Digital Photogrammetry

Deriving digital terrain models (DTMs) using digital stereo-photogrammetry is a well-established method described by, for example, Reference SchenkSchenk (1999) and Reference KääbKääb (2005). Overlapping stereo images are recorded. The recording geometry, as camera position and orientation angles, is estimated from ground-control points (GCPs) and tie points registered in the images. Identification of identical terrain features is conducted using cross-correlation calculations on subsections of the image pairs. This method requires contrast in the images. Snow and shadow areas typically have poor contrast and give inaccurate results. It can also be difficult to identify terrain features in steep areas, due to large geometrical distortions in the images. Quality control of the calculated elevations is conducted by visual inspection of the determined elevations in the stereoscopic viewing system on the screen. If the elevation looks incorrectly placed on the imaged terrain surface, the elevation is edited manually. In gridcells where no elevation can be measured, the elevation is determined from bilinear interpolation of surrounding points.

The accuracy of the determined recording geometry is crucial for the calculated elevations. It is an advantage that the GCPs are signalized on the ground using contrast colours so that they can be readily identified in the images. The GCPs and additional tie points should also be well distributed in the model.

The DTMs are constructed by stereophotogrammetry on vertical aerial photographs taken in 1968, 1985 and 2002 (Table 1) 1by Fjellanger Widerøe AS. The DTM from 1968 was constructed using analogue photogrammetry, while the two others are newly constructed using digital photogrammetry. The 1968 DTM was digitized earlier by the Norwegian Mapping Authority. A triangular irregular network (TIN) was constructed from the analogue contour map, and the DTM was interpolated from the TIN using bilinear interpolation. The DTM from 1985 is constructed by TerraTec AS, Oslo (formerly Fjellanger Widerøe AS), and the DTM from 2002 was constructed by the authors using SocketSet and Image Station software.

Table 1.

Photograph and DTM information. Ablation adjustment intervals and number of positive degree-days between date of photography and end of ablation season at the plateau are also shown

The coordinates of the points in the three DTMs do not coincide, both because of the grid size and the position of the grid nodes. The heights of the points in the 1985 DTM are interpolated to the positions of the points in the 1968 and 2002 DTMs using bilinear interpolation before the DTMs are differenced.

In the 1968 DTM the glacier front position and the elevation contours of the lower parts of Flatisen had been replaced with information from aerial photographs taken in 1993. This was done because Flatisen had retreated and thinned considerably during this period. A DTM with the 1968 heights for this area was not available.

The lake Storglomvatnet, north of Svartisen, is dammed by a power company. As three outlet glaciers terminate in this lake, their heights are influenced by the lake level in the reservoir. When the photographs were taken in 1968 the lake level was 510 ma.s.l., in 1985 it was 514 ma.s.l. and in 2002 580 ma.s.l.

Geodetic Mass Balance

The geodetic mass balances for the two periods are computed for both the whole ice cap and for Engabreen drainage basin. We used the drainage divides determined by Reference Kennett, Rolstad, Elvehøy and RuudKennett and others (1997). The boundaries for the Engabreen drainage basin are shown in Figure 21.

The DTMs are differenced using ESRI ArcInfo software. When a part of a drainage basin is not covered by the DTMs, the whole drainage basin is left out because the emergence velocity is not known. The different drainage basins can be seen in Figure 2. The 1968 DTM did not cover the entire Northern part drainage basin, so this part is left out in the first period. Because the Flatisen drainage basin contains many smaller outlet glaciers, one of these is excluded when differencing the 1985 and 1968 DTMs due to the updating of the glacier front position. We mapped a total of 88% in the first period. The 2002 DTM did not cover the entire Memorgebreen or Fonndalsbreen basins. These drainage basins are therefore excluded and we mapped a total of 91% in the second period. The elevations of the three outlet glaciers terminating in Storglomvatnet are influenced by the changing lake level in the reservoir. This affects the lowermost parts of the outlets, so that the elevation differences in the second period did not stem from glacier thinning or thickening only. However, this only affected the first 100m of the outlets, so excluding this area is assumed to be unnecessary.

