Table 1.

A Summary of Selected Recent Studies Using Volume‐Area (V‐S) Scaling, Modifications of Volume‐Area Scaling, or Alternative Methods^a

Reference		Application	Strengths/New Insights	Weaknesses/Errorsa	Relevant Sections
Adhikari and Marshall [2012]		V‐S scaling evaluated in comparison to numerical model	Experiments compared to full 3‐D model; restriction to ensembles of glaciers (not individuals) recognized. Correctly identifies glacier shape and slope as drivers of volume (closure conditions in this work and in Bahr et al. [1997]).	Both c and γ treated as variables, or c is fixed and γ treated as variable; some derived values (e.g., γ > 2) fall outside theoretically meaningful limits; Bahr et al. [1997] V‐S scaling theory incorrectly claimed to be limited to shallow‐ice approximation and valid only for steady state conditions.	8.1, 8.2, 8.5, 8.9
Agrawal and Tayal [2013]		Volume change assessment in Himalayas.	Correctly specifies γ = 1.375 as a constant. Compares scaling results to other methods.	V‐S scaling applied to a single glacier.	8.2, 8.5
Arendti [2006]		PhD thesis, volume changes of Alaskan glaciers	Scaling exponent γ treated as fixed at appropriate value; multiplier c as variable.	V‐S scaling hypothesized to be restricted to perfect plastic media. Time derivative incorrectly applied to V‐S scaling relationship with finite‐sized changes in area.	8.1, 8.7
Arendt et al. [2006]		Regional glacier mass balance assessment	Scaling exponent γ treated as fixed at appropriate value; multiplier c as variable. Uncertainty of application to single glaciers is recognized and discussed.	Time derivative incorrectly applied to V‐S scaling relationship. Analysis applied to single glacier.	8.1, 8.7,
Barrand and Sharp [2010]		Regional glacier mass balance assessment	Significant error in subdividing glaciers into subunits is recognized; very small fractions of glaciers considered are subdivided.	V‐S scaling exponent treated as variable; Scaling applied to artificial subunits of glaciers spanning political boundaries. Uncertainties judged relative to spurious error arising from assumed variability of scaling exponent.	8.2, 8.6
Basagic and Fountain [2011]		Volume estimates of small sample of glaciers	V‐S scaling recognized as highly uncertain for individual glacier volume estimate and characterized as “rough estimate.”	V‐S scaling used with a small sample (though correctly noted as a rough estimate).	8.5
Binder et al. [2009]		Ice volume estimate for two glaciers	Correctly specifies γ = 1.375 as a constant.	V‐S scaling applied to an individual glacier. Incorrectly specifies c = 1. Assumes V‐S theory is invalid for small retreating glaciers.	8.2, 8.5
Bliss et al [2013]		Regional glacier volume estimate	Proper values of γ for glacier/ice cap used except in case of BEDMAP (bed topography of the Antarctic) data only.	Assumes exponent γ is a variable with associated uncertainty; exponent γ treated as spatially variable. Values of γ for glacier/ice cap not distinguished when using BEDMAP data only.	8.2
Farinotti et al. [2009]		Global glacier volume estimate. Alternate method.	Method is applicable to single glaciers. An alternate numerical method for calculating volumes is compared to scaling.	V‐S scaling wrongly assumed to be limited to steady state conditions. V‐S scaling applied to individual glaciers.	8.1, 8.5
Farinotti and Huss [2013]		Estimates accuracy of V‐S scaling	Performs a rigorous statistical analysis of the accuracy of V‐S scaling. Correctly concludes that scaling should be applied to large populations of glaciers. Correctly notes that allowing γ to vary with time is less accurate than treating γ as a constant.	Assumes exponent γ is a spatial and temporal variable or a constant other than 1.375. Estimates γ from data. The variability of γ is built into the estimate of V‐S accuracy. A balance gradient is specified for the 3‐D Stokes model, but its value in equilibrium may be inconsistent with the V‐S closure condition.	8.2, 8.3, 8.7, 8.9, 8.11
Fischer [2009]		Ice volume estimate for single glacier	Recognizes that V‐S scaling should only be applied to populations of glaciers.	V‐S scaling applied to individual glaciers.	8.5
Grinsted [2013]		Global glacier volume estimate	Good compendium of previously published analyses; theoretical insights on role of surface slope; acknowledges that treating glacier complexes will degrade accuracy.	Attempts to assign multiple values to fixed exponent γ by empirical means and introduces additional parameters with no theoretical basis; treats glacier complexes as single entities.	8.2, 8.6, 9.1
Haeberli et al. [2007]		Regional/global glacier mass balance assessment	Points out that some glacier volume estimates may contain measure of area.	Claims invalidity of V‐S scaling on grounds that all volume measurements intrinsically contain measure of area.	7.1, 7.2, 8.8
Hagg et al. [2013]		Regional glacier mass balance assessment	Applicability of V‐S scaling to nonequilibrium conditions recognized.	V‐S scaling applied to sample of only 7 glaciers. Exponent γ treated as a variable with multiplier c held constant.	8.2, 8.4, 8.5
Harrison [2013]		Insights into the characteristic response of glaciers to climate	A macroscopic scaling‐type derivation with a nondimensional parameter consistent with this review. Correctly notes that the V‐S exponent is a constant. Combines volume scaling with response time scaling.	Theory is limited to an idealized geometry.	4.2, 8.4
