Mass balance observations are very useful to assess climate change in
different regions of the world. As opposed to glacier-wide mass balances
which are influenced by the dynamic response of each glacier, point
mass balances provide a direct climatic signal that depends on surface
accumulation and ablation only. Unfortunately, major efforts are required to
conduct in situ measurements on glaciers. Here, we propose a new approach that
determines point surface mass balances from remote sensing observations. We
call this balance the geodetic point surface mass balance. From observations
and modelling performed on the Argentière and Mer de Glace glaciers over the
last decade, we show that the vertical ice flow velocity changes are small
in areas of low bedrock slope. Therefore, assuming constant vertical
velocities in time for such areas and provided that the vertical velocities
have been measured for at least 1 year in the past, our method can be used
to reconstruct annual point surface mass balances from surface elevations
and horizontal velocities alone. We demonstrate that the annual point
surface mass balances can be reconstructed with an accuracy of about
0.3 m of water equivalent per year (m w.e. a-1) using the vertical velocities observed over the previous years
and data from unmanned aerial vehicle images. Given the recent improvements
of satellite sensors, it should be possible to apply this method to high-spatial-resolution satellite images as well.
Introduction
Glacier surface mass balance observations are widely used to assess climate
change in various climatic regimes because of their sensitivity to climate
variables (e.g. Zemp et al., 2019; Marzeion et al., 2014; Kaser et al.,
2006; Gardner et al., 2013; Huss and Hock, 2018; IPCC, 2019). In situ surface mass
balance measurements have been conducted on only a few of the 200 000
mountain glaciers worldwide (WGMS, 2017; Zemp et al., 2019). In the European
Alps, about a dozen annual surface mass balance time series from in situ
measurements extending over more than 50 years are available (WGMS, 2017).
Recently, considerable efforts have been made to assess ice volume changes
at the mountain-range scale over long time periods using geodetic
measurements obtained from remote sensing techniques (e.g. Paul and
Haeberli, 2008; Abermann et al., 2011; Gardelle et al., 2012; Gardner et
al., 2013; Berthier et al., 2014; Brun et al., 2017). These geodetic methods
determine glacier-wide volume changes, or glacier-wide mass balances, by
differencing repeated determinations of glacier surface elevations obtained
from airborne and spaceborne surveys usually over multi-year to decadal
periods (e.g. Vincent, 2002; Bauder et al., 2007; Soruco et al., 2009;
Berthier et al., 2014; Dussaillant et al., 2019). These methods are
effective for estimating the overall glacier mass change and quantifying the
related hydrological impacts or sea level contribution (e.g. Hock et al.,
2005; Kaser et al., 2010; Huss, 2011; Immerzeel et al., 2013; Zemp et al.,
2019). However, the meaningfulness of a climatic interpretation of these
results is questionable. Indeed, glacier-wide mass balances are not solely
driven by changes in climate but also by changes in glacier geometry
controlled by the dynamic response of each glacier (Vincent, 2002; Fischer, 2010; Abermann et al., 2011; Huss et al., 2012; Vincent et al.,
2017). Consequently, they do not provide a direct climatic signal. On the
other hand, point surface mass balances provide a direct climatic signal
which depends only on local ablation and accumulation (Huss and Bauder,
2009; Thibert et al., 2013; Vincent et al., 2004, 2017, 2018b). Ablation is
related directly to the surface energy balance. Accumulation is related to
precipitation but is also strongly influenced by valley topography. Indeed,
glaciers are generally surrounded by very steep non-glacial slopes which
capture precipitation over a larger area than that of the glacier itself. In
this way, high accumulation values are due to downhill transportation and
strong wind actions (Vincent, 2002). Statistical modelling enables us to
extract a climatic signal from heterogeneous in situ observations of point
mass balance networks independently of effects related to ice flow dynamics
and glacier area changes (Vincent et al., 2018b). However, these previous
studies showed that it is crucial to perform observations of annual point
surface mass balance at the same locations every year. Unfortunately, the
only way to presently obtain point mass balance data is to make in situ
measurements. In particular, the net annual ablation in the ablation zone is
usually obtained from ablation stakes. These point surface mass balance
measurements require huge efforts involving field campaigns and the
collection of data from stake measurements scattered over the glacier. This
explains why so few in situ measurements are performed, especially on glaciers
located in remote areas with very difficult access (e.g. Azam et al., 2018;
Wagnon et al., 2013; Hoezle et al., 2017).
The objective of this paper is to propose an approach to determine point
surface mass balances from measurements obtained by remote sensing
techniques. In this way, we aim to determine point surface mass balances
in ablation areas without setting up ablation stakes each year. We will
develop this method using a comprehensive dataset of in situ measurements and
analysis of ice motion, elevation changes and point surface mass balance
data in the ablation area of the Argentière Glacier (French Alps). We will
then validate our method in other areas of the ablation zone of this glacier
and of the Mer de Glace glacier.
Study area
The Argentière Glacier is located in the Mont Blanc range, French Alps
(45∘55′ N, 6∘57′ E). Its surface area was about 10.9 km2 in 2018 (Fig. 1). The glacier extends from an altitude of
about 3400 m a.s.l. (above sea level) at the upper bergschrund down to 1600 m a.s.l. at the
snout. The length of this glacier is about 10 km. It faces north-west
except for a large part of the accumulation area (south-west facing
tributaries). The annual surface mass balance ranges roughly from 2 m
of water equivalent per year (m w.e. a-1) in the accumulation area to
about -10 m w.e. a-1 close to the snout. This glacier is free of
rock debris except for the lowermost part of the tongue below the ice fall
located between 2000 and 2300 m a.s.l. In the region studied in detail at
2350 m, the ice is generally free of debris. The debris cover can be 5 to 10 cm thick in some locations. The field observations of the Argentière
Glacier (i.e. mass balance, thickness variations, ice-flow velocities and
length fluctuations over 50 years) come from the French glacier monitoring
programme called GLACIOCLIM (Les GLACIers, un Observatoire du CLIMat;
https://glacioclim.osug.fr/, last access: 25 February 2021). For the present study, additional
detailed observations were carried out in the framework of the SAUSSURE
programme (Sliding of glAciers and sUbglacial water preSSURE; https://saussure.osug.fr, last access: 25 February 2021). The main part of our study focuses on a small
area of the Argentière Glacier (∼0.2 km2) located between 2320
and 2400 m a.s.l. in the ablation zone (Figs. 1 and 2). In this area, the
glacier is ∼ 600 m wide, the horizontal ice flow velocity is ∼ 55 m a-1 (Vincent et al., 2009), and the maximum ice thickness is 250 m
(Rabatel et al., 2018). Experiments conducted through boreholes (Hantz and
Lliboutry, 1983) indicate that the bed is composed of hard rock with no
thick or deforming sediment layer.
