In this paper we describe how recent high-resolution digital elevation
models (DEMs) can be used to extract glacier surface DEMs from old aerial
photographs and to evaluate the uncertainty of the mass balance record
derived from the DEMs. We present a case study for Drangajökull ice cap,
NW Iceland. This ice cap covered an area of 144 km
Mountain glaciers and ice caps accounted for more than half of the land ice runoff contribution to global mean sea-level rise during the 20th century (Vaughan et al., 2013). Understanding how these glaciers respond to a changing climate is essential to close the budget of the sea-level rise over the last decades and project the sea-level rise in the near future. In recent years an increased part of our knowledge about how these glaciers are changing has been based on remote sensing. The majority of these studies describe current or recent glacier changes in different parts of the globe by applying geodetic methods (e.g., Gardelle et al., 2012; Berthier et al., 2010). Others have presented results on the geodetic mass balance extending further back (e.g., Fischer et al., 2015; Nuth et al., 2007; Soruco et al., 2009); these studies are particularly important since they indicate how the glaciers responded to 20th century climate variability. Such observations can be used to constrain or correct glacier mass balance models that are used to estimate how the glaciers will respond to future climate changes (e.g., Clarke et al., 2015).
Studies on long-term geodetic mass balance are generally based on digitized
contour maps, with some exceptions where mass balance records have been
derived from digital elevation models (DEMs) extracted from old archives of
aerial photographs by applying digital photogrammetry (e.g., James et al., 2006,
2012). The applicability of geodetic mass balance records as a key to
predicting future glacier changes depends on the accuracy of such records
and their resolution. To maximize both the accuracy and the resolution we
should rather focus, if possible, on archives of aerial photographs,
because
these archives often span more epochs than the published topographic
maps; with new and rapidly improving tools in digital photogrammetry the
potential to produce more accurate and detailed DEMs than those deduced
by interpolating elevation contours from old maps has increased
significantly; the availability of high-resolution DEMs has opened a new source of
ground control points (GCPs) for constraining the orientation of
photogrammetric DEMs (James et al., 2006; Barrand et al., 2009). Like (ii),
this will lead to more accurate DEMs from aerial photograph archives in
future studies. New spaceborne sensors such as Worldview and Pléiades
may allow such studies in remote areas without conducting expensive field
campaigns to survey GCPs (Papasodoro et al., 2015).
In order to maximize the value of geodetic mass balance records, realistic
uncertainty assessments are required. If the uncertainty is overestimated,
the value of the information that we can extract from the geodetic data will
be diminished, and the results will be neglected by the scientific community or
not even be published. If, however, the uncertainty is underestimated,
geodetic mass balance records with significant errors will be interpreted as
reliable observations. When extracting volume change from two different DEMs, a
common approach is to use the standard deviation of the DEM difference in
the unglaciated part of the DEMs as a proxy for the uncertainty of the
average elevation change (e.g., Cox and March, 2004). This method corresponds
to an extreme case, assuming that the errors of the surface elevation change
are totally correlated between all grid cells within the glacier. The
opposed extreme case assuming that the errors of surface elevation change
are totally uncorrelated between all grid cells has also been applied in the
literature (e.g., Thibert et al., 2008). This approach results in an
estimated uncertainty reduced by a factor
Location of study area. Blue lines in
Here, we present a case study of Drangajökull ice cap in NW Iceland
(Fig. 1) based on seven sets of aerial photographs in 1946–2005 and a lidar
DEM obtained from an airplane in 2011 (Jóhannesson et al., 2013). The
glacier covered an area of 144 km
Dates, main parameters and notes describing the data sets used in the study.
