Introduction
High altitude and arctic environments are exposed to strong winds and
drifting snow can create a small-scale pattern of highly variable snow
depths. Seasonal snow cover is a crucial factor for the ground thermal
regime in these areas (e.g. Goodrich, 1982; Zhang et al., 2001). This
small-scale pattern of varying snow depths results in highly variable ground
surface temperatures on the metre scale of up to 6 ∘C in areas of
less than 1 km2 (e.g. Gubler et al., 2011; Gisnås et al., 2014).
Grid-based numerical land surface and permafrost models operate on scales
too coarse to resolve the variability of snow depths and are not capable of
representing such small-scale variability. For the Norwegian mainland,
permafrost models have been implemented with a spatial grid resolution of
1 km2 (Gisnås et al., 2013; Westermann et al., 2013) and
therefore only represent the larger-scale patterns of ground temperatures.
As a consequence, they usually represent the lower limit of permafrost as a
sharp boundary, where the average ground temperature of a grid-cell crosses
the freezing temperature (0 ∘C). In reality, the lower permafrost
boundary is a fuzzy transition. Local parameters, such as snow cover, solar
radiation, vegetation, soil moisture and soil type, cause a pronounced
sub-grid variation of ground temperature. Different approaches have been
developed to address this mismatch of scales, such as the TopoSub
(Fiddes and Gruber, 2012), accounting for the variability of
a range of surface parameters using k-means clustering. At high latitudes
and altitudes, one of the principal controls on the variability of ground
temperature is the effect of sub-grid variation in snow cover
(e.g. Langer et al., 2013). The observed variability in ground
surface temperatures within 1 km × 1 km areas is to a large degree
reproduced by only accounting for the variation in snow depths
(Gisnås et al., 2014). Therefore, procedures capable
of resolving the small-scale variability of snow depths could considerably
improve the representation of the ground thermal regime.
The spatial variation of snow is a result of several mechanisms operating on
different scales in different environments (Liston, 2004). In tundra
and alpine areas, wind-affected deposition is the dominant control on the
snow distribution at distances below 1 km (Clark et al.,
2011). Physically based snow distribution models are useful over smaller
areas but are not applicable on a regional scale. The coefficient of
variation (CV), defined as the ratio between the standard deviation and the
mean, is a measure of the extent of spread in a distribution. The
CV of snow depths (CVsd) typically range from low
spread at 0.2 to high spread at 0.8, which suits snow distributions in a
range of environments (e.g. Liston, 2004; Winstral and Marks, 2014).
Liston (2004) assigned individual values of CVsd to
different land use classes in order to address sub-grid variability of snow
in land surface schemes. According to this scheme, non-forested areas in
Norway, as well as most of the permafrost areas in northern Europe
(“high-latitude alpine areas”), would have been allocated a CVsd of 0.7. A review of observed
CVsd from a large number of snow surveys in the northern hemisphere shows
a large spread of CVsd values, in particular within this land use class,
ranging from 0.1 to 0.9 (Clark et al., 2011). This illustrates the need for
improved representation of snow distribution within this land use class.
An accurate representation of the small-scale snow variation influences the
timing and magnitude of runoff in hydrological models, and a detailed
picture of the sub-grid variability is of great value for the hydropower
industry and flood forecasting. Adequate representations of the snow covered
fraction in land surface schemes improve simulated near-surface air
temperatures, ground temperatures and evaporation due to the considerable
influence of snow cover on the duration of melt season and the surface albedo.
In this study we derive functional dependencies between distributions of
snow depth within 1 km × 1 km grid cells and CVsd, based on an extensive
in situ dataset from Norwegian alpine areas. In a second step, we employ
the resulting snow distributions as input to the permafrost model CryoGRID1,
a spatially distributed, equilibrium permafrost model
(Gisnås et al., 2013). Using a sub-grid representation of
ground temperatures, permafrost probabilities are derived, hence enabling a
more realistic, fuzzy permafrost boundary instead of a binary, sharp
transition. With this approach, we aim to improve permafrost distribution
modelling in inhomogeneous terrains.
Modelled distribution of permafrost in Norway. Sites mentioned in
the text: (1) Finse, south of Hallingskarvet; (2) Juvvasshøe in
Jotunheimen; (3) Dovrefjell; (4) the Lyngen Alps; and (5) Finnmark.
Setting
The model is implemented for the Norwegian mainland, extending from
58 to 71∘ N. Both the topography and climate in Norway
is dominated by the Scandes, a mountain range stretching south–north through
Norway, separating the coastal western part with steep mountains and deep
fjords from the eastern part where the mountains gradually decrease in
height. The maritime climate of the west coast is dominated by low-pressure
systems from the Atlantic Ocean resulting in heavy precipitation, while the
eastern parts of the Scandes have a more continental and drier climate.
Mountain permafrost is present all the way to the southern parts of the
Scandes, with a gradient in the lower limit of permafrost from ∼ 1400 to
1700 m from east to west in central southern Norway and from ∼ 700 to
1200 m from east to west in northern Norway (Gisnås et al.,
2013). While permafrost is also found in mires at lower elevations both in
southern and northern Norway, most of the permafrost is located in exposed
terrain above the treeline where strong winds result in heavy redistribution of snow.