Data
In the selected area, annual point surface mass balances and ice flow
velocities were monitored with a high positioning accuracy at the end of
each ablation season between 2016 and 2019 from 19 ablation stakes (Fig. 2).
Our study also used surface mass balance and ice flow velocity observations
from a small part of the ablation zone of the Mer de Glace glacier at the
location named “Tacul glacier” (Fig. 1).
The ablation stakes are 10 m long and made of five 2 m long sticks tied
together with metallic chains. We performed the observations of annual point
surface mass balance at the same locations every year. Errors in ablation
measurements mainly come from the mechanical play of the jointed sticks. The
uncertainties in the annual surface mass balance measurements performed in
this ablation zone have been assessed at ±0.14 m w.e. a-1 (Thibert et
al., 2008). Topographic measurements were performed to obtain the 3D
coordinates of the ablation stakes. For this purpose, we used a Leica 1200
Global Navigation Satellite System (GNSS) receiver, running with dual
frequencies. Occupation times were typically 1 min with 1 s
sampling, and the number of visible satellites (GPS and GLONASS) was more
than seven. The distance between fixed and mobile receivers was less than 1 km.
The Differential Global Positioning System (DGPS) positions have an intrinsic accuracy of ±0.01 m. However,
given the size of the holes drilled to insert the stakes, we estimate that
the stake positions have an uncertainty of ±0.05 m.
Both velocity components are required. The vertical velocity is the vertical
component of the surface velocity obtained from measuring altitude
differences of the bottom tip of stakes. For this purpose, the emergence
measurement is required to obtain the buried length of the stake. Thus, the
purpose of emergence observations is two-fold. They enable us (i) to calculate
the surface mass balance from two field campaigns and (ii) to obtain the
altitude of the bottom tip of the stake using the altitude of the surface.
In practice, the DGPS measurements are performed simultaneously with the
emergence measurements in order to obtain the exact position of the bottom
tip of the stake buried in ice. In this way, it is possible to monitor ice
velocity along the three coordinate directions. Depending on the tilt of the
ablation stakes, the size of the drilling hole and the mechanical play of
the jointed stakes, we assume that the annual horizontal and vertical
velocities are known with an uncertainty of ±0.10 m a-1.
Aerial photographs of the glacier surface were taken on 5 September 2018 and
13 September 2019 using the senseFly eBee X unmanned aerial vehicle (UAV).
A total of 720 photos in 2018 and 673 photos in 2019 were collected with the
on board senseFly S.O.D.A. camera (20 MP RGB sensor with a 28 mm focal
length from an average altitude of 140 m above the glacier surface). Prior
to the survey flights, we collected GNSS measurements of ground control
points (GCPs) that consist of rectangular pieces of red fabric (100×60 cm)
with white painted circles (40 cm diameter) on the glacier (10 in 2018, 20
in 2019) and ten 40 cm diameter white circles painted on rocks on the sides
of the glacier. The original horizontal resolutions of the orthophoto
mosaics and digital elevation models (DEMs) are 10 cm and 1.00 m,
respectively. The photos from the survey were processed using the structure
for motion (SfM) algorithm that is implemented in the Agisoft Metashape
Professional version 1.5.2 software package (Agisoft, 2019). The SfM stereo
technique was then used to generate a dense point cloud of the glacier
surface. This dense point cloud was used to construct the DEMs using the
GCPs surveyed during the field campaigns. A detailed description of the
processing steps can be found in Kraaijenbrink et al. (2016) and Brun et al. (2016). The horizontal resolutions of the orthophoto mosaics and DEMs are 10 cm and 1.0 m, respectively.
To calculate horizontal ice flow velocities over the studied area, we used
the UAV orthophoto mosaics with COSI-Corr (Co-registration of Optically
Sensed Images and Correlation), a software tool developed for image
correlation (Leprince et al., 2007; Ayoub et al., 2009). Due to the
velocities of the Argentière Glacier in this region (∼ 55 m a-1), we resampled the UAV orthophotos at 1.0 m resolution because
the correlation was too noisy even with very large window sizes (i.e. 512
pixels). The surface velocities were computed using an initial window size
of 256 pixels, a final window size of 64 pixels and a step of 4 pixels. The
output velocity field was filtered using signal-to-noise ratios (SNRs)
provided by COSI-Corr. Using an SNR threshold greater than 0.9 provides a
good compromise between output details, noise and computing time. A detailed
description of the correct choice of the window size for correlation can be
found in Kraaijenbrink et al. (2016).
To establish the possible errors in the correlation process, horizontal
displacements on stable off-glacier areas were evaluated over 25 random
points and provided a maximum horizontal error of ∼ 0.55 m.
Map of the Argentière and Mer de Glace glaciers. The red dots on the Argentière Glacier are the ablation stakes used in this study for annual surface mass balance and ice flow velocity measurements in three regions of the glacier (at approximately 2400, 2550 and 2700 m a.s.l.). Aerial photo from the French National Geographical Institute, 2015 (https://www.geoportail.gouv.fr/, last access: 25 February 2021).
Map of the studied area in the ablation zone of the Argentière Glacier. The contour lines of surface topography correspond to the surface in 2018. The green, blue and red dots are the positions of the ablation stakes used for surface mass balance and ice flow velocity measurements when they were set up in 2016, 2017 and 2018, respectively. Aerial photo from unmanned aerial vehicle survey (5 September 2018).
