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\begin{document}
\volume{54}
\issueno{}
\pubyear{2009}
  
\title[Virutal network for estimating daily new snow water equivalent]{A
virtual network for estimating daily new snow water equivalent and snow
depth in the Swiss Alps}
\author[Egli and others]{Luca EGLI,$^1$\footnote{Present address:
  WSL Institute for Snow and Avalanche Research SLF,  Fl\"{u}elastrasse
11, 7260 Davos-Dorf, Switzerland}~ Tobias JONAS$^1$ and Jean-Marie
BETTEMS$^2$}
  \affiliation{%
  $^1$WSL Institute for Snow and Avalanche Research SLF, Davos,
Switzerland\\
  E-mail: egli@slf.ch\\
    $^2$Federal Office of Meteorology and Climatology MeteoSwiss,
Z\"urich, Switzerland}
  
\abstract{Daily new snow water equivalent (HNW) and snow depth (HS) are of
significant practical importance in cryospheric sciences such as snow
hydrology and avalanche formation. In this study we present a virtual
network (VN) for estimating HNW and HS on a regular mesh over Switzerland
with a grid size of 7\,km. The method is based on the HNW output data of
the numerical weather prediction model (COSMO-7), \textbf{[AUTHOR: IS
COSMO an acronym? Please define]} driving an external accumulation/melting
routine. The verification of the VN shows that, on average, HNW can be
estimated with a mean systematic bias close to 0 and an averaged absolute
accuracy of 4.01\,mm. The results are equivalent to the performance
observed when comparing different automatic HNW point estimations with
manual reference measurements. However, at the local scale, HS derived by
the VN may significantly deviate from corresponding point measurements. We
argue that the VN presented here may introduce promising cost-effective
options as input for spatially distributed snow hydrological and avalanche
risk management applications in the Swiss Alps.
} 
\maketitle
  
\section{Introduction}
In cryospheric sciences such as snow hydrology, avalanche
formation/dynamics and snow climatology, daily new snow water equivalent
(HNW, mm) and snow depth (HS, m) are important measurement categories.
Together with wind speed, HNW is of significant practical importance for
avalanche hazard estimation \citep{mcclung93,egli08} and represents a key
input parameter for spatially distributed snow models
\citep[e.g.][]{lehning06,liston06}. The monitoring and modelling of water
resources, represented by HS (respectively, SWE), \textbf{[AUTHOR: What is
SWE? (snow water equivalent?) Is HNW meant? Please clarify!]} is the basic
aim of snow hydrology \citep[e.g.][]{grayson00,anderton02}. In particular,
estimates of the spatial and temporal distribution of HS (or SWE)
\textbf{[AUTHOR: See previous comment]} is essential
\citep{luce98,skaugen07} for water resources management
\citep[e.g.][]{schaefli07} and flood prevention.

Since the early 1990s, \textbf{[AQ: Have reworded; please confirm
correct]} various expensive networks of manual and automatic point HS (and
occasionaly HNW) measurements have been set up in the Swiss Alps. Automatic
methods have the advantage of providing data at high temporal resolution in
terrain that is difficult to access. \cite{durand93} presented a first
analysis method for relevant meteorological parameters for snow models,
while \cite{eglietal09} investigated different methods to reveal a general
feasibility benchmark in automatic estimation of HNW.

In this study, we present a virtual network (VN) to estimate HNW and HS on
a regular grid (grid size 7\,km) covering the entire Swiss Alps. The method
is based on the HNW output data of the numerical weather prediction model
(COSMO-7). \textbf{[AUTHOR: Is COSMO an acronym? Please define]} The raw
snow precipitation output of COSMO-7 drives a simple external
accumulation/melting routine that allows HNW and HS to be computed for
each grid cell. This routine does not account for individual physical
processes (e.g. settling of snow cover and fresh snow). Instead, it
evaluates bulk accumulation and melting rates based on COSMO-7 HNW data
calibrated using observed HS data from existing snow monitoring networks. 

We believe that the VN presented here may introduce promising
cost-effective options as input for spatially distributed snow
hydrological models and avalanche risk management applications. If a
numerical weather model and HS point measurement stations are available, a
VN network can be developed for other mountainous regions that are not as
densely instrumented as the Swiss Alps.

\section{Data}\label{Data}
\subsection{COSMO-7 model output data}
A numerical weather prediction system is operated by MeteoSwiss
(www.meteoswiss.ch) for a wide range of applications. This system is based
on the COSMO-7 model (www.cosmo-model.org), which is a primitive equation
model with {non-hydrostatic, fully compressible dynamics}. The prognostic
variables include precipitation (separated into snow and rain) and many
other meteorological parameters. COSMO-7 is used in two modes: (1) a free
forecast mode, where the temporal evolution of the atmosphere is computed
without any constraints other than the lateral boundary conditions from
the driving model (European Centre for Medium-range Weather Forecasts,
www.ecmwf.int/research), and (2) an assimilation mode to produce the best
gridded representation of the atmosphere by blending available current
observations with the model dynamic (the so-called data assimilation
process). Note that no observed precipitation is assimilated in COSMO-7.
The data assimilation process is not a simple interpolation of available
observations, but instead produces a consistent \textbf{[AUTHOR: Is
continuous (or constant) meant?]} state of the full atmosphere.