The in situ records of snow depth data used to establish the snow
distribution scheme were collected at the Hardangervidda mountain plateau in
the southern part of the Scandes (Fig. 1). It is
the largest mountain plateau in northern Europe, located at elevations from
1000 to above 1700 m a.s.l., with occurrences of permafrost in the
highest mountain peaks. The terrain is open and slightly undulating in the
east, while in the west it is more complex with steep mountains divided by
valleys and fjords. The mountain range represents a significant orographic
barrier for the prevailing westerly winds from the Atlantic Ocean, giving
rise to large variations in precipitation and strong winds, two agents
promoting a considerably wind-affected snow distribution. Mean annual
precipitation varies from 500 to more than 3000 mm over distances of a few
tens of kilometres, and maximum snow depths can vary from 0 to more than
10 m over short distances (Melvold and Skaugen, 2013).
Model description
A statistical model for snow depth variation
The Winstral terrain-based approach (Winstral et al., 2002)
is applied over the entire Norwegian mainland using the 10 m national
digital terrain model from the Norwegian Mapping Authority (available at
Statkart.no), with wind data from the NORA10 dataset (Sect. 4.1) used to indicate the distribution of
prevailing wind directions during the accumulation season.
The terrain-based exposure parameter (Sx), described in detail in
Winstral et al. (2002), quantifies the extent of shelter or
exposure of the considered grid-cell. Sx is determined by the slope between
the grid-cell and the cells of greatest upward slope in the upwind terrain.
The upwind terrain is defined as a sector towards the prevailing wind
direction d constrained by the maximum search distance (dmax = 100 m) and a
chosen width (A) of 30∘ with the two azimuths extending
15∘ to each side of d (see Fig. 2). The
cell of the maximum upward slope is identified for each search vector,
separated by 5∘ increments. This gives in total seven search
vectors for each of the eight 30∘ wide sectors. Sx for the given
grid-cell is finally calculated as the average of the maximum upward slope
gradient of all seven search vectors:
Sxd,A,dmaxxiyi=maxtanZxv,yv-Zxi,yixv-xi2+yv-yi20.5,
where d is the prevailing wind direction, (xi, yi) are the coordinates of
the considered grid-cell, and (xv, yv) are the sets of all cell
coordinates located along the search vector defined by (xi, yi),
A and dmax. This gives the degree of exposure or shelter in the range -1 to 1,
where negative values correspond to exposure.
The area accounted for in each of the eight runs of the Winstral
terrain-based parameter, each of them with a prevailing wind direction
dn. The area accounted for when calculating the exposure of a grid cell
is constrained by the search window (A) and the search distance dmax
being 100 m upwind.
To estimate a realistic degree of exposure based on the observed wind
pattern at a local site, Sx was computed for each of the eight prevailing wind
directions d = [0, 45, 90, 135, 180, 225, 270, 315∘] and
weighted based on the wind fraction (wfd). wfd
accounts for the amount of different exposures in the terrain at various
wind directions and represents the fraction of hourly wind direction
observations over the accumulation season for the eight wind directions. The
selected period of wind directions influencing the redistribution of snow is
January to March. Wind speeds below a threshold of 7 ms-1 are excluded,
as this threshold is considered a lower limit required for wind drifting of
dry snow (Li and Pomeroy, 1997; Lehning and Fierz, 2008). We assume that
the snow distribution at snow maximum is highly controlled by the terrain
and the general wind exposure over the winter season, and we do not account
for the variation in snow properties over the season that controls how much
snow is available for transport at a given time.
The calculated Sx parameter values are used as predictors in different
regression analyses to describe the CVsd within 1 × 1 km derived from an
airborne laser scanning (ALS) of snow depths (see Sect. 4.1). The
coefficient of variation of exposure degrees (CVSx) within each 1 km × 1 km grid
cell is computed by aggregating the Sx map from 10 m to 1 km resolution
according to
CVSx=stdeSx/meaneSx.
Sx values below the 2.5th and above 97.5th percentiles of the
Sx distributions are excluded, giving Sx ≈ [-0.2, 0.2]. Three
regression analyses were performed to reduce the root mean square error (RMSE) between
CVSx and observed CVsd, where additional predictors such as
elevation above treeline (z) and maximum snow depth (μ) have been
included (Table 1). Ideally, wind speed should be
included as predictor. However, the NORA10 dataset (Sect. 4.1) does not sufficiently reproduce the local
variations in wind speeds over land, especially not at higher elevations and
for terrain with increased roughness. Elevation above treeline is chosen as
predictor to account for the increased wind exposure with elevation. There
is a strong gradient in treeline and general elevation of mountain peaks
from high mountains in the south to lower topography in the north of Norway.
Therefore, applying only elevation, not adjusted for the local treeline, as
predictor would result in an underestimation of redistribution in the north.
CryoGRID 1 with an integrated sub-grid scheme for snow variation
The equilibrium permafrost model CryoGRID 1 (Gisnås
et al., 2013; Westermann et al., 2015) provides an estimate for the MAGST (mean
annual ground surface temperature) and MAGT (mean annual ground temperature at
the top of the permafrost or at the bottom of the seasonal freezing layer)
from freezing (FDDa) and thawing (TDDa) degree days in the air according to
MAGST=TDDa×nT-FDDa×nFP,
and
MAGT=TDDa⋅nT⋅rk-FDDa⋅nFPforKtTDDs≤KfFDDsTDDa⋅nT-1rk⋅FDDa⋅nFPforKtTDDs≥KfFDDs,
where P is the period that FDDa and TDDa are integrated
over, rk is the ratio of thermal conductivities of the ground in thawed
and frozen states (assuming that heat transfer in the ground is entirely
governed by heat conduction), while nT and nF are semi-empirical
transfer functions which aim to capture a variety of key processes in one
single variable (see Gisnås et al. (2013) and Westermann et al. (2015) for details).