Method
We will now introduce the mathematical framework used further on.
Emergence velocities
The emergence velocity is the upward or downward flow of ice relative to the
glacier surface. This flow compensates for the surface mass balance exactly if
the glacier is under steady state conditions. The surface elevation change
equation (Cuffey and Paterson, 2010, p. 332) expresses the surface mass
balance as a function of surface velocity and surface gradient:
bs=∂S/∂t-(ws-us∂S/∂x-vs∂S/∂y),
with bs the surface mass balance expressed in metres of ice, firn or
snow (m a-1), S the surface elevation (m), us, vs and ws the
components of ice flow velocity at the surface (m a-1), ∂S/∂x the surface gradient in the x direction, and ∂S/∂y the surface gradient in the y direction.
The term ws-us∂S/∂x-vs∂S/∂y is
called the emergence velocity. If the horizontal x axis is taken in the flow
direction, vs=0, and the emergence velocity is written as
ve=ws-us∂S/∂x.
Note that under steady state conditions, ∂/∂t=0 and
bs=-ve. The emergence velocities can be calculated for each
ablation stake from horizontal and vertical velocities and the slope of the
surface ∂S/∂x. In this way, we assume that the downslope
direction is the flow direction. The slope of the surface can be obtained
from GNSS field measurements and calculated over a distance similar to that
travelled by the stake over 1 year. In the ablation zone, the emergence
velocities are positive, which corresponds to an upward flow of ice relative
to the glacier surface. Note also that the vertical velocity can be positive
or negative in any region of the glacier. The emergence velocity is a
classical way to relate the surface mass balance to the thickness changes
(Eq. 1). Unfortunately, as shown later in our study (Sect. 6.1), even if
the horizontal and vertical velocities are accurately measured, the large
uncertainties related to the slope and thickness changes prevent us from
calculating the point surface mass balance from the emergence velocities.
At the yearly scale, according to Eq. (1) and Fig. 3 and
considering that the x axis is taken along the flow line direction (i.e.
vs=0), the annual surface mass balance Bs between the years t
and t+1 is obtained from
Bs=Δh1+Ustanαt+1-Ws=Δh2+Ustanαt-Ws,
where us⋅∂S/∂x is replaced by Us tanαt
or Us tan αt+1, Us is the annual surface
horizontal velocity, and tanαt and tan αt+1 are
the slopes for the years t and t+1, respectively. Ws is the annual
vertical velocity. ∂S/∂t is replaced by Δh1
and Δh2, which are the annual thickness changes observed at the
ends of the annual ice flow vectors.
Figure 3 illustrates the components of Eq. (3).
Note that the slope of the surface may change from year t to year t+1 and
the expression depends on the selected slope and thickness changes Δh1 or Δh2 (Fig. 3). Obviously, the results are the same.
Diagram illustrating horizontal, vertical and emergence velocities
(m a-1) observed from an ablation stake (orange). Us (m a-1)
and Ws (m a-1) are the components of horizontal and vertical
velocities, αt and αt+1 are the slopes for the
years t and t+1, respectively, Zs1,t and Zs2,t+1 are the
elevations of the surface at each end of the ice flow vector, and Δh1 and Δh2 are the elevation changes (m) at each
end of the ice flow vector.
Calculation of the geodetic point surface mass balance
Let us reconsider the emergence velocity formulation in order to express the
point surface mass balance as a function of vertical velocity and altitude
changes at the ends of the annual displacement vector.
According to Eq. (3) and given that Δh1+Ustanαt+1=Δh2+Ustanαt=Zs2,t+1-Zs1,t (Fig. 3), we can write the following:
Bs=Zs2,t+1-Zs1,t-Ws.
This expression has a great advantage in that it does not depend on the
surface slope that can change from one year to the next. It is also
independent of thickness changes that can change from one site to another.
The term geodetic point surface mass balance refers to the value of Bs
obtained from Eq. (4). Once the vertical velocity is known, Bs can
be obtained from topographical surface measurements alone. Note that even if
the horizontal velocity is not included in Eq. (4), it is needed to
estimate the positions at which Zs2,t+1 and Zs1,t should be
measured.
ResultsAnnual horizontal and vertical velocities over the 3 years
Annual horizontal and vertical velocities were measured from a network of 19
ablation stakes over 3 years between 2016 and 2019 (Fig. 2). The stakes
were replaced each year and were always set up at the same locations using
a handheld GPS device, allowing a relevant comparison, except for stakes 1
and 11 which were located in areas with large crevasses, preventing the
possibility of drilling stakes at the chosen location. In addition, for the
year 2018/2019, stake 12 was accidentally replaced at a distance of more
than 30 m from its initial position due to both a lack of rigour and
the uncertainty in the handheld GPS measurement. This difference in
locations led to a difference in the horizontal velocity of 3 m a-1 in
a region with a strong horizontal gradient (left edge of the area in Fig. 4). However, it does not change the pattern of horizontal velocities or
horizontal velocity changes with time. This is not the case for vertical
velocities as described in the next paragraph. In the area of this network,
the annual horizontal velocities range from 35 to 60 m a-1. The annual
ice flow velocities have been interpolated from kriging over the entire
coloured areas shown in Fig. 4. In this way, we can accurately compare the
ice flow velocities over 3 years, 2016/2017, 2017/2018 and 2018/2019, at
the locations of each stake (Fig. 5a). Strong deceleration in horizontal ice
flow velocities can be observed over these 3 years. On average, ice
flow velocity decreased by 2.4 and 1.8 m a-1 over the two periods
2016/2017–2017/2018 and 2017/2018–2018/2019, which corresponds to an
average decrease of about 4.8 % and 3.6 % per year, respectively. Note that
the regression lines shown in Fig. 5a are almost parallel, which means
that the change in velocities is homogeneous in space.
