TCThe CryosphereTCThe Cryosphere1994-0424Copernicus PublicationsGöttingen, Germany10.5194/tc-10-1021-2016On the assimilation of optical reflectances and snow depth observations into a detailed snowpack
modelCharroisLucluc.charrois@meteo.frCosmeEmmanuelDumontMariehttps://orcid.org/0000-0002-4002-5873LafaysseMatthieuMorinSamuelhttps://orcid.org/0000-0002-1781-687XLiboisQuentinPicardGhislainhttps://orcid.org/0000-0003-1475-5853Université Grenoble Alpes-CNRS, LGGE, UMR 5183, Grenoble, FranceMétéo-France/CNRS, CNRM UMR 3589, CEN, Grenoble, FranceLuc Charrois (luc.charrois@meteo.fr)13May20161031021103820October201514December20157April201625April2016This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://tc.copernicus.org/articles/10/1021/2016/tc-10-1021-2016.htmlThe full text article is available as a PDF file from https://tc.copernicus.org/articles/10/1021/2016/tc-10-1021-2016.pdf
This paper examines the ability of optical reflectance data assimilation to
improve snow depth and snow water equivalent simulations from a chain of
models with the SAFRAN meteorological model driving the detailed multilayer
snowpack model Crocus now including a two-stream radiative transfer model for
snow, TARTES. The direct use of reflectance data, allowed by TARTES, instead
of higher level snow products, mitigates uncertainties due to commonly used
retrieval algorithms.
Data assimilation is performed with an ensemble-based method, the Sequential
Importance Resampling Particle filter, to represent simulation uncertainties.
In snowpack modeling, uncertainties of simulations are primarily assigned to
meteorological forcings. Here, a method of stochastic perturbation based on
an autoregressive model is implemented to explicitly simulate the
consequences of these uncertainties on the snowpack estimates.
Through twin experiments, the assimilation of synthetic spectral reflectances
matching the MODerate resolution Imaging Spectroradiometer (MODIS) spectral
bands is examined over five seasons at the Col du Lautaret, located in the
French Alps. Overall, the assimilation of MODIS-like data reduces by 45 % the
root mean square errors (RMSE) on snow depth and snow water equivalent. At
this study site, the lack of MODIS data on cloudy days does not affect the
assimilation performance significantly. The combined assimilation of
MODIS-like reflectances and a few snow depth measurements throughout the
2010/2011 season further reduces RMSEs by roughly 70 %. This work suggests
that the assimilation of optical reflectances has the potential to become an
essential component of spatialized snowpack simulation and forecast systems.
The assimilation of real MODIS data will be investigated in future works.
Introduction
Seasonal snowpack modeling is a crucial issue for a large range of
applications, including the forecast of natural hazards such as avalanches or
floods, or the study of climate change (e.g.,
). The most sophisticated detailed
snowpack models represent the evolution of snow microstructure and the
layering of snow physical properties
in response to
meteorological conditions. Despite constant efforts to improve these models,
large uncertainties remain in the representation of the snow physics, as well
as in the meteorological forcings
. These uncertainties are highly
amplified when propagated to avalanche hazard models . For
operational applications, the assimilation of observations can help reduce
the impact of the model and forcing uncertainties in the snowpack
simulations
e.g.,.
Satellite observations are becoming an essential component of snow modeling
and forecasting systems. In situ measurements are the most detailed
and accurate observations of the snowpack, but their spatial distribution is
far too scarce to capture the high spatial variability of the seasonal
snowpack properties and improve snowpack simulations through their
assimilation. For this reason, the assimilation of satellite observations of
snow is an active area of research.
Snow remote sensing is primarily performed in the microwave (passive and
active), visible and near-infrared spectra. Since the direct assimilation of
such data requires the use of radiative transfer models, a common and simple
approach consists in using satellite-based snow products. In particular, the
assimilation of snow cover fraction (SCF) estimates derived from optical
sensors (such as MODIS) and snow water equivalent (SWE) or snow depth (SD)
estimates derived from passive microwave sensors (such as AMSR-E) has been
investigated extensively
.
These studies have suggested that, most of the time, assimilating snow
observations may be useful to improve snowpack estimation. SWE or SD
assimilation generally outperforms the assimilation of SCF only, except from
because of large errors in the AMSR-E SWE products. The
assimilation of both combined revealed larger benefit by mitigating sensors
limitations. Recently, investigated the assimilation of
(synthetic) ice surface temperature while also
experimented the assimilation of albedo retrievals, both from optical
sensors. obtained a mass balance RMSE decrease of up to
40 % assimilating albedo data. However, satellite snow products are derived
using retrieval algorithms which are not perfect and, perhaps more
importantly, not physically consistent with the snowpack model used for the
data assimilation. For this reason, and as advocated by
who tested the assimilation of in situ microwave radiance observations,
assimilating the original satellite radiance data should be preferred when
possible.
The potential of assimilating passive microwave radiances (in the form of
brightness temperature) collected by AMSR-E satellite have been examined by
and . Significant improvements in the
SWE/SD predictions occurred but only during the accumulation period. Though
the melt period, when the snowpack is wet, liquid water alters the microwave
signal resulting in a lower performance of the assimilation. Moreover, for
small-scale applications in mountainous areas, the coarse spatial resolution
of these data considerably reduces their usefulness
. As for active microwave
measurements, several tests have been conducted to assimilate the satellite
signal (e.g., ). These tests were however limited by
the accuracy of the forward electromagnetic models and by the current lack of
satellite data at a daily or even weekly time frequency.
Visible and near-infrared reflectances from satellite observations have never
been assimilated into snowpack models despite their great sensitivity to the
snowpack properties . Even if cloud cover might limit their
utility, medium and high spatial resolution data are available at daily
resolution from several optical sensors (e.g., MODerate resolution Imaging
Spectrometer, Visible Infrared Imaging Radiometer Suite) and seem to be quite
suitable for complex topography . In particular, the MODIS
sensor, onboard the TERRA and AQUA satellites, offers a daily coverage and
provides reflectance measurements in seven bands distributed in the visible
(at 250 to 500 m spatial resolution), near and short-wave infrared
wavelengths. Surface bi-hemispherical reflectances corrected from complex
topographic effects in mountainous areas can be computed
and have been evaluated and used in several rugged areas
.
The work presented in this article examines the possibility, the relevance
and the limitations of assimilating visible and near-infrared satellite
reflectances into a multilayer snowpack model. A convenient approach, known
as twin experiment, uses synthetic data in the same spectral bands than the
real data, to examine the content of information of the observations, and the
impacts we can expect from their assimilation. In twin experiments, the model
used to create the synthetic data is the same as the model used for the
assimilation. The synthetic observations are extracted from a member of the
ensemble considered as the true state. Twin experiments are preferred in this
first study in order to focus on the information content of the observations
and to avoid the problem of observational biases. Data assimilation is
performed with a particle filter and a Sequential Importance Resampling (SIR)
algorithm . The particle
filter is easy to implement, free of hypotheses about the nature of the model
and the observations, and provides uncertainties in the estimation of the
snowpack state.
For a comprehensive snow simulation evaluation, as recommended by
, our study is based on five hydrological seasons (2005/2006,
2006/2007, 2009/2010, 2010/2011, 2011/2012) which represent a wide range of
possible snow cover conditions in the Alpine area. Moreover, two experimental
sites were used in this work in virtue of a long, continuous time series of
meteorological data and an area suitable for remote sensing measurements. The
Col de Porte (CdP) area, located in the Chartreuse area, near Grenoble,
France (1325 m a.s.l.) provides a data set from 1993 to present
from which meteorological statistics can be estimated, but
the instrumentation and surrounding forest at this site may affect satellite
measurements. For this reason, assimilation experiments are carried out at
the Col du Lautaret (CdL) located (2058 m a.s.l.) in the Ecrins area, France,
which exhibits a large flat open area, above treeline, more suitable for
remote sensing. Consequently, an ensemble of perturbed forcing was generated
in order to represent the range of possible weather conditions at the CdL
area. To this end, we developed a stochastic method using a first-order
autoregressive model based on the estimated meteorological uncertainty.
In Sect. , the SURFEX/ISBA – Crocus snowpack
model used in this study is described with an emphasis on the characteristics
that affect the implementation of the data assimilation method. In
particular, we consider the meteorological forcings as the only source of
uncertainties. Section presents in
detail how these forcings are perturbed to take the uncertainties into
account in the design of the ensemble simulations. The experimental setup and
the data assimilation implementation are presented in Sect. . The results of the reference assimilation experiment
(baseline experiment) using synthetic reflectance observations at one point
are presented and discussed in Sect. . In close relation
to this baseline experiment, results of different sensitivity tests are
addressed in Sect. .
SURFEX/ISBA – CrocusA brief overview
The unidimensional detailed multilayer snowpack model Crocus
simulates the evolution of the snowpack physical
and microstructural properties driven by near-surface meteorology and
includes a representation of snow metamorphism. A detailed description of
Crocus is provided by ; here we only emphasize aspects that
are key to data assimilation. The snowpack is vertically discretized into
snow layers with different physical properties and a dynamic layering
scheme handles its evolution (see details in Sect. ). The
evolution of the snow cover is a function of energy and mass transfer between
the snowpack, the atmosphere and the ground. The model simulates the major
physical processes of snowpack evolution such as heat conduction, light
penetration, water percolation and refreezing, settlement and snow
metamorphism.