In this study, seven sets of aerial photographs covering Drangajökull
ice cap in 1946, 1960, 1975, 1985, 1986 and 1994 from the archives of the
National Land Survey of Iceland and in 2005
from Loftmyndir ehf were used. Negative films were scanned with a
photogrammetric scanner in a resolution of 15 and 20
During the International Polar Year 2007–09, a major effort was
initiated to produce accurate DEMs of all the major Icelandic glaciers and
ice caps (Jóhannesson et al., 2013). In July 2011 Drangajökull ice
cap was surveyed with airborne lidar model Optech ALTM 3100. The lidar DEM
covers the entire ice cap as well as the close vicinity of the glacier,
which provides a useful reference to constrain and validate the other DEMs
produced in this study. Specifications of the survey are described in
Jóhannesson et al., 2013. The average density of the point cloud
measured with the lidar corresponded to 0.33 hits m
The DEMs were created from the aerial photographs using the software bundle IMAGINE Photogrammetry (©Intergraph). The photogrammetric processing is carried out in four steps: orientation of the images, automatic stereo matching, manual editing of the DEMs and orthorectification of aerial photographs.
The coverage of aerial photographs (black line boxes) at different epochs with the lidar DEM as background. The GCPs used for orientation of each series of aerial photographs are marked with triangles.
Each series of aerial photographs was oriented individually by means of a
rigorous bundle adjustment (Wolf and Dewitt, 2010). The glacier is covered
by a single series of images for all years except in 1960 when the glacier
was covered by three tiles, one per date (Table 1). Tie points were
automatically measured in the images and semiautomatically revised,
ensuring a good connection between all the adjacent photographs and between
strips. The exterior orientation was constrained by using series of GCPs extracted from the lidar DEM (2 m
The orientation of the 1960 images was carried out using the focal length and lens distortion information obtained from the calibration report of the DMA Cameras (Spriggs, 1966). The 1946 images included information of the focal length written at the margin of the first image of each strip. Both cases needed auxiliary pre-calibration, and therefore pseudo-fiducial marks were created to allow the location of a pseudo-principal point (see, e.g., Kunz et al., 2012, for details). The orientation of both sets of images included additional parameters in the bundle adjustment for refinement of the camera geometry. Bauer's model (Bauer and Müller, 1972) was used for the images of 1946 and Jacobsen's model (Jacobsen, 1980) was used for the images of 1960.
Once oriented, we produced the elevation point clouds from stereo-matching
of the images. The routine eATE (enhanced Automatic Terrain Extraction) of
the software allows for a pixel-wise evaluation in the matching process,
thus obtaining a high density of points. The low contrast in firn- and snow-covered areas
caused failures in the matching process. The point clouds for
low-contrast areas were therefore created from reduced resolution of the
stereo images and a larger window size and lower correlation coefficient of
the stereo matching. This resulted in an improved coverage of points
automatically measured in the snow-covered areas. A first edition of the
point clouds was carried out with the software CloudCompare (GPL Software);
automatic outlier removal was performed using the routine “Statistical
Outlier Removal” (Rusu and Cousins, 2011). The dense point clouds were then
subsampled in regular density of points corresponding to
To delineate the glacier margin and mask out snow-covered areas (Sect. 2.2
and 2.4) orthorectified photographs were required. The orthorectification
was carried out using preliminary DEMs linearly interpolated from the point
clouds as grids with 10 m
We use the high-resolution lidar DEM obtained in 2011 to assess the quality of the photogrammetric DEMs. The photogrammetric DEMs are expected to be of significantly worse quality in terms of accuracy than the lidar data and we therefore assume for simplicity that statistical parameters derived from the difference between the photogrammetric DEM and the lidar DEM (in areas assumed stable) describe errors in the photographic DEM. This is likely to produce a minor underestimate of the actual quality of the photographic DEMs. As described below, all photogrammetric DEMs were bias corrected relative to the lidar DEM. A possible bias in the absolute location of the lidar DEM does not affect our result since the bias is canceled out when calculating the difference between the DEMs.