The vertical velocities were obtained from the altitude changes in the
bottom tip of the stakes from one year to the next (Fig. 3). In the studied
area, the vertical velocities can be positive or negative and range from -4
to 4 m a-1 (Fig. 4). The vertical velocities have been interpolated
over the entire coloured areas shown in Fig. 4 using kriging. The patterns
of vertical velocities are very similar for the years 2016/2017 and
2017/2018. We note some differences with the 2018/2019 pattern. As mentioned
previously, stakes 1, 11 and 12 set up in 2018/2019 are located at distances
of more than 30 m from their initial positions. In addition, stakes 17,
18 and 19 were replaced in 2018 at distances ranging between 25 and 30 m
from their initial positions. These six stakes are shown with small dots in
Fig. 5b. If we exclude the velocity values of 2018/2019 for these stakes,
we can conclude that the measured vertical velocities are very similar over
this 3-year period. The differences do not exceed 0.5 m a-1. The
average of the differences is 0.01 m a-1, and the standard deviation
is 0.29 m a-1. These differences barely exceed the measurement
uncertainty. Note also that the vertical velocity changes could be affected
by the horizontal motion changes or vertical strain rate changes as
discussed in Sect. 6.
Horizontal (top panels) and vertical (bottom panels) ice flow velocities (m a-1) measured over 3 years from the ablation stakes. Note the different colour scales. Distances in metres.
Comparison of horizontal ice flow velocities (a) and vertical velocities (b) between the years 2016/2017, 2017/2018 and 2018/2019. The black dots correspond to the comparison between the 2016/2017and 2017/2018 periods. The red dots correspond to the comparison between the 2016/2017 and 2018/2019 periods. The thick dashed line corresponds to the bisector and the thin dashed lines to the regression lines. The small dots in the figure on the right correspond to the stakes that were set up in 2018 at distances of more than 25 m from their initial positions.
Emergence velocities
The emergence velocities have been calculated from Eq. (2) for each stake
and reported in Fig. 6. The slope was determined from the digital
elevation model using UAV measurements. We compared the emergence velocities
obtained each year at each stake location (Fig. 7). Unlike the vertical
velocities, the differences between emergence velocities calculated over the 3 years reveal a standard deviation of 0.8 m w.e. a-1. The value of
emergence velocities is affected by large uncertainties related to the
slope.
Combined with the measured thickness changes, the emergence velocity should
make it possible to estimate the surface mass balance. However, our study
shows that the uncertainties in the emergence velocity prevent us from
calculating the point surface mass balance accurately. Indeed, the
dispersion of 0.8 m w.e. a-1 is large compared to the spatial
variability of about 1 m w.e. a-1 for point surface mass balance in the
ablation zone of alpine glaciers (Vincent et al., 2018b).
For this reason, to calculate the surface mass balance, we suggest using the
geodetic point surface mass balance described earlier rather than the
emergence velocity.
Emergence velocities between the years 2016/2017, 2017/2018 and 2018/2019 (m a-1).
Comparison of emergence velocities between the years 2016/2017, 2017/2018 and 2018/2019. The black dots correspond to the comparison between the 2016/2017 and 2017/2018 periods. The red dots correspond to the comparison between the 2016/2017 and 2018/2019 periods. The red small dots correspond to the stakes that were set up in 2018 at distances of more than 25 m from their initial positions.
Geodetic point surface mass balances using in situ GNSS measurements
The geodetic point surface mass balance is calculated according to Eq. (4). We first tested the method in the studied region of the Argentière
Glacier at 2400 m a.s.l. using the in situ GNSS measurements. For this purpose, we
used the altitudes of the surface at the stake locations for the years 2017
and 2018 and the vertical velocities observed in 2016/2017. The resulting
point surface mass balances for the hydrological year 2017/2018 are compared
with the observed surface mass balances and plotted in Fig. 8a. Note that
the surface mass balances are in m of ice per year. The comparison shows
very good agreement. The maximum difference is 0.39 m of ice per year and
the standard deviation is 0.20 m of ice or 0.18 m w.e. per year. In
addition, we calculated the surface mass balances of 2018/2019 from the
vertical velocities observed in 2016/2017 and 2017/2018 (Fig. 8b). In this
case, the comparison with the observed surface mass balances shows large
discrepancies. However, a more detailed analysis reveals that the calculated
and observed surface mass balances are very similar if the vertical
velocities observed in 2016/2017 and 2017/2018 were measured exactly at the
same location as the stakes measured in 2018/2019. In Fig. 8b, the large
dots show the calculated and observed surface mass balances for the stakes
located within a distance no greater than 15 m. From this comparison, the
differences are less than 0.5 m a-1 of ice, and the standard deviation
is 0.17 m of ice or 0.15 m w.e. a-1.
From this analysis, we conclude that the geodetic point surface mass balance
can be obtained with an accuracy of about 0.2 m w.e. a-1 using the
vertical velocities observed over the previous years. It requires the
measurement of the horizontal ice flow velocity and the altitudes of the
ends of the velocity vector exactly at the same location within a radius of
less than 15 m compared to that of vertical velocity determination. In
practice, the vertical velocities should be observed accurately between two
years, t and t+1, from stakes and GNSS measurements. Then, for the following
or previous years, the point surface mass balance can be obtained from
surface measurements only (without drillings and setting new stakes) using
the horizontal velocity and the altitudes of the surface measured at each
end of the horizontal vector. In the next section, we examine how such
measurements obtained from remote sensing data can also be used effectively
to determine the point surface mass balance.
Observed and calculated point surface mass balances at 2350 m a.s.l. at Argentière Glacier. The point surface mass balances have been calculated (a) for the year 2017/2018 using the vertical velocities measured in 2016/2017 and elevations from GNSS measurements, (b) for the year 2018/2019 using the vertical velocities measured in 2016/2017 (black dots) and 2017/2018 (red dots) and elevations from GNSS measurements and (c) for the year 2018/2019 using elevations from remote sensing data (UAV data) and the vertical velocities measured in 2018/2019 (red dots), 2017/2018 (blue dots) and 2016/2017 (green dots). The large dots shown in Fig. 8b correspond to the stakes which were set up within a radius of less than 15 m.