In this study we have chosen to use data from the assimilation mode of
COSMO-7 between September 2005 and February 2008. For each day, snow
precipitation (i.e. solid precipitation only) cumulated from 0000\,UTC to
2400\,UTC for the automatic reference stations (and from 0800\,UTC to
0800\,UTC on the following day for the manual reference stations)
\textbf{[AUTHOR: Is this correct?]} have been extracted from COSMO-7 and
used to derive $\mathrm{HNW_{VN.RAW}}$ at the surface. Introduced
abbreviations are summarized in Table~\ref{tab1}. 

%table 1 here

\begin{table}%table1
\centering
\caption{Contingency table for the validation of the HNW estimations of
the virtual network. The point measurements at the control sites
$\mathrm{HNW_{MEAS}}$ are observed and the estimations of the virtual
network $\mathrm{HNW_{VN}}$ are forecast}\label{tab1}
\begin{tabular}{@{}lcc}\hline
 & \multicolumn{2}{c}{$\mathrm{HNW_{MEAS}}$ (observed)}   \\ 
$\mathrm{HNW_{VN}}$ (forecast) 
  & NO
  & YES\\ \hline
NO   & a & b\\
YES & c & d\\ \hline
\end{tabular}
\end{table}

\subsection{Point measurements}
%figure 1 here

\begin{figure*}%fig1
\centering
\includegraphics[width=90mm, angle=-90]{fig_1.ps}
\caption{Measurement stations: 141 point measurements of snow depths in
the Alpine region over Switzerland. The stations are located at altitudes
800--3130\,m\,a.s.l., where the three different elevation zones are
indicated with squares, points and triangles. Reference stations are
marked with a cross.}\label{fig1}
\end{figure*} 

For this study, 141 stations spread over the entire range of the Swiss
Alps (Fig.~\ref{fig1}) were available to provide daily measurements of HS
during the period September 2005 to February 2008. The sites have been
carefully selected in flat open terrain, with as little wind influence as
possible, to ensure a regional representativeness of the HS/HNW estimation
\citep{egli08}. In order to derive HNW from HS measurements, we used a
simple parameterization proposed by \cite{eglietal09}:
\begin{eqnarray}
&& \mathrm{HNW_{MEAS}}(\delta \mathrm{HS_{24h}})\,(\mathrm{mm}) \nonumber
\\
&=& \begin{cases} 1+1.09\,\delta \mathrm{HS_{MEAS.24h}}\,(\mathrm{cm}) &
\text{for~} \delta \mathrm{HS_{MEAS.24h}} >0 \\ 
0 & \text{for~} \delta \mathrm{HS_{MEAS.24h}} \leqslant 0,
\end{cases}\label{eqn1}
\end{eqnarray}
where $\delta \mathrm{HS_{MEAS.24h}}$ denotes the 24\,h difference between
an HS reading (measured here in centimetres) and the respective
observations from the previous day. Possible limitations of using
Equation~(\ref{eqn1}) will be discussed later. \textbf{[AUTHOR: Please
provide section number]} Note that for automatic measurements (for ENET
and IMIS \textbf{[AUTHOR: Please define acronyms]} network, see
\citealp{rhyner02}), HS estimations were provided at 0000\,am while manual
HS readings (for observations, see \citealp{marty08}) at around 0800\,am.
The values of HS were later checked for plausibility and single
missing/faulty values were corrected manually by interpolation. Finally,
HNW was only calculated for the period 1 November--30 April for each
season (analogous to \citealp{eglietal09}), while HS is considered during
the entire seasons of 2005--08 if HS$>$0.

\section{Methodology} 
\subsection{Virtual network (VN)}
\subsubsection{HNW estimated by VN}
The cumulated snow precipitation output of COSMO-7
($\mathrm{HNW_{VN.RAW}}$) constituted the basis of the VN. Thirty-five
snow stations (of 141) were set aside for adjusting
$\mathrm{HNW_{VN.RAW}}$ to measured HS ($\mathrm{HNW_{MEAS}}$). These
stations are referred to as reference stations (Fig.~\ref{fig1}, crosses),
while the remaining 106 stations are referred to as control stations. All
stations were separated into three different elevation bands
(800--1500\,m, 1501--2200\,m and 2201--3200\,m; see legend of
Fig.~\ref{fig1}). 

Reference stations were selected to cover all regions and elevation bands
approximately evenly, in the horizontal as well as in the vertical space.
COSMO-7 mesh points were assigned to corresponding snow stations by
searching the grid point with centre nearest to the station. The
horizontal station coordinates deviated about $\pm$2.7\,km in average
(maximum $7\,\mathrm{km}\times \sqrt{2}/2=4.9$\,km) from the centre of
$\mathrm{HNW_{VN.RAW}}$, while the vertical difference between stations
and respective $\mathrm{HNW_{VN.RAW}}$ grid cells deviated by about
$\pm$250\,m \citep{schaub07}. The VN therefore covers an area of
approximately 7\,km\,$\times$\,7\,km on the surface (horizontal). In
comparison with the point measurement stations, a vertical distention
\textbf{[AUTHOR: Is resolution meant? accuracy?]} of about 500\,m is
provided.

