require cloud cover data,
which are collected operationally at most airports within the US. If soil
moisture is explicitly simulated within the hydrologic model the modCFGI
model can also be used with the modified Berggren equation to simulate frost
depth, which requires information on soil type and an estimate of the
thermal conductivity of the soil.
Five main avenues are available for future research. First, the modCFGI
model should be generalized to include the effects of wind (as it relates to
the snowpack) and more completely consider the role of soil moisture. Soil
moisture is not considered when calculating the frost index, so it does not
impact the initiation or duration of frozen ground. This limitation results
from using a degree-day approach and may be important in some cases
(Kurganova et al., 2007; Willis et al., 1961). Second, the modCFGI model
should be tested further. Additional testing should consider other areas
where snow and frozen ground are known to affect runoff, such as the upper
Midwest region of the US. Additional testing should also better
characterize the insulation properties of ground cover under different
management scenarios. Third, the calculation of Trad is simple and
applicable in data-sparse environments, but other approaches for adjusting a
temperature value based on topography and land cover are available (Fox,
1992; Kang, 2005; Webster et al., 2017) and could be further tested. Fourth,
future research should also determine the effects of spatial heterogeneity
of snow and frost depth on runoff and streamflow at both the local and
watershed scales. Similar to Campbell et al. (2010), the RTI and modCFGI
models could be used in data-sparse watersheds to investigate how changes in
historic and future climate affect snow, frozen ground, and runoff. Finally,
although this paper focuses on the simulation of frost depth in the context
of watershed modeling, the methods described could also be used for
agriculture, overland mobility modeling, and infrastructure where snow and
frost depth are major concerns.
Data availability
Hourly precipitation and temperature data used in this
study are available by contacting the Vermont Field Office of the USGS
(https://nh.water.usgs.gov/project/sleepers/index.htm; USGS, 2018a). Cloud cover,
relative humidity, wind speed, and atmospheric pressure data used in this
study are available from the National Centers for Environmental Information
(NCEI, 2018; https://www.ncdc.noaa.gov/; last access: 7 November 2016). Land
cover data (2006 National Land Cover Database; Fry et al., 2011) and
elevation data (National Elevation Dataset; Gesch et al., 2002) used in this
study are available from the USGS (https://nationalmap.gov/; USGS, 2018b). Soil data
used in this study 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
https://websoilsurvey.sc.egov.usda.gov/App/HomePage.htm; last
access: 10 August 2016). Snow and frozen-ground data used in this study are
available by contacting the Vermont Field Office of the USGS
(https://nh.water.usgs.gov/project/sleepers/index.htm; USGS, 2018a).
Information regarding the GSSHA model is available at
https://gsshawiki.com/Gridded_Surface_Subsurface_Hydrologic_Analysis (GSSHA, 2018).
Additional information and output data sets are available by contacting the
corresponding author.
Competing interests
The authors declare that they have no conflict of
interest.
Acknowledgements
This project was funded by the Flood and Coastal Research
Program at the U.S. Army Corps of Engineers, Engineering Research and
Development Center, Coastal and Hydraulics Laboratory in Vicksburg,
Mississippi. We thank the Vermont Field Office of the USGS (specifically
James Shanley and Ann Chalmers) for their diligent efforts to collect,
process, and analyze numerous hydrologic data for research. We also thank
three anonymous reviewers and the editor who provided helpful comments that
substantially improved this article. Edited
by: Markus Weiler Reviewed by: three anonymous referees
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