Crocus has been run operationally at Météo-France in support of avalanche
risk forecasting over the last 20 years . It has been also
successfully used for various applications such as climate studies or
hydrology e.g.,. Recently, Crocus has
been integrated into the SURFEX externalized surface modeling system
as one of the snow schemes within the Interactions between
Soil, Biosphere and Atmosphere (ISBA) land surface model .
Thus the integrated system simulates the energy fluxes between the snow cover
and the multilayer soil component of the land surface model (ISBA-DIF,
).
Layering
In Crocus, the snowpack is vertically discretized in order to realistically
simulate the time evolution of a stratified snowpack. The layering scheme is
dynamic in order to preserve snowpack history and maintain the possible thin
and weak snow layers within the snowpack. The number of layers ranges from 0
(bare soil) to a maximum of 50, typically. Layering is updated at the
beginning of each time step. It consists in adding, removing, or merging
layers depending on their physical properties and thicknesses. The procedure
basically follows this set of rules:
For a snowfall on an existing snowpack, fresh snow is incorporated into the
top layer if (i) snow microstructure characteristics are similar, (ii) the
top layer is thinner than 1 cm and (iii) the snowfall intensity is inferior
to 0.03 kg m-2 h-1. If one of these criteria is not met or change
during the time step, a new top layer is created.
A snowfall on bare soil forms a snowpack with a set of identical layers,
the number of which depends on the quantity of fallen snow.
In absence of snowfall, the model first seeks to merge two thin and adjacent
layers with similar microstructure characteristics, or inversely, split the
thick ones. When the number of layers has reached a maximum of 50, layers
that are too small relative to a prescribed optimal vertical profile are
aggregated with adjacent ones. This idealized thickness profile depends on
the current snow depth and on the user-defined maximal number of layers. To
reach the optimal vertical profile, the model first seeks to thin the top
layers, most subject to the exchange of energy, and then to keep an
appropriate thickness ratio between adjacent snow layers to prevent numerical
instabilities in the resolution of the heat diffusion equation through the
snowpack.
Most of the time, compaction makes layers thinner without grid resizing.
Dynamic layering adds an extra challenge in the assimilation of
observations with Crocus. Data assimilation methods commonly used in
geophysics are well designed for fixed-grid models. For example, the Ensemble
Kalman filter involves the averaging of different snow profiles. This
specificity of Crocus largely determines our data assimilation method, as it
will be discussed in Sect. .
Penetration of solar light in the snowpack
Given that satellite observations indirectly relate to the quantities of
interest, an observation operator is required to link the satellite
observation and the model state variables . This operator
transforms the model variables into diagnostic variables to allow a direct
comparison with satellite observations, preserving the physical consistency
of the satellite signal with the snow model.
To this end, a new radiative transfer model was recently implemented in
Crocus to calculate spectral reflectances that can be used for the comparison
and the assimilation of satellite observations data such as MODIS data. This model, named
TARTES (Two-streAm Radiative TransfEr in Snow,
), simulates the absorption of solar radiation
within the stratified snowpack using the δ-Eddington approximation,
with a spectral resolution of 20nm. This contrasts with the original version
of Crocus, where albedo was computed for three large spectral bands only and
from the properties of the first two layers .
TARTES is implemented as an optional module to be called instead of the
original Crocus albedo scheme. This implementation has no significant impact
on the model structure but increases the computation time of roughly a factor
10 depending on the number of snow layers and the snow depth. TARTES makes
use of four Crocus prognostic variables (specific surface area – SSA,
density, snow layer thickness, impurity content) and the
angular and spectral characteristics of the incident radiance (e.g., the solar
zenith angle and the presence of cloud cover). The computation of SSA has
recently been implemented by .
The use of a full radiative transfer model embedded within the snowpack model
enables the assimilation of the satellite reflectance data, therefore
avoiding the introduction of uncertainties from an external retrieval
algorithm. And beyond its use for the assimilation of reflectances, TARTES
also provides a more accurate calculation of light absorption parameters,
leading to better simulations of the snowpack.
Snow impurities
Snow surface reflectance in the visible spectrum depends on the content of
light-absorbing impurities in the snowpack . The impurity
content can have a major impact on the snowpack simulations
. Despite efforts to improve the knowledge and the modeling
of impurities in snow e.g.,, snow impurity
deposition and evolution remain poorly quantified.
Currently implemented in a version of SURFEX/ISBA – Crocus, the radiative
model TARTES (introduced in Sect. ) calculates the impurity content as an equivalent black carbon
content . This impurity content evolves
according to (i) the impurity content in fresh snow, c0, (ii) the time of
exposure of the layer at the surface and (iii) the dry deposition flux of
impurity, τdry as described in the equation below.
c(t+Δt)=c(t)+Δtτdrye-D/href,
where c(t) is the impurity content at time t, D is the depth of the
middle of the considered snow layer and href=5 cm is the
e-folding of the exponential decay rate for the deposition of snow impurities
ensuring that only the top layers are influenced by dry deposition.
Atmospheric forcings
The snowpack evolution strongly depends on near-surface meteorological
forcings. These forcings are provided by the meteorological downscaling and
analysis tool SAFRAN (Système d' Analyse Fournissant des
Renseignements Atmosphériques à la Neige; ).
SAFRAN is used to drive snowpack simulations in the French mountains because
it is designed to operate at the geographical scale of meteorologically
homogeneous mountain ranges, varying from 400 to 2000 km2. The model
combines vertical profile estimates from the ERA-40 re-analysis with observed
weather data from the automatic surface observations network at different
elevations, the French Snow/Weather network, rain radars, and rain gauges. As
outputs, SAFRAN provides meteorological data to the snowpack model with an
hourly time step for all slopes and aspects, and a 300 m-elevation step.
Design of Crocus ensemble simulationsGeneral strategy
In view of assimilating observations to reduce snowpack simulation
uncertainties, we first need to represent them. As shown in
, the meteorological forcings are the major source of
uncertainty in snowpack simulations (when a meteorological model is used to
drive the snow model). In the present study, air temperature, wind speed,
snowfall and rainfall rates, shortwave and longwave radiative fluxes, and the
deposition rate of impurities will thus be considered as the only sources of
uncertainty. Snowpack model errors introduced by metamorphism and other
parameterizations of physical laws are not taken into account here. The
characterization and representation of these errors, notably in the
perspective of real data assimilation, will be addressed in a future and
dedicated work. An identified option is to use multi-physics ensemble
simulations.
We implement an ensemble method to represent the uncertainties in the
forcings and their impact on snowpack simulations. An ensemble of possible
realizations of the atmospheric forcings is formed and used to compute an
ensemble of snow profiles representing the probability distribution of the
model simulation. The following section describes the construction of the
ensemble of meteorological forcings and the response of the model to this
source of uncertainty, without assimilation.
Quantification of meteorological forcing uncertainties
To quantify and calibrate the meteorological forcing uncertainties, we
compare 18 years of surface meteorology from SAFRAN reanalysis with
in situ observations at the CdP site. A long time-series from 1993
to present being available at this site, uncertainties in
the SAFRAN meteorological reanalysis can be estimated.
Table (left column) reports the bias and the
standard deviation (STD) of the difference between SAFRAN and the observations
carried out at the CdP site, for each meteorological variable (values in
brackets and the right column report other data discussed later). The table
reflects differences between SAFRAN and in situ observations,
resulting, from the different spatial representativities of both sources, the
intrinsic errors of the analysis system and measurement errors.
As highlighted by who conducted an extended evaluation
of SAFRAN reanalysis but over a shorter period (one year), the large
discrepancies between the model and the observations can be explained by
local effects due to orography and vegetation and, for the precipitation and
wind speed, by the time interpolation necessary to obtain hourly forcing
fields from the daily analysis. For example, the precipitation analysis is
performed on a daily basis in order to include in the analysis the numerous
rain gauges observations. Radiation fluxes uncertainty might be attributed to
biases in cloud coverage and altitude estimates, effects of vegetation and
surrounding slopes that are not taken into account for longwave estimates.
Finally, the shading mask for shortwave radiation does not account for
vegetation evolution that can also lead to shortwave flux
discrepancies. carried out, only on a limited set of
variables, a more systematic evaluation of SAFRAN for the 1958–2002 period
using 43 sites in the French Alps. Averaged over all locations, the RMSE on
air temperature are similar to the one computed in our study. However, their
results also highlight the spatial variability of SAFRAN performance (site
RMSE ranges from -0.8 to +1.5 ∘C). Nevertheless, this will not
have a strong impact in this study since it is based on twin experiments.
Building the ensemble of meteorological forcings
The sample of meteorological forcings is formed by perturbing the original
SAFRAN reanalysis with a random noise commensurate with the actual
uncertainty. We thus build an ensemble of meteorological forcings with a
negligible bias with respect to the SAFRAN reanalysis and a standard
deviation close to the one computed from CdP statistics (Table , left column).
To keep the procedure simple and preserve physically consistent time
variations of the forcings, the random perturbations are computed using a
first-order autoregressive – AR(1) – model for each
variable:
Xt=φXt-1+ϵt,
with X being the perturbation value at time t and t-1. φ
is the AR(1) model parameter and can be written φ=e-Δtτ, Δt being the time step and τ
the decorrelation time. Parameter τ is adjusted for each variable, so
that the perturbed variable exhibits the same frequency of temporal
variations than the original variable (Fig. S1 bottom, in the Supplement, in
blue).