Fig. 2.

Elevation contours for the three years. The contour interval is 20m. The drainage basins are also shown: 1. Memorgebreen; 2. Fonndalsbreen; 3. Engabreen; 4. Dimdal–Frukosttindbreen; 5. Northern part; 6. Storglombreen; 7. Flatisvatnet; 8 . Nordfjordbreen.

In order to express the elevation changes as water equivalents, the thickness and density of the firn area at the time of photography should be known. However, as these data are not available, Sorge’s law (Reference BaderBader, 1954) is assumed. This states that the density is unchanged in an unchanging climate. By assuming this, the elevation change given in water equivalents can be found by multiplying the elevation change by the density of ice. A density of 900 kgm ^{− 3} is commonly used (Reference AndreassenAndreassen, 1999; Reference KrimmelKrimmel, 1999; Reference Cox and MarchCox and March, 2004), and is also used in this study.

The geodetic mass balances in both periods are adjusted for ablation. The sums of positive degree-days are calculated from temperature data between the date of photography and the end of the ablation season at the glacier plateau for each of the three years (Table 1). The end of the ablation season is determined from the weather station data from Skjæret. The temperature records from the nearest meteorological station, Glomfjord (39ma.s.l.), provided by the Norwegian Meteorological Institute (met. no), are shown in Figure 31, together with NVE’s temperature record for 2002 from Skjæret (1364ma.s.l.), a nunatak at the glacier plateau. The locations of the meteorological stations are shown in Figure 1. The latter station has been operated by NVE since 1995. As temperature records fromSkjæret in 1968 and 1985 are not available, the temperature at Skjæret is estimated using the temperature at Glomfjord and a temperature lapse rate between the stations (Reference SchulerSchuler and others, 2005). Reference SchulerSchuler and others (2005) determined an average lapse rate of −0.0074 ^˚ Cm− ¹ for the period 1999–2002. The sums of positive degree-days are compared, and the ablation adjustments are estimated based on the glacier melt rate calculated for Engabreen (Reference SchulerSchuler and others, 2005). This melt rate takes into account the different albedo for snow and ice surfaces. In the period 1968–1985 the conducted ablation adjustment is 0.2 mw.e.; for the period 1985–2002 it is −0.3mw.e.

Fig. 3.

Temperature at Glomfjord in 1968, 1985 and 2002 and temperature at Skjæret in 2002. Minor melt events may have occurred later than our estimated end of ablation season.

End-of-balance-year adjusted point elevation changes are not adjusted for emergence velocity. According to Reference Geist, Elvehøy, Jackson and StötterGeist and others (2005), the emergence at Engabreen in the period 24 September 2001 to 28 May 2002 varied from −4m at the glacier plateau to 7m at the glacier tongue for the whole period (246 days). This gives emergence velocities ranging from 0.028 md ^{− 1} at the tongue to −0.016md ^{− 1} at the plateau. As the timespans were 6 days in the period 1968–85 and 1 day in the period 1985–2002, no corrections are made. It is assumed that the emergence velocity is the same for all three years, and that this assumption does not cause additional uncertainty. For point balances this is a reasonable assumption because the uncertainty from inaccurate elevations will be large.