The winter nF factor relates the freezing degree days at the surface to the
air and thus accounts for the effect of the winter snow cover, and likewise
the nT factor relates the thawing degree days at the surface to the air and
accounts for the surface vegetation cover:
FDDs=nFFDDaandTDDs=nT⋅TDDa.
Variation in observed n factors for forests and shrubs are relatively small,
with nT factors typically in the range 0.85 to 1.1, and nF factors in the
range 0.3 to 0.5 (Gisnås et al., 2013). Forest, shrubs and mires
are assigned nT factors 0.9/1.0/0.85 and nF factors 0.4/0.3/0.6 respectively
(Gisnås et al., 2013).
Observed variations in nT and nF within the open non-vegetated areas are
comparably large, with values typically in the range 0.4–1.2 for nT and
0.1–1.0 for nF. The variability is related to the high impact and high spatial
variability of snow depths (Gisnås et al., 2014).
While nF accounts for the insulation from snow due to low thermal conductivity,
nT indirectly compensates for the shorter season of thawing degree days at the
ground surface in areas with a thick snow cover. Relationships between
n factors for open areas and maximum snow depths are established based on air
and ground temperature observations together with snow depth observations at
the end of accumulation season at the 13 stations in southern Norway,
presented in Hipp (2012) and at arrays of nearly 80 loggers at Finse
and Juvvasshøe (Gisnås et al., 2014) (Fig. 3):
nF=-0.17⋅ln(μ)+0.25,nT=-0.13⋅μ+1.1.
The relationships between n factors and snow cover in open areas are shown to
be consistent within the two sites in southern Norway
(Gisnås et al., 2014). Due to lack of field
observations including all required variables at one site in northern
Norway, the relation is not tested for this area. However, it fits very well
with a detailed study with 107 loggers recording the variation in ground
surface temperature at a lowland site in Svalbard (Gisnås et al., 2014).
Other factors, such as solar radiation and soil moisture, have minor effects
on the small-scale variation in ground surface temperatures in these areas.
Gisnås et al. (2014) demonstrated that most of the sub-grid variation in
ground temperatures within 1 km × 1 km areas in Norway and Svalbard was
reproduced by including only the sub-grid variation of snow depths. In other
areas, other parameters than snow depth might have a larger effect on the
ground surface temperatures and should be accounted for in the derivation of n factors.
nF and nT related to maximum snow depth observed at more
than 90 sites located above 1000 m a.s.l. in southern Norway.
We assume that the distribution of maximum snow depths within a grid cell
with a given CVsd and average maximum snow depth (μ) follows a
gamma distribution with a probability density function (PDF) given by
f(x;α,β)=1βαΓ(α)xα-1e-xβ,
with a shape parameter α = CVsd-2 and a rate parameter
β = μ ⋅ CVsd2 (e.g. Skaugen et al., 2004; Kolberg
and Gottschalk, 2006). The average maximum snow depth corresponds to the
coarse-scale snow observation, and the original coarse-scale snow depth is
therefore conserved in the sub-grid snow distribution. Corresponding
n factors are computed for all snow depths (x) based on Eqs. (6) and (7) and
related to the PDF (Eq. 8). The model is run for each nF from 0 to 1 with 0.01
spacing, giving 100 model realisations. Each realisation corresponds to a
unique snow depth, represented with a set of nF and nT factors. Based on the
100 realisations, distributions of MAGST and MAGT are calculated for each grid cell,
where the potential permafrost fraction is derived as the percentage of
sub-zero MAGT. A schematic overview of the model chain and the evaluation is
shown in Fig. 4. To assess the sensitivity of the
choice of the theoretical distribution function, the model was also run with
PDFs following a lognormal distribution (e.g. Liston, 2004):
f(x;λ,ζ)=1xζ2e-12ln(x)-λζ2,
where
λ=ln(μ)-12ζ2,ζ2=ln1+CVsd.
Schematic of the model chain, including input data, calibration and
evaluation procedures.
Model evaluation
The CVsd was derived for 0.5 km × 1 km areas based on the ALS snow depth
data (Sect. 4.1) resampled to 10 m × 10 m
resolution. Each 0.5 × 1 km area includes 500 to 5000 grid cells 10 m × 10 m,
depending on the area masked out due to lakes or measurement errors. There
were > 4000 grid cells in 70 % of the areas. Goodness-of-fit
evaluations for the theoretical lognormal and gamma distributions applying the
Anderson–Darling test in MATLAB (adtest.m, Stephens, 1974)
were conducted for each distribution. Parameters for gamma (shape and rate) and
lognormal (μ, σ) distributions were estimated by maximum likelihood as
implemented in the MATLAB functions gamfit.m and lognfit.m.
The results of the permafrost model are evaluated with respect to the
average MAGST and MAGT within each grid cell, as well as the fraction of sub-zero
MAGST. For the evaluation runs, the model is forced with climatic data for the
hydrological year corresponding to the observations. The performance in
representing fractional permafrost distribution is evaluated at two field
sites where arrays of 26 (Juvvasshøe) and 41 (Finse) data loggers have
measured the distribution of ground surface temperatures at 2 cm depth
within 500 × 500 m areas for the hydrological year 2013
(Gisnås et al., 2014). The general lower limits of
permafrost are compared to permafrost probabilities derived from BTS (basal
temperature of snow) surveys (Haeberli, 1973; Lewkowicz and Ednie,
2004) conducted at Juvvasshøe and Dovrefjell (Isaksen et al.,
2002). The model performance of MAGST is evaluated with data from 128 temperature
data loggers located a few centimetres below the ground surface in the period 1999–2009
(Farbrot et al., 2008, 2011, 2013; Isaksen et al., 2008, 2011; Ødegaard et al., 2008). The
loggers represent all vegetation classes used in the model and cover
spatially large parts of Norway (Fig. 1, black
dots). Four years of data from 25 boreholes (Isaksen et al., 2007, 2011;
Farbrot et al., 2011, 2013) are used
to evaluate modelled MAGT (Fig. 1, red dots). Tables of
ground surface temperature loggers (Table S1) and boreholes used for
validation (Table S2) are included in the Supplement.
