Supraglacial, moraine-dammed and ice-dammed lakes represent a potential
glacial lake outburst flood (GLOF) threat to downstream communities in many
mountain regions. This has motivated the development of empirical
relationships to predict lake volume given a measurement of lake surface
area obtained from satellite imagery. Such relationships are based on the
notion that lake depth, area and volume scale predictably. We critically
evaluate the performance of these existing empirical relationships by
examining a global database of glacial lake depths, areas and volumes.
Results show that lake area and depth are not always well correlated
(
Globally, there is a general trend of mountain glacier recession and thinning in response to climatically controlled negative mass balances (Zemp et al., 2015). In most mountain ranges, glacier shrinkage since the Little Ice Age has been accompanied by the development of proglacial, ice-marginal and supraglacial lakes impounded by moraine and outwash fan head structures (e.g. Röhl, 2008; Janský et al., 2009; Thompson et al., 2012; Carrivick and Tweed, 2013; Westoby et al., 2014). The integrity of these structures often reduces over time as ice cores degrade and slopes are subject to mass wasting processes, raising the concern of dam failure. Further, the location of these lakes in valleys with steep, unstable slopes, often in tectonically active regions prone to earthquakes, means that rock and ice avalanches are common, adding a further threat of displacement-wave overtopping if avalanche material were to impact the lake (e.g. Schneider et al., 2014). Dam failure, breach or overtopping can lead to glacial lake outburst floods (GLOFs) that pose a significant threat to lives, industry and infrastructure (Richardson and Reynolds, 2000; Westoby et al., 2014). Other potentially dangerous lakes are dammed by ice, either in ice-marginal locations where surface meltwater or water from tributary valleys ponds against the glacier margin (e.g. Merzbacher Lake – Mayer et al., 2008; Lac de Rochemelon – Vincent et al., 2010) or where advancing (often surging) glaciers block river drainage (e.g. Kyagar Glacier – Haemmig et al., 2014). In these situations, water may escape through subglacial tunnels, along the ice margin between the glacier and valley side or by mechanical failure of the ice dam (Walder and Costa, 1996; Clague and Evans, 2000).
Crucial to the management of GLOF hazards is the ability to assess the likelihood and magnitude of any such event. In most cases, this requires an understanding of the volume of water impounded in the lake, the structural integrity and longevity of the dam, potential external trigger mechanisms and the likely flow path of the flood (e.g. Richardson and Reynolds, 2000; McKillop and Clague, 2007; Westoby et al., 2014). There are a number of challenges for anyone interested in estimating or calculating lake volume. Field studies are complicated by the fact that many glacial lakes are located in relatively inaccessible or physically challenging and dangerous environments, making bathymetric surveys of lake basins difficult. As yet, there is no reliable technique available for measuring lake bathymetry or volume from satellite imagery where turbidity precludes the derivation of reflectance–depth relationships (e.g. Box and Ski, 2007). Consequently, a number of studies have adopted an empirical approach to volume calculation from satellite imagery based on known relationships between lake depths, areas and volumes (e.g. Evans, 1986; O'Connor et al., 2001; Huggel et al., 2002; Yao et al., 2012; Loriaux and Cassassa, 2013; Carrivick and Quincey, 2014). This allows rapid and simple calculation of lake volumes from widely available satellite imagery, whilst avoiding the necessity for often challenging fieldwork.
Two key empirical approaches have become adopted for lake volume estimation.
First, O'Connor et al. (2001) derived a relationship between lake area and
volume for moraine-dammed lakes of the Central Oregon Cascade Range. Lake
volumes were derived from detailed bathymetric surveys. The relationship
takes the form
An alternative relationship was derived by Huggel et al. (2002). First,
Huggel et al. demonstrated that lake depth and area were correlated for a
combination of ice-dammed, moraine-dammed and thermokarst lakes at a number
of locations globally. This relationship takes the form
Summary of relationships derived from measured lake area and depth data.