Error Assessment

The relative elevation errors are important for the accuracy of the geodetic mass balance. The absolute errors of the three DTMs are of less importance for geodetic mass-balance determination, as they only indicate how accurate the DTMs are relative to a datum (Reference Cox and MarchCox and March, 2004). Elevation differences over bedrock and spatial autocorrelation results have previously been used to assess the elevation uncertainty (Reference Nuth, Kohler, Aas, Brandt and HagenNuth and others, 2007). By differencing the DTMs over bedrock, the relative errors are estimated. The standard deviation of the elevation difference over bedrock gives the uncertainty of the point measurements. For assessing the spatial averaged uncertainty, the elevation differences are first detrended. Presence of spatial autocorrelation due to, for example, orientation errors is evaluated using semivariograms, and the results are used to evaluate spatial averaged uncertainty for the drainage areas. Details of the spatial analysis will be presented elsewhere. Only areas that are matched automatically in the 2002 DTM are used for determining the relative error. These areas exclude mountainsides steeper than 30˚ and shadow areas where few identical features are found in the stereo pairs. As steep mountainsides are excluded, relative elevation differences that may stem from horizontal shifts of the elevation grids due to uncertainty in the absolute orientation are reduced. We choose a threshold of 30 ^˚ because areas with a slope under this threshold are assumed to be most representative for the glacier area. Only 2% of the ice cap has a slope steeper than 30 ^˚ . This 2% is not clustered, and the largest area is 0.1 km².

Ideally the three DTMs should have been constructed at the same time by the same constructor using the same software and the same well-signalized GCPs. If so, the DTMs would have had a more similar absolute orientation, and hence the relative difference between the DTMs over bedrock would be smaller. Some of the GCPs were used for orienting both the 1985 and the 2002 DTM, but the GCPs were, unfortunately, not well signalized in the 2002 photographs and the DTMs were constructed by different people. The quality of the absolute orientation thus differs for the two DTMs.

The melt rate used to estimate ablation corrections is calculated specifically for Engabreen. This means that the topography of Engabreen and its surrounding area, such as slope, aspect and effective horizon, is taken into account (Reference HockHock, 1999). Hence, errors are introduced by using the same melt rate for the whole of western Svartisen. The numbers of positive degree-days for the three years are calculated only for Skjæret. However, as the corrections are small, due to the relatively small differences in positive degree-days, this error will not be large. Based on this, an error of ±0.3mw.e. is assumed.

Uncertainties introduced by assuming constant density can be due to both changes in the equilibrium-line altitude (ELA) and changes in the density of the firn layer. As no measurements of the density and thickness of the firn layer are available, the estimate of the uncertainty introduced by assuming constant density is based on information on the transient snowline altitude for the three years. Traditional mass-balance data are also investigated in order to reveal changes in the densification process of the firn layer. The densification process can change as the accumulation rate, the density of the newly accumulated snow and the temperature change (Reference Arthern and WinghamArthern and Wingham, 1998; Reference McConnellMcConnell and others, 2000). The transient snowline in 1968 was ∼1000ma.s.l., in 1985 ∼1400ma.s.l. and in 2002 ∼1250ma.s.l. Modelling results of the mass balance from 1962 to 1968 show that all these years had positive mass balance (Reference Andreassen, Elvehøy, Jóhannesson, Oerlemans, Beldring and van den BroekeAndreassen and others, 2006). The traditional mass-balance data show that the years prior to 1985 had positive mass balances, while 1985 had a negative mass balance. In 2001 there was a large negative mass balance, when almost all the snow melted. The mass balance in 2002 was positive. This indicates that the firn in 1968 was new and relatively little densified, and that snow was present. In 1985 the firn was also new, but could have been more densified due to the negative mass balance in 1985. In 2002 the firn was dense, due to the very negative mass balance in 2001, but new snow was present. Based on this information, the uncertainty introduced by assuming constant density is ±0.6mw.e. for 1968–85, and ±0.2mw.e. for 1985–2002.

The grid size of the DTMs must be adequate to represent the topographic variations of the glacier surface. Reference AndreassenAndreassen (1999) tested how the geodetic mass balance varied when using grid sizes of 5, 10, 20, 50 and 100m for Storbreen, Norway, and found that as long as the grid size was 5–20m, the geodetic mass balance showed little variation. Storbreen glacier is a valley glacier with an area of 5.4 km² and an average slope of 14˚. The three DTMs have different grid sizes, as shown in Table 1. Because the area of western Svartisen is 190 km² and the average slope is 8 ^˚ , the grid sizes are assumed to represent the topographic variations of the icecap surface.