The amplitudes of the meteorological uncertainties are introduced with ϵt,
a white noise process with zero mean and constant variance
σ2. Variance σ2 is computed from each standard deviation of
the residuals between the reanalysis and observations at CdP (σCdP:
Table , left column) following this equation:
σ2=σCdP×(1-φ2).
Finally, for each meteorological variable, the selection of an additive or
multiplicative perturbation method is driven by (i) the nature of the
variable (ii) the dependency of the model–measurement difference to the
measured values as detailed below.
For precipitation rates, shortwave radiation and wind speed, the choice of a
multiplicative method is motivated by the following reasons:
SAFRAN reanalysis effectively captures the occurrence of precipitation (since it assimilates
surface observation network) but are more subject to errors in the amount of
precipitation;
Regarding shortwave flux and wind speed, the model biases exhibit a linear dependency
to the value of the variable (not shown). Consequently, a multiplicative
method was selected.
For longwave radiation and air temperature, given that there is no dependency
between the model biases and the field values, an addition method is chosen.
At every time step the perturbation Xt is applied as follows.
For the additive method, variablet=variablet+Xt. For the
multiplicative method, the perturbation is centered on 1 (Yt) before
multiplying the variable.
Yt=Xt+1variablet=variablet×Yt.
For the multiplicative method, the perturbations are bounded by 0.5 and 1.5
to avoid extreme values. The result from this perturbation method is
illustrated by Fig. S1 which shows the SAFRAN snowfall rates over a 1-week period, a realization of the
perturbed analysis and the full ensemble of perturbed analysis.
To maintain further physical consistency between the meteorological
variables, snowfall is changed to rainfall if air temperature is higher than
274.5 ∘K and the shortwave radiation is bounded to
200 W m-2
in case of rain or snow fall due to the inherent cloud cover. This behavior
is consistent with the CdP statistics where over 18 years, during a
precipitation period, the measured in situ shortwave radiation rarely exceeds
200 W m-2.
Bias and standard deviations (STDs) of the differences between SAFRAN
reanalysis and in situ observations (left) and the differences between SAFRAN
reanalysis and the ensemble built up in the present study (right), for the
perturbed meteorological forcings. The first set of statistics is derived
from 18 years (1993–2011) of observations and reanalysis at the CdP and the
second set is derived from our 300-members ensemble over the 2010/2011
hydrological season. The values in brackets correspond to the adjusted
standard deviation used to generate the ensemble at CdL site.
Ensembles are generated with model errors coming from the statistics of the
CdP site but as explained previously, the assimilation framework is based on
the CdL area. Some adjustments in the building of ensembles are also required
to account differences between these two areas.
In particular, the forest at CdP affects the local wind field and the
radiative fluxes , which explains a large part of the
variability of SAFRAN errors at CdP. At CdL, an open meadow area, such
variability is unlikely. To limit the overspreading of the forcing ensemble,
the standard deviation used in the Eq. () for wind
speed, short and longwave radiation are reduced to 0.6 m s-2, 70 and 7 W m-2, respectively, against 1.12 m s-2, 79 and 24.5 W m-2
(Table , left column, values in brackets). As
shown in Table 1, the standard deviations computed from the generated
ensemble (right column) are close to the ones prescribed to generate it (left
column).
In the end, this stochastic method of perturbations makes possible the
construction of an ensemble of perturbed forcings which are required when
using ensemble methods. The calibration of the perturbations are based on the
CdP statistics while their temporal correlation is ensured by the AR(1)
model. The perturbation method exhibits some obvious limitations.
Inter-variable correlations are indeed not taken into account in the ensemble
except from the precipitation phase and the maximum value of shortwave
radiation in case of precipitation. Adjustments to CdL are somewhat
subjective, but this is not crucial in our twin experiment context since the
considered truth will be simulated running Crocus with one forcing member
drawn from this generated ensemble. A more physically consistent ensemble
will be required when real data assimilation is investigated.
Perturbation of impurity deposition rate
In this study, the deposition fluxes of impurities are also considered as a
meteorological forcing but unlike meteorological variables previously
mentioned (Sect. ), the deposition fluxes of
impurities are not provided by the SAFRAN model. Instead, the impurity
content in fresh snow c0 and the dry deposition flux τdry
are perturbed online during a model run.
The parameters c0 and τdry are subject to multiplicative
perturbations drawn from lognormal distributions. The perturbations are
constant in time, but are reinitialized at each observational update when
data assimilation is performed. For c0, the probability density function
(pdf) parameters are σ= 0.8 and μ= 0. c0 is bounded at 0 and 500 ng g-1 and the mode
value of the pdf is 100 ng g-1. As for
τdry, the pdf parameters are σ= 1.2 and μ= 0.
τdry is bounded at 0 and 0.5 ng g-1 s-1 with a mode
value of 0.015 ng g-1 s-1. These values have been selected to obtain
the same order of magnitude of albedo decrease with snow age as in the
original Crocus formulation .
Ensemble simulation with 300 members at the Col du Lautaret site
over the 2010/2011 hydrological season. (a) – reflectance at 640 nm (center of
band 1 of MODIS), (b) SD, and (c) SWE. On each graph, the red solid line is
the simulation forced by the unperturbed SAFRAN analysis. The blue patterns
represent the envelopes including the 300 members which are shown by the
black lines.
Ensemble simulations
To investigate the impact of the stochastic perturbations, an ensemble of 300
simulations of the snowpack, forced by the 300 forcings of the meteorological
ensemble, is run over the 2010/2011 hydrological season without data
assimilation. Figure presents the result of the ensemble
simulation with 300 members (represented by the black lines). The simulation
forced by the unperturbed reanalysis (red line) is included within the
envelope of the ensemble. The spread of the ensemble reflects the
consequences of possible overestimations and underestimations of
meteorological data by the reanalysis.
The spread of the SD and SWE ensembles (Fig. 1b–c) is the largest at the
end of the season, leading to a range of 24 days from the first to the last
member to fully melt. The maximum dispersion range of SWE
(ΔSWE ≈ 300 kg m-2) occurs in early April. At this time,
the snowpack in some ensemble members has just started to melt, while in
other cases, the snowpack has already disappeared.
Snowfalls reset all members to high reflectance values (at 640 nm, 0.98 for a
significant event, Fig. a) and drastically reduce the spread
of the reflectance ensemble. Concomitantly, the SD and SWE ensemble spreads
can increase due to the uncertainties in the precipitation rates. After a
snowfall, impurity content and grain size increase along with the age of
snow, decreasing the surface reflectance. This evolution is also influenced
by atmospheric forcings, which are slightly different from one ensemble
member to another, enlarging the spread of the ensemble. We can therefore
expect that the timing of the available reflectances will strongly affect the
impact of their assimilation on the snowpack ensemble simulations.
Dispersion of the ensemble of Crocus simulations
Here we assess whether our ensemble represents a realistic spread of SD over
time with respect to previous evaluations of the model through a spread-skill
plot.
Given that no SD measurements were systematically carried out at the CdL
site, we were not able to evaluate our ensemble spread from SAFRAN-Crocus
simulations with a time series of in situ measurements at this site.
But, as demonstrated by , the ability of the ensemble spread
to depict the simulation error can be evaluated by the comparison of the RMSE
and the ensemble spread (Spd) with respect to the ensemble mean.
Firstly, using the method previously described, an ensemble of Crocus
simulations was carried out at the CdP site, with no adjustment on CdP
statistics, to evaluate the relevance of our perturbation method by comparing
the RMSE between SAFRAN-Crocus simulation and in situ measurements with the
Spd of the CdP ensemble. Then, we compare the Spd of our ensemble simulation
at the CdL site with a SAFRAN-Crocus RMSE computed from the difference
between SD Crocus estimates with in situ SD measurements across multiple
stations (at the same elevation than CdL). We used roughly 60 daily snow
depth measurements stations from the Météo-France observation stations
network (only stations within the same altitude range as the CdL site (1800–2200 m a.s.l.).
The multiple station RMSE and Spd terms are defined as follows, for a variable X,
Spd(X)=1M∑t=1M1Ne∑n=1NeXt,n-Xt¯21/2,RMSE(X)=1M∑t=1M1Nk∑k=1NkXt,kmodel-Xt,kin situ21/2,
where M represents the number of time steps, Ne the size of the ensemble
and Nk the number of in situ measurements. The SD value of the ensemble
member n at the date t is Xt,n and Xt¯ is the mean of the
ensemble at the date t. The value from SAFRAN-Crocus simulation at the
measurement site k and at the date t is given by
Xt,kmodel, and Xt,kin situ is the
value from the in situ SD measurement. RMSE and Spd are computed at
observation times. For comparisons based on only one point, the RMSE equation
for a variable X becomes
RMSE(X)=1M∑t=1MXtmodel-Xtin situ21/2.
Time evolution over the 2010/2011 season of (in red) the SD ensemble
Spd with respect to the ensemble mean and (in blue) the SD RMSE between
SAFRAN-Crocus estimates and in situ observations, (a) for the CdP site
and (b) for the CdL ensemble compared to the multiple Alps stations at the
same elevation as CdL.