The correction routine for $\mathrm{HNW_{VN.RAW}}$ first identified
possible outliers of $\mathrm{HNW_{VN.RAW}}$ if $\mathrm{HNW_{VN.RAW}}$
exceeded the maximum of $\mathrm{HNW_{MEAS}^{r}}$ at the reference
stations (index r). Outliers were replaced by the maximum of
$\mathrm{HNW_{MEAS}^{r}}$ for each elevation band. Secondly, a correction
matrix $\mathrm{CorrMat^{r}}$ was calculated from the difference between
$\mathrm{HNW_{MEAS}}$ and $\mathrm{HNW_{VN.RAW}}$ for each reference
station and for each elevation band:
\begin{equation}
\mathrm{CorrMat^{r}} = \mathrm{HNW_{MEAS}^{r}} -
\mathrm{HNW_{VN.RAW}^{r}}.
\end{equation}
For all remaining locations (index c) of the respective elevation band,
the values of the correction matrix were then interpolated by a inverse
distance interpolation:
\begin{equation}
\mathrm{CorrMat^{c}} = \frac{\sum_{i=1}^{N}w_{i}\times
\mathrm{CorrMat^{r}}_{i}}{\sum_{i=1}^{N} w_{i}},
\end{equation}
where
\begin{equation}
w_{i} = \frac{1}{d(|r_{i}-c|)}
\end{equation}
is the inverse distance weighting function. $d$ represents the Euclidean
metric distance operator and $N$ is the total number of reference
locations ($N=35$ here). The corrected COSMO-7 HNW estimation for the
control stations ($\mathrm{HNW_{VN.CORR}^{c}}$) were then defined by:
\begin{equation}
\mathrm{HNW_{VN.CORR}^{c}} = \mathrm{CorrMat^{c}} +
\mathrm{HNW_{VN.RAW}^{c}}
\end{equation}
where negative values of $\mathrm{HNW_{VN.CORR}^{c}}$ were set to zero.

\subsubsection{HS estimation by the VN}
HS is basically calculated from the summation of HNW during the
accumulation period, while a simple melting procedure is used to
calculated HS during the ablation period. To calculate a temporary term
$\mathrm{TEMP.\mathrm{HS_{VN}^{cr}}}(t)$ for the COSMO-7 grid points, the
values of $\mathrm{HNW_{VN.CORR}^{cr}}$ were added iteratively day by day:
\begin{equation}
\mathrm{TEMP.\mathrm{HS_{VN}^{cr}}}(t) = \mathrm{HNW_{VN.CORR}^{cr}}(t) +
\mathrm{HS_{VN}^{cr}}(t-1).
\end{equation}

Note that instead of a simple cumulation of
$\mathrm{HNW_{VN.CORR}^{cr}}(t)$ for each day, this iterative procedure
also allows HS to be determined for snowfall events during the melting
period. However, at this point $\mathrm{TEMP.\mathrm{HS_{VN}^{cr}}}(t)$
does not indicate a real snow depth measure. In order to convert
$\mathrm{TEMP.\mathrm{HS_{VN}^{cr}}}(t)$ to corresponding HS data, we
employed data from the reference stations to calibrate
$\mathrm{TEMP.\mathrm{HS_{VN}^{cr}}}(t)$ using a simple linear
parameterization:
\begin{equation}
\mathrm{HS_{VN}^{r}}(t) = \mathrm{TEMP}\times \mathrm{HS_{MEAS}^{r}}(t)
\times \mathrm{slope} + \mathrm{intercept}.\label{eqn7}
\end{equation}
Again, such a parameterization was obtained for each elevation band
separately. The resulting parameters of Equation~(\ref{eqn7}) (slope and
intercept) were subsequently used to convert TEMP.HS to HS at the control
stations:
\begin{equation}
\mathrm{HS_{VN}^{c}}(t) = \mathrm{TEMP}\times \mathrm{HS_{VN}^{c}}(t)
\times \mathrm{slope} + \mathrm{intercept}.
\end{equation}

The above procedure was applied for all days (considered here as the
period of accumulation) except during ablation, which is defined as all
days if $\mathrm{mean(HS_{MEAS}^{r}}(t)) -
\mathrm{mean(HS_{MEAS}^{r}}(t-4)) < 0$ and
$\mathrm{mean(HS_{MEAS}^{r}}(t)) < 0.75 \times
\max(\mathrm{mean(HS_{MEAS}^{r}}(t)))$. Note that this criterion does not
take into account any information about the physical processes of melting
(e.g. liquid water content). It approximately defines the point where the
mean snow depth at the reference stations for each elevation band
decreases due to melting conditions \citep[c.f.][]{egli09}.
In case of melting conditions, we calculated a melting matrix from
$\mathrm{HS^{r}_{MEAS}}(t)- \mathrm{HS^{r}_{MEAS}}(t-1)$ for every
elevation band at the reference stations:
\begin{equation}
\mathrm{MeltMat^{r}}(t) = \mathrm{HS^{r}_{MEAS}}(t)-
\mathrm{HS^{r}_{MEAS}}(t-1).
\end{equation}