Geodetic point surface mass balances using remote sensing measurements
Here, we used the same method described in the previous section. However,
the in situ GNSS measurements used to determine the altitudes and horizontal
velocities are replaced by remote sensing measurements. For this purpose, we
used the horizontal velocities (Fig. 9) and the DEMs (Fig. 10) obtained
from UAV surveys in 2018 and 2019. The vertical velocities are those
observed in 2018/2019, 2017/2018 and 2016/2017. The horizontal velocities
have been neglected for stakes 9 and 10 given the poor quality of the
correlation and the opening and/or closing of crevasses in the ice (close to
stake 10) that caused a drastic change between the photos, which
subsequently affected the image correlation (Fig. 9).
Some details on the procedure are given below for the sake of clarity. The
horizontal velocities retrieved from the UAV surveys were determined at
positions where vertical velocities were measured. In this way, the
coordinates XY of each vector end have been calculated (green dots in Fig. 9). Then we used the DEMs from 2018 and 2019 (Fig. 10) to determine the
elevations of these points, Zs1,2018 and Zs2,2019 (see
Eq. 4 and Fig. 3). The comparison between the in situ horizontal velocities and
the velocities obtained from the UAV surveys reveals a standard deviation of
0.7 m a-1.
The reconstructed point surface mass balances are compared with the observed
surface mass balances in Fig. 8c. For this reconstruction, we used the
vertical velocities observed in 2018/2019 (red dots), 2017/2018 (blue dots)
and 2016/2017 (green dots). For the reconstructions using the vertical
velocities of 2016/2017 and 2017/2018, we excluded data from sites 1, 11, 12,
17, 18 and 19 for which the stakes were measured at distances of more than
30 m from those of 2018/2019.
The differences between the observations and the reconstructed surface mass
balances using the 2018/2019 vertical velocities are less than 0.45 m of ice
per year, and the standard deviation is 0.24 m of ice or 0.22 m w.e. a-1. The differences between the observations and the reconstructed
surface mass balances using the 2016/2017 and 2017/2018 vertical velocities
show standard deviations of 0.42 and 0.40 m w.e. a-1, respectively.
From these results, we conclude that the point surface mass balances can be
obtained with an accuracy of about 0.3 m w.e. a-1 using remote sensing
measurements, assuming that the vertical velocities have been observed
accurately over the previous years.
Horizontal velocities obtained from feature tracking (Cosi-Corr) using UAV images. The black crosses show the locations where the vertical velocities were observed. The red dots correspond to the ends of horizontal vectors for 2018/2019, determined from UAV images.
DEMs obtained from the UAV survey in 2018 (blue contour lines) and 2019 (red contour lines). The black dots correspond to the positions of the stakes in 2018 and 2019 observed from GNSS measurements. The red dots correspond to the ends of the horizontal velocity vectors obtained from UAV images.
Validation of the method: geodetic surface mass balances obtained in other
regions
In order to establish that the results are neither accidental nor
site-dependent, we tested the method on other areas of the Argentière
Glacier and on another glacier, which is the Mer de Glace located
approximately 10 km away (Fig. 1) and for which vertical velocities were
available. Here, we used GNSS in situ measurements given that accurate elevation
observations from remote sensing data are not available.
First, we selected two ablation stakes in a sector of the Argentière Glacier
located at 2530 m a.s.l. These stakes were replaced within a radius of
±35 m each year between 2001 and 2018 (Fig. 1). Note that these
measurements were not intended for vertical velocity determination but
rather for point surface mass balance measurements. This explains why the
stakes were not set up at exactly the same locations over the whole period.
Note also that the region is not debris-covered and consequently the surface
roughness is lower compared to the studied area at 2350 m. Using Eq. (4)
and the method described in the previous section, we calculated the point
surface mass balances at these two stakes over the period 2001–2018. For
this purpose, we used the average vertical velocities calculated over this
period. In addition, the altitudes of each stake for each year of this
period have been observed. These two stakes (named stake 2 and stake 3) are
located about 120 m apart. The average calculated vertical velocities are
-0.24 m a-1 (±0.44 m a-1) and -0.79 m a-1 (±0.33 m a-1), respectively, and they did not show strong temporal changes
(Fig. 11b). Note that the horizontal velocity decreased from 75 to 50 m a-1 in this region between 2002 and 2018 (Fig. 11a). The geodetic point
surface mass balances are compared to the observations (Fig. 12a). The
standard deviations of the calculated and observed surface mass balance
differences are similar to those of the vertical velocities (0.44 and 0.33 m a-1, i.e. 0.4 and 0.3 m w.e. a-1).
Second, we tested the method in another sector of the Argentière Glacier
close to the equilibrium line which is located close to 2800 m a.s.l. For
this purpose, we selected six stakes (stakes 7, 8, 9, 10, 11 and 12) which were
measured along a longitudinal section between 2650 and 2750 m a.s.l. (Fig. 1) over the period 2005–2018. In this region, the horizontal ice flow
velocity is about 50 m a-1 (Fig. 11a). Here again, the network of
stakes was mainly designed for point surface mass balance measurements.
Thus, given that the stakes were set 10 m deep in the ice and the surface
mass balance ranges between -4 and 0 m w.e. a-1 depending on the year,
the ablation stakes were not replaced each year. As the ablation stakes move
with the ice flow, we selected only the measurements that were performed at
the same locations. Indeed, after the first year following installation, the
location of each stake was far from its initial position, and we cannot
assume that the vertical velocity was similar. Consequently, 5 years are
available to calculate the vertical velocities and to make the comparison
between calculated and observed point surface mass balances (Fig. 12b). The
standard deviations of calculated and observed point surface mass balance
differences are 0.22 m of ice a-1, i.e. 0.20 m w.e. a-1.
Finally, we tested the method on another glacier, Mer de Glace (Fig. 1). On
this glacier, we selected one stake at 2100 m a.s.l that was measured over
15 years between 2003 and 2018 (Vincent et al., 2018a). This ablation stake
was set up each year at the same location within a radius of about 30 m. Using the method described in the previous sections, we calculated
the point surface mass balances at this stake over the period 2003–2018. The
average calculated vertical velocity is -1.10 m a-1. Note that the
horizontal velocity decreased from 80 to 50 m a-1 and the thickness by
55 m in this region between 2003 and 2018. The results are plotted in Fig. 12c. The standard deviation of the calculated and observed point surface
mass balance differences is 0.40 m w.e. a-1.