The first step in estimating the quality of a DEM derived from the aerial
photographs was calculating the difference between the photogrammetrically
derived point clouds (Fig. S1 in Supplement) and the lidar DEM with 2 m
Extraction of geodetic mass balance requires co-registered DEMs prior to
calculation of glacier volume changes. This usually includes estimates of
relative vertical and horizontal shift between the DEMs, using areas where
the elevation change is expected to be insignificant (Kääb, 2005;
Nuth and Kääb, 2011; Guðmundsson et al., 2011). In this study
the GCPs used during the orientation of the photographs were extracted from
the lidar DEM in maximum resolution (2 m
The series of DEMs of Drangajökull ice cap created from the
aerial photographs. The shaded relief images and contour maps indicate the
glaciated part of each DEM. The elevation difference off ice (after masking
out outliers and areas with slope > 20
The elevation difference between DEMs covering stable areas is commonly used
to estimate zero order (bias correction, see, e.g., Nuth and Kääb,
2011; Guðmundsson et al., 2011) or higher order correction (e.g.,
Rolstad, 2009; Nuth and Kääb, 2011) to compensate for slowly varying
errors in DEM difference over glaciated areas. The result from such approach
is, however, sensitive to the area chosen as the reference area. One can
choose to use the entire area covered by both DEMs outside the glacier or an
area limited by a certain distance from the glacier. In this study we apply
geostatistical methods for deriving bias correction of each photogrammetric
DEM within the glacier and an estimate of the uncertainty in the derived
bias correction. These calculations consisted of the following five main steps.
Preparation of DEM error input data (derived from the comparison with the
lidar), explained below. Resulting error data from ice- and snow-free areas
are shown in Fig. 3. Transformation of the derived DEM errors into a new variable with the
nscore function (Deutsch and Journel, 1998) in WinGSlib V.1.5.8 (©Statios LLC). The histogram of the new variable fits a normal distribution,
with zero mean and Calculation of semivariogram for the nscored input data, in which the
semivariogram describes the variance, Calculation of a spherical semivariogram model, fitting the derived
semivariogram. Use of the derived spherical model and the nscored data that constrain
the semivariogram to run 1000 SGSims of the
nscored errors in the glaciated areas using the sgsim function (Deutsch and
Journel, 1998) in WinGSlib. The sgsim function includes reversed
transformation from the nscored variable to the derived DEM error. SGSims are
commonly applied in errors assessments of geostatistical studies (e.g., Lee
et al., 2007; Cardellini et al., 2003). The results from the sgsim runs were
used to estimate both the most likely bias of each photogrammetric DEM
within the glacier and 95 % confidence level of this bias, as explained
further below.
The approach adopted here requires that the statistics of the DEM errors
outside the glacier be descriptive for the errors in the photogrammetric
DEM within the glacier margin. This should be kept in mind both during the
photogrammetric processing and in the preparation of input data (step 1)
used in the geostatistical calculation. The photogrammetric processing
requires fairly even spatial distribution of GCPs; otherwise artificial dip
or rise in the photogrammetric DEM are likely to be produced in areas far
from a GCP (Kraus, 2007). Such errors would not be represented in a
semivariogram based on DEM error in areas where distribution of GCPs is
adequate.
The horizontal RMSE of the GCPs (no. of GCPs within brackets),
glacier coverage and error assessment of the photogrammetric DEMs, using
four different approaches: (i) direct comparisons of ice-free areas (mean and
standard deviation (
The low contrast of snow-covered glacier surface may also result in a
difference in error statistics between the glacier and the ice- and snow-free
areas (Rolstad et al., 2009). The low contrast should mostly produce high-frequency errors,
whereas low-frequency errors are mostly caused by an
inaccurate orientation. The eATE configuration used resulted in fewer but
better matching points in the low-contrast areas (Sect. 2.1) and the
thorough manual 3-D revision likely removes most of the high-frequency noise
in the resulting DEM. A semivariogram of the difference between the point
cloud in 1946 at low-contrast glacier areas and the lidar DEM (blue crosses
in Fig. 4c) reveals the variance with distance for the elevation error plus
the elevation changes in 1946 to 2011. The variance of elevation changes
over a short distance should be small for smooth glacier surface. At short
distances the semivariogram should therefore mainly represent the DEM
errors. For
A difference in terrain slope between areas can produce a significant
difference in the calculated semivariogram (Rolstad et al., 2009). Local
horizontal shift between DEMs can produce significant artificial elevation
difference in steep areas. The average slope on the glacier in 2011 was
6.2
The glaciated parts of the photogrammetric DEMs were all manually revised
using 3-D vision, securing removal of significant outliers within the
glacier. A thorough revision was not carried out for the unglaciated areas.
