Fractional snow-covered area (SCA) is a key parameter in large-scale hydrological, meteorological and regional climate models. Since SCA affects albedos and surface energy balance fluxes, it is especially of interest over mountainous terrain where generally a reduced SCA is observed in large grid cells. Temporal and spatial snow distributions are, however, difficult to measure over complex topography. We therefore present a parameterization of SCA based on a new subgrid parameterization for the standard deviation of snow depth over complex topography. Highly resolved snow depth data at the peak of winter were used from two distinct climatic regions, in eastern Switzerland and in the Spanish Pyrenees. Topographic scaling parameters are derived assuming Gaussian slope characteristics. We use computationally cheap terrain parameters, namely, the correlation length of subgrid topographic features and the mean squared slope. A scale dependent analysis was performed by randomly aggregating the alpine catchments in domain sizes ranging from 50 m to 3 km. For the larger domain sizes, snow depth was predominantly normally distributed. Trends between terrain parameters and standard deviation of snow depth were similar for both climatic regions, allowing one to parameterize the standard deviation of snow depth based on terrain parameters. To make the parameterization widely applicable, we introduced the mean snow depth as a climate indicator. Assuming a normal snow distribution and spatially homogeneous melt, snow-cover depletion (SCD) curves were derived for a broad range of coefficients of variations. The most accurate closed form fit resembled an existing fractional SCA parameterization. By including the subgrid parameterization for the standard deviation of snow depth, we extended the fractional SCA parameterization for topographic influences. For all domain sizes we obtained errors lower than 10 % between measured and parameterized SCA.

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

A few studies previously tackled subgrid snow distributions.

Numerous studies analyzed catchment snow depth distributions by relating
measured snow depth data to small-scale terrain parameters

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,

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

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 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.

To account for the influence of different climates on the spatial snow
distribution, we used snow depth data from three large alpine areas in two
distant geographical regions. Two alpine areas, called Wannengrat and
Dischma, are located in eastern Switzerland around Davos (Fig.

The third alpine catchment, called Val de Núria, is located in the
eastern part of the Spanish Pyrenees (Fig.

For the Wannengrat and Dischma sites, spatial snow depth data were obtained
using an opto-electronic line scanner (Sensor ADS80, Leica Geosystems)
mounted on a plane. Photogrammetric image correlation techniques were applied
for summer and winter aerial imagery to calculate
DSMs in 2 m horizontal resolution

For the Val de Núria site, point clouds of snow depth values were
obtained by ALS measurements

For Wannengrat and Dischma, we neglected all measurements that coincided with
trees, buildings, rivers and glaciers. Negative snow depth values were set to
zero. In total we obtained about

Maps of

Probability density functions (pdf) of measured snow depths are shown for the three areas.

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

For building domain averages, all data points were spatially averaged in a
domain size

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

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

One example probability density function (pdf) of measured snow
depths HS for each domain size

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

Mean root mean square errors (RMSE) between theoretical probability
density functions (pdf) and measured pdfs as function of domain size

We found mostly unimodal distributions of snow depths in all domain sizes

Standard deviation of snow depth

Pearson correlation coefficients

We analyzed our ensemble of snow depth data grids to relate mean and standard
deviation of each snow depth distribution, HS and

Standard deviation of snow depth

We found weaker correlations between mean snow depth HS and
terrain parameters, than between

Measured mean snow depth HS as function of mean measured
flat field

In order to develop a parameterization for

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

Measured standard deviation of snow depth

The mean snow depth HS is obtained from

We followed the procedure of

We extend the fractional SCA

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. (

Error in fractional SCA

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

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

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

Overall, our subgrid parameterization for the standard deviation of snow
depth

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

By employing the new subgrid parameterization for the standard deviation of
snow depth

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

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

We summarize that the subgrid parameterization for

We thank the two reviewers, Juraj Parajka and Davide Bavera for helpful comments on the manuscript. Furthermore, we acknowledge Henning Löwe for valuable discussions and Yves Bühler and Thomas Grünewald for data preparation. This study was partly funded by the Federal Office of the Environment FOEN. Edited by: C. De Michele