Idealized glacier scenarios

Figure 2 displays the seven idealized glacier retreat scenarios tested. Each scenario was constructed so that a terminus undergoes a width-averaged retreat of R. For simplicity, R is taken to be equal to a single glacier width, W. All results are referred to in terms of glacier retreat, though the inverse of the results from these tests will be relevant for equivalent glacier advances. The experiment scenarios constructed are:

Fig. 2. Experiments showing the idealized scenarios, where R is an identical width-averaged retreat for each experiment. The letters of each experiment conform to those indicated in the text. The grey dashed line indicates a width-averaged retreat of R parallel to the original terminus. Where the rectilinear box method is applied to experiments (e)–(g), the box is orientated so that the upstream edge of the box is both parallel to, and centred on, the black dashed line indicated. Unless specified in the text, the curvilinear box used for experiments (e)–(g) is of width W and tracks the fjord width.

• (a) Retreat of a linear terminus by R to form a new parallel terminus in a straight fjord (Fig. 2a).

• (b) Width-averaged retreat of a linear terminus by R to form a new symmetric terminus with a calving bay of 0.25R in a straight fjord (Fig. 2b).

• (c) Width-averaged retreat of a linear terminus by R to form a new asymmetric terminus, with the asymmetry hinged on the centre line (at 0.5W), and a calving bay of 0.25R in a straight fjord (Fig. 2c).

• (d) Width-averaged retreat of a linear terminus by R to form a new asymmetric terminus, with the asymmetry offset from the centre line by 0.25W, and a calving bay of 0.5R in a straight fjord (Fig. 2d).

• (e) Width-averaged retreat of a linear terminus by R to form a new parallel terminus in a fjord that has changed orientation. The retreated terminus is offset to the right by 0.5W (Fig. 2e).

• (f) As for (e), but the retreated terminus is positioned perpendicular to the fjord wall (Fig. 2f).

• (g) As for (e), but the retreated terminus has a calving bay of 0.25R symmetric along the glacier centre line (Fig. 2g).

In reality, the challenge of tracking glacier terminus change reflects a combination of some or all of the above scenarios

Real glacier examples

To allow an intercomparison of results from the different methods, six Greenlandic tidewater glaciers exhibiting a range of retreat behaviour were selected for analysis (Fig. 3). Together they exemplify a number of potentially problematic scenarios that can arise when tracking change of real termini. The glaciers selected were Narssap Sermia (NS; 64.648 N, 49.978 W), Jakobshavn Isbræ (JI; 68.178 N, 49.858 W), Petermann Glacier (PG; 80.788 N, 60.618 W), Helheim Glacier (HG; 68.618 N, 32.938 W), Qalerallit Sermia West 1 (QSW1) and Qalerallit Sermia West 2 (QSW2) (61.048 N 46.728 W). Some of the issues with the retreat of these glaciers include: the terminus of NS changes orientation within the fjord, and demonstrates asymmetry and increasing complexity across the time series analysed; the width of JI changes significantly during its retreat, while embayments with relatively stagnant ice also exist; the terminus of PG is an ice shelf, meaning that periodically large calving events of tabular icebergs occur, and identification of the terminus itself can be problematic; HG is a fast-flowing glacier retreating through a straight fjord that experiences significant changes in terminus geometry; and QSW retreats from being a single glacier to form two separate termini that are analysed separately (named here as QSW1 and QSW2).

Fig. 3. Landsat-based observations of terminus change at six Greenlandic glaciers, and the results given by different methods of tracking terminus change (inset). Glaciers shown are (a) Narssap Sermia (64.648 N, 49.978 W), (b) Jakobshavn Isbræ (68.178 N, 49.858 W), (c) Petermann Glacier (80.788 N, 60.618 W), (d) Helheim Glacier (68.618 N, 32.938 W) and (e) Qalerallit Sermia West 1 and Qalerallit Sermia West 2 (61.048 N, 46.728 W). The rectilinear and curvilinear boxes used to track terminus change are overlaid on the images. Box width was limited to that of the narrowest terminus observation.

For each glacier, five terminus positions acquired between 1992 and 2012 were delineated from Landsat panchromatic band (15 m pixel resolution) images, and terminus change quantified using each method. For JI and HG, images were preselected for the period after the melange in their fjord had broken up to allow easier definition of their termini. Termini were delineated at a level of image magnification that allowed individual pixels to be visually resolved. This allowed a consistent level of detail to be maintained when delineating individual termini from different glaciers.