Figure a shows that at the CdP site the SD dispersion
(Spd) of the ensemble is consistent with the RMSE between SAFRAN-Crocus
simulation with respect to in situ measurements at this site. This suggests
that our perturbation method is able to represent the forcing uncertainties
on snowpack simulations. Nevertheless, concerning the CdL area over the
2010/2011 season, the SAFRAN-Crocus RMSE is roughly 2 times higher than the
SD dispersion (Spd) of our ensemble (Fig. b). This means
that our ensemble is under-dispersive in terms of SD. This may be partly
explained by the calibration of perturbations, based on statistics at a
location (CdP) which is not highly affected by wind erosion/accumulation in
contrast to many other measurement sites. In addition, only meteorological
errors are considered in our ensemble, whereas the other model errors also
contribute to the simulation error.
Nonetheless, given that experiments in the present work are twin and that the
observations are selected within the ensemble (synthetic observations), the
impact of this under dispersion is not crucial, but will be considered when
using real data.
Data assimilation setup
This section describes the assimilation framework and the assimilation
strategies designed for this study prior to presenting results of
assimilation experiments (Sect. and ). First of all, the experimental setup and
diagnostics applied in this study are detailed before describing the two
synthetic observational data sets used for assimilation. An overview of the
SIR filter is given at the end of this section and further details are
provided in the Appendix .
General settings and diagnostics
The assimilation experiments are twin, meaning that the observations are
synthetics and come from a single model simulation. They are performed over
five winter seasons at the CdL area.
A synthetic truth simulation is first obtained by running Crocus, through the
use of the radiative transfer model TARTES, forced by one perturbed
meteorological forcing, as detailed in Sect. . The synthetic
observations used in all the assimilation experiments reported in Sects. and are extracted from
this synthetic truth simulation. The synthetic truth simulation is also
considered as the truth to evaluate the performance of data assimilation in terms of SD and SWE variables.
Data assimilation performances are evaluated by comparing RMSE for ensembles
with and without assimilation, and by comparing the synthetic true simulation
to the 33rd, 50th, and 67th quantiles from the ensembles with assimilation.
For a variable X, the ensemble RMSE is defined as
RMSE(X)=1Ne∑n=1NeXn-Xtruth21/2,
where Ne represents the size of the ensemble, Xn the value from the
ensemble member n, and Xtruth the value from the
synthetic truth. RMSEs are computed at observation times. The uncertainty on
the melt-out date is quantified as the difference (in days) between the first
and the latest full melted member.
Nature of the assimilated observations
The first set of synthetic observations is composed of surface reflectances
of the first seven bands of MODIS (central wavelengths: 460, 560, 640, 860,
1240, 1640, 2120 nm; ). In twin context, these synthetic
observations are provided from the synthetic truth simulation running Crocus
with its radiative TARTES model. Snow surface reflectances in the visible and
near-infrared spectra are sensitive to the properties of the first
millimeters to the first centimeters of the snowpack for a given wavelength
. They mainly vary with snow microstructure (near-infrared
part) and impurity content (visible part) . The reflectance
observations error variances, necessary for the assimilation, are defined
according to . They are prescribed to 7.1× 10-4, 4.6× 10-4,
5.6× 10-4, 5.6× 10-4, 2.0× 10-3, 1.5× 10-3 and
7.8× 10-4, for the seven bands, respectively. In the framework of our
twin experiments, the covariance matrix of observation errors is diagonal.
Note that the TARTES model calculates bi-hemispherical reflectances while the
satellite measurements provide directly hemispherical-conical top of
atmosphere reflectances .
The second set of observations is composed of synthetic snow depth (SD)
observations. Previous studies have indeed reported that the assimilation of
snowpack bulk variables such as SD greatly improve snow estimations
. However, SD observations are only available at
one point. In our study, the observation error variance of SD is taken to be
0.003 m (corresponding to a standard deviation of about 5 cm). The impact of
synthetic SD assimilation is detailed in Sect. .
The setup designed in our study (one point, twin experiments) allows relevant
comparisons of the benefits of assimilating separately or jointly the two
above mentioned types of observations. For future works assimilating real
data, the difference in the geometrical configuration between the simulated
TARTES reflectances and satellite observations will be
addressed.
Assimilation method: the particle filter
The data assimilation method has been chosen after considering the
requirements and the possible degrees of freedom that our problem imposes or
offers.
Firstly, we require that the method quantifies uncertainties. This plays in
favor of ensemble methods e.g.,. Secondly, we
prefer an already existing and well tested method. This argues for the
Ensemble Kalman Filter (EnKF, ) or the particle filter
. Thirdly, the method should not rely on
assumptions about the physical system, such as linearity or weak
nonlinearity, because the physics of our model are nonlinear. This draws us
toward the particle filter. Fourthly, the method should be easy to implement
for this first study. assessed the effectiveness
assimilating streamflow data using an EnKF sequential procedure but
implemented in a simpler snow scheme than Crocus. The fact that the EnKF
involves state-averaging operations, to which Crocus hardly complies due to
its varying number of snowpack layers, argues in favor of the particle
filter. Note that also chose the SIR filter for the
assimilation of microwave radiances in a snowpack model. The major drawback
of the particle filter is that it is not applicable to high-dimensional
systems because it quickly degenerates (all ensemble
members converge toward a unique and spurious model trajectory). But our
model, with hardly more than a few hundreds of variables, is not
high-dimensional. Our experiments show it indeed does not degenerate if a
well-tested resampling method is used, with ensembles of a few hundreds of
members only. Thus, we choose the sequential importance resampling (SIR)
filter , which is a particular type of the particle
filter. Our ensembles are composed of 300 members.
The SIR filter seeks to represent the probability density function (pdf) of
the model state by a discrete set (an ensemble) of states commonly called
particles. The propagation over time of all particles, through the nonlinear
model equations, describes the evolution of the model pdf. When observations
are available, the ensemble is updated following two steps: (i) the particles
are weighted according to their respective distances from the observations,
and (ii) the pdf defined by the newly weighted particles is resampled by
ruling out particles with negligible weights, and duplicating particles with
large weights, so that the updated pdf is again represented by an ensemble of
equally weighted particles. The new ensemble is then ready to be propagated
in time by the model. As long as a particle is not removed, it keeps its
original perturbed forcing to be propagated. Inversely, a new perturbed
forcing is assigned to a duplicated particle for propagation to the next
analysis. The governing equations of the data assimilation scheme are given
in the Appendix and more details are presented in
.
Assimilation of MODIS-like reflectances
In this section, we assess to what extent the assimilation of the available
MODIS-like reflectance observations allows the accurate estimation of
snowpack properties throughout the season. This experiment will be considered
as our baseline experiment.
Evolution of the ensemble over the 2010/2011 season, (a) and (b) – reflectance at 640 and 1240 nm
(first and fifth MODIS band, respectively), (c) SD and (d) SWE. The blue
shading represent the envelopes of the ensemble assimilating MODIS-like
reflectances and the grey shading the envelopes of the ensemble without
assimilation. The red lines represent the control simulation (synthetic
truth). In graph (a) and (b), the red dots show the assimilated observations.
In both (c) and (d), the black solid line shows the 50 % quantiles
(median of the ensemble) and the black dotted lines the 33 and 67 %
quantiles for the baseline experiment.
Data assimilation results for the 640 and 1240 nm reflectance (first and
fifth MODIS bands) and for SD and SWE over the hydrological season 2010/2011
are shown in Fig. . To mimic real cloud conditions,
reflectances are assimilated at 34 clear sky days of the season. We define a
clear sky date according to the real cloud mask from MODIS data computed with
the method of . The corresponding 640 and 1240 nm
synthetic reflectance observations are shown by the red dots in Fig. a and b. The control simulation (from which the
synthetic observations are drawn) is shown by the red lines.
Throughout the season, the envelopes of SD and SWE ensembles for the baseline
experiment (Fig. , blue envelopes) include the control
simulation, which is a prerequisite for the good behavior of the
assimilation. Overall, the assimilation of reflectance observations reduces
the uncertainties in the estimation of the snowpack characteristics
throughout the season. This is observed in Fig. , where
the baseline experiment envelopes (blue shading) are narrower than those of
the ensemble without assimilation (grey shading). In particular, the snow
melt-out date is estimated much more accurately with the assimilation of
reflectances: the uncertainty range drops from 24 days without assimilation
to 9 days with assimilation.
SD and SWE seasonal averaged RMSE computed with respect to the
synthetic truth for all experiments over the 2010/2011 season. Results
reported in Fig. are the RMSE computed over the five
selected seasons.
Time evolution of the ensemble RMSEs on (a) reflectance at 640 nm,
(b) reflectance at 1240 nm, (c) SD and (d) SWE, over the 2010/2011 season,
for the run without assimilation (red lines), and the baseline assimilation
experiment (blue solid line: forecast; blue dotted line: analysis). Dots
indicate analysis steps.