We calculated melt rates for all remaining locations using the inverse
distance weighting method, in the same way as for Equations~(3) and (4):
\begin{equation}
\mathrm{MeltMat^{c}}(t) = \frac{\sum_{i=1}^{N}w_{i}\times
\mathrm{MeltMat^{r}}_{i}(t)}{\sum_{i=1}^{N} w_{i}},
\end{equation}
where
\begin{equation}
w_{i} = \frac{1}{d(|r_{i}-c|)}.
\end{equation}

Finally, HS (for melting conditions) was derived for the control stations
as
\begin{equation}
\mathrm{HS_{VN}^{c}}(t) = \mathrm{MeltMat^{c}}(t) +
\mathrm{HS_{VN}^{c}}(t-1).
\end{equation}

Note that at the very end of the winter season the reference stations
would eventually meltout, preventing any melting rates from being derived.
In this case, the final melting rates before meltout have been used for the
interpolation. In addition, negative values of $\mathrm{HS_{VN}}$ have been
set to 0.

\subsection{Parameters of verification}\label{sect3.2}
\subsubsection{HNW verification}
In order to validate the performance of $\mathrm{HNW_{VN}}$ at the control
stations (in the following the index $c$ is not included) by comparing to
the point measurements $\mathrm{HNW_{MEAS}}$,  and to compare the results
to the performance of other automatic methods, parameters of comparison
were used \citep{egli09}. The parameters are briefly summarized below.

%table 2 here

\begin{table*}%table2
\centering
\caption{Comparative statistics to assess the performance of VN relative
to competing methods tested in \cite{eglietal09}; notation is described in
section~\ref{sect3.2} \textbf{[AUTHOR: Will the reader know what all these
methods are/how acronyms are defined?]}}\label{tab2}
\begin{tabular}{lcccccc}  \hline
Method & $n_\mathrm{stations}$ & $n_\mathrm{days}$ & $\overline{\delta
\mathrm{HNW}}$& $\sigma({\delta \mathrm{HNW}})$ & $R^{2}_{\log}$ & Ranking
points \\ 
\vspace{4.5pt} \\
 & & & mm & mm & \\ \hline
{\bf VN.CORR} & {\bf 106} & {\bf 484} & {\bf --0.03 ($\pm$ 0.35)} & {\bf
4.01 ($\pm$ 0.62)} & {\bf 0.78 ($\pm$ 0.037)} & {\bf 51} \\  
GAUGE & 1 & 725 & --1.15 & 4.12 & 0.89 & 50 \\ 
SNOWPILLOW & 1 & 719 & 0.6 & 4.41 & 0.79 & 49 \\ 
SNOWPACK & 1 & 726 & 0.6 & 5.07 & 0.82 & 48 \\   
SIMPLE-HNW & 1 & 723 & --0.68 & 5.04 & 0.8 & 45 \\  
{\bf VN.RAW} & {\bf 106} & {\bf 484} & {\bf 0.22 ($\pm$ 0.47)} & {\bf 4.89
($\pm$ 0.65)} & {\bf 0.72 ($\pm$ 0.035)} & {\bf 30} \\ 
COSMO forecast & 1 & 722 & 0.09 & 6.59 & 0.74 & 30 \\  
SNOWPOWER & 1 & 348 & 2.59 & 15.37 & 0.41 & 5 \\ \hline 
\end{tabular}
\end{table*}

\begin{enumerate}
\item Systematic bias $\overline{\delta \mathrm{HNW}}$:
\begin{equation}
\overline{\delta \mathrm{HNW}} = \mathrm{mean(HNW_{VN}-HNW_{MEAS})}.
\end{equation}

\item The absolute accuracy $\sigma(\delta \mathrm{HNW})$:
\begin{equation}
\sigma(\delta \mathrm{HNW}) =
\mathrm{standard~deviation(HNW_{VN}-HNW_{MEAS})}.
\end{equation}

\item $R^{2}_{\log}$: The coefficient of correlation ($R^{2}_{\log}$)
between log-transformed $\mathrm{HNW_{VN}}$ and $\mathrm{HNW_{MEAS}}$.

\item POD and FAR: Parameters adopted from severe weather forecast theory
\citep{murphy86} have been used to detect certain classes of snowfall
events, namely the Probability of Detection (POD) and the False Alarm Rate
(FAR). To calculate POD and FAR, we used contingency tables
(Table~\ref{tab2}) for four classes of HNW intensity (Snow-NoSnow: 
$\mathrm{HNW_{MEAS}}\geqslant 0$\,mm  and $<1$\,mm; Low:
$\mathrm{HNW_{MEAS}}\geqslant 1$\,mm and $< 15$\,mm; Medium:
$\mathrm{HNW_{MEAS}}\geqslant 15$\,mm and $<30$\,mm; High:
$\mathrm{HNW_{MEAS}}\geqslant 30$\,mm).

The POD and FAR were calculated for every class using
\begin{equation}  
   \mathrm{POD} = \frac{d}{b+d}
\end{equation}
and
  \begin{equation}  
   \mathrm{FAR} = \frac{c}{c+d},
\end{equation}
where $b, c$ and $d$ were derived from the contingency table
(Table~\ref{tab2}).

