Introduction
At the peak of winter, a snow cover resembles a sparkling, smooth
blanket. However, it is well known that the spatial distribution of snow
depths underneath is heterogeneous. Complex topography adds extra spatial
variability due to spatial patterns of wind (sheltering/exposure),
precipitation (e.g., mountain luv/lee), shortwave radiation (shading, sky view, terrain
reflections) and longwave radiation (sky view, terrain emission).
Furthermore, in complex topography, snow relocation can occur due to snow
avalanches. To complicate matters, these processes operate at different
spatial scales cf.. The result is a patchy snow
cover consisting of snow-free and snow-covered areas (SCAs). In various scientific
and operational applications, knowledge about spatial snow depths plays a key
role. Hydrologists are interested in predicting the timing of snowmelt
runoff as well as the overall amount of snow in a catchment to estimate the
water stored, allowing one to forecast available water resources. This is a
relevant issue, e.g., in controlling the drinking water supply, in hydropower
production planning or in warning of spring flooding. Climatologists,
studying present and future climates, are interested in the snow coverage in
a large-scale model grid cell which forms a key parameter in general
circulation models e.g.,. For instance, from
fractional SCAs, coarse-scale surface albedos can be
derived by weighting snow-free and snow-covered albedos
. Since snow has a high surface albedo, it alters the
energy and moisture fluxes on the earth and thus the surface energy budget
. Knowing the actual spatial snow depth distribution,
especially in mountainous terrain, is therefore a relevant topic in
large-scale hydrological, meteorological and regional climate models. Due to
computational constraints, large-scale models often have to simplify physical
processes over snow surfaces and within snow. Frequently, they lack a subgrid
snow distribution representation which is a shortcoming that deteriorates
atmospheric interaction simulations cf.. In general,
the purpose of subgrid parameterizations is to account for subgrid scale
processes, i.e., unresolved processes, with analytical approximations in
large-scale model systems. The considered subgrid snow
distributions as important for simulating observations of seasonal snow
cover.
A few studies previously tackled subgrid snow distributions.
improved a regional climate model by performing
separate surface energy balance calculations over snow-covered and snow-free
fractions of each model grid cell. Similar, calculated
vertical and horizontal energy fluxes between the atmosphere and snow,
snow-free and vegetation grid cell portions and found a warming feedback
through decreases in surface albedo and increases in sensible heat fluxes to
the atmosphere. computed SCAs by assuming lognormally
distributed snow depth and by introducing a dichotomous key for coefficient
of variations for snow depth (CV is standard deviation divided by mean)
depending on topographic variability, air temperature and wind speed.
introduced a snow cover protruding vegetation fraction
for grid cell portions covered by shrubs or grass.
validated previously published ad hoc closed forms of SCA over non-forested
terrain with those derived from a peak of winter lognormal distribution that
undergoes homogeneous melt. They found the closest snow-cover depletion (SCD)
curves using a functional form proportional to tanh, similar to what was
proposed by and . Instead of a
roughness length of the surface or the standard deviation
of the summer digital surface model (DSM) ,
included the peak of winter standard deviation of snow
depth in the SCA parameterization. However, peak of winter standard
deviations of snow depth are rarely available.
Numerous studies analyzed catchment snow depth distributions by relating
measured snow depth data to small-scale terrain parameters for a
recent literature overview see. Until now, multiple linear
regressions were frequently applied to relate mean snow depth, standard
deviation of snow depth or deviations of the mean to small-scale terrain
parameters such as elevation, slope or aspect. Others found linear
or power-law
relationships for the accumulation period, solely between standard deviation
of snow depth and mean snow depth using constant fit parameters. While the
CVs presented by depend on topographic variability,
the relationships of , and
result in CVs which neglect varying complexities of
terrain. Even though previous parameterizations for the snow distribution
parameters provide good descriptions for the investigated regions, they might
easily fail in a different geographic region with other terrain
characteristics. Recently, analyzed snow depth data
from seven mountainous catchments around the world. For each catchment, their
developed multiple regression equations for the relative snow depth
(HS – catchment mean) using subgrid topographic parameters showed
good performance. However, a similar performance for a global model, based on
all data sets, could not be achieved, and argue
that the snow depth and topography are less universally related than
hypothesized by .
A poorer performance of a subgrid parameterization for the snow distribution
can also arise from the different scales on which the spatial variability of
snow depths is created in complex topography. Recently,
and therefore investigated
the influence of scale on aggregated snow depth data. By analyzing snow depth
data in differently sized grid cells up to 800 m for several catchments,
found a lower limit of 400 m for the grid cell size
to explain most of the remaining larger-scale spatial variability. By
analyzing snow depth data from a large mountainous area in Norway in grid
cell sizes up to 1 km, , however, determined a larger
lower limit of 1 km to eliminate most of the spatial variability such that
the mean adequately represents the average grid cell snow depth. A reason why
a global parameterization might not be derivable at one certain horizontal
resolution is that too many different snow-cover shaping processes are still
active, at that scale, making it a challenge to parameterize the subgrid snow
distribution.