Horizontal (a) and vertical (b) velocities observed at the different stakes at 2550 m a.s.l. (stakes 2 and 3) and 2700 m a.s.l. (stakes 7, 8, 9, 10, 11 and 12).
Observed and calculated point surface mass balances from (a) two ablation stakes located at 2550 m a.s.l. at Argentière Glacier measured between 2002 and 2018, (b) six stakes located at around 2700 m a.s.l. at Argentière Glacier measured between 2006 and 2017 and (c) one stake located at 2100 m a.s.l. on Mer de Glace glacier measured between 2003 and 2018.
DiscussionsPoint surface mass balance obtained from emergence velocities vs. vertical
velocities
A classical approach to relate the point surface mass balance to thickness
change is to use the emergence velocity (Cuffey and Paterson, 2010; Kääb and
Funk, 1999). From this approach, the point surface mass balance is obtained
from the sum of the emergence velocity and the thickness change (Eq. 1).
However, the value of the mass balance reconstructed from the emergence
velocity depends strongly on the selected surface slope and on thickness
change, which both vary considerably with space and time. The value of the
slope depends on the choice of the selected distance for the slope
calculation and on the roughness of the surface.
In addition, the slope can change significantly from one year to the next.
The emergence velocity is therefore not well-defined given that it depends
strongly on the spatial and temporal changes in surface roughness,
preventing an accurate determination of point surface mass balance, as shown
in our study.
In contrast, in our analysis, we find that the changes in vertical velocity
are insignificant over the 3 years of observations at 2350 m. From the other
long series of observations, one can also see that they are small over decadal
timescales. Thus, we propose to reformulate the emergence velocity
formulation (Eq. 1) in order to express the point surface mass balance as a
function of vertical velocity and altitude changes at the ends of the annual
displacement vector (Eq. 4). In this way, provided that the vertical
velocity has been assessed from in situ measurements over previous years, the point
surface mass balance can be determined from remote sensing measurements
alone outside the period of field measurements with no need of prescribing
surface slope or elevation changes as required when using emergence
velocities, which introduces significant uncertainties. Our results from
the detailed studied area at Argentière Glacier (2350 m a.s.l.), for
which the observations were designed to accurately determine the vertical
velocity, demonstrate that the surface mass balance can be obtained from
this method with an accuracy of about 0.2 m w.e. a-1 from in situ GNSS
measurements and about 0.3 m w.e. a-1 using elevations and horizontal
velocities obtained from very high-resolution remote sensing data acquired
from UAV surveys.
Spatial and temporal variability in the vertical velocitiesAnalysis from observations
Our dataset shows that vertical velocities strongly vary in space over the
glacier surface. Our detailed observations from the network used between
2016 and 2018 at the Argentière Glacier (2350 m) showed that the
vertical velocity change can exceed 0.3 m a-1 if the stakes are located
at distances of more than 25 or 30 m (Sect. 5.1). We showed that the
surface mass balance can be reconstructed with an accuracy of about 0.2 m w.e. a-1 using the vertical velocities observed within a radius of less
than 15 m. Records from the whole network suggest that the vertical velocity
spatial gradient can exceed 1.5 m a-1 per 100 m in this region. As a
consequence, a horizontal deviation of 10 m could lead to a vertical
velocity change exceeding the measurement uncertainty (0.15 m a-1). To
better assess the vertical velocity spatial gradient over length scales of
20 to 100 m, the vertical velocities have been calculated from 10 stakes set
up in 2018/2019 on a longitudinal profile located between stakes 3 and 13
(Fig. 2). Note that the distances between these stakes are small, and they enable us
to assess the vertical velocity variations at small scales. According to
measurements shown in Fig. A1, the spatial gradient can reach up to 0.02 a-1, which is slightly more important than what we found previously
(0.015 a-1). We can conclude that reconstructing surface mass balance
from remote sensing requires measurements of the horizontal ice flow
velocity and the altitudes of the ends of the velocity vector exactly at the
same locations, i.e. within a radius of less than 15 m compared to that of
vertical velocity determination.
The analysis of temporal changes also deserves particular attention. The 3
years of detailed observations performed at 2350 m at Argentière Glacier
does not reveal temporal changes exceeding the measurement uncertainties, as
shown in Fig. 5b. Note that the longer series of observations available to
study the temporal changes over decadal timescales were not designed to
measure the vertical velocities. However, from the longer series of
observations performed at Argentière Glacier at 2550 and 2700 m a.s.l.
(Fig. 11b), we assessed a general temporal trend of about 0.07 m a-2.
We can conclude that the past period over which the vertical velocities are
determined should not exceed 4 years in order to not exceed an uncertainty
of 0.3 m w.e. a-1 on the reconstructed surface mass balance. This
conclusion could be different with stronger temporal changes in vertical
velocities. Further observations and analysis are needed to better estimate
the temporal changes.
Analysis from numerical modelling
To analyse the spatial and temporal variabilities in the vertical velocities
over the entire glacier, we performed 3D full-Stokes ice-flow simulations
for two different glacier geometries using a surface DEM measured in 1998
and 2015 and reconstructed bedrock topography (Rabatel et al., 2018). The
calculation is solved using the Elmer/Ice model (Gagliardini et al., 2013).
The linear basal friction parameter is inferred from surface velocity and
topography measurements made in 2003 (Berthier et al., 2005) using the
adjoint-based inverse method (Gillet-Chaulet et al., 2012). For each given
glacier geometry, we compute the corresponding flow solution and assume
constant friction over time. Therefore, changes in velocity are only induced
by changes in the glacier geometry between 1998 and 2015. We used an
unstructured mesh with a 100 m horizontal resolution refined down to 10 m in the stake network monitoring area at 2400 m a.s.l.