Figure shows the time evolution of the RMSE with assimilation
at every observation time, at the end of the forecast step (blue solid line)
and just after the filter analysis (blue dotted line). These results are
compared to the RMSE without assimilation (red lines). The RMSE of the
ensemble with assimilation is always lower than the RMSE without
assimilation. Averaged over the season, a reduction of 46 % was obtained for
SD and 44 % for SWE, (Table – Baseline: seasonal RMSE for SD: 0.07 m;
SWE: 19.7 kg m-2 compared to 0.13 m and 35.4 kg m-2 from the
ensemble without assimilation). These results indicate the usefulness of
using spectral optical radiance rather than albedo data since
obtained an improvement in SD estimate of only 14 % when
assimilating albedo retrievals from MODIS sensor. It is remarkable that,
despite the significant RMSE reduction in our experiment, there is most of
the time no strong reduction of the RMSE from a single analysis. The reduced
RMSEs with assimilation are consequently due to the successive observations
throughout the season, highlighting the role of model dynamics.
The strongest RMSE reductions occur right after extended periods without
precipitation and without available observations, when the reflectance
ensemble spread is particularly pronounced (e.g., Fig. a). During these periods (e.g., from 7 to 14 December 2010, or from
11 to 21 January 2011), the ensemble uncertainties on reflectances, SD and
SWE grow under the influence of the perturbed forcings including the
perturbed impurity deposition rate. Observations of reflectances have a large
impact when they are used. However, since reflectance observations are not
very sensitive to the inner snowpack hidden by recent snowfalls, the
uncertainties on SD and SWE accumulated earlier and not corrected by past
analysis remain, which ultimately results in limited corrections on SD and
SWE (Fig. , for example, on 28 January 2011), and sustained
ensemble spreads and RMSE throughout the season.
After a significant snowfall, the uncertainties in SD and SWE may increase,
and the assimilation of reflectances generally has a very small impact on
these two variables. Indeed, the uncertainty in the amount of snowfall
(translated here in perturbations on the snowfall rate) tends to increase the
ensemble spread and RMSE on SD and SWE. Moreover, whether it be in the
visible range of wavelengths sensitive to the impurity content or in the
infrared part where changes on the microstructure dominates, a snowfall
resets all ensemble members to the same set of reflectance values. This makes
the discrimination between members using reflectances alone impossible, and
the subsequent analysis provides a rather small uncertainty reduction for SD
and SWE. This is illustrated on Fig. on 10 November and
on 1 December 2010, for example.
The remarks stated above for the season 2010/2011 hold for the other seasons.
Figure reports the time evolution of the SD and SWE RMSEs
for all the selected seasons, in the experiments without assimilation (red
lines) and with assimilation of reflectances (blue line; the experiments
shown in green and black are discussed in the next section). On average, SD
and SWE RMSEs are reduced by 45 and 48 %, respectively. This is comparable
with results of , who assimilate radiances in the microwave
spectrum from AMSR-E, and reduce the SD RMSE by 50 %. However, passive
microwave observations are very sensitive to liquid water. Consequently, the
performance of the assimilation during the melting period is reduced
( reduce the SD RMSE up to 61 % from January to March, during
only the dry snow period). In contrast, our results show a well-marked
reduction of errors near the end of the seasons (Fig. ,
red lines and blue dotted lines). Our results are also consistent with those from
assimilating MODIS-derived snow cover fractions (SCFs), after a
processing of the retrieval to improve accuracy of cloud coverage and snow
mapping. Without this processing, the performance of SCF assimilation falls,
with a SWE RMSE reduction near 10–20 %, similarly to .
Consequently, our ability to control the seasonal evolution of the snowpack
with the assimilation of reflectances is demonstrated, though it exhibits
limitations. In particular, the reduction of the snowpack SD and SWE ensemble
spread greatly depends on the timing of the assimilated observations.
Sensitivity to the nature and the timing of observationsImpact of cloud coverage on the experiment
The presence of cloud coverage strongly reduces the number of optical data
available for assimilation. To investigate impact of limiting the number of
available observations, an experiment similar to the baseline experiment (see
Sect. ) is carried out, but assimilation is performed
every day, (134 days) instead of 34 days in the baseline experiment.
Figure S2 presents the results with the blue shading representing
the envelopes of the ensemble assimilating daily MODIS-like observations and
the grey shading representing the envelopes of the baseline experiment, reported from
Fig. .
Time evolution
of ensemble RMSEs on SD (left) and SWE (right) for the five seasons under
study, for the run without assimilation (red lines), the baseline experiment
(assimilating reflectances, blue lines), the experiment assimilating SD data
(green lines) and the experiment assimilating combined reflectances and SD
data (black lines). Crosses indicate analysis steps. Seasonal averages are
displayed in the upper left corner of each graph. The model control
simulation is represented by the grey lines, scaled by the “Synthetic
truth”
y axes.
Obviously, in this second experiment, concerning the 640 nm reflectance
variable, the spread of the ensemble is greatly reduced, efficiently fitting the
observations (red dots) and its envelope does not show any extended periods
with a large range of reflectance values anymore (Fig. S2a). Compared to the baseline experiment (grey envelopes), the uncertainty
in the snow melt-out date is also reduced to 3 days. However during the major
part of the winter, the SD and SWE ensemble spreads (Fig. S2b–c: blue envelopes) are comparable to the spreads obtained in the
baseline experiment (Fig. S2b–c: grey envelopes). This
is also reflected in Table – All days: The seasonal RMSEs on SD and
SWE are 0.05 m and 14.4 kg m-2, respectively, against 0.07 m and 19.7 kg m-2
in the baseline experiment. This shows that assimilating a
limited number data due to realistic cloud conditions is not necessarily
harmful to the estimation of the snowpack state. Note that this conclusion
holds here for bulk variables such as SD and SWE. The estimation of other
physical properties of the snowpack will be addressed in a future work using
real observations.
On the timing of observations
The baseline experiment suggests that the timing of observations may largely
determine the quality of the assimilation process. To explore the role of the
timing, four additional assimilation tests are designed for which MODIS-like
reflectances are assimilated (i) only at the beginning of the season (before
31 December 2010, Fig. S3: Accu), (ii) only in the second part of
the snow season (after 31 December 2010, Fig. S4: Melt), (iii) only
after several day-long periods without precipitation (Fig. S5: Before Snowf) and (iv) only right after snowfall events
(Fig. S6: After Snowf).
In case i (Accu), results show that even if the SD and SWE spreads are
reduced during the assimilation period, the assimilation has almost no effect
on the snow estimates during the snow melt period. The ensemble spread at the
end of the season returns to almost the same value than the experiment
without assimilation. The uncertainty of the snow melt-out date is reduced to
only 22 days, compared to 24 days without assimilation. As for case ii (Melt), the spread reduction becomes quite discernible roughly 2 months
after the first assimilation date and never reaches the range of the baseline
experiment. The uncertainty of the snow melt-out date is however reduced to
11 days. This demonstrates that it is essential to assimilate reflectances
over the entire season to compensate the fast growth of the snowpack ensemble
in response to the uncertainties in the meteorological forcing.
In both cases iii (Before Snowf) and iv (After Snowf), reflectances are
assimilated at only seven dates of the season. Case iii (Before Snowf) exhibits
a more pronounced SD and SWE spreads reduction compared to case iv (After Snowf). The uncertainty on the snow melt-out date drops to 9 days in case iii (Before Snowf), while it stays at 23 days in case iv (After Snowf). In
absence of precipitation, the snow surface is aging, leading to a decrease of
reflectance values and a spread of the reflectance ensemble (Fig. S5a). Therefore, an observation after such a period
provides a significant amount of information and produces an efficient
analysis. On the contrary, solid precipitation resets the reflectance to high
values and limits the spread of the reflectance ensemble (Fig. S6a) leading to a limited efficiency of the ensemble
analysis. Assimilating only a few synthetic observations well distributed in
time nearly leads to the same uncertainty of SD and SWE estimates as the
baseline experiment assimilating 34 observations (Table –
Before Snowf: seasonal RMSE SD: 0.07 m; SWE: 21.8 kg m-2 compared to
baseline experiment 0.07 m and 19.7 kg m-2, respectively).
Consequently, the time distribution of the observation turns out to be a key
element in the expected success of the assimilation of reflectance
observations. The end of an extended period without precipitation, when the
surface snow layer is aging, is the best time to assimilate reflectances.
Assimilation of snow depths
To better evaluate the impact of the reflectance assimilation, we here
compare the baseline experiment to an experiment assimilating synthetic SD
observations keeping the same time distribution of the observations. Apart
from the different nature of the observations, the assimilation setup is the
same as the one described in Sect. including the time
frequency of observations. The results are displayed in Fig. S7.
The assimilation of synthetic SD observations greatly improves the estimates
of SD and SWE (Fig. S7b–c). The spread reduction is much
stronger than with the assimilation of reflectance observations (Table – SD, Clear sky days:
the seasonal RMSE on SD and SWE are 0.03 m and 7.4 kg m-2, respectively,
against 0.07 m and 19.7 kg m-2 in the
baseline experiment) and is maintained throughout the season. The uncertainty
range on the snow melt-out date is decreased to 8 days compared to 9
assimilating MODIS-like reflectances and 24 days without assimilation for the
2010/2011 season. Note that the spread reduction of the reflectance ensemble
is very limited compared to the baseline experiment. This is consistent with
the fact that while SD and SWE are better estimated in the case of SD
simulation, the surface and inner physical properties of the snowpack are
less impacted than in the case of assimilating reflectance observations.
Figure S7 shows that, at the beginning of the snow season
(before 16 November 2010) and for a thin snowpack (less than 20 cm), SD
assimilation seems to have less impact than reflectance assimilation. Indeed,
with a thin snowpack, visible wavelengths penetrate down to the ground, and
reflectance contains information on the whole snowpack. In this case,
reflectance contains more information than SD. This could explain the better
performance of the baseline experiment.