How can we acquire snow depth data spatially in order to better investigate
subgrid snow depth distributions? Measuring snow distribution, both
temporally and spatially, is a challenging task in mountainous terrain. To
overcome the limitations of point measurements of automated stations or hand
probing, terrestrial laser scanning (TLS) was introduced to continuously
measure snow depths in very high resolutions
. Airborne laser scanning (ALS) can
cover larger regions in a shorter time without the limitations of TLS
. ALS measurements
are, however, quite expensive and for larger regions they require large
investments to gather snow depths in adequate temporal and spatial
resolutions e.g.,. Visible satellite remote sensing
provides information on snow coverage in various horizontal and temporal
resolutions. However, the interpretation of satellite signals is difficult
and requires complex algorithms extracting clouds
e.g., and the influence of topography on the signal
e.g.,. Small-scale distributed snow surface
modeling e.g., over complex topography could fill
the gap of missing temporal and spatial snow depth data. However, for large
regions this is rarely feasible due to computational constraints and/or the
lack of small-scale input data. Erroneous input data could easily blur
modeled distributed snow depths. For now, we therefore prefer spatially and
temporally measured snow depth data to investigate subgrid snow depth
distributions.
To our knowledge, a systematic analysis of snow depth data from a large
region, aggregated in grid sizes comparable to those of large-scale models,
is still missing. Here, we are aiming for grid cell sizes where the subgrid
variability is deducible from the underlying characteristic terrain lengths.
We assume that the smoothing out of small-scale snow depth heterogeneities
originating from processes such as snowdrift or avalanches reveals the
large-scale topographic influences on precipitation and the shortwave
radiation balance. Our hypothesis is motivated by the observation of
, in that, in contrast to summer convective-precipitation
systems, the spatial distribution of winter precipitation is more influenced
by topographic distributions. Furthermore, it is motivated by the results of
and , which confirmed that
the snow depth distribution is dominated by topography at scales of several
hundred meters.
In this study our principal goal is thus to develop a subgrid
parameterization of SCA for large-scale model grid cell sizes of a few
kilometers that account for varying levels of complex, treeless topography.
For this, we relate snow depth data to terrain parameters in view of a
subgrid parameterization of the standard deviation of snow depth. We use
easily
accessible, computationally cheap terrain parameters calculated from the
summer DSM. We employ highly resolved spatial snow depth data from alpine
terrain of two large areas in the eastern Swiss Alps as well as from one in
the eastern part of the Spanish Pyrenees, i.e., from two distinct climates.
The snow depth data resolves for all small-scale variability of the snow
cover. We analyze the probability density functions (pdf) of snow depth and
the two defining parameters, mean and standard deviation, and examine the data
both within and between domain sizes of various dimensions. Finally, we point
out the limitations of our subgrid parameterizations originating from using
measured snow depth data sets.
Method
Aggregating snow depth data
Analyzing a sufficiently large number of differently sized domains from a
large mountainous region allows one to study snow distributions at different
scales. By randomly selecting different grid origins, we aggregated the snow
depth data sets in different squared domain sizes L. Note that L can be
seen as a coarse grid cell size Δx in a large-scale model (cf. Fig. a).
We chose domain sizes of L=50, 100, 200, 500, 750,
1000, 1250, 1500, 1750, 2000, 2500 and 3000 m covering the range
of typical grid cell sizes from hydrologic measurement campaigns to the
smallest grid cell sizes in meteorological models. For each domain size we
used 50 realizations allowing for overlap between domain sizes L (cf. Fig. a).
In total we generated ensembles of 600 snow depth grids for each
Swiss site. In Val de Núria we could not aggregate snow depth data in
domain sizes L larger than 1500 m, resulting in 400 snow depth grids at
this site.
For building domain averages, all data points were spatially averaged in a
domain size L. However, we only used domain sizes L with at least 75 %
valid snow depth measurements (including zero values). For larger domain
sizes L≥1 km in Val de Núria we had to allow for a maximum of 40 %
of missing values due to the irregular perimeter of that catchment (cf.
Fig b). In the following, we omit the normally used overbars for
domain-averaged variables.