We have compiled a data set of glacial lake areas, average depths and bathymetrically derived volumes from published articles and reports (Tables S1 and S2 in the Supplement). The data set comprises 42 lakes with measured lake areas and mean depths (Table S1), most of which (36) were reported in the publications themselves. The remainder were derived by the current authors from published bathymetric maps, which were georeferenced in ArcMap and then digitised; mean depth measurements were then interpolated from the contour data. Some of these data represent duplicate readings from individual sites where repeat measurements have been made over several years. When these duplicates are removed, the data set comprises 30 lakes (Table 1). Lake area and depth data presented in Huggel et al. (2002) represent a further 15 data points, and we derive empirical relationships between lake area and depth with and without duplicates and with and without the data of Huggel et al. (2002) included (Table 1). Empirical relationships are derived by fitting power-law functions to the area–depth data plotted on logarithmic scales. We have not used depth data derived from dividing bathymetrically derived volumes over measured areas to avoid the issue of auto-correlation.
Summary of relationships derived from measured lake area and bathymetrically derived volume data.
Plot of lake area vs. depth for the data compiled in this study
(including duplicate measurements of individual lakes) and the data
presented by Huggel et al. (2002). Best-fit lines and corresponding
equations and
There are 69 lakes with measured areas and volumes calculated from bathymetric data (Table 2). As with the area–depth data, most of these data points (63) were reported directly in the literature; the remainder were derived from interpolated bathymetric map data by the current authors. Removal of duplicate sites reduces the number of data points to 49. The area and volume data of O'Connor et al. (2001) represent a further six sites and, again, empirical relationships are derived with and without the duplicate sites and data from O'Connor et al. (2001) by fitting a power-law function to the data.
Plot of lake area against volume for the data compiled in this
study and for the data presented by O'Connor et al. (2001). Best-fit lines
and corresponding equations and
Derivation of power-law functions for area–depth and area–volume data is
performed in conjunction with a calculation of the coefficient of
determination,
High
Figure 1 presents all of the lake area against measured mean depth data from
Huggel et al. (2002) and from the range of data compiled in this study, with
best-fit line equations and
The re-plot of data presented in Huggel et al. (2002) differs from that
presented in their study (their Fig. 1). Indeed, the one significant outlier
in their graph actually plots very close to the best-fit line for their
data, and two points that appear in their Table 2 do not appear in their
Fig. 1. Hence, overall, the
Plotting all available data compiled in this study (including duplicate
readings for some sites where there are data for two or more measurement
periods) reveals a low
Since the data of Huggel et al. (2002) plot with a high
O'Connor et al. (2001) derived their lake area–volume relationship (Eq.
A re-plot of the O'Connor et al. (2001) data reveals a high
Despite the visually close association of most of the data points in Fig. 2
and the relatively high
Comparison of bathymetrically derived lake volumes with those
calculated using existing empirical relationships. Errors are calculated
according to Huggel et al. (2004) and coded such that the error between
bathymetrically derived and modelled volumes of
Continued.
Continued.
Continued.
Continued.
Table 3 presents a measure of error between bathymetrically derived volumes
and the volumes calculated using Eqs. (
Table 3 demonstrates that the use of O'Connor et al.'s (2001) volume
calculation leads to very large errors in most cases. The relationships of
Huggel et al. (2002) and Evans (1986) perform better in general, although
there are exceptions. For ease of interpretation, we ascribe error scores in
the right-hand columns. For any individual estimate, errors beyond
Since the method of O'Connor et al. (2001) seems to overestimate greatly lake volumes in most cases, even when the other methods are reasonable predictors, the furthest right-hand column presents error scores based only on Huggel et al. (2002) and Evans (1986). Combined scores of 5–6 are considered “highly unpredictable”, and scores of 3–4 are considered “unpredictable”. Scores of 0–2 are considered to be “reasonably predictable”. The results of these two right-hand columns are broadly comparable, identifying the same lakes in most cases.
