Many methods exist to model snow densification in order to calculate the depth of a single snow layer or the depth of the total snow cover from its mass. Most of these densification models need to be tightly integrated with an accumulation and melt model and need many forcing variables at high temporal resolution. However, when trying to model snow depth (HS) on climatological timescales, which is often needed for winter tourism-related applications, these preconditions can cause barriers. Often, for these types of applications, empirical snow models are used. These can estimate snow accumulation and snowmelt based on daily precipitation and temperature data only. To convert the resultant snow water equivalent (SWE) time series into snow depth, we developed the empirical model SWE2HS. SWE2HS is a multilayer densification model which uses daily snow water equivalent as sole input. A constant new snow density is assumed and densification is calculated via exponential settling functions. The maximum snow density of a single layer changes over time due to overburden and SWE losses. SWE2HS has been calibrated on a data set derived from a network of manual snow stations in Switzerland. It has been validated against independent data derived from automatic weather stations (AWSs) in the European Alps (Austria, France, Germany, Switzerland) and against withheld data from the Swiss manual observer station data set which was not used for calibration. The model fits the calibration data with root mean squared error (RMSE) of 8.4

Seasonal snow cover is an important variable with regard to ecology, water resource management, and the tourism industry. Accordingly, a large range of
models of different complexity exist to calculate various properties of the snow cover. Traditionally, snow models were emerging from the hydrological
community in order to estimate water resources from snow. Therefore, the focus was set on snow water equivalent (SWE) of the snow cover for the first
simple approaches such as the empirical temperature index models in which the amount of melt is estimated by the sum of positive air temperatures

Snow depth is the result of SWE and the bulk snow density (

All densification models need to initialize the density of a snow layer or of the whole snowpack. Since there is no simple method yet for deriving new
snow density from a physical snowfall model, new snow density is either parameterized or kept at a fixed value in snow models. Various
parameterizations exist and are usually based on estimating new snow density as a function of wind speed, temperature, and relative humidity

Several methods exist to model snow densification, either per layer or for the entire snowpack, which can roughly be classified into three
categories. The first category is purely empirical whereby densification dynamics are described via exponential settling functions. This approach has
first been proposed by

To our knowledge, none of the densification models described above can easily be used as a standalone model independently of the snow model for transferring daily SWE to snow depth; however, many approaches exist to do the opposite, i.e. convert HS into SWE

Illustrative snowpack evolution within the SWE2HS model calculated from 35

The remainder of the paper is structured as follows. In Sect.

The density model SWE2HS calculates snow depth at a daily resolution and is driven by the daily snow water equivalent of the snow cover only. In the
following, we use the unit millimetre water equivalent (

The density of a layer at day

where

The maximum density to which the density of a snow layer converges,

If a layer experiences an overburden

SWE losses are defined by

At the end of every time step, the snow depth of the snowpack is calculated by summing up the thickness of all

We provide an implementation of the model as a Python package under GNU General Public License v3.0 (GPLv3). One-dimensional station data and two-dimensional model grids of daily SWE time series can be transformed to snow depth with the snowpack evolution described above. Additionally, a step-by-step processing mode with caching of the model state variables for two-dimensional SWE grids of consecutive days is available for operational
applications. Python, being a high-level, interpreted general-purpose programming language has been chosen due to its easy-to-read syntax, growing
user base, and community support for scientific computing and data analysis. Our implementation is using the just-in-time Python compiler

As for every empirical model, parameters in our density model need to be calibrated. Calibrated parameters may differ depending on the station, snow
type, and snow climatological setting. Here, we try to find one single generic optimal parameter set which suits most snow climatological conditions in Switzerland and the European Alps in general. We do so by calibration of a data set which covers a large range of different altitudes and
climatologic settings in Switzerland (see Sect.

Our model has 6 model parameters which need to be calibrated. Before calibration, we define upper and lower bounds of possible values for each model parameter (see Table

For parameter calibration, we use the differential evolution algorithm which is a stochastic population-based method for minimizing nonlinear and
non-differentiable continuous space functions as implemented in SciPy

We optimize the model by minimizing the root mean squared error (RMSE) which is a measure of the distance between the predicted values from the
model

In order to assess the importance of individual model parameters on the result, we perform a sensitivity analysis on the validation data set by
calculating

To calibrate the SWE2HS model, we use data from 58 Swiss manual observer stations between 1080 and 2620

While we are aware that it might be preferable to calibrate a model on measured data instead of output from another model, we still chose the approach described above in order to have an exhaustive calibration data set which (a) covers a wider range of altitudes, expositions, and snow climatic
settings in our target region; (b) does not have problems of potential over- and under-measurement from automatic SWE measurement devices

As a first validation data set, we use the remaining years of the long station records which were shortened to 15 water years for the calibration data set (see above). This calibration set of manual observer station data contains 1279 station years from 42 different stations.