Terrain characteristics
To relate the snow depth distribution parameters to topographic features, we
computed several terrain parameters from the summer DSMs. For selecting
terrain parameters, we exploited the fact that real topographic slope
characteristics are reasonably well described by Gaussian statistics
. Gaussian random fields with a Gaussian covariance such
that topography is reduced to only two underlying large length scales in a
model domain of size L, were previously used to systematically investigate
radiative transfer in complex terrain via the radiosity approach
as well as to develop a
parameterization for domain-averaged sky view factors in complex terrain
. Assuming a Gaussian covariance for the summer
topography, the two underlying characteristic length scales are a
valley-to-peak elevation difference σ (typical height of topographic
features), which is the standard deviation of the elevation model, and a
lateral extension ξ (typical width of topographic features), which is the
correlation length of the elevation model. We use a terrain parameter
μ=2σ/ξ, which is related to the mean squared slope and
which can be derived from first partial derivatives ∂xz and
∂yz (slope components) in orthogonal directions:
μ=[(∂xz)2+(∂yz)2]‾21/2,
using 2μ2=(∂xz)2+(∂yz)2‾=tan2ζ‾=4σ/ξ2
as outlined by . We also use the L/ξ ratio where a
large ratio indicates that more topographic features are included in a domain
size L. Note that the typical width of topographic features ξ in a
domain size L can be obtained via ξ=2σz/μ, with
the standard deviation of the summer DSM σz.
showed that to minimize influences of (subgrid) grid
size Δx and domain size L on domain-averaged shortwave terrain
reflected radiation, the condition Δx≪ξ≪L must be
fulfilled. The relevance of including enough terrain in a domain, here
L×L, was confirmed by , where errors of a subgrid
parameterization for the sky view factor over complex topography decreased
with increasing L/ξ ratio. We believe that in complex terrain for
domain-averaged snow depths, the above condition should always be met in
order to accurately capture the predominant subgrid processes shaping the
snow distribution at the corresponding scale. Consequently, we need to
detrend the summer DSMs in order to obtain the correct characteristic length
scales for the corresponding domain size L. Linearly detrending reveals the
dominant processes that shape the scale dependent characteristic snow depth
distribution by shifting the scaling parameters. For small domain sizes L
this leads to smaller correlation lengths ξ and thus to larger
L/ξ ratios.
Parameterizing spatial variability of snow depth
In order to specify the spatial variability of snow depth over mountainous,
treeless topography for large-scale grid cells, we first need to define the
pdf of snow depths in a domain size L.
Commonly applied snow depth distributions at the peak of winter range from
lognormal for complete snow cover
to gamma
to normal in forests
. Second, we need to scale the defining parameters
mean and standard deviation of the snow depth distribution, HS and
σHS, respectively, with the underlying subgrid terrain
characteristics. Previously published linear or power-law relationships, solely between σHS
and HS, lead to snow depth coefficients of variation CV which do
not depend on varying topography. Yet, we computed a mean CV for L≥1 km
of 0.63 for the Wannengrat and 0.48 for the Dischma region. The CV for the
catchment in the eastern Spanish Pyrenees for L≥1 km is 1.04, i.e.,
considerably larger than for the two large areas in the eastern Swiss Alps.
Deriving the CVs from the power-law relationship (via
σHS=HS0.84) results in overall larger but
similar CV values among the three regions: 0.91 for Wannengrat, 0.89 for
Dischma and 1.01 for Val de Núria. The CV of the eastern Swiss Alps
compares well to the CV categories of the dichotomous key in that
geographic region of 0.5 to 0.7, which was based on topographic variability,
air temperature and wind speed . However, for the area
in the eastern Spanish Pyrenees the CV of the dichotomous key of
is about 0.06, i.e., completely different to our 1.04.
One example probability density function (pdf) of measured snow
depths HS for each domain size L (in color) in each
area.
Given that we use snow depth data sets from two distinct climate regions, we
can focus on the development of a subgrid parameterization of the standard
deviation of snow depth σHS which is not constrained to
one specific geographic area but is more widely applicable. For this, we
employ the mean snow depth HS as a climate indicator variable for
each domain size L. However, mean HS is generally not easily
measured. We therefore investigate if mean snow depth HS can be
approximated by averaged flat field measurements HSflat. A
flat field was defined as a 22 m × 22 m (for Wannengrat and Dischma) or a
11 m × 11 m (for Val de Núria) area where each slope angle was lower than or
equal to 10∘. We computed the average flat field snow depth from all
snow depth values within a flat field. To obtain an average flat field snow
depth HSflat for each domain size L, we averaged all
mean snow depths of flat fields within each L. Note that in the following
we will use the superscript m for measured, mean quantities when opposed to
parameterized quantities.
Mean root mean square errors (RMSE) between theoretical probability
density functions (pdf) and measured pdfs as function of domain size L.
Error bars indicate standard deviation of RMSEs.