Data
Forcing and evaluation of the snow distribution scheme
Wind speeds and directions during the snow accumulation season are
calculated from the boundary layer wind speed and direction at 10 m
above surface in the Norwegian Reanalysis Archive (NORA10) wind dataset.
NORA10 is a dynamically downscaled dataset of ERA-40 to a spatial
resolution of 10–11 km, with hourly resolution of wind speed and direction
(Reistad et al., 2011). The dataset is originally produced for wind
fields over sea and underestimates the wind speeds at higher elevation over
land (Haakenstad et al., 2012). A comparison with weather station
data revealed that wind speeds above the treeline are underestimated by
about 60 % (Haakenstad et al., 2012). For these areas wind
speeds in the forcing dataset have been linearly increased by 60 %.
The snow distribution scheme is derived from an ALS snow depth over the
Hardangervidda mountain plateau in southern Norway (Melvold and
Skaugen, 2013). The ALS survey is made along six transects, each covering a
0.5 × 80 km area with nominal 1.5 × 1.5 m ground point spacing. The survey
was first conducted between 3 and 21 April 2008 and
repeated in the period 21–24 April 2009. The snow cover was at
a maximum during both surveys. A baseline scan was performed 21 September 2008
to obtain the elevation at minimum snow cover. The ALS data
are presented in detail in Melvold and Skaugen (2013).
Distributions of snow depth, represented as CVsd, are calculated for each
0.5 × 1 km area, based on the snow depth data resampled to 10 m × 10 m
resolution. About 400 cells of 0.5 km × 1 km exist for each year, after lakes
and areas below treeline are excluded.
The snow distribution scheme is validated with snow depth data obtained by
ground penetrating radar (GPR) at Finse (60∘34′ N,
7∘32′ E;
1250–1332 m a.s.l.) and Juvvasshøe (61∘41′ N,
8∘23′ E; 1374–1497 m a.s.l.). The two field sites are both located
in open, non-vegetated alpine landscapes with major wind re-distribution of
snow. They differ with respect to mean maximum snow depth (∼ 2/∼ 1 m),
average winter wind speeds (7–8/10–14 m s-1) and
topography (very rugged at Finse, while steep, but less rugged, at
Juvvasshøe). Snow surveys were conducted late March to April (2009,
2012–2014) around maximum snow depth, but when the snow pack was still dry.
The GPR surveys at Finse are constrained to an area of 1 × 1 km, while at
Juvvasshøe they cover several square kilometres, but with lower
observation density. The GPR data from the end of the accumulation season
in 2013 are presented in Gisnås et al. (2014), and the
data series from the other years are obtained and processed following the
same procedures, described in detail in Dunse et al. (2009). The
propagation speed of the radar signal in dry snow was derived from the
permittivity and the speed of light in vacuum, with the permittivity
obtained from snow density using an empirical relation (Kovacs et
al., 1995). The snow depths were determined from the two-way travel time of
the reflection from the ground surface and the wave speed. Observations were
averaged over 10 m × 10 m grid cells, where grid cells containing less
than three samples were excluded. The CVsd for 1 × 1 km areas are computed
based on the 10 m resolution data.
Permafrost model setup
The climatic forcing of the permafrost model is daily gridded air
temperature and snow depth data, called the seNorge dataset, provided by the
Norwegian Meteorological Institute and the Norwegian Water and Energy
Directorate. The dataset, available for the period 1961–2015, is based on
air temperature and precipitation data collected at the official
meteorological stations in Norway, interpolated to 1 km × 1 km resolution
applying optimal interpolation as described in Lussana et al. (2010).
Snow depths are derived from the air temperature and precipitation
data, using a snow algorithm accounting for snow accumulation and melt,
temperature during snow fall and compaction
(Engeset et al., 2004; Saloranta, 2012). Freezing (FDDa) and
thawing (TDDa) degree days in the air are calculated as
annual accumulated negative (FDD) and positive (TDD) daily mean air temperatures,
and maximum annual snow depths (μ) are derived directly from the daily
gridded snow depth data. The CryoGRID 1 model is implemented at 1 km × 1 km
resolution over the same grid as the seNorge dataset.
The three regression models for CVsd with in increasing number
of predictors are calibrated with observed snow distributions from the ALS
snow survey (left columns). P values are < 10-6. The isolated
snow distribution scheme is validated with independent snow distribution
data collected with GPR snow surveys (right columns). Root mean square error (RMSE),
coefficient of determination (R2) and Nash–Sutcliffe model efficiency (ME)
are given for each model evaluation.
CVsd =
Fit of regression
CVsd, GPR survey
RMSE
R2
ME
RMSE
R2
ME
Model 1
0.39 + 3.4 × CVSx
0.14
0.36
0.36
0.20
0.04
-0.71
Model 2
0.31 + 3.1 × CVSx + 4.05 × 10-4 × z
0.12
0.52
0.52
0.12
0.59
0.36
Model 3
0.40 + 3.1 × CVSx + 4.95 × 10-4 × z - 0.0713 × μ
0.12
0.55
0.55
0.09
0.62
0.61
Scores from the Anderson–Darling test statistics for goodness of fit
between theoretical gamma and lognormal distributions and the observed
distribution within each 1 × 1 km area in the ALS snow survey. Lower scores
indicate better fit.