Locations of the snow measurement sites in the European Alps from which snow water equivalent and snow depth data were used to compile the calibration and two validation data sets. Background colouring resembles elevation. GMTED2010 elevation data, courtesy of the U.S. Geological Survey, was used to create the map

Automatic weather stations from which we used snow water equivalent and snow depth data for validation of the model. The number of years refers to complete hydrological years (September–August) included after data cleaning, the average snow depth (

As a second validation data set, we gathered data of 10 different automatic weather stations (AWSs) in Austria, France, Germany, and Switzerland that
automatically measure SWE with either a snow pillow or a snow scale and measure snow depth with an ultrasonic measurement device at subdaily
resolution (see Table

Schematic modelled snowpack evolution for 6 different station years from the validation data set from automatic stations with different altitudes and snowpack thicknesses (see Table

Parameters of the model, lower and upper bounds during calibration, and optimized value.

The model calibration on the data set described in Sect.

Scatterplots of modelled versus measured snow depth values for

Score values of RMSE,

With the optimized parameter set, the model is able to fit the calibration data with RMSE of 8.4

According to the sensitivity analysis, the settling resistance factor

Boxplots comparing the distributions of measured and modelled data in the months from October to June for

Boxplots of the scores

Global

On our way towards the model presented here, we tried models of different complexity and we included and removed processes while iterating back and
forward. Some prototype model versions additionally included daily temperature as input forcing, which we tried to use for parametrization of new snow
density and onset of the wetting from the top of the snowpack by using the cold content parameterizations used in

We assumed that it would be beneficial to use temperature in the beginning of model conceptualization and one could argue that when using SWE from an
accumulation and ablation model, there is always at least daily mean temperature available used to drive a melt model. However, when quantitatively
assessing the model versions with parameterized new snow density or cold content parameterizations, we did not see model improvement from the daily mean
temperature inclusion and thus decided to only use SWE. This additionally comes with the asset that the model can be plugged in as a post-processing
tool to any snow model which outputs daily SWE. Besides the best performance, another important factor to keep the model simple was to reduce the risk
of equifinality, meaning that an optimal solution can be achieved through different states, i.e. parameter combinations of the model

As shown in Sect.

Other sources of uncertainty are due to inherent limitations of our empirical modelling approach. As mentioned above, the model is not able to
represent rain-on-snow events. In the exemplary snowpack evolution of the winter of 2020–2021 at station Kühroint, an increase in SWE causes modelled
snow depth to increase although the measured snow depth is constantly decreasing during this time (middle of March 2021, Fig.

Since the model is of empirical nature, the parameter set which is presented here for the European Alps might not be suited for other regions on earth
with different climatologic conditions. If applied to other regions, the model parameters need to be calibrated again. However, as we never tested the
model in e.g. Arctic regions, we cannot make any statements about whether the model is able to represent settling dynamics in these snow climatologic conditions
even if it would be calibrated on data from there. Although the calibrated parameter set presented in this paper is thought to be representative for
the European Alps, it may not be suitable for some stations in the validation data set with bias of up to 22.7

Simple empirical models that try to conceptualize processes in a non-physical way are often subject to the risk of potential model equifinality

The model is less sensitive to changes in the model parameter

The SWE2HS model is tailored for use with daily resolution SWE data. When attempting to use the model with higher temporal resolutions such as hourly,
additional processes to those considered in the model become increasingly important and additional parameters such as radiation and temperature are
likely to be required to satisfactorily represent densification. For example, the new snow density will be much more variable on shorter timescales,
and it is likely that the fixed new snow density approach used in the SWE2HS model will not be sufficient at hourly resolutions. In addition, the
empirical transition rate from dry to wet snow (

We present a simple snow density model which can be used to transfer continuous daily snow water equivalent data to snow depth. The empirical
multilayer model uses exponential settling equations, a fixed new snow density, and assumes a changing maximum snow density over time based on
overburden and SWE losses. The model was calibrated with a gradient-free evolutionary algorithm on a data set from the Swiss Alps that was generated
from biweekly SWE and daily HS records. Prior to calibration, the biweekly SWE records were converted to daily values with the

Schematic modelled snowpack evolution for 6 different station years from the manual station validation data set. Winters from stations with different elevations and with differing snowpack thicknesses are shown. For an explanation of the figure, the reader is referred to the caption of Fig.

Schematic modelled snowpack evolution for 6 different station years from the manual stations calibration data set. Winters from stations with different elevations and with differing snowpack thicknesses are shown. For an explanation of the figure, the reader is referred to the caption of Fig.

The current version of the SWE2HS model source code, including documentation and examples is available at

JA compiled the calibration and validation sets, developed the methodology and software code, and wrote the initial paper draft. CM, TJ, and AM gave input to the methodology and reviewed different model versions. CM acquired funding and supervised the study. All authors reviewed and commented on the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We want to thank the Bavarian avalanche warning centre (Lawinenwarnzentrale Bayern) in Munich and the Austrian Research Centre for Forests BFW in Innsbruck for contributing data from their measurement stations to the calibration data set.

This work was financially supported by the internal project “Climatological maps for snow depth” of the Swiss Federal Institute for Forest, Snow and Landscape Research (WSL).

This paper was edited by Fabien Maussion and reviewed by two anonymous referees.