Results
Snow depth distribution
We found mostly unimodal distributions of snow depths in all domain sizes L
ranging from 50 m to 3 km in all three areas (Fig. ). We tested
three, previously published theoretical pdfs on our ensembles of gridded
snow depth data: normal, lognormal and gamma density functions. While for
small domain sizes a gamma distribution best described the measured snow
depth distributions, for larger domain sizes (L≥500 m) a normal
distribution worked as well or better (Fig. ). The mean RMSE between
theoretical pdfs and measured snow depths decreased with increasing domain
size L for all three areas. A comparison of computed quantiles for the
theoretical and measured snow depth distribution also resulted in decreasing
mean RMSE with increasing L. Note that our domain sizes do include subgrid
snow-free values.
Standard deviation of snow depth σHS as function
of domain size L for all three areas. The squares represent mean
σHS.
Pearson correlation coefficients r for mean snow depth
HS and standard deviation of snow depth σHS
with terrain parameters for pooled data of all three catchments as well as
for each catchment separately. Gaussian covariance parameters σ
(σz) and ξ are obtained as described in Sect. .
For mean slope μ, see Eq. (). Values in bold
indicate statistically significant correlations (p values <0.05).
All regions
Wannengrat
Dischma
Val de Núria
Terrain parameter
HS
σHS
HS
σHS
HS
σHS
HS
σHS
μ
0.20
0.65
0.01
0.72
0.16
0.62
0.09
0.63
σz
0.14
0.38
-0.01
0.59
0.03
0.25
–0.16
0.37
ξ
0.08
0.32
0.01
0.52
-0.03
0.15
–0.17
0.35
L/ξ
0.17
0.22
-0.09
0.37
0.11
0.23
-0.06
–0.19
L
0.17
0.38
-0.01
0.49
0.05
0.25
–0.17
0.35
Scaling of snow depth data grids
We analyzed our ensemble of snow depth data grids to relate mean and standard
deviation of each snow depth distribution, HS and
σHS, to terrain parameters. An interesting result is that
the mean of σHS increased with increasing L. For domain
sizes of L≥1 km the overall changes in the mean of
σHS became small (Fig. ). Similar to
and we found that overall,
with larger domain size L, the scatter in standard deviation of snow depth
σHS decreased (Fig. ). However, in comparison to
and to , we also included
L>1 km and found that for L ≥1 km the scatter in
σHS still somewhat decreased. Note that we obtained
similar trends and magnitudes of σHS as a function of domain
size L for both climates, which allowed us to pool the data of all three
areas. Furthermore, similar trends in σHS were found with
terrain parameters in all three areas, suggesting that a parameterization can
be developed which can be applied under a broad range of topographic
characteristics. For example, Fig. shows the standard deviation of
snow depth σHS of the three areas as function of the
standard deviation of the summer DSM, σz. In all areas
σHS increased similarly with increasing
σz and with increasing domain size L. Furthermore, the
scatter or the standard deviation of σHS among the same
domain sizes L decreased with increasing L and σHS. A
correlation analysis between terrain characteristics and standard deviation
of snow depth σHS revealed significant Pearson correlation
values ranging from 0.22 to 0.65 for pooled snow depth data from all
catchments (Table ). The overall larger scatter in snow depths for
all L in the Dischma catchment (cf. Figs. and ) resulted in
lower correlation values r when looking at the correlations coefficients of
each area separately (cf. Table ).
Standard deviation of snow depth σHS as a
function of detrended valley-to-peak elevation difference σ (indicated
by σz) of the underlying topographic features. Colors indicate
corresponding domain size L.
We found weaker correlations between mean snow depth HS and
terrain parameters, than between σHS and terrain
parameters (Table ). For the correlation between terrain parameters
and pooled snow depth data from all catchments, the significance was
marginally lower than for σHS. However, the correlation
analyses between HS and terrain parameters conducted for each
catchment separately often showed statistically insignificant correlations,
i.e., p values ≥0.05 (Table ). Yet, we observed an
approximately linear relationship between HS and mean flat field
snow depths HSflat when we pooled snow depth data of all
areas, especially for domain sizes larger than 1500 m (Fig. ). The overall
deviations between HS and HSflat decreased with
increasing domain size L. For the overall relationship of HS and
HSflat, we obtained a Pearson correlation coefficient r
of 0.86, a squared correlation coefficient R2 of 0.65, a RMSE of 36.7 cm,
a normalized root mean square error NRMSE of 5.4 % and a mean squared error
(MSE) of 13.4 cm.
Measured mean snow depth HS as function of mean measured
flat field HSflat for all three areas. Colors indicate
corresponding domain size L.