\item Ranking points: A ranking point scale for the overall assessment
relative to other HNW estimation methods was tested in \cite{eglietal09}.
For each comparative measure as described above, the best performance was
rated with 7 points and the lowest performance with 0 points. The
different ranking points for each parameter and class were added.
\end{enumerate} 

\subsubsection{HS verification}
HS data from the VN ($\mathrm{HS_{VN}}$) was validated against
$\mathrm{HS_{MEAS}}$ by applying the following parameters of comparison
\citep{egli09}, restricted to the control stations.

\begin{enumerate}
\item Systematic bias ($\alpha$): The percent systematic bias $\alpha$ was
determined by a least squares fit to
 \begin{equation} 
\mathrm{HS_{VN}} = \alpha\times \mathrm{HS_{MEAS}},
\end{equation} 
where the standard error of $\alpha$ was also determined
($\mathrm{error}_{\alpha}$). Here, $\mathrm{error}_{\alpha}$ is a measure
of the spread of the data around the fitting line.

\item Absolute error (RMSE): The averaged absolute difference between
$\mathrm{HS_{VN}}$ and $\mathrm{HS_{MEAS}}$, the root mean squared error
(RMSE), was included in the comparison statistics:
 \begin{equation}
\mathrm{RMSE} =  \sqrt{\frac{1}{N}\sum_{i=1}^{N}\left(\mathrm{HS_{VN}}^i-
\mathrm{HS_{MEAS}}^i\right)^2}
\end{equation} 
where $i$ indexes one day of the analysed periods if either
$\mathrm{HS_{VN}}^i>0$ or $\mathrm{HS_{MEAS}}^i>0$. Since HS is
investigated instead of SWE \citep{eglietal09}, a direct comparison of
these studies by a ranking point score is not appropriate.
\end{enumerate} 

\section{Results and discussion}
\subsection{HNW}\label{sect4.1}
Results of the validation of $\mathrm{HNW_{VN}}$ and $\mathrm{HNW_{MEAS}}$
are listed in Table~\ref{tab3} and presented in Figure~\ref{fig2}. The
parameters of comparison were calculated for both $\mathrm{HNW_{VN.CORR}}$
and $\mathrm{HNW_{VN.RAW}}$ compared to $\mathrm{HNW_{MEAS}}$. For
reference, they are listed together with corresponding results of other
HNW estimation methods which were tested in \cite{eglietal09}. The
parameters for VN.CORR and VN.RAW are averaged over all 106 control
stations (Table~\ref{tab3}: $n_\mathrm{stations}$), where the lines below
and above the end of the vertical bars in Figure~\ref{fig2} (as well as
the numbers in brackets in Table~\ref{tab3}) represent $\pm$half of the
standard deviation.

%table 3 here

\begin{table}[h]%table3
\centering
\caption{Table of nomenclature}\label{tab3}
\begin{tabular}{ll} \hline 
{Abbreviation} & {Description}  \\ \hline  
HNW & Daily new snow water equivalent  \\ 
HS & Snow depth  \\ 
$\mathrm{HNW_{MEAS}}$ & Observed daily new snow water equivalent  \\ 
VN.RAW & Uncorrected COSMO-7 output  \\ 
VN.CORR & Corrected COSMO-7 (virtual network)\\ 
$\overline{\delta \mathrm{HNW}}$ & Systematic bias \\ 
$\sigma(\delta \mathrm{HNW})$ & Absolute accuracy \\ 
POD & Probability of detection \\ 
FAR & False alarm rate \\ \hline
\end{tabular}
\end{table}

%figure 2 here

\begin{figure*}%fig2
\centering
\includegraphics[width=90mm, angle=-90]{fig_2.ps}
\caption{POD and FAR values are presented for the four different classes
of HNW intensity (diamond, point, triangle, star).  The values are listed
either for single point measurements or the average of 106 control
stations, where the error bars denote the deviation from the
mean.}\label{fig2}
\end{figure*} 

Table~\ref{tab3} summarizes the parameters for HNW comparison
($\overline{\delta \mathrm{HNW}}$, $\sigma(\delta \mathrm{HNW})$ and
$R^2_{\log}$) \textbf{[AUTHOR: $R^2_{\log}$ not in Table 3]} in order of
the highest ranking points. While $\overline{\delta \mathrm{HNW}}<0$
indicates a general underestimation of $\mathrm{HNW_{VN}}$ to
$\mathrm{HNW_{MEAS}}$, $\overline{\delta \mathrm{HNW}} >0$ denote a
statistical overestimation. The results for VN.CORR and VN.RAW reveal a
systematic bias close to 0. However, the considerably large deviation
around the mean values of VN.CORR and VN.RAW indicates that some stations
exhibit a large under-/overestimation of $\mathrm{HNW_{VN}}$. 

A more detailed analysis of the frequency distribution showed that the
individual $\overline{\delta \mathrm{HNW}}$s at each control station are
approximately equally distributed around the mean value of --0.03\,mm.
This indicates an arbitrary character which stems from the comparison of
the point measurements $\mathrm{HNW_{MEAS}}$ with data representing a much
larger domain of estimation for $\mathrm{HNW_{VN.CORR}}$ or
$\mathrm{HNW_{VN.RAW}}$. This is supported by the fact that the stations
are randomly distributed in a cuboid of 7\,km\,$\times$\,7\,km by 500\,m
\citep{schaub07}, which also explains why the deviations are generally
similar for VN.CORR and VN.RAW. 