An additional experiment (not shown here) was also conducted assimilating
daily synthetic SD observations because such measurements are usually available daily at about 60 different stations in the French Alps. This shows
that, contrary to reflectance assimilation, for SD assimilation, the more
frequent the observations, the greater the spread reduction (seasonal RMSE
SD: 0.02 m; SWE: 4.7 kg m-2).
Except for thin snow cover, the assimilation of SD observations outperforms
reflectance assimilation in terms of SWE and SD estimates and seems to be
less affected by the time distribution of the observations. When assimilating
reflectance data, the ensemble needs to sufficiently spread (from an extended
period without precipitation) to observe an impact of the assimilation (Fig. a). Inversely, and even if the reduction may be very small,
every SD observations assimilation reduces the SD ensemble independently of
the precipitation events (Fig. S7, excepted for thin snow
cover).
Figure also shows that, all these findings obtained for
the 2010/2011 season are also verified for the five studied seasons. All
assimilation experiments of synthetic SD observations reduce the RMSE with
respect to both the model run without assimilation (red lines) and the
experiments assimilation synthetic reflectances data (blue lines). However,
in case of shallow snowpack, better performance is obtained using
reflectance data.
Combining reflectance and snow depth assimilation
Though the assimilation of synthetic SD observations generally outperforms
MODIS-like reflectance assimilation, spatially distributed SD measurements
are rarely available over large areas on a daily basis. In situ SD
observations give information only at the measurement point and many studies
attest to the strong spatial variability of the snow cover
(e.g., ; ; ). Airborne lidar or ground-based laser lidar provide accurate SD measurements with fine resolution, but
their low temporal frequency limits their utility for operational
applications. So, one can imagine that over a mountain range, SD measurements
are available at several locations for only a few dates in the season (e.g.,
occasional snow course, crowd-sourcing, ski resorts observations, …). This
scenario motivates the set-up of the following experiment. The experimental
setup is the same as the baseline reflectance assimilation scheme previously
described with an extra synthetic SD observation the 10th of each month.
Results are shown in Fig. S8 and compared to the
previous experiments in Fig. .
Combining the assimilation of MODIS-like reflectances with the assimilation
of synthetic SD observations provides a benefit compared to assimilating
reflectance only (Fig. , black and blue lines
respectively). (i) In presence of a thin snow cover, the SD and SWE RMSEs of
the combined reflectances and SD ensembles are reduced as the ones from the
assimilation of the reflectance only. (ii) Almost all along the season, SD
and SWE RMSEs remain below the reflectance assimilation RMSE thanks to SD
assimilation. The combined assimilation leads to SWE seasonal RMSE of 9.6 kg m-2 to be compared
to 7.4 kg m-2 for the experiment
assimilating synthetic SD observations and 19.7 kg m-2 for the baseline
reflectances assimilation experiment (Table ).
These results indicate the usefulness of combining these two data sets in
operational applications. reached a similar conclusion by
combining the assimilation of SCF and SD (with a SWE RMSE reduction up to
72 %; up to 74 % in our study). However, given the strong spatial variability
of the snow cover, the spatial representativity of punctual SD measurements
may make their assimilation questionable. This issue should be addressed with
experiments over two-dimensional, realistic domains.
Conclusions
This study investigates the assimilation of MODIS-like reflectances from
visible to near-infrared (the first seven bands) into the multilayer snowpack
model Crocus. The direct use of reflectance data instead of higher level snow
products limits the introduction of uncertainties due to retrieval
algorithms. For the assimilation, we implement a particle filter. A particle
filter is chosen because (i) it is an ensemble method providing uncertainty
estimates, and (ii) it is easily implemented (in comparison with other
assimilation methods) with Crocus model, characterized by strong
nonlinearities and its lagrangian representation of the snowpack layering.
Given that the major source of error in snowpack simulations can be
attributed to meteorological forcings, a stochastic perturbation method is
designed to generate an ensemble of possible meteorological variables. This
algorithm uses a first-order autoregressive model to account for the temporal
correlations in the meteorological forcing uncertainties. This ensemble of
meteorological forcings is then applied to generate the ensemble of snowpack
simulations for the assimilation. Twin experiments are conducted at one point
in the French Alps, the Col du Lautaret, over five hydrological years. The
assimilated reflectance data corresponds to the first seven spectral bands of
the MODIS sensors.
Reflectance assimilation using only data from clear-sky days reduces the SD
and SWE seasonal RMSE by a factor close to 2. The uncertainty range on the
snow melt-out date drops to 9 days compared to 24 without assimilation.
Additional assimilation tests using different time distributions of the
observations show that (i) reflectance assimilation greatly improves snowpack
estimates if the observation comes after an extended period without
precipitation, (ii) the assimilation has almost no impact if it comes right
after a snowfall, and (iii) using only a few observations with the
appropriate timing, i.e., after extended periods without precipitation,
reduces RMSE almost as much as assimilating reflectances on a daily basis.
The assimilation of synthetic SD observations leads to a decrease of SD and
SWE RMSE by a factor of more than 4. The uncertainty range on the snow
melt-out date is reduced to 8 days. The assimilation of SD observations
generally outperforms reflectance assimilation except for shallow snowpacks,
typically less than 20 cm. However, whereas optical reflectance maps can be
obtained daily thanks to spaceborne sensors such as MODIS or VIIRS, SD
measurements are rarely available either over large areas or at the same time
frequency. Combining reflectance assimilation with SD assimilation at four dates
during the snow season leads to a decrease of SD and SWE RMSE by a factor
close to 3.
This study provides a general theoretical framework to test the efficiency of
several kinds of data assimilation in a snowpack model. This also highlights the
benefit of using remotely sensed optical surface reflectance in the
assimilation scheme to provide significant improvements of the snowpack SD
and SWE estimates. Even if the assimilation of SD outperforms the
assimilation using reflectance data, the sparsity of in situ
measurements in space and/or time strongly reduces their utility in real data
assimilation systems. Nevertheless, given their complementary features,
combining remotely sensed reflectances and SD data, when available, would
definitely improve snowpack simulations.
This study presents a first attempt to assimilate snow observations into the
Crocus snowpack model with the overarching objective of improving operational
snowpack forecasting. The next steps to proceed toward operational
applications must include the assimilation of actual satellite data and the
spatialization of the assimilation on larger domains. These steps include
several challenges such as the increased calculation costs and degrees of
freedom, and the need for a physically consistent 2-D meteorological ensemble,
which will be addressed in future work.
Particle filter and sequential importance resampling, definitions (Gordon et al., 1993; Van Leeuwen, 2009, 2014)
In a discrete-time space model, the state of a system evolves according to
xk=fk(xk-1,vk-1),
where xk is the state vector of the system at time k, vk-1 is the
state noise vector and fk is the non-linear and time-dependent function
describing the evolution of the state vector.
Information about xk is obtained through noisy measurements, yk, which
are governed by the observation operator equation:
yk=hk(xk,nk),
where hk is a possibly non-linear and time-dependent function linking the
state vector to the observation (observation operator) and nk is the
measurement noise vector.
The filtering problem is to estimate sequentially the values of xk, given
the observed values y0, …, yk, at any time step k. In a Bayesian
setting, this problem can be formalized as the computation of the
distribution p(xk|y1:k), which can be done recursively in the
following two steps.
Updating step to estimate p(xk|y1:k) using Bayes' rule:
p(xk|y1:k)∝p(yk|xk)p(xk|y1:k-1).
In the particle filter, the prior pdf is represented by equally weighted delta functions centered on the ensemble members or particles:
p(xk-1|y1:k-1)=1N∑i=1Nδ(xk-1-xk-1i),
where N is the ensemble size. With this representation, the propagation step Eq. () provides
p(xk|y1:k-1)=1N∑i=1Nδxk-xki,
where xki=f(xk-1i,vk-1i); vk-1i is a realization of the noise vk-1. Then the analysis step follows with
p(xk|y1:k)=∑i=1Nwkiδ(xk-xki),
where the wki are the particle weights, normalized to sum up to 1, and given by
wki∝p(yk|xki).
To compute the weights, the error nk of the observation operator hk
(Eq. ) is often considered additive and Gaussian with mean 0 and
covariance matrix Rk, so that the likelihood p(yk|xki) writes
p(yk|xki)∝exp-12(yk-h(xki))TR-1(yk-h(xki)).
After the computation of the weights, the ensemble is resampled: particles
with zero or negligible weights are ruled out; particles with large weights
are duplicated a number of times commensurate with their weights. Several
algorithms exist for this resampling step; we use the one of Kitagawa
.
The Supplement related to this article is available online at doi:10.5194/tc-10-1021-2016-supplement.
Acknowledgements
We wish to thank the two anonymous reviewers for their detailed comments. We
gratefully acknowledge funding from Labex OSUG@2020 (Investissements d'avenir
– ANR10 LABX56) and Fondation Eau Neige et Glace. This work has also been
supported by the INSU/LEFE/MANU program. The development of the radiative
model TARTES has been funded by the French ANR MONISNOW project, number
ANR-11-JS56-0005.