Parameterization for the standard deviation of snow depth
In order to develop a parameterization for σHS, we pooled
the snow depth data of all three areas. We derived the following subgrid
parameterization for the standard deviation of snow depth
σHS over mountainous terrain from snow depth data
aggregated in domain sizes ranging from L=50 m to 3 km:
σHS(μ,L/ξ,HS)=HSaμbexp-(ξ/L)2,
with a=0.549 and b=0.309 and HS, ξ and L in meters. When
fitting for each area separately, the parameters changed slightly. The
standard deviation of snow depth σHS in Eq. ()
has three scaling parameters: a terrain parameter μ (Eq. ),
related to the mean squared slope in each domain size L, the mean snow
depth HS and the L/ξ ratio, roughly describing how many
subgrid topographic features are in a domain size L. The functional form of
our subgrid parameterization was motivated by the result that we consistently
obtained the largest correlation coefficients for σHS with
the terrain parameter μ (cf. Table ). The third scaling
parameter, the L/ξ ratio, accounts for the uncertainty that in fixed,
finite domain sizes L with varying correlation lengths of topographic
features ξ the condition L/ξ≫1 is not always fulfilled and
corrections are required. Naturally, the correction factor decreases with
increasing L/ξ ratio. We chose a Gaussian factor e-(ξ/L)2 based
on our result that in large-scale grid sizes the snow depth distribution can
be described by a Gaussian distribution. Assuming that topography is the
major driver for the snow distribution, the Gaussian factor is also a
consequence of previously found Gaussian slope statistics for real
topographies . Mean snow depth HS has to be
included in a parameterization of σHS to account for
varying surface climates. We performed the nonlinear regression analysis to
optimize the parameters in Eq. () by robust M-estimators using
iterated reweighted least squares; see R v2.15.2 statistical programming
language and its robustbase v0.9-7 package
. Our subgrid parameterization, as in Eq. (),
predicts the observed σHS well (cf. Fig. a).
The performance of the parameterization improves with increasing
domain sizes L. Our subgrid parameterization for the standard deviation of
snow depth σHS is statistically significant
(Pearson r= 0.70, p value <0.001, R2 of 0.45, RMSE of 22.9 cm, NRMSE of 7.6 % and
MSE of 5.2 cm). The performance of parameterized σHS (Eq. ) also improved compared to previously published
parameterizations of σHS, which did not explicitly
account for subgrid topography (Fig. b and c). Note, that the subgrid
parameterization for σHS was developed for peak of winter
snow depth data.
Parameterization of fractional snow-covered area
Snow-covered area is an important parameter in the energy balance of
large-scale models, e.g., to weight energy flux components and surface albedos
for snow-covered and snow-free fractions. Fractional SCA f in
a large-scale grid cell is, however, reduced due to subgrid topographic effects
on the snow depth distribution. Here, we showed that the standard deviation
of snow depth σHS at the peak of winter over complex
topography scales with the underlying terrain characteristics combining
previously published observations. We therefore suggest including
σHS, as in Eq. (), in a closed form
parameterization of the fractional SCA f. When deriving a functional form for f,
concentrated on homogeneous surface units where the
peak of winter snow depth distribution could be described by a lognormal
distribution. We are focussing on large-scale grid cell sizes over complex
topography where we employ our result that the simpler normal distribution
describes the snow depth distribution equally well or better (cf. Fig. ).
We start the derivation from a normal distribution at the peak of
winter over alpine terrain (including snow-free sub-pixels):
p(HS)=12πσHS0exp-12HS-HS0σHS02,
with σHS0 as the standard deviation of snow depth
and HS0 as the mean snow depth at the peak of winter, both
indicated here with the subscript 0. The SCA f is obtained by assuming a
homogeneous melt amount M and by integrating over the peak of winter snow
depth distribution from M to ∞:
f=121-erfM-HS02σHS0.
Measured standard deviation of snow depth
σHSm as function of parameterized standard
deviation of snow depth σHS for all three areas.
(a) Parameterized via Eq. (), (b) parameterized via
and (c) parameterized via . Colors indicate
corresponding domain size L. NRMSEs are given for each
parameterization.
The mean snow depth HS is obtained from
HS=∫M∞(HS-M)p(HS)dHS=∫M∞HSp(HS)dHS-fM,
leading to
HSHS0=CV2πexp12MσHS0-1CV2+f-M2HS0f.
We followed the procedure of to derive a more practical
closed form of f than Eq. (). For this we also assumed
homogeneous melt for our peak of winter normal snow depth distribution
(Eq. ). Note that showed that the concept of
spatially uniform melt can even be applied over mountainous terrain when
starting from a measured snow distribution. In contrast to
we included a larger range for coefficients of
variations CV to derive a closed form of f. We chose the CV values of
defining snow distribution categories around the world but
added a maximum CV value of 1: 0.06, 0.09, 0.12, 0.17, 0.4, 0.5, 0.6,
0.7, 0.85, 1. Dashed lines in Fig. show the fitted f(HS) by means
of
f(HS)=tanh1.30HSσHS0
using σHS0, the standard deviation of snow depth
at the peak of winter. We obtain the closest fit with the same functional form as
who started with a lognormal snow depth distribution
(Fig. ). Our pre-factor in Eq. () varies slightly from the
one presented by . For our data and the fit parameter we
computed a 95 % confidence interval ranging from 1.27 to 1.35. For the fit
in Eq. () we obtain a mean RMSE of 0.02, and a mean NRMSE of 2.5 %
for all CV. Similar to the fit of our RMSEs increase
with increasing CV with the largest RMSE of 0.04 for a CV =1.