For VN.CORR, $\sigma(\delta \mathrm{HNW})$ demonstrates considerably small
values for the mean ($\sigma({\delta \mathrm{HNW}})=4.01$\,mm) with a
deviation from the mean of about $\pm$0.62. 
This low absolute error $\sigma({\delta \mathrm{HNW}})$ implies that, in
principle, $\mathrm{HNW_{VN.CORR}}$ can also be calibrated for each single
station of the measuring network. The systematic bias of each station can
therefore be reduced, resulting in a lower deviation ($\pm$) in
$\overline{\delta \mathrm{HNW}}$. However, due to the comparison of a
point measurement with a cuboid area, a calibration of
$\mathrm{HNW_{VN.CORR}}$ is not meaningful. In fact, the values of
$\mathrm{HNW_{VN.CORR}}$ may be considered as an average measurement of
HNW over a large area, therefore representing a complementary network to
the point measurements. 

Taking the performance of SNOWPACK for $\sigma({\delta \mathrm{HNW}})$ as
a benchmark, about 82\% of the 106 control stations of the VN exhibit a
$\sigma(\delta \mathrm{HNW})< 5.07$\,mm. The absolute accuracy between
$\mathrm{HNW_{MEAS}}$ and $\mathrm{HNW_{VN.CORR}}$ is therefore comparable
to the operational SNOWPACK-HNW estimations for manual reference
measurements investigated at one single location. Note that SNOWPACK
calculations are in operational use for Swiss avalanche warning
\citep{rhyner02} and provide different types of snowpack properties
\citep{lehning02}. However, they are restricted to the locations of the
IMIS network \citep{rhyner02}. In contrast, VN.CORR constitutes a
complementary network over the entire Swiss Alps (although restricted to
the areas of HNW and HS estimation).

The POD/FAR statistics show that VN.CORR performs for all classes of
intensities with better POD/FAR values than VN.RAW. Note that POD denotes
the percentage of $\mathrm{HNW_{MEAS}}$, also measured by
$\mathrm{HNW_{VN}}$, while FAR represents the percentage of
$\mathrm{HNW_{VN}}$ which are not observed by $\mathrm{HNW_{MEAS}}$.
VN.CORR has similar features to the POD/FAR statistics of SNOWPACK, in
particular for the two highest classes of intensities (Medium and High)
and for the Snow-NoSnow class, most important for avalanche risk
management decisions. \textbf{[AUTHOR: Have reworded; please confirm
correct]} The good performance of VN.CORR in POD/FAR statistics with
respect to the Snow-NoSnow class also implies that COSMO-7 is capable of
distinguishing between snow and rain. Note that results do not deteriorate
if evaluated for the three elevation bands separately (data not shown).

Furthermore, the calibration routine to derive VN.CORR implies that at the
reference station $\mathrm{HNW_{VN.CORR}} = \mathrm{HNW_{MEAS}}$,
independent of the processes leading to $\mathrm{HNW_{MEAS}}$ such as
combinations of solid, liquid precipitation and evaporation.

The low performance of VN.CORR in POD ($\sim$0.45) and FAR ($\sim$0.35)
for the Low class diverges from the SNOWPACK results. In this class,
VN.CORR is comparable to the performance of the SIMPLE-HNW method. This is
unsurprising given the fact that HNW are determined using SIMPLE-HNW at the
control stations. This limits the potential of our approach to deal with
situations in which the change in snow depth is strongly influenced by
both snowfall and settling. While Equation~(\ref{eqn1}) implicitly
includes the effect of settling of new snow and the entire snow cover, it
may become inaccurate after the first day of a multi-day snowfall event.
In such situations VN.CORR therefore may not perform as well as a physical
snowpack model. Note also that the difference in the time of measurements
between the manual and automatic measurements (8\,h; see
section~\ref{Data}) may lead to erroneous HNW estimates in a few cases
(e.g. if a significant precipitation event occurs between 0000 and 0800).
However, an HNW analysis of manual and automatic stations separately (data
not shown) revealed very similar results with respect to POD/FAR
statistics, which implies that this effect is minor.

%figure 3 here

\begin{figure*}%fig3
\centering
\includegraphics[width=140mm]{fig_3.eps}
\psfrag{Date}{{\LARGE $\alpha$}}
\caption{The temporal development of snow depths derived by the virtual
network (black lines) and the point measurements (grey lines) for three
selected stations. The systematic bias ($\alpha$) and the abolute error
(RMSE) are indicated in the legend. \textbf{[AUTHOR: Can a new version of
better quality be provided?]}}\label{fig3}
\end{figure*} 

Overall, VN.CORR demonstrates an advantage over VN.RAW. In particular, the
calculation described by Equations~(1)--(5) demonstrated the largest
improvement. The  method of removing the outliers has a minor impact on
the results of VN.CORR. Moreover, in comparison to the other methods,
VN.CORR yields the highest number of ranking points and therefore
constitutes an adequate automatic estimation for HNW over the entire Swiss
Alps.