Soil properties and surface cover is kept as in Gisnås et
al. (2013), with five land cover classes: forest, shrubs, open non-vegetated
areas, mires and no data, based on CLC level 2 in the Norwegian Corine Land
Cover map 2012 (Aune-Lundberg and Strand, 2010). Sub-grid distributions of snow
are only implemented for open non-vegetated areas.
Results
Observed snow distributions in mountain areas of Norway
CVsd within 1 × 1 km areas in the ALS snow survey at Hardangervidda
ranged from 0.15 to 1.14, with mean and median of respectively 0.58 and
0.59. According to the Anderson–Darling goodness-of-fit evaluations, 70 out
of 932 areas had a snow distribution within the 5 % significance interval
of a gamma distribution, while only 1 area was within the 5 % significance
interval of a lognormal distribution. Although the null hypothesis rejected more than
90 % of the sample distributions, the Anderson–Darling test score was all over lower for the gamma
distribution, indicating that the observed snow distributions are closer to
a gamma than to a lognormal theoretical distribution (Fig. 5).
For lower lying areas with less varying topography and shallower snow
depths, in particular in the eastern parts of Hardangervidda, the observed
snow distributions were similarly close to both distributions. In higher
elevated parts with more snow to the west of the plateau the snow
distributions were much closer to a gamma distribution. Based on these findings a
gamma distribution was used in the main model runs, while a model run with
lognormal distributions of snow was made to evaluate the sensitivity towards the
choice of the distribution function (Sect. 3.2).
Left panel: fit of the regression Model 3 for CVsd, calibrated with CVsd
derived from the ALS snow survey. Right panel: the model performance is evaluated
with independent ground penetrating radar (GPR) snow surveys from at Finse
and Juvvasshøe.
Left panel: distribution of modelled CVsd in non-vegetated areas of
Norway with Model 3. CVsd increases in areas of rougher topography (western side
of Norway) and higher elevations (central part following the Scandes).
Right panel: standard deviation of modelled MAGT for areas of modelled permafrost.
Sites mentioned in the text: (1) Finse, south of Hallingskarvet; (2) Juvvasshøe
in Jotunheimen; (3) Dovrefjell; (4) the Lyngen Alps; and (5) Finnmark.
Evaluation of the snow distribution scheme
Three regression models for CVsd as a function of the terrain-based
parameter Sx, elevation (z) and mean maximum snow depth (μ) were calibrated
with the snow distribution data from the ALS snow survey over the
Hardangervidda mountain plateau (Table 1). Model 1 results
in a RMSE of only 0.14. However, the correlations of the
distributions are significantly improved by including elevation as predictor (Model 2; R2 = 0.52).
Including maximum snow depth as additional predictor (Model 3) improves the model slightly to
R2 = 0.55 (Fig. 6). The distribution of CVsd (example of Model 3 in Fig. 7, left panel) shows increased
values in areas of rougher topography (western side of Norway) and higher
elevations (central part following the Scandes), with maximum CVsd up
to 1.2 in the Lyngen Alps and at peaks around Juvvasshøe (Fig. 1, site 2
and 4). The lowest values of 0.2–0.3 are modelled in larger valleys in south
eastern Norway, where elevations are lower and topography gentler.
The regression models for CVsd are validated with data from GPR snow
surveys at Juvvasshøe and Finse (Table 1). The
correlation for Model 1 is poor, with R2 = 0.04 and Nash–Sutcliffe model efficiency
(ME) = -0.7 (Table 1). Model 2 improves the correlation
significantly, while the best fit is obtained with Model 3
(Fig. 6, RMSE = 0.094, R2 = 0.62 and ME = 0.61). The improvement
in Model 3 compared to Model 2 is more pronounced in the validation than in the fit of the
regression models and is mainly a result of better representation of the
highest CVsd values. The validation area at Juvvasshøe is located at
higher elevations than what is represented in the ALS snow survey dataset
and undergoes extreme redistribution by wind. The representation of extreme
values, therefore, has a high impact in the validation run.
Modelled ground temperatures for mainland Norway
The main results presented in this section are based on the model run with
100 realisations per grid cell, applying gamma distributions over the CVsd
from Model 3. The main results are given as averages over the 30-year period 1981–2010.
According to the model run, in total 25 400 km2 (7.8 %) of
the Norwegian mainland is underlain by permafrost in an equilibrium
situation with the climate over the 30-year period 1981–2010
(Fig. 1). Of the land area, 12 % features
sub-zero ground temperatures in more than 10% of a 1 km grid cell and is
classified as sporadic (4.4 %), discontinuous (3.2 %) or continuous (4.3 %)
permafrost (Fig. 1). In comparison, the
model run without a sub-grid variation results in a permafrost area of only
13 460 km2, corresponding to 4.1% of the model domain
(Table 2). The difference is illustrated for
Juvvasshøe (Fig. 8a) and Dovrefjell (Fig. 8c), where
the sub-grid model reproduces very well the observed lower limit of
permafrost based on borehole temperatures and BTS surveys. In contrast, the
model without sub-grid variability indicates a hard line for the permafrost
limit at much higher elevations (Fig. 8b and d).