The flying height influences the accuracy of the determined elevations; flying lower gives a larger scale and a smaller standard error. The scales of the photographs listed in Table 1 show that the photographs from 2002 are likely to have the lowest errors.

Digitizing the analogue 1968 DTM introduces horizontal random errors, depending on the accuracy of the digitizer and the condition of the analogue manuscript (Reference Andreassen, Elvehøy and KjøllmoenAndreassen and others, 2002). According to Reference Kjekshus, Edvardsen, Kristensen, Jaren, Ø.Olafsen and TallhaugKjekshus and others (2007), the standard deviation of this DTM is within 4–6m.

In aerial photographs it is sometimes difficult to determine the glacier outline. Rocks and moraine material on top of the ice can make the glacier area erroneously defined, and also snow close to the glacier can make it difficult to draw the glacier outline. There is little moraine material on western Svartisen, so this is considered to be a minor problem. Snow is also considered to be a minor problem.

Flatisen terminates in the lake Flatisvatnet. Without estimates of the water depth, the actual volume change is difficult to obtain. This is considered to be a minor problem.

Elevation uncertainty, melt uncertainty and uncertainty from applying Sorge’s law are uncorrelated errors. According to standard error propagation, the total error of the geodetic mass balance is the square root of the sum of each individual squared uncertainty.

Photogrammetry

The root-mean-square (rms) errors for the GCPs of the two DTMs constructed using digital photogrammetry, are shown in Table 21. The elevation contours for the three years are shown in Figure 2.

Table 2.

The root-mean-square errors given in ground coordinates for the GCPs in the newly constructed DTMs

Geodetic Mass Balance

The geodetic mass balance for the period 1968–85 is −2.6mw.e. A total of 167 km² (88%) was mapped in this period. The westernmost and northernmost parts thinned considerably (Fig. 4)1. For Engabreen drainage basin, with an area of 40 km², the value is −2.1mw.e.

The geodetic mass balance for the period 1985–2002 is −2.0mw.e. A total of 170 km² (91%) was mapped in this period. Flatisen and the northernmost areas thinned considerably (Fig. 4). For Engabreen drainage basin the value is −0.3mw.e.

Fig. 4.

Surface elevation change adjusted to the end of the balance year (in mw.e.) of western Svartisen in 1968–85 (left) and 1985–2002 (right). The black line marks the Engabreen drainage basin.

In order to compare the geodetic mass balance in the two periods, it is also computed for the drainage basins covered by all three DTMs. This area covers 80% of the ice cap and is 150 km². In the first period the geodetic mass balance for this area is −2.6mw.e. and in the second it is −1.0mw.e.

The ice-cap area decreased from 190 km² in 1968, to 187 km² in 1985 and to 184 km² in 2002. Between 1968 and 1985 Engabreen was the only outlet glacier that advanced. Between 1985 and 2002 Engabreen and two small outlet glaciers on the eastern side advanced slightly. Flatisen retreated 1700m between 1968 and 2002.

Figure 5 1shows the surface elevation change for western Svartisen as a function of surface elevation for the two periods. In the period 1968–85, the positive elevation changes below 250ma.s.l. can be fully attributed to the advance and thickening of Engabreen, as Engabreen is the only part of the ice cap that extends down to below 250 ma.s.l. Between 250 and 650 ma.s.l. the variation in the elevation changes reflects the thickening of Engabreen and the thinning and/or disappearance of the three outlet glaciers from Storglombreen and smaller outlets. Above 650ma.s.l. the mean is below zero for all elevations. Unfortunately the elevation changes for Flatisen could not be obtained for the period 1968–85, due to the updating of the 1968 DTM mentioned earlier. In the period 1985–2002 the positive elevation changes below 100ma.s.l. can be attributed to the advance of Engabreen. However, during this period no thickening of the Engabreen tongue could be observed. Between 350 and 550ma.s.l. the retreat and thinning of Flatisen is the prevailing feature. The elevation changes are negative up to 1100ma.s.l.