\subsection{HS}

Figure~\ref{fig3} shows the temporal development of the modelled HS
($\mathrm{HS_{VN}}$, black lines) and the measured HS
($\mathrm{HS_{MEAS}}$, grey lines) during the season of winter 2006/07 for
three locations of automatic measurement stations (ELM~2, DAV~2 and DAV~3).
A temporal development of HS is first displayed where $\mathrm{HS_{VN}}$
describes a curve consistently below $\mathrm{HS_{MEAS}}$ (ELM~2),
resulting in a systematic underestimation of $\alpha= 0.6$ and a mean
absolute error (RMSE) of 0.42\,m. Secondly, a trajectory is shown (DAV~3)
where the systematic bias ($\alpha$) is close to 1 and RMSE\,=\,0.19\,m,
indicating a strong congruence between the curves. Finally, a station
(DAV~2) is shown which describes a path constantly above
$\mathrm{HS_{MEAS}}$, resulting in an $\alpha=1.54$ and RMSE\,=\,0.45\,m
(i.e. similar to that of ELM~2). Despite the fact that $\mathrm{HS_{VN}}$
shows considerable deviation from $\mathrm{HS_{MEAS}}$ for some stations,
all stations generally reproduce the temporal progression qualitatively
well. This is particularly true for the peaks during accumulation and the
following settling (produced by Equations~(7) and (8)). The same result is
obtained for the ablation derived by Equations~(9)--(12). 

Basically, the ablation period is generated by the VN only by means of
differences of HS, where the melt rate matrix (Equation~(8)) actually
refers to a combination of some fresh snow accumulation, settling and
melting. As the peaks are attributed to snow accumulation by daily new
snowfall ($\mathrm{HNW_{VN.CORR}}$), the deviation from
$\mathrm{HS_{MEAS}}$ stem from possible under-/overestimations of
$\mathrm{HNW_{VN.CORR}}$, as discussed in section~\ref{sect4.1}. Analogous
to the results and discussion of the systematic bias of
$\mathrm{HNW_{VN.CORR}}$, the specific values of $\alpha$ for each station
are approximately equally distributed around the mean value of
$\overline{\alpha}=1.06 \pm 0.286$. Accordingly, while $\alpha$ is
reasonably close to 1, $\overline{\mathrm{error}_{\alpha}}=0.022 \pm
0.026$ and $\overline{\mathrm{RMSE}}=0.40 \pm 0.12$\,m are considerably
large (see Table~\ref{tab4}).
 
%table 4 here

\begin{table*}%table 4
\centering
\caption{The parameters of $\mathrm{HS_{VN}}$ evaluation statistics
($\alpha$, $\mathrm{error}_{\alpha}$ and RMSE) as described in
section~\ref{sect3.2} calculated either for the analysis per station
averaged over 106 stations ($\mathrm{HS_{VN}}$ per station), or for the
analysis of $\mathrm{HS_{VN}}$ averaged over all stations (average of
$\mathrm{HS_{VN}}$)}\label{tab4}
\begin{tabular}{lccccc} \hline
 & $n_\mathrm{stations}$ & $n_\mathrm{days}$ & $\overline{\alpha}$ &
$\overline{error_{\alpha}}$  & $\overline{RMSE}$ \\ 
$\mathrm{HS_{VN}}$ per station & 106  & 782 ($\pm$ 73)  & 1.06 ($\pm$
0.286) & 0.022 ($\pm$ 0.026) & 0.40 ($\pm$ 0.12)   \\ \hline
 & $n_\mathrm{stations}$ & $n_\mathrm{days}$ & $\alpha~ of~ \overline{HS}$
 & $error_{\alpha}~ of~ \overline{HS}$  & $RMSE~ of~ \overline{HS}$ \\ 
Average of $\mathrm{HS_{VN}}$ & 106  & 809 & 0.96  & 0.004  & 0.08 \\
\hline
\end{tabular}
\end{table*}

\subsection{Spatial analysis}
Since the analysis of the single point measurements of $\mathrm{HS_{VN}}$
is limited due to its random nature, an investigation of
$\mathrm{HS_{VN}}$ and $\mathrm{HNW_{VN.CORR}}$ considering the Swiss Alps
as an entire area is also considered. First, the mean of all snow depths of
point and VN estimations ($\overline{\mathrm{HS_{MEAS}}}$ and
$\overline{\mathrm{{HS_{VN}}}}$) and their standard deviation
($\sigma(\mathrm{HS_{MEAS}})$ and $\sigma(\mathrm{HS_{VN}})$) are
calculated. Figure~\ref{fig4} displays the evolution of
$\sigma(\mathrm{HS_{VN}})$ with $\overline{\mathrm{{HS_{VN}}}}$ (black
symbols) and $\sigma(\mathrm{{HS_{MEAS}}})$ with
$\overline{\mathrm{HS_{MEAS}}}$ (grey symbols), where each point of the
trajectory represents one day of the year 2006/07. The curves show a
characteristic hysteretic dynamics as discussed in \cite{egli09}. The
trajectories of the period of accumulation (points) and ablation
(triangles) are clearly separated. The quasi-linear increase of
$\sigma(\mathrm{HS_{VN}})$ with increasing $\overline{\mathrm{{HS_{VN}}}}$
highlights that accumulation of snow leads to an increase in the
differences between sites. The path during ablation, on the other hand, is
mainly attributed to the spatial distribution of melting rates in the
Alpine region. 