Table 3 reveals several lakes with “highly unpredictable” lake volumes including Hooker, Ivory Lake, Laguna Safuna Alta, Lake No Lake, Nef and Ngozumpa 4. A group with `unpredictable' volumes includes Checquiacocha, Gelhaipuco, Hazard/Steele Lake, Imja (in 1992), Maud Lake, Mt Elbrus, Mueller, Ngozumpa, Petrov, Quitacocha and Tam Pokhari.
The relationship of O'Connor et al. (2001) outperforms those of Huggel et al. (2002) and/or Evans (1986) in a few cases including, including many of the “highly unpredictable” lake volumes. Specifically, these are Hooker, Imja (in 1992), Ivory, Laguna Safuna Alta, Lake No Lake, Miage, MT Lake, Ngozumpa 4, Quitacocha and Tam Pokhari.
We have compiled a data set of Alpine glacial lake areas, depths and volumes in order to evaluate critically the use of existing empirical relationships for the estimation of glacial lake volumes. The plot of lake area against mean lake depth (Fig. 1) reveals a significant degree of scatter, indicating that lake area and depth do not always scale predictably. Hence, empirical relationships for estimating lake volume that are founded upon a strong correlation between lake area and depth (e.g. that of Huggel et al., 2002) should be used with caution. Equally, Fig. 2 shows that there are also significant outliers in the data set of measured areas against bathymetrically derived volumes, even though one might expect some degree of auto-correlation between area and volume (Huggel et al., 2002; Mergili and Schneider, 2011).
In general, the empirical relationships derived by Evans (1986) and Huggel et al. (2002) perform better at estimating lake volumes than the relationship of O'Connor et al. (2001) (Table 3). These relationships are also more robust because they are derived from a relationship between lake depth and area and hence are not affected by auto-correlation (Huggel et al., 2002; Mergili and Schneider, 2011). The re-plotting of lake depth and area data from Huggel et al. (2002) reveals a slightly different relationship to that reported in the original study (Table 1), although it will make little difference to calculated volumes if either the original or revised relationship is used. As McKillop and Clague (2007) explain, the O'Connor et al. (2001) relationship is derived from a data set of lakes whose volumes are large for their relatively small areas. This is a consequence of moraine dam emplacement on steep slopes, giving comparatively large depths and volumes. Hence, the relationship of O'Connor et al. (2001) should be expected to overestimate lake volume with increasing lake area in most situations. Table 3 reveals that the relationship of O'Connor et al. (2001) outperforms the other empirical relationships for Hooker, Imja (in 1992), Ivory, Laguna Safuna Alta, Lake No Lake, Miage, MT Lake, Ngozumpa 4, Quitacocha and Tam Pokhari. These lakes may be unusually deep for their respective surface areas, as were the lakes investigated by O'Connor et al. (2001).
Figure 1 shows that glacial lakes can be exceptionally deep or exceptionally shallow for any given surface area. There are several reasons that may account for this depth variability. First, glaciers achieve different levels of erosion and sediment flux, meaning that the depth of erosion of glacial basins (overdeepenings) within which lakes sit, and the height of moraine dams that impound lakes, can be highly variable (e.g. Cook and Swift, 2012). Second, shallow lakes may develop on top of stagnant or stagnating ice (Yao et al., 2012), or where lake basins become progressively filled with sediment (Allen et al., 2009), meaning the evolution of such lakes can vary widely even if their starting morphology is the same. Third, the presence or absence of a lake outlet, and the elevation of that outlet or notch with respect to the glacier terminus bed elevation, will have a significant control on the depth of water that is allowed to accumulate in any lake basin.