Fig. 5.

Surface elevation change for western Svartisen as function of surface elevation in the two periods. The surface elevation change is adjusted to the end of the balance year. Note that the scale is different for the two periods.

Error assessment

The individual and geodetic mass-balance uncertainties are shown in Table 31. The uncertainties in the geodetic mass balance for the whole of western Svartisen and for Engabreen are estimated based on the error estimates and using standard propagation of individual uncertainties.

Table 3.

Individual uncertainties and the geodetic mass-balance uncertainties for western Svartisen, Engabreen and single points

Photogrammetry

The georeferencing of the photographs affects the final result. All three DTMs were constructed at different times and by different constructors, hence they have different absolute orientation. This is unfavourable for the accuracy. The 1985 and the 2002 DTMswere constructed using some of the same GCPs, which were not signalized in the 2002 photographs. The large differences in rms errors of the 1985 and 2002 DTMs are because the 2002 GCPs were not signalized. The absolute orientation of the 1985 DTM is considerably better than expected. In this study, the unsignalized GCPs in the 2002 DTM caused a larger scale of autocorrelation in the bedrock differences, which made the elevation uncertainties large in the second period. In order to reduce this uncertainty in future studies the GCPs need to be well signalized.

Varying the grid size in the low-contrast area during matching can improve the results. This was not done in the present study, but should be done in future studies. When the grid size is increased, spatially infrequent features in the images may be identified and successfully matched. With a reduced grid size, spatially frequent features may be identified.

In order to avoid interpolation for a shift of the DTM to common point coordinates, the DTMs should be of the same grid size, with the gridpoints located at the same coordinates.

Geodetic Mass Balance

The geodetic mass balances for the two periods for the drainage basins covered completely by all three DTMs are –2.6±0.8mw.e. for the period 1968–85 and –1.0±1.6mw.e. for the period 1985–2002. The mass balances are therefore not different in the two periods.

The results obtained in this study are not in agreement with the traditional mass-balance measurements of Engabreen that NVE has obtained annually since 1970. Reference Kjøllmoen, Andreassen, Engeset, Elvehøy and M. JacksonKjøllmoen and others (2003) report a positive cumulative mass balance of 22 mw.e. from 1970 to 2002 for the Engabreen drainage basin. Using the same drainage divides, we find the geodetic mass balance for the 1968–85 to be –2.1±0.9mw.e., and –0.3±2.4mw.e. for the period 1985–2002. Other authors have also found the geodetic mass balance to be more negative than the traditional mass balance (Reference KrimmelKrimmel, 1999; Reference Østrem and HaakensenØstrem and Haakensen, 1999). As Reference Østrem and HaakensenØstrem and Haakensen (1999) point out, there are several sources of systematic errors in the traditional mass balance which can make the cumulative traditional mass balance too positive. Sinking of stakes in the firn area, snow probes deviating from the vertical, and snow probes penetrating the previous year’s summer surface, are all possible sources of systematic errors. As Engabreen is an outlet glacier from an ice cap, the determination of drainage divides can also cause systematic errors which can make the cumulative traditional mass balance too positive (Reference Elvehøy, Jackson and AndreassenElvehøy and others, in press).

In the period 1968–85 the positive elevation changes below 250 ma.s.l. exceed the uncertainty. The negative elevation changes at elevations between 650 and 1500ma.s.l. also exceed the uncertainty. In the period 1985–2002 both the positive elevation changes below 100ma.s.l. and the negative elevation changes found at elevations between 300 and 1050 ma.s.l. exceed the uncertainty.

The advances of Engabreen in both periods are in agreement with registrations of the front position (Reference Kjøllmoen, Andreassen, Engeset, Elvehøy and M. JacksonKjøllmoen and others, 2003).