%figure 4 here

\begin{figure*}%fig4
\centering
\includegraphics[width=100mm, angle=-90]{fig_4.ps}
\caption{The evolution of the mean snow depths ($\overline{\mathrm{HS}}$)
and the standard deviation of snow depths ($\sigma{(\mathrm{HS})}$)
derived by the virtual network ($\mathrm{HS_{VN}}$, black symbols) and the
point measurements ($\mathrm{HS_{MEAS}}$, grey symbols). The trajectory
during the period of accumulation is indicated by points; the trajectory
during ablation is indicated by triangles.}\label{fig4}
\end{figure*}  

It has been shown \citep{egli09} that the curves of the hysteretic
dynamics are similar between years and therefore characterize the seasonal
development of HS in the Swiss Alps. Figure~\ref{fig4} shows that both the
periods of accumulation and ablation are well reproduced by
$\mathrm{HS_{VN}}$ when compared to $\mathrm{HS_{MEAS}}$. The seasonal
characteristic of the HS development is therefore also displayed by the
VN. We therefore speculate that $\mathrm{HS_{VN}}$ correctly reproduces
the total amount of snow depths during a season at Swiss Alpine scale.
Moreover, when evaluating $\alpha$, $\mathrm{error}_{\alpha}$ and RMSE for
$\overline{\mathrm{HS_{MEAS}}}$ and $\overline{\mathrm{{HS_{VN}}}}$ (HS is
averaged day by day over all stations), the percentage bias is as small as
--4\% (Table~\ref{tab4}).

Furthermore, the variogram analysis of HNW \citep{egli08} showed that the
statistical correlation length of HNW over the entire Swiss Alps is about
50--60\,km. The same correlation length was found in \citep{schaub07},
analysing the raw HNW output of COSMO ($\mathrm{HNW_{VN.RAW}}$). This
additionally supports the assumption that $\mathrm{HNW_{VN}}$ reproduces
the regional precipitation patterns of the Swiss Alps appropriately and
can be used as an input parameter for spatially distributed snow models.

Finally, the POD statistics of point HNW measurements as a function of the
distance between two measuring sites \citep{egli08} showed that, for the
smallest scale (5--10\,km), the POD is about 60\% for HNW intensities
$>30$\,mm (High). The same value has been derived from a comparison of
$\mathrm{HNW_{VN.CORR}}$ and $\mathrm{HNW_{MEAS}}$. Again, a better
performance of the VN HNW estimation in the POD statistics of two point
measurement stations about 7 km distant would have been astonishing, since
the large area of $\mathrm{HNW_{VN.CORR}}$ covers a grid cell of
7\,km\,$\times$\,7\,km. As the POD decreases rapidly with the distance
from a point measurement station \citep{egli08}, the VN may also be
applied for avalanche risk management warning for the locations between
the point measurements where no HNW estimation is available.

\section{Conclusion and outlook}
In this study, the HNW output of the numerical weather prediction model
COSMO-7 was coupled with a simple snow accumulation/melting model in order
to provide HNW/HS grids of 7\,km resolution for the entire Swiss Alps. The
results for the $\mathrm{HNW_{VN}}$ estimation showed that, on average,
its performance is comparable to the performance of different automatic
point measurements. $\mathrm{HNW_{VN}}$ therefore represents a
complementary network for avalanche risk management applications and can
also be used as input data for spatially distributed snow hydrological
models where HNW is required. 

This is also the case for the HS estimation by the VN, where the
statistical dynamics of the mean of HS ($\overline{\mathrm{{HS_{VN}}}}$)
and its standard deviation ($\sigma(HS_{VN})$) are congruent to the
measured dynamics. As a consequence, the estimations of HS by the VN over
the entire Swiss Alps may be used to estimate the total amount of snow and
snow water equivalent for larger catchments. At a local scale, however, a
direct comparison of the point measurement to the corresponding grid cell
of the VN may result in considerable deviation. This may stem from the
fact that a measurement over a large area (7\,km\,$\times$\,7\,km for the
VN) is compared to a single point measurement. 

Future effort is necessary to investigate the spatial scales over which
the VN is capable of providing a good performance for estimation of
HNW/HS. For this purpose we will extend the application to COSMO-2, a new
high-resolution version of COSMO available from February 2008 for a mesh
size of 2.2\,km. Additionally, the differences in the VN between the
assimilation mode and the forecast mode of COSMO-2 will be investigated.
Finally, the principle of the VN presented here can be applied to other
regions which are less densely equipped with automatic or manual point
measurement stations than the Swiss Alps. If point measurement stations
and a numerical weather prediction model are available, HNW and HS can be
estimated with a refined accuracy for spatially larger extended regions
for avalanche risk and water resources management.

\section{Acknowledgements}
The authors thank D.~Schaub and F.~Schubiger for their support with the
data acquisition and processing and the external reviewer E.~Brun for his
constructive suggestions which helped to improve the manuscript.
  
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\end{document}

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