Instead we apply automatic removal of outliers. This was carried out by
calculating the standard deviation of the DEM error (photogrammetric DEM
The semivariograms obtained with (step 3) and without the nscore
transformation of the 1946 DEM error in ice- and snow-free areas are shown in
Fig. 4a–b. The spherical semivariogram model calculated in step 4 is given
as a function of
The size of the DEM error grid in full resolution (20 m
The semivariograms of the 1946 DEM error before
Each SGSsim, constrained by the input data and the spherical semivariogram
model and calculated in resolution corresponding to 100 m
The photogrammetrically derived point clouds are typically much less dense
for the snow-covered glacier surface than for bare ice or ground (see
Supplement). The typical distance between points on the snow-covered glacier
surface in the 1946 point cloud (the worst data set in terms of noise and
point density) is
The resulting grids of elevation changes relative to lidar contained some larger gaps due to lack of contrast, cloud cover or incomplete coverage of aerial photographs for all data sets except the one of 2005 (Table 2). To complete the difference maps, two main interpolation methods were used. For relatively small gaps, spanning a short elevation range, kriging interpolation with data search radius > 500 m was applied using the derived elevation difference at the boundary of the data gap as input. For larger areas spanning significant elevation range we estimated a piecewise linear function for the elevation change as a function of the 2011 elevation (at 100 m elevation intervals) using the elevation difference between the point cloud and the lidar DEM as input (see Supplement). For data gaps covering an area at both the east and west side of the glacier the two different interpolations were carried out, one for the area west of the ice divides and another for the area east of it. In four cases neither of the above interpolation methods were considered applicable. The approaches adopted for each of these cases is described in the Supplement. The location of data gaps are shown in Fig. S1 and the interpolation method applied in each case is shown in the Supplement.
The uncertainties associated with interpolation of data gaps in the DEMs were
approximated independently of the uncertainties of measured
photogrammetric DEMs (Sect. 2.2). It is difficult to quantify these errors,
but since these areas are generally small relative to the measured areas we
adopted a generous approximation of the uncertainty roughly based on the
scatter of the elevation change with altitude (point clouds compered to
lidar DEMs). We assign three values of elevation uncertainty (95 %
confidence level) to the interpolated areas,
The glacier margin and nunataks at each time were delineated manually using the orthorectified aerial photographs at given time as well as the derived elevation difference compared to the lidar DEM. For 2011 the glacier outlines were drawn based on a shaded relief image of the 2011 DEM in maximum resolution and the intensity image of the lidar measurements. All glacier margins were delineated by the same person. The glacier margin was therefore interpreted in similar manner for all years, in areas where the outlines are uncertain. This working procedure minimizes variations in relative area changes of the ice cap. Due to numerous firn patches in the vicinity of Drangajökull, some of which are connected to the ice cap, it is actually a matter of definition whether these connected patches should be included as part of Drangajökull or not. We follow the approach of Jóhannesson et al. (2013) and exclude these patches. In a few areas the aerial photographs do not always reveal the glacier margin. This includes the southernmost part of Drangajökull in 1946. In this area the location of the glacier margin has been very stable since 1960. We therefore adopted the outermost glacier margin in the 1960–2011 data sets at each location as the 1946 margin in this area. Data used to approximate the location of the glacier margin in other areas where data are absent are described in the Supplement. The evolution of the glacier area is shown in Fig. 5. Also shown in Fig. 5 is the area of the eastern and western sections of the glacier, when Drangajökull is divided in two along the ice divides from north to south (see Fig. 6).
The relative area change of the entire Drangajökull ice cap, the western and eastern sections of the ice cap. The purple lines in Fig. 6 show the ice divides; they are used to define the east and west sections of the glacier. Labels give the glacier area in square kilometers at each epoch.
The average annual elevation change of Drangajökull during six intervals since 1946. Red indicates thinning and blue thickening.