NS was subject to extra experiments (below) to demonstrate how the bow and box methods deal with issues pertaining to terminus asymmetry, width and fjord orientation when applied to real glaciers. Its terminus retreated significantly between 2009 and 2012, having previously maintained an approximately stable terminus position since the end of the Little Ice Age (Fig. 4; Weidick and others; 2012). To allow the relative effects of a change in fjord orientation to be fully evaluated, a straight terminus of semi-arbitrary orientation was also positioned upstream of the 2012 terminus (terminus A; Fig. 4).

Fig. 4. Location map of Narssap Sermia showing the reference points used in the bow method indicated by the red crosses, and their associated offsets from a semi-arbitrary 08 position centred on terminus A. Each reference point is positioned at least 6W from terminus A. The background Landsat image shown was acquired on 15 September 1987. Dates are day/month/year.

Bow method experiment

The distance between the reference and measurement points used for the bow method is known to exert control on the accuracy and consistency of its results (Reference BjørkBjørk and others, 2012). However, the effect of the reference point positions relative to fjord and terminus orientations has not been established. Consequently, the range of effects that could result from changes in fjord orientation upstream of the termini was evaluated by quantifying terminus change for experiments (a)– (g) using 19 different reference points positioned 6W behind the terminus distributed evenly along a 1808 arc at 108 intervals. This experiment was also conducted over the same range of reference points for NS to demonstrate how reference point position can affect results for real termini (Fig. 4). Although the majority of fjord orientations will change by significantly less than 908 from their original terminus orientation, testing a large range of reference point positions allows the effects of a comprehensive range of potential changes in fjord orientation to be evaluated. The points on the termini used in the calculation of overall position were spaced along each terminus at equidistant 0.01W intervals for scenarios (a)–(g) and 10 m intervals for NS.

Box method experiment

One of the primary advantages of the box methods is that they are capable of accounting for the natural asymmetry of glacier termini (Reference Moon and JoughinMoon and Joughin, 2008). However, since it is necessary to use boxes of fixed width for the method, it is inevitable that parts of the fjord terminus margins will fall either outside or inside the predefined box. To test the relative importance of this, boxes of 0.25–2W width (with 0.25W increments) were applied to scenarios (a)–(g), while boxes of 1–7 km width (with 1 km increments) were applied to NS. This was done for both the rectilinear and curvilinear box methods. Where the terminus end of the box was wider than the delineated glacier terminus, the box was closed by extending the terminus edges with straight lines perpendicular to the upstream edge of the box for the rectilinear box method, and with lines perpendicular to the centre line of the box for the curvilinear box method.

he aim of this experiment is to test the sensitivity of both box methods to the proportion of the glacier terminus that is included in the calculation of terminus change.

Extrapolated centre-line experiment

To allow the method to be evaluated in a non-dimensionalized manner for experiments (a)–(g), the search radius that defines the number of x_n values used in the calculation of each t_k value needs to be defined in a clear and consistent way. This experiment aims to determine over what scales of search radii calculated terminus positions begin to converge, so that the value of the search radius can be standardized. Different search radii were tested, from a minimum using individual x_n values in the calculation of t_k , to ranges of x_n values taken over a centre-line distance of 3W. These are then applied to the scenarios (e)–(g) where termini retreat a distance of R through a fjord that has changed orientation and exhibit different terminus geometries. Terminus position was calculated from points positioned on the terminus at 0.01W intervals, while points representing values of x_n on the centre line were also separated by 0.01W.

Centre-line method

While simple and quick to implement, the centre-line method does not account for the full complexity of terminus geometries, and hence cannot provide an accurate value of width-averaged retreat. This is highlighted in experiments (b), (c), (d) and (g), where the method overestimates terminus retreat (Table 1). Terminus geometry therefore exerts a strong control on the accuracy and applicability of the centre-line method.