Some of the lakes with “highly unpredictable” or “unpredictable” volumes
(Table 3) share common characteristics, which may prove instructive when
deciding upon an appropriate empirical relationship with which to estimate
the volume of different lake types. First, Mueller, Ngozumpa, Petrov and
Mt Elbrus are all lakes that are either situated (partly or wholly) on top
of stagnant or relict glacier ice or have large subaqueous ice bodies that
protrude into the lake from the glacier terminus. At Mueller Glacier,
Robertson et al. (2012) detected an exceptionally long (510 m) subaqueous
ice ramp that covered
The second grouping includes lakes situated within basins with complex bed topography, some of which may be related to focussing of glacial erosion. Hooker Lake had a greater than predicted volume in 1995 and 2002 but not in 2009. Comparison of glacier terminus position and bathymetric maps in Robertson et al. (2013) indicates that in 1995, the glacier terminus was retreating out of a deep basin. By 2002, the glacier had retreated to the position of a deep notch in the bed profile. At Ivory Glacier, lake volume was significantly underpredicted for 1976 and 1986, although less so for 1980. Examination of lake long profiles in Hicks et al. (1990) indicates that in 1976 and 1986, the glacier had recently retreated into a deep basin. The lake in these situations is disproportionately deep at one end and shallower toward the moraine dam, which means that the lake volume is not well predicted. Ivory Glacier in 1986 terminated in a nested overdeepening (a basin within a basin). This complex lake basin morphometry may thus yield lake volumes that are underpredicted by existing empirical relationships. Tam Pokhari, Checquiacocha, Maud Lake and arguably Ivory Lake all appear in places where glacial erosion may have been particularly intense and hence might be expected to generate particularly deep basins with lake volumes that are not well predicted by existing empirical relationships (Table 3). Tam Pokhari, Checquiacocha and Ivory Lake appear at the base of what would have been steep icefalls with greater potential for erosion and sediment transfer (cf. Cook et al., 2011). Maud Lake is located in what would have been a tributary glacier junction where erosion would have been intense as a consequence of enhanced ice flux (cf. Cook and Swift, 2012).
A third identifiable situation is represented by Hazard/Steele Lake, which formed when a glacier advanced across a valley (Collins and Clarke, 1977; Clarke, 1982). Table 3 reveals that empirical relationships underestimate its volume. We make the tentative suggestion that the morphometry of lake basins such as this, where the host valley has been shaped to some extent by fluvial and mass movement processes before glacier advance, means that their volumes are not well predicted by empirical relationships based on measurements of lakes that occupy basins of purely glacial origin. Lake No Lake may also fit within this category because it occupies a valley situated between two glaciers (Geertsema and Clague, 2005).
The remaining outliers from Table 3 are lakes with a range of site-specific characteristics that make their volumes hard to predict or represent situations where there is no clear reason for their unusual volumes. Some of these outliers are related to apparently unusual situations (compared to lakes upon which empirical relationships have been based). Specifically, Ngozumpa 4 is an ice-marginal moraine-dammed lake that is reported by Sharma et al. (2012) to have a deep crevice at its base, giving it an unusually deep bed; Laguna Safuna Alta has a complex history of lake level change, involving modification by engineering works, and a suspected increase in moraine dam permeability as a consequence of an earthquake in 1970 (Hubbard et al., 2005), although it is not clear why it should be unusually deep. Quitacocha and Gelhaipuco lakes are both moraine-dammed and their volumes are underestimated by empirical relationships. Again, it is unclear why this should be the case.
An intriguing result from our analysis is that lakes within similar
geographical areas do not necessarily have equally predictable lake volumes.
A number of studies have adapted existing empirical relationships by adding
data from specific regions (e.g. Loriaux and Cassassa, 2013) or by
generating completely new relationships from known lake properties for
specific regions in favour of adopting existing empirical relationships
(e.g. Yao et al., 2012). There is some merit in this approach because, for
example, the volumes of many of the Himalayan glacial lakes listed in Table 3 are consistently underpredicted by existing empirical formulae,
indicating regional controls on lake volumes. Yet, the data set compiled in
this study reveals a number of examples where lakes in the same region can
have very different degrees of volume predictability. For example, the
Hooker and Mueller lakes are only
Plots of lake area–volume data according to different lake dynamic
contexts:
Conceptual consideration of glacial lake evolution and its impact
on volume–area relationships:
In order to better understand lake growth and the application of empirical
relationships, we have re-plotted the data according to lake context (Fig. 3) and developed a corresponding conceptual model for each (Fig. 4). One of
the striking results of our error analysis (Table 3) was that growing
supraglacial lake volumes are not well predicted by existing empirical
relationships. Supraglacial lake evolution has been examined in a number of
studies (e.g. Kirkbride, 1993; Sakai et al., 2000, 2003, 2009; Benn et al.,
2001; Thompson et al., 2012) with small ponds developing through melting of
exposed ice faces and large lakes expanding primarily through calving.