Edited by: X. Fettweis
References
Abaza, M., Anctil, F., Fortin, V., and Turcotte, R.: Exploration of sequential
streamflow assimilation in snow dominated watersheds, Adv. Water Resour., 80, 79–89, 2015.Andreadis, K. M. and Lettenmaier, D. P.: Assimilating remotely sensed snow
observations into a macroscale hydrology model, Adv. Water Resour., 29, 872–886, 10.1016/j.advwatres.2005.08.004, 2006.Bartelt, P. and Lehning, M.: A physical SNOWPACK model for the Swiss avalanche
warning: Part I: numerical model, Cold Reg. Sci. Technol., 35,
123–145, 10.1016/S0165-232X(02)00074-5, 2002.Bavay, M., Grünewald, T., and Lehning, M.: Response of snow cover and
runoff to climate change in high Alpine catchments of Eastern Switzerland,
Adv. Water Resour., 55, 4–16,
10.1016/j.advwatres.2012.12.009, 2013.
Blayo, É., Bocquet, M., Cosme, E., and Cugliandolo, L. F.: Advanced Data
Assimilation for Geosciences: Lecture Notes of the Les Houches School of
Physics: Special Issue, June 2012, Oxford University Press, Oxford, UK, 2014.Boone, A. and Etchevers, P.: An intercomparison of three snow schemes of
varying complexity coupled to the same land surface model: Local-scale
evaluation at an Alpine site, J. Hydrometeorol., 2, 374–394,
10.1175/1525-7541(2001)002<0374:AIOTSS>2.0.CO;2,
2001.
Brun, E., Martin, E., Simon, V., Gendre, C., and Coléou, C.: An energy
and mass model of snow cover suitable for operational avalanche forecasting,
J. Glaciol., 35, 333–342, 1989.Brun, E., David, P., Sudul, M., and Brunot, G.: A numerical model to simulate
snow-cover stratigraphy for operational avalanche forecasting, J. Glaciol.,
38, 13–22, http://refhub.elsevier.com/S0165-232X(14)00138-4/rf0155,
1992.Brun, F., Dumont, M., Wagnon, P., Berthier, E., Azam, M. F., Shea, J. M.,
Sirguey, P., Rabatel, A., and Ramanathan, Al.: Seasonal changes in surface
albedo of Himalayan glaciers from MODIS data and links with the annual mass
balance, The Cryosphere, 9, 341–355, 10.5194/tc-9-341-2015, 2015.Bühler, Y., Marty, M., Egli, L., Veitinger, J., Jonas, T., Thee, P., and
Ginzler, C.: Snow depth mapping in high-alpine catchments using digital
photogrammetry, The Cryosphere, 9, 229–243, 10.5194/tc-9-229-2015, 2015.Carmagnola, C. M., Morin, S., Lafaysse, M., Domine, F., Lesaffre, B.,
Lejeune, Y., Picard, G., and Arnaud, L.: Implementation and evaluation of
prognostic representations of the optical diameter of snow in the
SURFEX/ISBA-Crocus detailed snowpack model, The Cryosphere, 8, 417–437,
10.5194/tc-8-417-2014, 2014.Carpenter, T. M. and Georgakakos, K. P.: Impacts of parametric and radar
rainfall uncertainty on the ensemble streamflow simulations of a distributed
hydrologic model, J. Hydrol., 298, 202–221, 10.1016/j.jhydrol.2004.03.036, 2004.Castebrunet, H., Eckert, N., Giraud, G., Durand, Y., and Morin, S.: Projected
changes of snow conditions and avalanche activity in a warming climate: the
French Alps over the 2020–2050 and 2070–2100 periods, The Cryosphere, 8,
1673–1697, 10.5194/tc-8-1673-2014, 2014., 2014.
Che, T., Li, X., Jin, R., and Huang, C.: Assimilating passive microwave remote
sensing data into a land surface model to improve the estimation of snow
depth, Remote Sens. Environ., 143, 54–63, 2014.
Clark, M. P., Slater, A. G., Barrett, A. P., Hay, L. E., McCabe, G. J.,
Rajagopalan, B., and Leavesley, G. H.: Assimilation of snow covered area
information into hydrologic and land-surface models, Adv. Water Res., 29, 1209–1221, 2006.Cordisco, E., Prigent, C., and Aires, F.: Snow characterization at a global
scale with passive microwave satellite observations, J. Geophys. Res.-Atmos. (1984–2012), 111, D19102, 10.1029/2005JD006773, 2006.Dechant, C. and Moradkhani, H.: Radiance data assimilation for operational snow
and streamflow forecasting, Adv. Water Resour., 34, 351–364,
10.1016/j.advwatres.2010.12.009, 2011.De Lannoy, G. J. M., Reichle, R. H., Arsenault, K. R., Houser, P. R., Kumar,
S., Verhoest, N. E. C., and Pauwels, V. R. N.: Multiscale assimilation of
Advanced Microwave Scanning Radiometer–EOS snow water equivalent and
Moderate Resolution Imaging Spectroradiometer snow cover fraction
observations in northern Colorado, Water Resour. Res., 48, W01522, 10.1029/2011WR010588, 2012.Deodatis, G. and Shinozuka, M.: Auto-regressive model for nonstationary
stochastic processes, J. Eng. Mech.-ASCE, 114, 1995–2012,
10.1061/(ASCE)0733-9399(1988)114:11(1995), 1988.Doherty, S. J., Grenfell, T. C., Forsström, S., Hegg, D. L., Brandt, R. E.,
and Warren, S. G.: Observed vertical redistribution of black carbon and other
insoluble light-absorbing particles in melting snow, J. Geophys. Res.-Atmos., 118, 5553–5569, 10.1002/jgrd.50235, 2013.Domine, F., Sparapani, R., Ianniello, A., and Beine, H. J.: The origin of sea
salt in snow on Arctic sea ice and in coastal regions, Atmos. Chem. Phys., 4,
2259–2271, 10.5194/acp-4-2259-2004, 2004.Dong, J., Walker, J. P., Houser, P. R., and Sun, C.: Scanning multichannel
microwave radiometer snow water equivalent assimilation, J.
Geophys. Res.-Atmos. (1984–2012), 112, D07108, 10.1029/2006JD007209, 2007.Dumont, M., Durand, Y., Arnaud, Y., and Six, D.: Variational assimilation of
albedo in a snowpack model and reconstruction of the spatial mass-balance
distribution of an alpine glacier, J. Glaciol., 58, 151–164,
10.3189/2012JoG11J163, 2012.Dumont, M., Brun, E., Picard, G., Michou, M., Libois, Q., Petit, J., Geyer, M.,
Morin, S., and Josse, B.: Contribution of light-absorbing impurities in snow
to Greenland/'s darkening since 2009, Nat. Geosci., 7, 509–512, 10.1038/ngeo2180, 2014.Durand, M., Kim, E. J., and Margulis, S. A.: Radiance assimilation shows
promise for snowpack characterization, Geophys. Res. Lett., 36, L02503, 10.1029/2008GL035214, 2009.Durand, Y., Brun, E., Mérindol, L., Guyomarc'h, G., Lesaffre, B., and
Martin, E.: A meteorological estimation of relevant parameters for snow
models, Ann. Glaciol., 18, 65–71,
http://www.igsoc.org/annals/18/igs_annals_vol18_year1993_pg65-71.html, 1993.
Durand, Y., Giraud, G., Brun, E., Mérindol, L., and Martin, E.: A
computer-based system simulating snowpack structures as a tool for regional
avalanche forecasting, J. Glaciol., 45, 469–484, 1999.Essery, R., Morin, S., Lejeune, Y., and Menard, C. B.: A comparison of 1701
snow models using observations from an alpine site, Adv. Water
Res., 55, 131–148, 10.1016/j.advwatres.2012.07.013, 2013.Etchevers, P., Golaz, C., and Habets, F.: Simulation of the water budget and
the river flows of the Rhone basin from 1981 to 1994, J. Hydrol.,
244, 60–85, 10.1016/S0022-1694(01)00332-8, 2001.
Evensen, G.: Data assimilation: the ensemble Kalman filter, Springer Science
and Business Media, Berlin, Germany, 2009.Fortin, V., Abaza, M., Anctil, F., and Turcotte, R.: Why should ensemble spread
match the RMSE of the ensemble mean, J. Hydrometeorol., 15,
1708–1713, 10.1175/JHM-D-14-0008.1, 2014.Foster, J. L., Sun, C., Walker, J. P., Kelly, R., Chang, A., Dong, J., and
Powell, H.: Quantifying the uncertainty in passive microwave snow water
equivalent observations, Remote Sens. Environ., 94, 187–203,
10.1016/j.rse.2004.09.012, 2005.Gabbi, J., Huss, M., Bauder, A., Cao, F., and Schwikowski, M.: The impact of
Saharan dust and black carbon on albedo and long-term mass balance of an
Alpine glacier, The Cryosphere, 9, 1385–1400, 10.5194/tc-9-1385-2015,
2015.
Gordon, N. J., Salmond, D. J., and Smith, A. F.: Novel approach to
nonlinear/non-Gaussian Bayesian state estimation, IEE Proc.-F, 140, 107–113, 1993.Hall, D. K. and Riggs, G. A.: Accuracy assessment of the MODIS snow
products, Hydrol. Process., 21, 1534–1547, 10.1002/hyp.6715, 2007.