At Juvvasshøe, the model without sub-grid distribution still reproduces
the permafrost limit to some extent because of the large elevation gradient.
At Dovrefjell, where the topography is much gentler, the difference between
the models is much larger and the approach without sub-grid distribution is
not capable of reproducing the observed permafrost distribution. The
modelled permafrost area for model runs applying the other models for
CVsd and theoretical distribution functions are summarised in Table 2.
Distribution of permafrost at Juvvasshøe in Jotunheimen (a, b)
and at Dovrefjell (c, d), modelled as permafrost zones applying the
sub-grid approach (left panels) compared to the modelled mean annual ground
temperature (MAGT) without a sub-grid approach (right panels). Lower limits of 50 %
and 80 % probability of permafrost derived from BTS surveys are shown as
black and red contour lines respectively. Borehole locations with permafrost
(red) and seasonal frost (green) are shown as dots in the map at Juvvasshøe.
The model performance is evaluated with respect to the mean annual
ground surface temperature (MAGST) and the mean annual temperature at the depth
of the active layer or seasonal freezing layer (MAGT). Modelled average MAGST or
MAGT over a grid cell is compared to more than 100 GST logger locations and
25 boreholes. The location of the GST loggers and boreholes are shown in
Fig. 1. Modelled permafrost distribution is given in total areas and as
percentage of the model domain, corresponding to the Norwegian mainland area.
Permafrost model evaluation
MAGST, GST loggers
MAGT, boreholes
Modelled permafrost area
RMSE
R2
ME
RMSE
R2
ME
(km2)
(%)
No sub-grid variation
1.57
0.65
-0.56
1.19
0.62
-1.90
13 462
4.1
GAMMA
CVsd = 0.6
1.37
0.64
0.06
0.77
0.66
0.22
23 571
7.3
Model 1
1.36
0.63
0.12
0.77
0.66
0.11
23 147
7.1
Model 2
1.29
0.65
0.31
0.65
0.71
0.62
23 674
7.3
Model 3*
1.29
0.65
0.38
0.67
0.71
0.68
25 407
7.8
LOGN
Model 1
1.40
0.64
-0.06
0.87
0.67
-0.25
19 975
6.2
Model 2
1.38
0.65
0.01
0.82
0.69
0.09
20 067
6.2
Model 3
1.36
0.65
0.06
0.78
0.69
0.22
20 889
6.2
* Reference model run.
Observed and modelled values for the coefficient of variation for
maximum snow depth (CVsd) and spatial distributions of mean annual
ground surface temperature (MAGST) at the field sites at Finse and
Juvvasshøe. The MAGST modelled without a sub-grid distribution of snow is
given in parenthesis.
Juvvasshøe
Finse
Observed
Modelled
Observed
Modelled
CVsd
0.85
0.80
0.71
0.77
MAGST < 0 ∘C
77 %
64 %
30 %
32 %
MAGSTmin
-1.8 ∘C
-2.6 ∘C
-1.9 ∘C
-1.6 ∘C
MAGSTmax
1.0 ∘C
0.8 ∘C
2.7 ∘C
1.0 ∘C
MAGSTavg
-0.5 ∘C
-0.5 ∘C (0.8 ∘C)
0.8 ∘C
0.2 ∘C (1.3 ∘C)
The standard deviations of the modelled sub-grid distribution of MAGT range
from 0 to 2.5 ∘C (Fig. 7, right panel). The highest standard deviation values are found in the Jotunheimen area, where
modelled sub-grid variability of MAGT is up to 5 ∘C. Also, at
lower elevations in southeastern parts of Finnmark standard deviations
exceed 1.5 ∘C. Here, the CVsd values are below 0.4, but because of cold
(FDDa < -2450 ∘C) and dry (max SD < 0.5 m) winters
even small variations in the snow cover have large effects on the ground temperatures.
Close to 70 % of the modelled permafrost is situated within open,
non-vegetated areas above treeline, classified as mountain permafrost
according to Gruber and Haeberli (2009). This is the major part of the
permafrost extent both in northern and southern Norway. In northern Norway
the model results indicate that the lower limit of continuous and sporadic
mountain permafrost decreases eastwards from 1200 and 700 m a.s.l. in
the west to 500 and 200 m in the east respectively. In southern
Norway, the southernmost location of continuous mountain permafrost is in
the mountain massif of Gaustatoppen at 59.8∘ N, with continuous
permafrost above 1700 m a.s.l. and discontinuous permafrost down to
1200 m a.s.l. In more central southern Norway the continuous mountain
permafrost reaches down to 1600 m a.s.l. in the western Jotunheimen and
Hallingskarvet, and down to 1200 m a.s.l. in the east at the Swedish
border. The sporadic mountain permafrost extends around 200 m further
down both in the western and eastern parts.
Evaluation of CryoGRID 1 with sub-grid snow distribution scheme
The observed and modelled CVsd values at the field sites were 0.85
and 0.80 at Juvvasshøe, and 0.71 and 0.77 at Finse. At Juvvasshøe the
observed fraction of loggers with MAGST below 0 ∘C was 77 %, while
the model result indicates an aerial fraction of 64 %. Similarly, at
Finse the observed negative MAGST fraction was 30 %, while the model indicates
32 %. The measured ranges of MAGST within the 1 km × 1 km areas were
relatively well reproduced by the model (Table 3). The average MAGST within
each field area was also improved compared to a model without a sub-grid
representation of snow (Table 3, in parenthesis).