Sakai et al. (2009) suggested that wind-driven currents of relatively warm
water were important for lake growth and calving, and hence, lake fetch
(defined as the maximum lake length along the axis of glacier flow)
represents a primary control on lake evolution. Their work demonstrated that
supraglacial lakes expand by calving once lake fetch exceeds
In contrast, lakes that have evolved toward the moraine-dammed endmember appear to have more predictable volumes. Figure 3b illustrates that most moraine-dammed lake volumes scale linearly with increasing area. Likewise, the available data indicate that ice-dammed lakes may evolve predictably, such that lake volume grows exponentially with increasing lake area (Figs. 3c and 4).
The ability to estimate accurately the volume of glacial lakes is important for the modelling of GLOF magnitudes and run-out distances. Direct estimation of lake volume in the field through detailed bathymetric surveying is a potentially difficult and dangerous undertaking. Hence, many studies rely on empirically derived relationships that allow the estimation of lake volume from a measurement of lake area, which is readily gained from satellite imagery. However, there has been no systematic assessment of the performance of these existing empirical relationships or the extent to which they should apply in different glacial lake contexts. In this study, we have compiled a comprehensive data set of glacial lake area, depth and volume in order to evaluate the use of three well-known empirical relationships, namely those of Huggel et al. (2002), Evans (1986) and O'Connor et al. (2001).
Our first key finding is that lake depth and area are only moderately
correlated (with an
Our second key finding is that two of the three existing empirical relationships (those of Huggel et al., 2002 and Evans, 1986) give reasonable approximations of lake volume for many of the lakes examined in this study, but there are several lakes whose volumes are over- or underestimated by these relationships, sometimes with errors of as much as 50 to over 400 %. The relationship of O'Connor et al. (2001) is only reliable in a handful of cases, seemingly where lakes are unusually deep.
Many of the lakes whose volumes are not well predicted by empirical relationships fall into distinct groups, meaning that it is possible to identify situations where it could be inappropriate to apply empirical relationships to estimate lake volume, important for robust assessments of GLOF risk. Specifically, these groups include (i) lakes that are developing supraglacially, which tend to grow areally by calving and edge melting, but are shallow due to the presence of ice at the lake bed or of ice ramps protruding from calving faces; (ii) lakes that occupy basins with complex bathymetries comprising multiple overdeepenings or are particularly deep due to carving by intense erosion (e.g. at the base of an icefall or at former tributary glacier junctions); and (iii) lakes that form in deglaciated valleys (e.g. when glaciers advance to block valley drainage). Other outliers represent a range of unusual cases where site-specific factors complicate the relationship between lake area and volume.
Ultimately, we develop a conceptual model of how volume should be expected to change with increasing area for a range of lake contexts, based on re-plotting of the data according to lake type. Specifically, these include moraine-dammed, ice-dammed, supraglacial ponds and supraglacial lakes. We suggest that further measurements of the bathymetry of growing supraglacial ponds and lakes would be very valuable in developing robust relationships for the prediction of their evolving volumes.
We gratefully acknowledge the interactive comment from Wilfried Haeberli, which made us think critically about our use of terminology as well as about empirical lake area–volume relationships. Likewise, we thank Jürgen Herget and an anonymous reviewer for their thoughtful and constructive reviews. We are particularly grateful to Jürgen Herget for providing electronic and hard copies of an edited volume that we would not otherwise have been able to access. We thank Vladimir Konovalov for providing further reports pertinent to this study. Edited by: J. Turowski