By integrating the mass conservation equation for an incompressible fluid
along the vertical axis, we can write the following:
ws=wb-∫zbzs∂u∂x+∂v∂ydz,
where wb is the vertical velocity at the bed, zs is the surface
elevation, and zb is the bed elevation. Vertical velocity at the surface
can therefore be viewed as a sum of a component coming from sliding along
the bedrock and a component coming from convergence/divergence of the ice
flow integrated over the glacier thickness. For example, local depression in
the bedrock topography creates negative vertical velocity wb at the
glacier base but also flow convergence that creates positive vertical
velocity resulting in a smoothing of surface vertical velocity ws by
the ice deformation. Figure 13a shows the modelled vertical surface velocity
in 2015. At the scale of the glacier, vertical surface velocities are
spatially heterogeneous due to a combination of bedrock slope and the ice
flux divergence/convergence (Fig. 13a). In the model, the basal vertical
velocity wb produced by ice flow along the bedrock can lead to small-scale variability in the basal vertical velocity that can be visible at the
surface when sliding velocity is significant, as modelled around 2400 m a.s.l. in the studied stake network (Fig. 13a). Bedrock topography is
therefore likely the origin of the observed pattern at 2400 m a.s.l. (Fig. 4). The pattern differences between the observations and the modelling
results are likely due to bedrock elevation errors. Although the pattern of
horizontal velocities is reproduced well (Fig. 13b), it seems difficult to
properly reconstruct the vertical velocities.
Vertical (a) and horizontal (b) surface velocities modelled at Argentière Glacier in 2015. Red dots show the locations of the ablation stakes set up at 2400 and 2650 m a.s.l.
Our numerical experiments were used to analyse the temporal changes in
vertical velocities. We found that the response of the vertical velocities
at the glacier surface to changes in glacier thickness over time is
sensitive to the bedrock slope (averaged over a distance greater than the
ice thickness). Consequently, a decreasing vertical velocity magnitude
should be associated with decreasing horizontal velocities when bed slopes
are significant (Fig. 14). However, the magnitude of small-scale
(length-scale inferior to glacier thickness) spatial variations in vertical
velocity due to bedrock topography seems to be little affected by the large
change in horizontal velocities (Figs. 14 and 15). We show that reduced
amplitude of wb due to decreasing sliding speed is compensated for by the
reduced amplitude of the ice flux convergence/divergence produced by bedrock
anomalies (red arrows in Fig. 15). Bedrock depressions and bumps of sizes
comparable to the glacier thickness produce, respectively, convergence and
divergence in the ice flow, creating vertical velocities of opposite signs
compared to the velocities created by sliding at the glacier base. These two
components of the surface vertical velocity decrease in magnitude in
response to thickness changes resulting in a limited change in the sum of
the two components and therefore in surface vertical velocities. This
results in nearly constant vertical velocity in which large-scale averaged
bedrock slope is low, which explains why the observed pattern of surface
vertical velocity (Fig. 4) is conserved well over time.
Modelled changes in vertical (a) and horizontal (b) surface velocities between 1998 and 2015. Insets compare modelled velocities at the stake location (orange dots) between 1998 and 2015.
In summary, at large scales, the magnitude of surface vertical changes over
time are proportional to bedrock slope and changes in horizontal velocities,
while at small scales, the spatial patterns tend to be conserved over time
due to compensation between changes in bedrock vertical velocities and ice
flux convergence/divergence. These findings suggest that our method is
likely applicable only in areas of low bedrock slope.
Modelled changes in vertical velocities at the surface (a) and at the bedrock (b) between 1998 and 2015. The righthand figure (c) shows the change in vertical velocity at the surface due to changes in flow convergence/divergence. The red arrows indicate the locations where changes in basal vertical velocities are compensated for by flow convergence/divergence changes, resulting in constant surface vertical velocities.
Note that, in our study, we used the annual velocities measured at the end
of the ablation season (from September to September) such that potential
seasonal changes in the vertical and horizontal motion or in basal uplift
and bed separation (e.g. Sugiyama et al., 2004; Nienow et al., 2005) are not
expected to bias the geodetic annual surface mass balances obtained from the
vertical velocities.
Uncertainties in geodetic point surface mass balances
The uncertainty related to the point surface mass balance determination
results from uncertainties in the elevation measurements and in the vertical
velocity. Using Eq. (4) and assuming the independence of the different
sources of uncertainties, the overall uncertainty related to the
reconstructed point surface mass balance is obtained by applying the method
of error propagation and assuming uncorrelated errors:
σb2=2σz2+σw2,
in which σb, σz and σw are the
uncertainties relative to the point surface mass balance, elevation and
vertical velocity, respectively.
The uncertainty in elevation depends both on the method of XY positioning,
the surface slope or roughness, and the method of altitude determination.
Depending on the surface roughness, we can assess the elevations with an
accuracy ranging from 0.1 to 0.3 m from UAV measurements, as shown in this
study.
The uncertainty in vertical velocity is ±0.1 m a-1, as mentioned
in the Data section. However, additional uncertainty could come from the
method of elevation observations for the bottom of the stakes. Indeed, the
GNSS measurements are commonly related to the surface of the ice at the
location of the stakes and not to the summit of the stakes. Consequently,
the altitude of the bottom of each stake results from the difference between
the altitude of the surface and the buried height of the ablation stake.
Indeed, this determination is accurate only if the measurement of emergence
has been performed exactly from the point on which the GNSS measurement was
made. Unfortunately, in most cases, one operator held the stick of the GPS
antenna at the ice surface close to the ablation stake, and another operator
measured the emergence of the stake but not exactly from the surface
altitude that corresponds to the bottom tip of the GPS antenna. Except for
the measurements performed at 2350 m a.s.l between 2016 and 2019, which
were designed for this purpose, this gives an additional uncertainty of
±0.1 m for the altitude of the bottom of the stake, i.e. ±0.14 m a-1 for the calculated vertical velocity.