Of the observed MAGSTs, 58 % are captured by the modelled range of MAGST for the
corresponding grid cell and 87 % within 1 ∘C outside the range
given by the distribution. The overall correlation between observed MAGST and
average modelled MAGST for a grid cell is fairly good with RMSE, R2 and ME of
1.3 ∘C, 0.65 and 0.37 respectively (Fig. 9, left panel). The measured MAGT
was within the range of modelled MAGTs in all boreholes
except one, where MAGT deviates 0.2 ∘C outside the range. All the
average modelled MAGTs are within ±1.6 ∘C of observations, while
90 % are within 1 ∘C. The RMSE between the observed and modelled
average MAGT is 0.6 ∘C (Fig. 9, right panel).
Correlation between modelled and observed MAGST (left panel) and MAGT at the top
of permafrost (right panel). The dotted line indicates ±2 ∘C of
the 1 : 1 line (black line). The vertical bars indicate the variation of
modelled temperatures within the grid cell, and the red dots indicate the
mean temperature.
The evaluation of the model runs with all three CVsd models, as well as
lognormal instead of gamma distribution functions, is summarised in
Table 2. The highest correlation between observed
and mean MAGST and MAGT was obtained by Model 3, but Model 2 yielded similar correlations. All
three model runs capture 58 % of the observed MAGST and more than 98 % of
the observed MAGT within the temperature range of the corresponding grid cell.
The total area of modelled permafrost is 9 % less when applying the
simplest snow distribution model (Model 1) compared to the reference model (Model 3),
while the same model without any sub-grid distribution results in 47 %
less permafrost area. With a lognormal distribution the modelled permafrost area is
18 % less (Model 3) than with a gamma distribution.
Discussion
The effect of a statistical representation of sub-grid variability in a regional permafrost model
The total distribution of modelled permafrost with the sub-grid snow scheme
corresponds to 7.8 % of the Norwegian land area, while the modelled
permafrost area without a sub-grid representation of snow only comprises
4 %. This large difference in total modelled permafrost area stems
exclusively from differences in the amount of modelled permafrost in
mountains above the treeline. In these areas the snow distribution is highly
asymmetric and a majority of the area has below average snow depths.
Because of the nonlinearity in the insulating effect of snow cover, the
mean ground temperature of a grid cell is not the same as, or even far from,
the ground temperature below the average snow depth. Often, the majority of
the area in high, wind-exposed mountains is nearly bare blown with most of
the snow blown into terrain hollows. Consequently, most of the area
experiences significantly lower average ground temperatures than with an
evenly distributed, average depth snow cover. In mountain areas with a more
gentle topography and relatively small spatial temperature variations, an
evenly distributed snow depth will result in large biases in modelled
permafrost area, as illustrated at Dovrefjell in
Fig. 8. This study provides clear evidence that the
sub-grid variability of snow depths should be included in model approaches
targeting the ground thermal regime and permafrost distribution.
The model reproduces the large range of variation in sub-grid ground
temperatures, with standard deviations up to 2.5 ∘C, coincident
to the observed small-scale variability of up to 6 ∘C within a
single grid cell (Gubler et al., 2011; Gisnås et al., 2014).
Inclusion of sub-grid variability of snow depths in the model provides a
more adequate representation of the gradual transition from permafrost to
permafrost-free areas in alpine environments and, thus, a better estimation
of permafrost area. In a warming climate, a model without such a sub-grid
representation would respond with an abrupt decrease in permafrost extent.
In reality, bare blown areas with mean annual ground temperatures of
-6 ∘C need a large temperature increase to thaw. Increased
precipitation as snow would also warm the ground; however, bare blown areas
may still be bare blown with increased snow accumulation during winter. A
statistical snow distribution reproduces this effect, also with an increase
in mean snow depth.
CryoGRID1 is a simple modelling scheme delivering a mean annual ground
temperature at the top of the permanently frozen ground based on
near-surface meteorological variables, under the assumption that the ground
thermal regime is in equilibrium with the applied surface forcing. This is a
simplification, and the model cannot reproduce the transient evolution of
ground temperatures and is therefore not suitable for future climate
predictions. However, it has proven to capture the regional patterns of
permafrost reasonably well (Gisnås et al., 2013; Westermann et al.,
2013). Because of the simplicity, it is computationally efficient and
suitable for doing test studies like the one presented in this paper and in
similar studies (Westermann et al., 2015).
For the model evaluation with measured ground temperatures in boreholes
(Sect. 5.4), the modelled temperatures are forced
with data for the hydrological year corresponding to the observations.
Because of the assumption of an equilibrium situation in the model approach,
such a comparison can be problematic as many of the boreholes have undergone
warming during the past decades. However, with the majority of the boreholes
located in bedrock or coarse moraine material with relatively high
conductivity, the lag in the climate signal is relatively small at the top
of the permafrost. The lag will also vary from borehole to borehole,
depending on the ground thermal properties. Since we use data distributed
over larger areas and longer time periods, including a large range of
situations, the effect is mainly evident in terms of a larger statistical
spread and not a systematic error.
The large amount of field observations used for calibration and evaluation
in this study is mainly conducted in alpine mountain areas. The large
spatial variation in winter snow depths is a major controlling factor also
of the ground temperatures in peat plateaus and palsa mires and is a
driving factor in palsa formation (Seppälä, 2011). The
sub-grid effect of snow should therefore also be implemented for mire areas,
where comparable datasets are lacking.