We extend the fractional SCA f(HS) of Eq. () to
complex topography by employing standard deviation of snow depth at the peak of
winter parameterized for complex subgrid topography (cf. Eq. ).
Figure a shows that the mean errors between parameterized and
observed SCA f for all our areas decrease with increasing domain size L.
Also, the scatter per L decreases with increasing L (cf. error bars in
Fig. a). Note that the largest mean errors are still below 10 %.
When using previously derived parameterizations for σHS in
parameterized f(HS) (Eq. ) both mean errors and
scatter also decrease with increasing L, however, the overall errors are
larger and mean errors do not approach zero for the largest domain sizes
L≥1750 m (Fig. b and c).
Discussion and Conclusion
Scaling snow depth distribution parameters is a relevant issue for various
applications in large-scale hydrological, meteorological and regional climate
models. In this study, we derived a parameterization for the fractional
SCA over complex, treeless topography for large-scale
models with grid cell sizes of a few kilometers. This required developing a
subgrid parameterization for the standard deviation of snow depth over
mountainous terrain. For the parameterization we chose easy to derive subgrid
terrain parameters and the mean snow depth as a climate indicator variable.
We derived the subgrid parameterization from highly resolved snow depth data
sets in large areas gathered at the peak of winter.
Snow-cover depletion curves derived assuming normally distributed
snow depth and homogeneous melt via Eq. () as function of mean
HS normalized with the peak of winter mean snow depth
HS0 (indicated by the subscript 0) (solid lines). Dashed lines
represent parameterized fractional SCA f via Eq. (). Coefficient of
variations CV vary between 0.06 (upper left) and 1 (lowest
one).
Error in fractional SCA f between measured fm and parameterized f
(Eq. ) as function of the domain size L for all three areas.
(a) Parameterized using σHS from Eq. (),
(b) parameterized using σHS from and
(c) parameterized using σHS from . Mean
values are indicated by squares. Error bars show the standard deviation of the
error per L. NRMSEs are given for each parameterization.
Investigating a spatial distribution entails studying the distribution
parameters, mean and standard deviation. Furthermore, measured mean and
standard deviation of snow depths require to be analyzed as a function of
scale in order to reveal the scale at which the dominant shaping processes
can be reliably parameterized, i.e., when small-scale snow depth variations
are no longer resolved. We performed a scale dependent analysis by
creating data sets from randomly selecting differently sized squared domain
sizes L (equivalent to a coarse grid cell size of a large-scale model)
ranging from 50 m to 3 km within our three large areas with measured snow
depths. To ensure that local anomalies are eliminated we chose 50
realizations for each domain size. Evaluating the resultant snow depth
distributions, we found more unimodal distributions, including snow-free
values, the larger the domain size (Fig. ). While for small L a
gamma distribution best described the measured snow depth distributions, for
L≥500 m, a normal distribution showed similar or even better performance
(Fig. ). We therefore conclude that over alpine terrain, in
large-scale grid cells, the snow depth distribution can be approximated by a
simple normal distribution. We also found a strong dependency of the
distribution parameter, the standard deviation of snow depth
σHS, as a function of coarse grid cell size L for domain
sizes L≤ 1 km in each of the three data sets separately (Fig. ).
This indicated that there should be a lower limit for large-scale
grid cell sizes to minimize scatter, which we suggest being ≥ 1 km,
similar to . A scale analysis of domain-averaged snow
depth values with domain-averaged terrain parameters revealed similar trends
and magnitudes with terrain characteristics for each of the three catchments
(Fig. and Table ). Scattering within a domain size L
consequently decreased with increasing L. We therefore concluded that a
parameterization using terrain parameters can possibly predict the standard
deviation of snow depth at the peak of winter. Furthermore, this allowed us to
create a pooled data set in order to derive a subgrid parameterization
independently of one geographic region. Note that, despite similar trends
between the three catchments, the scatter in the standard deviation of snow
depth σHS varied, for which we assume two reasons. First,
there was increased overlap of the randomly picked domains in the smaller
catchments of Wannengrat area and Val de Núria (cf. Fig. ).
Second, an overall larger scatter in the Dischma area data set might stem
from a larger flight height resulting in higher measurement uncertainties
which was, however, necessary due to local topographic features.