The overall uncertainty in the geodetic point surface mass balance obtained
from remote sensing data is therefore estimated to range between ±0.20 and ±0.60 m a-1 using accurate DEMs from UAV photogrammetry
depending on the surface roughness and the method used for vertical velocity
determination.
Conclusions
The classical way to determine the point surface mass balance in the
ablation zone of a glacier is to set up ablation stakes and dig pits or
conduct drillings in the accumulation zone. Here, we showed that, in the
ablation zone, the point surface mass balances can be reconstructed from
surface altitudes and horizontal velocities only, provided that the vertical
velocities have been measured for at least 1 year in the past. Our method
first requires accurate measurement of the vertical velocities between two
years, t and t+1, from stakes and GNSS measurements. Then, for the following
or previous years, the point surface mass balances can be obtained easily
from surface measurements only using the horizontal velocity and the
surface elevation at each end of the horizontal displacement vector (Eq. 4).
These measurements can be obtained from remote sensing provided that the ice
flow velocity and altitude determinations are sufficiently accurate.
Our method assumes that the annual vertical velocities are almost constant
with time. We have used a numerical modelling study to show that this
approximation holds in areas of low bedrock slope (averaged over a
distance greater than the ice thickness). This is supported by our detailed
observations performed on the Argentière Glacier at 2400 m a.s.l. and
designed for this purpose. A comparison between the reconstructed point
surface mass balances and the observed values shows close agreement. Further
tests performed on datasets acquired in other regions of the Argentière
and Mer de Glace glaciers show standard deviations of ±0.2 to ±0.4 m w.e. a-1 between reconstructed and observed point surface mass
balances despite the fact that these measurements were not designed for
this purpose. For these tests, we used the averaged vertical velocities
obtained over the last decade.
From our results, we conclude that the point surface mass balances can be
obtained with an accuracy of about 0.3 m w.e. a-1 using remote sensing
measurements and assuming that the vertical velocities have been observed
accurately over the previous years within a radius of less than 15 m. We
also conclude from our datasets that the past period over which the
vertical velocities are determined should not exceed 4 years in order to not
exceed an uncertainty of 0.3 m w.e. a-1 on the reconstructed surface
mass balance, although further observations and analysis are needed to
better estimate these spatial and temporal changes. Note that, for
comparison, the measurement uncertainty related to the in situ measurements of
point surface mass balance is 0.14 m w.e. a-1 in the ablation zone
(Thibert et al., 2008).
Given the recent improvements in satellite sensors, it is conceivable to
apply our method using high-spatial-resolution satellite images like
Pléiades or WorldView (0.5 m resolution). For these point surface mass
balance reconstructions, note that, given the strong spatial variability in
vertical velocity, it is crucial to determine the altitudes of the surface
at each end of the horizontal displacement vector at the exact sites on
which the vertical velocities are known. We conclude that our method could
be useful to determine numerous point surface mass balances and reduce the
amount of effort required to conduct field measurements, especially in
remote areas.
Previous studies have shown that the point surface mass balance signal
reveals a climatic signal that is unbiased by the dynamic glacier response,
unlike the commonly used glacier-wide mass balance (Rasmussen, 2004; Huss et
al., 2009; Eckert et al., 2011; Thibert et al., 2018; Vincent et al., 2017).
In the glaciological community, there is growing awareness that point
surface mass balance measurements are important basic data to be shared for
mass balance and climate change analyses. In this respect, the World Glacier
Monitoring Service has started collecting such data on a systematic basis as
a complement to glacier-wide surface mass balances (WGMS, 2020). Our method
should open up new prospects to obtain more numerous point surface mass
balances in the future while reducing the amount of time and energy required
for in situ measurements.
Another line of research not explored in the present study could also be
examined. The method proposed in the present study requires the vertical
velocity to reconstruct the annual point surface mass balance. However, if
we derive Eq. (4) and assume that vertical velocity is constant with
time, we can determine the surface mass balance changes instead of the
absolute surface mass balances with the elevation determinations only.
Assuming that satellite sensors provide sufficient accuracy in elevation and
horizontal velocity, this method could be very helpful to reconstruct
changes in surface mass balance in remote areas for which in situ measurements are
very difficult. In this way, point surface mass balance changes in numerous
unobserved glaciers could be considered with remote sensing observations
only. This would make it possible to obtain climatic signals all over the
world which are unbiased by dynamic glacier response.
Vertical velocities obtained from 10 stakes measured in 2018/2019 on a longitudinal profile located between stakes 3 and 13 (see Fig. 2 for the locations of stakes 3 and 13).
Data availability
The surface mass balance, ice flow velocities and DEM
data can be accessed upon request by contacting Christian Vincent
(christian.vincent@univ-grenoble-alpes.fr).
Author contributions
DC, OL, DS, BJ, LA, UN, AW, LP, OG, VP, ET, FB and CV
performed the topographic measurements (photogrammetry, lidar, GNSS). OL,
DC, AG, FG, OG, FB, AR, and CV performed the numerical calculations and the
analysis. AG performed the numerical modelling calculations. CV supervised
the study and wrote the paper. All co-authors contributed to the discussion of
the results.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
This study was funded by the Observatoire des Sciences de l'Univers de Grenoble (OSUG) and the Institut des Sciences de l'Univers (INSU-CNRS) in the framework of the
French GLACIOCLIM (Les GLACIers, un Observatoire du CLIMat) programme. IGE
and ETGR are part of LabEx OSUG@2020 (Investissements d'avenir – ANR10
LABX56). We thank all those who conducted the field measurements. We are
grateful to Harvey Harder for reviewing the English. We thank the editor, Kenichi Matsuoka, the reviewer, Ben Pelto, and an anonymous reviewer for their
comments and suggestions that greatly improved the quality of the paper.
Financial support
This research has been supported by the Agence Nationale de la Recherche (grant no. ANR 18 CE1 0015 01) in the framework of the SAUSSURE programme (Sliding of glAciers and sUbglacial water preSSURE; https://saussure.osug.fr, last access: 25 February 2021).
Review statement
This paper was edited by Kenichi Matsuoka and reviewed by Ben Pelto and one anonymous referee.
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