To derive the volume change,
The volume change of the southernmost of Drangajökull, which is missing in the 1994 DEM (Fig. 3), plotted as a function of the volume change in the northern part covered by the 1994 DEM, for the periods available (shown with black labels). The thick dashed line shows linear fit for the data points with the 95 % confidence area shown as light red. The red dots are the corresponding volume change estimates for the southern part in 1985–1994 and 1994–2005.
The DEMs of Drangajökull were extracted from data acquired at different
dates during the summer or the autumn (Table 1). Deriving mass balance
records from DEM difference without taking this into account will skew
the results, particularly if the acquisition time of the DEMs differs much
from one DEM to another. Seasonal correction (sometimes referred to as date
correction) has been applied and discussed in numerous studies (e.g.,
Krimmel, 1999; Cox and March, 2004; Cogley, 2009). In this study the derived
volume change between DEMs (
The average elevation change during periods defined by the DEMs
before (
The glacier-wide mass balance rate,
When estimating
Table 2 gives values of several error estimation parameters for the
photogrammetric DEMs deduced by comparison with the 2011 lidar DEM in ice-
and snow-free areas. Some of these parameters can be used both to correct
the DEMs and to estimate the uncertainty of geodetic mass balance results.
In some cases significant difference is observed between the mean DEM error,
commonly used to correct for bias (zero order correction) of the DEM (e.g.,
Guðmundsson et al. 2011), and the bias derived from the SGSim. The
greatest difference is for the 1946 DEM, which after removal of outliers and
steep slopes the ice- and snow-free part of it has a mean error of
The parameters in Table 2 that can be used to estimate the uncertainty of geodetic mass balance show even more diversity. The crudest parameter would be the standard deviation of the DEM error derived from ice- and snow-free areas. Standard deviation is commonly interpreted as 68 % confidence level assuming normal error distribution and should therefore be multiplied by 1.96 to obtain 95 % confidence level as derived for the other two approaches shown in Table 2. This interpretation of the standard deviation as uncertainty proxy of the volume change implies the assumption that the DEM errors at different locations within the glacier are totally correlated (Rolstad et al., 2009). Since the confidence level of geodetic mass balance results is typically not mentioned in studies using the standard deviation as their uncertainty proxy, the conversion of the standard deviation to 95 % confidence level is omitted in Table 2. The values of standard deviation for the ice-free DEMs are 5–45 % lower after removal of outlier and steep slopes. The lower standard deviation values are, however, still by far higher than the uncertainty (95 % confidence level) of the bias correction derived with SGSim. The SGSim results in uncertainty between 0.21 m (in 2005) and 1.58 m (in 1946). The SGSim uncertainties correspond to 24–46 % of the standard deviation (after slope and outlier removal). If we exclude the three DEMs from 1960, covering only about one-third of Drangajökull each, the range is 24–33 %. The SGSim uncertainties correspond to 27–80 % of the uncertainties derived with method described by Rolstad et al. (2009) and the percentage seems to depend strongly on the range of the spherical semivariogram model used in both calculations (Fig. 8).
The ratio between uncertainties (95 % confidence level) from the
methods demonstrated in this work and the method demonstrated by Rolstad et al. (2009) as a
function of the range,
The glacier-wide mass balance rate (
The effects of seasonal correction and the estimated contribution of each
type of error to the total volume change are summarized in Table 3. The
importance of seasonal correction for Drangajökull is clearly revealed,
particularly for the first two periods, 1946–1960 and 1960–1975, due to the
early acquisition of the 1960 aerial photographs. The sum of the two
seasonal corrections for these periods corresponds to a larger value than the
derived total uncertainty of
The main source of uncertainties is different from one period to another,
but in no case is the highest contribution from the estimated uncertainty
of the DEM elevation (
The uncertainty percentage of
Figure 9 shows the derived
The middle panel of Fig. 9 shows
Figure 9 shows different evolution of the west and east glaciers. Both parts
suffered significantly negative mass balance rate in 1946–1960 and
1994–2011. The period in between was significantly negative on the east
side, apart from the period 1985–1994, when the upper 95 % confidence
level is slightly above 0, whereas the western part had
The high precision of the geodetic mass balance results presented can be primarily explained by (i) the use of the high-resolution and high-accuracy lidar DEM to extract evenly distributed GCPs for constraining the orientation of photogrammetric DEMs (obtaining equivalent distribution of GCPs in the field was not possible within the financial frame of this study) and (ii) the thorough uncertainty assessment of the results where the lidar data from ice- and snow-free areas are also key data since they enable assessment of geostatistical parameters of the photogrammetric DEMs. Both (i) and (ii) highlight the need for high-resolution and high-accuracy DEMs from the present in areas of interest to conduct studies of geodetic mass balance using aerial photographs from the past. The third important use of the lidar data in this study is the creation of DEMs from the photogrammetric point clouds within the glacier. Rather than interpolating the elevation point clouds directly we interpolate the difference between the point cloud and lidar DEM (much less high-frequency variability, and the difference is a smoother surface; Cox and March, 2004) and add the interpolated product to the lidar DEM. This results in more accurate DEMs in areas where the density of the photogrammetric point clouds is low.