Table 1. Summary of results from each method after application to each of the idealized scenarios shown in Figure 2. Results for rectilinear and curvilinear box methods are for box widths of W only. All results are given in terms of R

Bow method

The bow method accurately calculates width-averaged retreat for experiments (a) and (b), but underestimates terminus retreat in the remaining scenarios (Table 1). When compared with Figure 5a, underestimates for experiments (e)–(g) are caused by the position of the reference point relative to the change in fjord orientation. In these cases, the correct retreat value is obtained for reference points offset 408 from the original (Fig. 5a). A line from this reference point will be orientated perpendicular to the termini of experiments (e)–(g). The error produced by the method can be significant depending on the position of the reference point used to measure terminus retreat, with the –808 and –908 offsets of experiments (e)–(g) even showing the terminus to have advanced (Fig. 5a).

Fig. 5. Results of experiments testing the sensitivity of (a) the bow method to changes in the position of the reference point from which measurements are taken and (b) the rectilinear box method and (c) the curvilinear box method to different box widths. The results are from each method applied to the experiments shown in Figure 2.

The bow method also underestimates retreat for experiment (c), and more significantly for experiment (d), where the retreated terminus has a more pronounced asymmetry (Table 1). The results curves are also laterally offset towards the relative position of terminus asymmetry, compared with the results of experiments (a) and (b) (Fig. 5a). This offset is greater for the more asymmetric experiment (d), indicating that terminus asymmetry also exerts a control on the relative accuracy of the technique. However, the effects of terminus asymmetry are markedly less significant than that of the relative positioning of the reference point.

Rectilinear and curvilinear box methods

Where box width equals terminus width, the curvilinear box method successfully measures width-averaged retreat across all scenarios, while the rectilinear box method demonstrates a dependence on fjord orientation (experiment (e)) and terminus geometry (experiment (f)); Table 1). The results from the curvilinear box method for experiments (f) and (g) replicated the results from experiments (a) and (b) for the rectilinear method. This demonstrates that the curvilinear method accurately accounts for changing fjord orientation, provided that the box width is identical to that of the terminus (Fig. 5c; Table 1).

Where the boxes were less than the full glacier width, terminus geometry dictated the accuracy of both the rectilinear and curvilinear box methods (Fig. 5b and c). For box widths greater than the idealized terminus widths, the curvilinear box method consistently underestimated retreat, with the scale of this underestimation increasing with box width (Fig. 5c). The overall magnitude of underestimation by the curvilinear box method is dependent on terminus geometry, with only linear termini (experiments (a), (e) and (f)) being unaffected. By comparison, terminus geometry and changing fjord orientation can result in the rectilinear box method producing both under- and overestimates where the box is wider than the glacier terminus being measured.

Extrapolated centre-line method

The initial extrapolated centre-line method experiment shows that changing the ranges of x_n values used to calculate values of t_k has minimal effect, with a 2.5% range of results observed (Fig. 6). As the search radius increases towards W, the method becomes more accurate across all experiment scenarios, with search radii >W producing generally concordant results (Fig. 6). Consequently, for subsequent analyses, values of t_k were calculated using the nearest x_n values taken from a stretch of the centre line of W length. This was also used for the application of the extrapolated centre-line method to NS (where W 5 km).

Fig. 6. Results testing sensitivity of the extrapolated centre-line method to the different ranges of x_n values used in the calculation of t_k and how that affects overall calculated terminus position

The results of the extrapolated centre-line method demonstrate that it succeeds in accurately calculating terminus retreat where termini are symmetrical, but underestimates retreat where termini are asymmetric (Table 1). Comparing the results of experiment (c) with experiment (d), the error increases with asymmetry, though the magnitude of this is still 5% (Table 1). This demonstrates that the method is capable of accounting for changes in fjord orientation, though it is prone to small errors where complex terminus geometries occur.

Narssap Sermia experiments

As observed with scenarios (a)–(g), results given by the bow method for NS are also dependent on the positioning of the reference point relative to the termini (Figs 4 and 7a). For all box sizes tested, the general pattern of retreat calculated by both box methods is comparable (Fig. 7b and c). Where the box widths are less than the terminus width of NS (5 km), the two box methods generate different values since the boxes have different orientations. Therefore, even though the boxes may have identical widths, the two methods will be tracking change between different sections of the termini. Nevertheless, both the curvilinear and rectilinear box methods produce very similar results for the retreat of NS from 2009 to 2012 (Fig. 7b and c). However, where the fjord changes orientation for 2012-terminus A, the value from the rectilinear box method disagrees with that from the curvilinear box method, exceeding it by 339 m (Fig. 3a).