Jordan, R.: A One-Dimensional Temperature Model for a Snow Cover: Technical
Documentation for SNTHERM. 89., Tech. rep., Cold Regions Research and
Engineering Lab, Hanover, NH, USA, 1991.
Kitagawa, G.: Monte Carlo filter and smoother for non-Gaussian nonlinear state
space models, J. Comput. Graph. Stat., 5, 1–25,
1996.Lehning, M., Völksch, I., Gustafsson, D., Nguyen, T. A., Stähli, M.,
and Zappa, M.: ALPINE3D: a detailed model of mountain surface processes and
its application to snow hydrology, Hydrol. Process., 20, 2111–2128,
10.1002/hyp.6204, 2006.Li, W., Stamnes, K., Chen, B., and Xiong, X.: Snow grain size retrieved from
near-infrared radiances at multiple wavelengths, Geophys. Res.
Lett., 28, 1699–1702, 10.1029/2000GL011641, 2001.Libois, Q., Picard, G., France, J. L., Arnaud, L., Dumont, M., Carmagnola, C.
M., and King, M. D.: Influence of grain shape on light penetration in snow,
The Cryosphere, 7, 1803–1818, 10.5194/tc-7-1803-2013, 2013.Libois, Q., Picard, G., Dumont, M., Arnaud, L., Sergent, C., Pougatch, E.,
Sudul, M., and Vial, D.: Experimental determination of the absorption
enhancement parameter of snow, J. Glaciol., 60, 714–724,
10.1002/2014JD022361, 2014.Liu, Y., Peters-Lidard, C. D., Kumar, S., Foster, J. L., Shaw, M., Tian, Y.,
and Fall, G. M.: Assimilating satellite-based snow depth and snow cover
products for improving snow predictions in Alaska, Adv. Water
Res., 54, 208–227, 10.1016/j.advwatres.2013.02.005, 2013.López-Moreno, J. I., Fassnacht, S. R., Beguería, S., and Latron, J.
B. P.: Variability of snow depth at the plot scale: implications for mean
depth estimation and sampling strategies, The Cryosphere, 5, 617–629,
10.5194/tc-5-617-2011, 2011.Masson, V., Le Moigne, P., Martin, E., Faroux, S., Alias, A., Alkama, R.,
Belamari, S., Barbu, A., Boone, A., Bouyssel, F., Brousseau, P., Brun, E.,
Calvet, J.-C., Carrer, D., Decharme, B., Delire, C., Donier, S., Essaouini,
K., Gibelin, A.-L., Giordani, H., Habets, F., Jidane, M., Kerdraon, G.,
Kourzeneva, E., Lafaysse, M., Lafont, S., Lebeaupin Brossier, C., Lemonsu,
A., Mahfouf, J.-F., Marguinaud, P., Mokhtari, M., Morin, S., Pigeon, G.,
Salgado, R., Seity, Y., Taillefer, F., Tanguy, G., Tulet, P., Vincendon, B.,
Vionnet, V., and Voldoire, A.: The SURFEXv7.2 land and ocean surface platform
for coupled or offline simulation of earth surface variables and fluxes,
Geosci. Model Dev., 6, 929–960, 10.5194/gmd-6-929-2013, 2013.Morin, S.: Observation and numerical modeling of snow on the ground: use of
existing tools and contribution to ongoing developments, Habilitation à
diriger des recherches, Université Joseph Fourier, Grenoble, France, available at: https://tel.archives-ouvertes.fr/tel-01098576 (last access: 12 May 2016), 2014.Morin, S., Lejeune, Y., Lesaffre, B., Panel, J.-M., Poncet, D., David, P.,
and Sudul, M.: An 18-yr long (1993–2011) snow and meteorological dataset from
a mid-altitude mountain site (Col de Porte, France, 1325 m alt.) for driving
and evaluating snowpack models, Earth Syst. Sci. Data, 4, 13–21,
10.5194/essd-4-13-2012, 2012.Navari, M., Margulis, S. A., Bateni, S. M., Tedesco, M., Alexander, P., and
Fettweis, X.: Feasibility of improving a priori regional climate model
estimates of Greenland ice sheet surface mass loss through assimilation of
measured ice surface temperatures, The Cryosphere, 10, 103–120,
10.5194/tc-10-103-2016, 2016.Noilhan, J. and Planton, S.: A simple parameterization of land surface
processes for meteorological models, Mon. Weather Rev., 117, 536–549,
10.1175/1520-0493(1989)117<0536:ASPOLS>2.0.CO;2, 1989.Painter, T. H., Barrett, A. P., Landry, C. C., Neff, J. C., Cassidy, M. P.,
Lawrence, C. R., McBride, K. E., and Farmer, G. L.: Impact of disturbed
desert soils on duration of mountain snow cover, Geophys. Res.
Lett., 34, L12502, 10.1029/2007GL030284, 2007.Phan, X. V., Ferro-Famil, L., Gay, M., Durand, Y., Dumont, M., Morin, S.,
Allain, S., D'Urso, G., and Girard, A.: 1D-Var multilayer assimilation of
X-band SAR data into a detailed snowpack model, The Cryosphere, 8, 1975–1987,
10.5194/tc-8-1975-2014, 2014.Quintana Segui, P., Moigne, P. L., Durand, Y., Martin, E., Habets, F.,
Baillon, M., Canella, C., Franchisteguy, L., and Morel, S.: Analysis of near
surface atmospheric variables: validation of the SAFRAN analysis over
France, J. Appl. Meteorol. Clim., 47, 92–107,
10.1175/2007JAMC1636.1, 2008.Raleigh, M. S., Lundquist, J. D., and Clark, M. P.: Exploring the impact of
forcing error characteristics on physically based snow simulations within a
global sensitivity analysis framework, Hydrol. Earth Syst. Sci., 19,
3153–3179, 10.5194/hess-19-3153-2015, 2015.
Reichle, R. H.: Data assimilation methods in the Earth sciences, Adv. Water Resour., 31, 1411–1418, 2008.Sirguey, P., Mathieu, R., and Arnaud, Y.: Subpixel monitoring of the seasonal
snow cover with MODIS at 250 m spatial resolution in the Southern Alps of New
Zealand: methodology and accuracy assessment, Remote Sens. Environ., 113,
160–181, 10.1016/j.rse.2008.09.008, 2009.Snyder, C., Bengtsson, T., Bickel, P., and Anderson, J.: Obstacles to
high-dimensional particle filtering, Mon. Weather Rev., 136, 4629–4640,
10.1175/2008MWR2529.1, 2008.Stankov, B., Cline, D., Weber, B., Gasiewski, A., and Wick, G.: High-resolution
airborne polarimetric microwave imaging of snow cover during the NASA cold
land processes experiment, IEEE T. Geosci. Remote S., 46, 3672–3693, 10.1109/TGRS.2008.2000625, 2008.Sun, C., Walker, J. P., and Houser, P. R.: A methodology for snow data
assimilation in a land surface model, J. Geophys. Res.-Atmos., 109, D08108, 10.1029/2003JD003765, 2004.Tedesco, M., Reichle, R., Löw, A., Markus, T., and Foster, J. L.: Dynamic
approaches for snow depth retrieval from spaceborne microwave brightness
temperature, , IEEE T. Geosci. Remote S., 48,
1955–1967, 10.1109/TGRS.2009.2036910, 2010.Van Leeuwen, P. J.: Particle filtering in geophysical systems, Mon. Weather
Rev., 137, 4089–4114, 10.1175/2009MWR2835.1, 2009.Van Leeuwen, P. J.: Particle filters for the geosciences, Advanced Data
Assimilation for Geosciences: Lecture Notes of the Les Houches School of
Physics: Special Issue, June 2012, p. 291,
10.1093/acprof:oso/9780198723844.003.0013, 2014.Veitinger, J., Sovilla, B., and Purves, R. S.: Influence of snow depth
distribution on surface roughness in alpine terrain: a multi-scale approach,
The Cryosphere, 8, 547–569, 10.5194/tc-8-547-2014, 2014.Vernay, M., Lafaysse, M., Mérindol, L., Giraud, G., and Morin, S.: Ensemble
forecasting of snowpack conditions and avalanche hazard, Cold Reg. Sci. Technol., 120, 251–262, 10.1016/j.coldregions.2015.04.010, 2015.Vionnet, V., Brun, E., Morin, S., Boone, A., Faroux, S., Le Moigne, P.,
Martin, E., and Willemet, J.-M.: The detailed snowpack scheme Crocus and its
implementation in SURFEX v7.2, Geosci. Model Dev., 5, 773–791,
10.5194/gmd-5-773-2012, 2012.Warren, S.: Optical properties of snow, Rev. Geophys., 20, 67–89,
10.1029/RG020i001p00067, 1982.Warren, S. G. and Clarke, A. D.: Soot in the atmosphere and snow surface of
Antarctica, J. Geophys. Res.-Atmos. (1984–2012), 95,
1811–1816, 10.1029/JD095iD02p01811, 1990.Wright, P., Bergin, M., Dibb, J., Lefer, B., Domine, F., Carman, T.,
Carmagnola, C. M., Dumont, M., Courville, Z., Schaaf, C., and Wang, Z.:
Comparing MODIS daily snow albedo to spectral albedo field measurements in
Central Greenland, Remote Sens. Environ., 140, 118–129,
10.1016/j.rse.2013.08.044, 2014.