We developed a subgrid parameterization of snow depth distributions based on
spatial snow depth data sets acquired by aerial imagery and photogrammetric
image correlation techniques. Even though measurement errors can reach up to
33 cm cf. compared to small-scale modelings of
spatial snow depths, which require detailed input data and which sometimes
even rely on parameterizations, errors are clearly defined. Three snow depth
data sets from large, alpine areas were analyzed to develop the subgrid
parameterization of snow depth distributions. Two areas were located in
eastern Switzerland and one catchment in the eastern part of the Spanish
Pyrenees showing a somewhat dryer snow climate than the other areas (Fig. ).
We focussed on developing a subgrid parameterization for the
standard deviation of snow depth σHS independent of one
specific geographic area or winter season. For this, we introduced the mean
snow depth as a climate indicator variable. By analyzing flat field snow
depth measurements, gathered at the peak of winter, we found that the mean of all
average flat field snow depth measurements in a domain size L was
approximately linearly correlated with the mean snow depth in the same L
(Fig. ). This was especially true for domain sizes L larger than
about 1.5 km. It also has the interesting practical advantage that deriving
the mean snow depth for a large domain at the peak of winter can be conducted by
measuring snow depths on several flat field sites which are representative
for a specific geographic region. Since measuring snow depths on a few flat fields within each domain size for large-scale models covering a
wide area
is generally not feasible, we suggest that those can be replaced by an
automated flat field measurement, showing good climate representativeness for
the corresponding large-scale domain size L. Though the linear relationship
might have to be further verified in other geographic catchment areas and
during other seasons, using measured flat field snow depths as an easily
accessible climate descriptor allows one to develop a parameterization for the
standard deviation of snow depth independently of its geographical region.
The three snow depth data sets were gathered in two different winters, each
time at approximately the peak of winter (Fig. ). Until now we do not have
measurements during other seasons and a re-evaluation of the subgrid
parameterization for the standard deviation of snow depth
σHS (Eq. ) during other seasons might be
necessary. However, in principle, using the mean snow depth as a climate
indicator variable, Eq. () should also capture seasonal
differences.
To relate snow depth distributions, measured at the peak of winter, to terrain
characteristics we chose Gaussian statistics to approximate slope
characteristics of real summer topographies. Assuming that real topographies
can be described by a Gaussian covariance cf.
topography is reduced to two underlying characteristic length scales, namely,
a typical height of topographic features σ (standard deviation of the
summer DSM σz) and a typical width of topographic features
ξ. From these we computed the L/ξ ratio indicating how many
topographic features are included in a domain size L as well as a terrain
parameter μ, which is related to the mean squared slope (Eq. ).
Before deriving the terrain parameters we linearly detrended the summer DSM
to reveal the correct characteristic terrain length associated with the
shaping process of the snow depth distribution at the corresponding scale.
Detrending all summer DSMs then resulted in reasonably large L/ξ ratios
ranging from 2.7 to 15 for all domain sizes L. Without detrending overall
smaller L/ξ ratios prevailed, with the smallest L/ξ ratio of 1.7 for
L=50 m. In grid cells with small ratios the relevant shaping processes
might not be accurately resolved, and a subgrid parameterization could be
flawed. Domain averages were built by spatially averaging the data in a
domain size.
Overall, our subgrid parameterization for the standard deviation of snow
depth σHS (Eq. ) describes measured snow
distributions in the three different alpine areas very well (Fig. a).
As expected, the accuracy of parameterized σHS increased
with increasing domain size L (Fig. a). This is partly because at
small scales the shaping processes are more diverse (or random) which are,
however, smoothed out at larger scales, here for L≥ 1 km. The
parameterization in Eq. () describes the processes dominating
at larger scales. On the other hand, the accuracy of the subgrid
parameterization of σHS also increases with increasing
L/ξ ratios, i.e., the subgrid topographic features and their impact on
snow depth distributions are represented more accurately. In the following we
discuss the three scaling parameters in the subgrid parameterization of
σHS. First, it includes a terrain parameter, related to
the mean squared slope μ (Eq. ), describing the influence of
topography due to varying incident shortwave radiation and precipitation.
This terrain parameter was motivated by the result that we consistently
obtained the largest correlation coefficients for σHS with
μ (cf. Table ). For now, we assume that parameterized
σHS approaches zero for mean squared slopes μ of zero.