Fig. 7. Comparison of results tracking the change of Narssap Sermia, showing the sensitivity of (a) the bow method to changes in the position of the reference point from which measurements are taken and (b) the rectilinear box method and (c) the curvilinear box method to different box widths.

For the bow method no single reference point from the 1808 arc displays results comparable with the box or centre-line methods for the entire record from 2009-terminus A (Fig. 7). However, a similar pattern and magnitude of change can be extracted from the bow method by using the results of all the reference points in the 1808 arc and taking the single largest retreat value for each year to create a record of terminus change (Fig. 7).

Different box widths display no systematic behaviour of over- or underestimation between the different NS terminus observations (Fig. 7b and c). This is probably a result of differing terminus geometries between observations. A change of fjord orientation can be ruled out as a causal factor of this since, for the period 2009–12, retreat occurs in a relatively straight section of the fjord (Fig. 4). Where the boxes are less than the full terminus width (i.e. <5 km), the magnitude of terminus-change values is reduced. Boxes that overlap the fjord edges (i.e. >5 km) tend to display the extreme upper and lower values for terminus change (Fig. 7b and c).

Real glacier method intercomparison

Figure 3 shows the results of different methods applied to the termini of six Greenlandic tidewater glaciers. There is significant variability in the results, with no method displaying a consistent bias for under- or overestimation compared with others. In most cases each method produces the same general pattern of terminus change, though the absolute numbers can vary considerably, with examples of discrepancies up to 1.94 km at HG (2003–05) and 4.76 km at PG (2009–12; Fig. 3c and d). In some cases the methods identify differing patterns of retreat, an example being the bow and extrapolated centre-line method results contrasting with those of the other three methods at QSW1 after its terminus becomes diffluent from QSW2 (Fig. 3e). The bow method also estimates significantly less retreat compared with the others at NS (a difference of up to 1.67 km), after the termini change orientation by 2011 (Fig. 3a). Prior to this, terminus observations are positioned directly behind the bow method reference point, with these results being more concordant with other methods compared with subsequent observations. A further notable discrepancy occurs at JI, where the centre-line method estimates an advance (84 m between 2005 and 2012), with other methods estimating a retreat of 1.58–3.12 km.

Evaluation of methods

The centre-line method has been shown to be the least broadly applicable method over a range of scenarios (Table 2). Results from scenarios (a)–(g) demonstrate that it is incapable of determining width-averaged terminus change, and commonly displays little absolute agreement with any other method (Table 1; Fig. 3). Conversely, the method is possibly the quickest and easiest to implement, and mostly successfully identifies the general pattern of terminus change on the real-world glacier examples (Fig. 3). However, considerable caution should be exercised in the application of this method since it is prone to significant error in both absolute and relative terms. It is vulnerable to substantial under- and overestimation of terminus change depending on terminus geometry, recording an advance at JI of 84 m between 2005 and 2012, when the terminus had clearly substantially retreated (Fig. 3b). The ability of the method to account for changes in fjord orientation provides an advantage over the rectilinear box and bow methods, which are both constrained by fixed frames of reference. The centre-line method should therefore be considered useful as a method for gaining cursory insight into the large-scale changes occurring at a glacier terminus, though owing to its inability to reflect width-averaged change, its application to detailed studies of tidewater terminus dynamics should be avoided. It is more suited to tracking changes at land-terminating margins where terminus response is likely to be more symmetric (see, e.g., Reference Lopez, Chevallier, Favier, Pouyaud, Ordenes and OerlemansLopez and others, 2010; Reference Leclercq and OerlemansLeclercq and Oerlemans, 2012).

The sensitivity of the bow method to terminus shape, fjord orientation and reference point positioning is demonstrated by the results of both idealized (Table 1; Fig. 5a) and real glacier analyses (Figs 3 and 7a). While the general pattern of retreat can be extracted from an arc of reference points, as exemplified by results from NS (Fig. 7a), this adds complexity to the method, requiring extra calculation and user input, thus compromising its simplicity. Furthermore, since the bow method calculates terminus position using straight-line distances from the reference point, it is unable to accurately track terminus change directly along the glacier flow axis when fjord orientation changes. In these situations the method will underestimate the along-glacier distance between the reference point and positions on the terminus. This is likely to be even more significant for smaller non-ice-sheet glaciers where changes in fjord/valley orientation compared with their overall size can be considerable.