Model sensitivity
The sensitivity of the CVsd model to the modelled ground temperatures is
relatively low, with only 9 % variation in permafrost area, although the
performance of the snow distribution scheme varies significantly between the
models when evaluated with GPR snow surveys (Table 1). In comparison, a lognormal
instead of a gamma distribution function reduces the permafrost area by 18 % (Table 2). The choice of
distribution function therefore seems to be of greater importance than the
fine tuning of a model for CVsd. This result contradicts the conclusions
by Luce and Tarboton (2004), which suggest that the parameterization of
the distribution function is more important than the choice of distribution
model. With a focus on hydrology and snow cover depletion curves, equal
importance was given to both the deeper and shallower snow depths in the
mentioned study. In contrast, an accurate representation of the shallowest
snow depths is crucial for modelling the ground thermal regime. The low
thermal conductivity of snow results in a disconnection of ground surface
and air temperatures at snow packs thicker than 0.5–1 m, depending on the
physical properties of the snow pack and the surface roughness
(Haeberli, 1973). In wind-exposed areas prone to heavy redistribution,
large fractions of the area will be entirely bare blown
(Gisnås et al., 2014). These are the areas of greatest
importance for permafrost modelling. In order to reproduce the gradual
transition in the discontinuous permafrost zone, where permafrost is often
only present at bare blown ridges, shallow snow covers must be
satisfactorily represented. Compared to a gamma function, a lognormal distribution
function to a larger degree underestimates the fraction of shallow snow
depths, resulting in a less accurate representation of this transition.
Several studies include statistical representations of the sub-grid
variability of snow in hydrological models, most commonly applying a two- or
three-parameter lognormal distribution (e.g. Donald et al., 1995; Liston, 2004;
Pomeroy et al., 2004; Nitta et al., 2014). Observed snow distributions
within 1 × 1 km in the ALS snow survey presented in this paper are closer to a
gamma than to a lognormal distribution, supporting the findings by Skaugen (2007)
and Winstral and Marks (2014) which were both conducted in
non-forested alpine environments. However, the difference is not substantial
in all areas; the two distributions can provide near-equal fit in eastern
parts of the mountain plateau where the terrain is gentler and the wind
speeds are lower. We suggest that the choice of distribution function of
snow is important in model applications for the ground thermal regime and
recommend the use of gamma distribution for non-vegetated high alpine areas prone
to heavy redistribution of snow.
While a gamma distribution offers improvements over a lognormal distribution, the bare
blown areas are still not sufficiently represented. One attempt to solve
this is to include a third parameter for the “snow-free fraction”
(e.g. Kolberg et al., 2006; Kolberg and Gottschalk, 2010). We made an attempt to
calibrate such a parameter for this study, but no correlations to any
of the predictors were found. It is also difficult to determine a threshold
depth for “snow-free” areas in ALS data resampled to 10 m resolution,
where the uncertainty of the snow depth observations are in the order of 10 cm (Melvold and Skaugen, 2013).
In this study a high number of realisations could be run per grid cell
because of the low computational cost of the model. To evaluate the
sensitivity of sampling density, the number of realisations was reduced from 100
to 10 per grid cell. This resulted is a 2.6 % increase in total
modelled permafrost area relative to the reference model run. This
demonstrates that a statistical downscaling of ground temperatures as
demonstrated in this study is robust and significantly improves the model
results with only a few additional model realisations per grid cell.
Conclusions
We present a modelling approach to reproduce the variability of ground
temperatures within the scale of 1 km2 grid cells based on probability
distribution functions over corresponding seasonal maximum snow depths. The
snow distributions are derived from climatic parameters and terrain
parameterisations at 10 m resolution and are calibrated with a large-scale dataset of snow depths obtained from laser scanning. The model
results are evaluated with independent observations of snow depth
distributions, ground surface temperature distributions and ground
temperatures. From this study the following conclusions can be drawn.
The total modelled permafrost area in an equilibrium with the average
climate for the period 1981–2010 is 25 400 km2. This corresponds to
7.8 % of the Norwegian mainland.
The model simulation without a sub-grid representation of snow produces
almost 50 % less permafrost.
Due to the nonlinear insulating effect of snow in combination with
asymmetric snow distributions within each grid cell, the spatial average
ground temperature in a 1 km2 area cannot be determined based
on the average snow cover for that area.
Observed variations in ground surface temperatures from two logger arrays
with 26 and 41 loggers respectively are very well reproduced, with
estimated fractions of sub-zero MAGST within ±10 %.
Of the observed mean annual temperatures at top of permafrost in the
boreholes, 94 % are within the modelled ground temperature range for the
corresponding grid cell, and mean modelled temperature of the grid cell
reproduces the observations with an accuracy of 1.5 ∘C or better.
The sensitivity of the model to CVsd
is relatively low compared to the choice of theoretical snow distribution
function. Both are minor effects compared to the effect of running the model
without a sub-grid distribution.
The observed CVsd within 1 km2 grid cells in the
Hardangervidda mountain plateau varies from 0.15 to 1.15, with an average
CVsd of 0.6. The observed CVsd values are nearly identical at the end
of the accumulation seasons in 2008 and 2009.
The distributions are generally closer to a theoretical gamma distribution than
to a lognormal distribution, in particular in areas of very rough topography, thicker
snow cover and higher average winter wind speeds.
In areas subject to snow redistribution, the average ground temperature of a
1 km2 grid cell must be determined based on the distribution and not
the overall average of snow depths within the grid cell. Modelling the full
range of ground temperatures present over small distances facilitates a
better representation of the gradual transition from permafrost to
non-permafrost areas and most likely a more accurate response to climate
warming. This study demonstrates that accounting for the sub-grid
variability of snow depths can strongly improve model estimates of the
ground thermal regime and permafrost distribution alpine conditions.