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CC Attribution 3.0 License. Density assumptions for converting geodetic glacier volume change to mass change M. Huss Department of Geosciences, University of Fribourg, 1700 Fribourg, Switzerland Correspondence to: M. Huss ([email protected]) Received: 16 December 2012 – Published in The Cryosphere Discuss.: 11 January 2013 Revised: 21 April 2013 – Accepted: 2 May 2013 – Published: 27 May 2013 Abstract. The geodetic method is widely used for assess- be covered and the integrated changes of the whole glacier ing changes in the mass balance of mountain glaciers. How- system are captured. However, the differencing of digital el- ever, comparison of repeated digital elevation models only evation models (DEMs) provides a change in glacier vol- provides a glacier volume change that must be converted to ume instead of a mass change that is the relevant quantity a change in mass using a density assumption or model. This for climate impact assessments (e.g. sea-level rise contribu- study investigates the use of a constant factor for the volume- tion and mountain hydrology). In glaciological studies, ob- to-mass conversion based on a firn compaction model applied servable volume change 1V is usually converted to a mass to simplified glacier geometries with idealized climate forc- change 1M by assuming a conversion factor f1V as ing, and two glaciers with long-term mass balance series. It = · is shown that the “density” of geodetic volume change is not 1M f1V 1V. (1) a constant factor and is systematically smaller than ice den- The robustness of f1V in time and space is crucial for the sity in most cases. This is explained by the accretion/removal accuracy of geodetic mass balance determination but has re- of low-density firn layers, and changes in the firn density pro- ceived little attention so far. file with positive/negative mass balance. Assuming a value of It can be shown that the factor f varies from case to −3 1V 850±60 kgm to convert volume change to mass change is case, and is inherently difficult to determine. A temporal appropriate for a wide range of conditions. For short time in- change in glacier mass M is given by the change in glacier tervals (≤ 3 yr), periods with limited volume change, and/or volume V multiplied by its density ρ as changing mass balance gradients, the conversion factor can −3 ∂M ∂(ρV ) however vary from 0–2000 kgm and beyond, which re- = . (2) quires caution when interpreting glacier mass changes based ∂t ∂t on geodetic surveys. Integrated with time this yields 1M = 1ρV + 1Vρ, (3) 1 Introduction where ρ is the bulk density of the glacier including its firn coverage, and 1ρ is the change in average glacier density Determination of glacier mass balance via the comparison over the time interval considered. In practice, the exact in- of two terrain elevation models obtained from air- or space- dividual quantification of 1ρ, ρ and V is however close to borne observation is one of the most popular and accurate impossible. If we write Eq. (3) as methods to monitor glacier mass change over periods of 1ρV a few years to some decades (e.g. Rignot et al., 2003; Bam- 1M = + ρ · 1V = f1V · 1V, (4) 1V ber and Rivera, 2007; Cogley, 2009; Nuth et al., 2010; and Gardelle et al., 2012). Geodetic measurements of the glacier the terms 1ρ, ρ and V can be lumped into the conversion fac- mass budget are powerful as large and inaccessible areas can tor f1V which allows direct calculation of mass change from Published by Copernicus Publications on behalf of the European Geosciences Union. 878 M. Huss: Converting glacier volume change to mass change volume change. Nevertheless, it is evident from this formu- logical and the geodetic method are difficult to interpret lation that f1V will not be a straight-forward constant. as they might be explained by various poorly constrained Most studies make basic assumptions on the conversion sources of uncertainty (Fischer, 2011; Zemp et al., 2013). factor between volume change and mass change and often At the mountain-range scale, glacier mass change can also rely on Sorge’s law (Bader, 1954) that prescribes no changes be directly measured with satellite-based observations us- in the vertical firn density profile over time. If Sorge’s law ing gravimetry (e.g. Jacob et al., 2012), but the uncertainties −3 holds, 1ρ (Eq. 3) equals zero, and f1V is about 900 kgm . are still considerable and the spatial resolution is too low This number derives from the density of ice and has been for providing mass variations of individual glaciers. Ground- adopted in many previous assessments of geodetic mass bal- based gravimetry has also been applied for detecting local ance of mountain glaciers (e.g. Cox and March, 2004; Bauder mass changes (Breili and Rolstad, 2009). Application of the et al., 2007; Paul and Haeberli, 2008; and Cogley, 2009). method for determining the overall glacier mass budget is A significant change in volume however implies either pos- however not feasible. itive or negative mass balance, and hence a shift in the firn To date, detailed studies for mountain glaciers that con- line, as well as a change in firn thickness and average density, nect geodetically measured variations in overall ice volume thus directly contradicting Sorge’s law. Based on a simple to their mass balance by taking into account changes in firn analysis of firn area changes, Sapiano et al. (1998) estimated volume and density at a spatially distributed scale are not the average “density” of volume change as 850 kgm−3. Sev- available. In order to investigate the value and the temporal eral recent studies have adopted this value in order to ac- robustness of the conversion factor f1V (Eq. 1), an empiri- count for the effect of increasing or decreasing firn thickness cal firn densification model coupled to idealized surface mass on f1V (Huss et al., 2009; Zemp et al., 2010, 2013; Fischer, balance forcing is applied to a range of simplified glacier ge- 2010, 2011). Another approach is to use zonally variable con- ometries. Further, the evolution of firn thickness and den- version factors (Schiefer et al., 2007; Moholdt et al., 2010; sity since 1960 is simulated for two glaciers in the Swiss Gardelle et al., 2012; Kaeaeb et al., 2012). The uncertainty Alps based on long-term mass balance observations, and range due to the density assumption is explored by either set- the changes in mass and volume are jointly discussed. This −3 −3 ting f1V = 900 kgm for the entire glacier, or 900 kgm study provides recommendations for the “density” of volume −3 in the ablation area, and 500 to 600 kgm in the firn zone change f1V to be used in mass balance studies based on the for converting volume changes to mass changes. Sørensen geodetic method, and demonstrates in which cases the as- et al. (2011) and Bolch et al. (2013) refined this approach sumption of a straight-forward conversion factor fails. by assessing firn densities above the equilibrium line altitude (ELA) of the Greenland ice sheet or Arctic ice caps using firn compaction models. 2 Data and methods Based on a non-steady firn densification model for the per- colation zone of Greenland, Reeh (2008) concludes that the 2.1 Glacier geometry “straight-forward translation of observed short-term ice sheet surface-elevation variations into mass changes may be com- f1V is assessed at two stages: (1) for synthetic glacier ge- pletely misleading.” This is explained by the transient adap- ometries, and (2) in case studies with field data for the 50 yr tations of the firn density profile in response to temporally time series of surface mass balance for the Griesgletscher and varying melt and accumulation rates. Helsen et al. (2008) Silvrettagletscher, Swiss Alps. This has the advantage that, at show that surface elevation changes in the interior of Antarc- stage 1, the response of glacier volume to a prescribed ideal- tica are mainly due to firn depth variations and cannot be in- ized mass balance forcing can be discussed under controlled terpreted as a mass change. These findings related to ice sheet conditions, depicting the dominant features, and at stage 2, mass balance are also relevant regarding mountain glaciers; these findings are put into the applied context of a glacier surface elevation in the accumulation area, and hence glacier monitoring program. volume, is not coupled linearly to glacier mass. A strongly simplified synthetic glacier geometry is defined ◦ Assessing the robustness of the conversion factor f1V nec- at stage 1.