Other state-of-the-art high-resolution elevation data sets obtained with
airborne or spaceborne sensors are also suitable to replace the lidar data
in the work procedure described here. This probably includes Worldview and
Pléiades high-resolution stereo images, allowing extraction of DEM with
< 5 m cell dimensions and orthorectified photographs with < 1 m
In this study, the derived bias correction of the glaciated DEM section and the uncertainty of volume changes related to DEM errors are obtained from the probability distribution calculated by using SGSim. The bias correction corresponds to the probabilistic mean of the average error within the glacier. As shown in Table 2 the difference between the mean error in snow- and ice-free areas and the bias derived from the SGSim (the estimated probabilistic mean of the glacier DEM error) was up to 2.5 m (in 1946). This difference would presumably be lower if we would only calculate the mean error using areas within certain distance from the glacier margin, but it is not straightforward to select this distance without using some geostatistical approaches. The relation is also not obvious between the probabilistic mean of an average DEM error within the glacier and higher-order corrections of a glacier DEM obtained with least square fit (or similar) using deduced DEM errors in ice- and snow-free areas. If the average correction does not correspond to the probabilistic mean, the results of geodetic mass balance will be incorrectly centered even if the width of the error bars is realistic.
When comparing different proxies used for estimating the uncertainty of DEM difference derived volume change, it is no surprise that using the standard deviation of the DEM error in snow- and ice-free areas leads to great overestimate of the uncertainty (Table 2). This has been shown before by Rolstad et al. (2009). Other estimators that ignore information of the spatial dependency of the DEM errors, such as the NMAD value (Höhle and Höhle, 2009), should also be considered as incomplete for this purpose.
The difference in uncertainty estimates between the method described here
and the method of Rolstad et al. (2009) is especially noteworthy (Table 2
and Fig. 8). Rolstad et al. (2009) provided a simple and logical method to
estimate the uncertainty of derived volume change. The DEM errors (or
difference) in ice- and snow-free areas are used to calculate a semivariogram
that constrains a spherical semivariogram model. From the spherical
semivariogram model alone the expected variance of the DEM error (
Our study emphasizes the importance of including seasonal correction of DEMs for glacier with high mass turnover to avoid wrong interpretation of derived volume change. The most extreme case is the negative volume change derived from the difference between the 1960 and 1975 DEMs. The seasonal correction results in about three-quarters of this negative volume change being effectively transferred in to the period 1946–1960 due to large seasonal correction of the 1960 DEM resulting from relatively early acquisition of the aerial photographs (Table 1). The seasonally corrected volume change revealing the volume change between the start of different glaciological year obviously has higher uncertainty than the uncorrected volume change. We, however, consider this trade-off important for easy comparison with other data records, including meteorological data and in situ mass balance measurements. The uncertainty due to the seasonal correction as well as the uncertainty related to the interpolation of the data gaps should be considered as cautious estimates of the 95 % confidence level of the error associated with these two error sources. Effort should be made to constrain these uncertainties further, which could narrow the uncertainty estimates of this study and other similar even further, but it is beyond the scope of this paper.