Although scenarios (c) and (d) show that effects of terminus asymmetry on the accuracy of the bow method are small, the impacts of the relative positioning of the reference point on the end results are significant. The bow method should only be applied on straight glaciers, where the termini to be measured are near-linear and approximately parallel, otherwise the positioning of the reference point may significantly affect results. This is demonstrated by results from NS where termini change orientation and are highly asymmetric (Fig. 4).

Both the rectilinear and curvilinear box methods performed well in tracking the terminus change of glaciers of uniform width, though results from the idealized and real scenarios indicate that the latter has wider applicability (Table 1; Figs 3 and 5b and c). When tracking NS terminus change using different box widths, the larger spread of values given by the curvilinear box method compared with the rectilinear method can be accounted for by the two methods measuring different sections of the termini (Figs 4 and 7b and c). This is because the centre of the rectilinear box will differ from the curvilinear box, as the latter tracks the glacier centre line while the former is indirectly defined by the user when the rectilinear box is drawn. This demonstrates how terminus geometry can significantly affect box method results where the entire width of the glacier is not used to calculate terminus change (Fig. 5b and c). This was a significant problem for NS, JI, HG, QSW1 and QSW2 where box widths (limited in size to the narrowest terminus observation) were in some cases unable to cover significant fractions of the delineated termini. This is best exemplified at QSW1, where the pattern of terminus retreat differed significantly from the bow and extrapolated centre-line methods that account for full glacier widths (Fig. 3e).

Conversely, where termini are of roughly uniform width, the curvilinear box method is fully capable of tracking fjord orientation changes and accounting for complex terminus geometries (Tables 1 and 2; Fig. 3c). Owing to its inability to track changes in fjord orientation, the rectilinear box method should only be applied where glaciers retreat through a straight fjord.

It is worth noting that a further variation on the box method could potentially account for changes in glacier width and orientation. This would be achieved through dividing the terminus area into multiple adjoining boxes that are (1) orientated parallel with one another, (2) of equal ‘b –axis’ length (i.e. short axis approximately parallel with flow axis) and (3) of variable ‘a –axis’ length (i.e. long axis approximately perpendicular to flow axis), allowing changes in fjord width and orientation to be tracked. The accuracy of the method would also improve as the ‘b –axis’ length is decreased and the fjord geometry is more accurately captured by the boxes. Application of this method is complex compared with the other methods discussed, and hence beyond the confines of this study; however, if it could be automated then it would prove a highly accurate method of tracking area/width-averaged terminus change.

The extrapolated centre-line method performs well where termini are regular, and successfully tracks most asymmetric and complex termini with minimal error (Table 1; Fig. 3). It was anticipated that significant issues with terminus asymmetry and/or complexity may arise when tracking the termini of glaciers such as PG, where the margins are often comparatively fractured. This means that a significant fraction of the total length of their termini can be located near their margins (Fig. 3c). The effect of this would be to disproportionately weight the terminus position towards the terminus position of the ice margins. However, results for PG proved comparable with other methods (Fig. 3c). Nevertheless, this is an issue that should be considered before applying the method to termini with significantly fractured margins. Elsewhere the effect of terminus asymmetry will be less significant, with the idealized retreats showing an error of <5% (Table 1; Fig. 6). In most cases, this will lie within the operator delineation error.

Notable advantages of the extrapolated centre-line method include its ability to account for the entire terminus length irrespective of changes in glacier width, in addition to changes in fjord orientation. This proved to be especially useful where advance/retreat occurs through a glacier or fjord confluence such as at QSW, or where the terminus area of interest may change between observations.

Finally, the absolute uncertainty surrounding the delineation of each individual terminus position is worth brief consideration. If a given terminus observation is delineated multiple times, and the range of values determined, the absolute error will be different for each method for similar reasons to those outlined above. Nevertheless, where imagery such as Landsat or MODIS (Moderate Resolution Imaging Spectroradiometer) is used, the absolute error values in the delineation of individual termini are likely to be negligible compared with the absolute differences given by different terminus change methods (Fig. 3).