Even though, mean slope angles of all domains range from 2 to
58∘, the lowest domain-averaged slope angles only coincide with the
smallest domain sizes. Equation () can be extended for large-scale
grid cells showing slopes of zero, once the necessary snow depth data become
available. As for the second scaling parameter, the parameterization includes
the L/ξ ratio, a correction term for finite grid sizes which can show a
range of correlation lengths of subgrid topographic features ξ
cf. that might or might not be captured by the
domain size L. As a consequence of the overall good agreement of the pdf of
snow depths with a normal distribution at larger scales we used a Gaussian
factor e-(ξ/L)2. The Gaussian factor also follows from the assumption
that topography has a large impact on the snow distribution in large-scale
grid cell sizes and from the previously found Gaussian slope characteristics
for slope characteristics of real topographies cf..
The third parameter in the σHS parameterization includes
the mean snow depth which accounts for climate or seasonal differences.
Since the snow depth data sets were only acquired at approximately the peak of
winter slight hysteresis phenomena of the alpine, seasonal snow depth
distribution were introduced (cf. Fig. a). With
snow depth data gathered exactly at the peak of winter, constant parameters a and
b in Eq. () might change but overall errors are expected to
decrease. Note that we optimized a and b in Eq. () with a
nonlinear regression analysis. Our parameterization performed better than
previously published parameterizations for σHS, which did
not account for subgrid topography (Fig. b and c). Since the averaged
coefficient of variation for snow depth CV of all domain sizes in our
catchments of 0.63 resembles the one for alpine tundra of 0.43 which
used in a linear relationship, this parameterization
shows an overall better performance among the tested parameterizations (Fig. ).
By employing the new subgrid parameterization for the standard deviation of
snow depth σHS we developed a parameterization for the
fractional SCA over complex mountainous terrain
(Eq. ). For this large-scale model application we re-evaluated a
previously presented functional closed form for homogeneous landscapes
. To obtain a parametric fractional SCA f (Eq. ) we
similarly integrated the snow distribution assuming uniform melt but started
from a normal snow depth distribution (Fig. ). Fitting the resultant
parametric f (Eq. ) we obtained the same functional closed form
fit as which is proportional to tanh (Eq. ).
We assume that the slightly differing pre-factor stems from our
broader range for CV stretching from 0.06 to 1 compared to the one used by
with CV values from 0.1 to 0.5. We stress that the
parameterization for σHS (Eq. ) as a function of
terrain characteristics coincides well with previously presented dependencies
of f on terrain parameters such as the roughness length of the surface
or the standard deviation of the summer DSM
. Overall, we found decreasing errors between
parameterized and measured f, for our three areas at the peak of winter, with
increasing domain size L with the largest errors being below 10 % (Fig. a).
When applying previously derived parameterizations for
σHS we also found decreasing errors between parameterized
and measured f with increasing L. However, we obtained overall larger
errors and errors did not approach zero for the largest domain sizes
L≥1750 m (Fig. b and c). We emphasize that applying
Eq. () with parameterized σHS leads to a
NRMSE of only 4 % more than when applying measured
σHSm in Eq. (). Note that in line with
replacing exhausting snow depth measurements in large domain sizes L by
parameterized σHS via Eq. (), we investigated
the increase of error in the parameterization for f when applying averaged
flat field snow depths instead of mean snow depth HS. Applying
measured HSflat instead of mean snow depth
HS, but using measured snow depth distribution
σHSm, in the SCA parameterization
increased the NRMSE only by about 7 %.
We believe that the parameterization for fractional SCA is also applicable during the
accumulation or melt season, during other winters and in a different
geographic region. However, our assumption requires verification once
highly resolved spatial snow depth data become available, preferably in
different snow climates, at times other than at the peak of winter, and from less
topographical influenced regions. By performing snow depths measurements over
several winter seasons, persistent snow depth distributions at the peak of winter
were already found . These findings
suggest that our parameterization for σHS should be
applicable during other winters, and motivates to investigate the evolution
of spatial distributions of snow depth throughout the (melting) season.
Regarding grid cell size, horizontal resolutions of large-scale
meteorological and regional climate models can be much larger than our
largest tested grid cell size of 3 km. However, at these larger scales, the
presented parameterizations should also be applicable. To mimic the dominant
snow-cover shaping processes in a domain size L, the domain size has to be
substantially larger than the subgrid topographic correlation length ξ,
i.e., L≫ξ. Note that every grid cell shows a different terrain
correlation length ξ due to different subgrid topographies. A reliable
subgrid parameterization for σHS (Eq. ) was
therefore derived by including a scaling parameter that corrects for finite
L/ξ ratios as in.
We summarize that the subgrid parameterization for σHS
depends on both terrain length scales and on mean snow depth, which allowed
developing a parameterization for the SCA over complex topography independent
of a specific geographic region. A parameterization for the SCA over
mountainous terrain has several practical applications. For instance, from
SCA surface albedos can be derived to improve radiation balance estimates in
large-scale meteorological models. Moreover, a SCA parameterization can be
used to improve simulations of averaged snowmelt fluxes in large grid cells,
which is a relevant issue for flood warnings.