The presented geodetic mass balance records indicate slower volume decrease
for Drangajökull ice cap since the 1940s than for most other glaciers in
Iceland with geodetic mass balance record extending back to that period.
While we observe
The difference in the geodetic mass balance results between the east and
west part of Drangajökull highlights how difficult it is to extrapolate
mass balance records from one glacier to another, even over short distances.
The results, showing
The geodetic mass balance record on Drangajökull ice cap is the first
record revealing glacier volume change in Iceland on a decadal timescale
of the past
This paper highlights the opportunities that new high-resolution DEMs are opening to improve the procedure carried out to obtain geodetic mass balance records. Such DEMs are key data in three aspects of this study: (a) extracting GCPs from recent airborne lidar DEM to constrain photogrammetric DEMs at six different epochs; (b) interpolating over glacier surface the elevation difference of derived photogrammetric point cloud relative to the lidar DEM; (c) applying new geostatistical approaches based on comparison with the lidar data to estimate simultaneously a bias correction for the glacier DEMs along with its 95 % confidence level. The latter reveals the uncertainty associated with DEM errors in geodetic mass balance records.
The new geostatistical method applies SGSim using the DEM errors in ice- and snow-free areas and a spherical semivariogram model constrained by the DEM errors as input data. The resulting bias correction may differ considerably (in our case up to 2.5 m in 1946) from the simple approach of applying bias correction using the mean DEM error outside the glacier. The resulting uncertainty of the DEM (95 % confidence level) was typically estimated 20–35 % of the standard deviation derived from the DEM errors in ice- and snow-free areas after outliers and high slopes were masked out. The uncertainty contribution from DEM errors obtained with SGSim was 25–80 % of the uncertainty estimate obtained with the geostatistical method of Rolstad et al. (2009). We argue that methods typically carried out in uncertainty assessments of geodetic mass balance generally overestimate the uncertainty related to DEM errors, while the geostatistical approach described here results in more realistic uncertainty estimates.
This study also reveals the importance of seasonal corrections of geodetic
mass balance for glaciers with high annual turnover; Drangajökull is a
good example. The highest correction in our study was
During the whole period 1946–2011 we obtain
The writing of this paper and the research it describes was mostly carried out by the first two authors of this paper, with input from the other three co-authors. All photogrammetric processing and revision of the resulting point clouds was carried out by J. M. C. Belart. The interpolation of the point cloud differences compared to the lidar DEM and construction of glacier DEM based on that, interpolation of data gaps, delineation of glacier margin, seasonal correction of the volume change, the construction of the presented mass balance records and associated error analysis was carried out by E. Magnússon based on fruitful discussions with J. M. C. Belart and F. Pálsson. All figures in this paper were made by E. Magnússon and J. M. C. Belart as well as tables. P. Crochet and H. Ágústsson contributed to the handling and interpretation of the meteorological data.
This work was carried out within SVALI funded by the Nordic Top-level Research Initiative (TRI) and is SVALI publication number 70. It was also financially supported by alpS GmbH. This work is a contribution to the Rannís grant of excellence project, ANATILS. We thank the National Land Survey of Iceland and Loftmyndir ehf. for acquisition and scanning of the aerial photographs. This study used the recent lidar mapping of the glaciers in Iceland that was funded by the Icelandic Research Fund, the Landsvirkjun Research Fund, the Icelandic Road Administration, the Reykjavík Energy Environmental and Energy Research Fund, the Klima- og Luftgruppen (KoL) research fund of the Nordic Council of Ministers, the Vatnajökull National Park, the organization Friends of Vatnajökull, the National Land Survey of Iceland and the Icelandic Meteorological Office. We thank Cristopher Nuth and an anonymous reviewer for constructive reviews, as well as Etienne Berthier for helpful comments on the manuscript. We thank Alexander H. Jarosch for fruitful discussions on the subject of this paper. Auður Agla Óladóttir is thanked for introducing the tools and methodology used in the error analysis of this study to E. Magnússon. Edited by: I. M. Howat