Huss and Farinotti [2012]		Regional glacier volume estimate. Alternate method.	Method is applicable to single glaciers. V‐S scaling correctly noted to be inapplicable to multiple‐glacier complexes treated as single entity.	V‐S scaling wrongly assumed to be limited to steady state conditions and to be unsuitable for complex geometries such as dendritic glaciers. V‐S exponent treated as tunable parameter. V‐S exponent assumed spatially variable.	8.1, 8.2, 8.12
Meehl et al. [2007]		Global glacier mass balance assessment.	Scaling only applied to large populations of glaciers.	V‐S scaling wrongly assumed to be limited to steady state conditions	8.4
Kulkarni and Karyakarte [2014]		Regional glacier mass balance assessment	Calculates two different estimates of the volume for large populations of glaciers.	Polynomial volume‐area formula used without theoretical justification or discussion.	9.1
Leclercq et al. [2011]		Global historical glacier contribution to sea level.	Reconstruction of global glacier sea level contribution over past 2 centuries.	V‐S scaling wrongly assumed to be limited to single glaciers; method modified to extend to populations of glaciers.	8.5
Lüthi [2009]		Theoretical derivation of scaling parameters.	An alternative theoretical derivation of V‐S, volume‐length, and response time scaling for a limited geometry. Results are largely consistent with this review and with a 3‐D Stokes model. Contains a theoretical derivation of c for the specified geometry. Correctly applies response time and V‐S scaling simultaneously.	Theory is limited to an idealized geometry.	7.1, 7.2, 8.4, 8.10
Marzeione et al. [2012]		Scaling applied in numerical model of future global glacier mass balance.	Robust treatment of time scale of transients. Proper scaling parameters adopted.		8.2, 8.4
Moore et al. [2013]		Overview of projection methods for glacier mass balance	Recognizes that V‐S scaling cannot be applied to multiple‐glacier complexes treated as single entity or to subunits of a glacier.	V‐S scaling parameters wrongly assumed to depend on assumption of perfect plasticity and to be intrinsically variable.	8.1, 8.6
Möller and Schneider [2010]		Glacier mass balance modeling	Parameter γ correctly assumed to be a constant. Single‐glacier limitation avoided by using data for the same glacier at multiple times.	Parameter c = 1 incorrectly assumed to hold for equilibrium conditions; empirical value for c ≠ 1 sought for nonequilibrium conditions.	7.1, 7.2, 8.2
Radić and Hock [2010]		Global glacier volume estimate	Selects appropriate values for γ for glaciers and for ice caps.	Assumes parameter γ has associated uncertainty.	8.2
Radić and Hock [2011]		Global glacier volume estimate	Selects appropriate values for γ for glaciers and for ice caps; notes potential time variation in parameter c.	Assumes parameter γ has associated uncertainty.	8.2
Radić et al. [2007]		V‐S scaling exponents derived by numerical model	Acknowledges potential for errors from use of 1‐D model.	A 1‐D glacier model is used (V‐S parameters derived from this model will not necessarily apply to real glaciers); unrealistically wide range of values for parameter γ found. Assumes γ is time dependent for nonequilibrium conditions. Assumes V‐S scaling applies only to steady state conditions.	8.1, 8.2, 8.3, 8.9
Radić et al. [2008]		V‐S scaling evaluated in comparison to numerical model	V‐S scaling adapted to nonsteady conditions; application of multiple scaling relationships to single glacier.	Parameter γ treated as adjustable parameter; assumes parameter γ must vary to account for nonequilibrium conditions.	8.2, 8.4
Raper and Braithwaite [2006]		Global glacier volume estimate	One of the first papers to pioneer the use of V‐S scaling to estimate global glacier ice volume.	Treats glacier complexes as single entities; subdivides glacier along grid cell boundaries and applies scaling to subunits.	8.6
Salzman et al. [2013]		Regional glacier volume estimate	V‐S scaling properly applied to aggregate glacier in Peruvian Andes. Uses scaling as a test of a numerical model.		8.10
Schneeberger et al. [2003]		GCM‐forced regional/global glacier mass balance	Recognizes that V‐S scaling should only be applied to populations of glaciers.	V‐S scaling applied to individual glaciers.	8.5
Slangen and van de Wal [2011]		Uncertainty assessment of V‐S scaling in sea level rise projections	Fixes exponent γ at appropriate values for glaciers (γ = 1.375) and ice caps (γ = 1.25). Allows c to vary.	Incorrectly performs a sensitivity analysis on γ which should remain constant.	8.2
van de Wal and Wild [2001]		Glacier mass balance model	One of the first papers to pioneer the use of V‐S scaling to estimate sea level. Fixes exponent γ at appropriate values for glaciers (γ = 1.375) and ice caps (γ = 1.25); recognizes that V‐S scaling applies in nonequilibrium conditions	Incorrectly performs a sensitivity analysis on γ which should remain constant.	8.2
Wang et al. [2014]		Ice volume estimate for single glacier	Recognizes that V‐S scaling is highly uncertain when applied to single glacier.	V‐S scaling applied to single glacier. Scaling exponent assumed to depend on individual glacier characteristics and time.	8.2

^aIn some cases, the identified error or weakness only propagates a common misunderstanding without detracting from the overall value of the study. In many cases, the insights of the paper remain substantial even though the conclusions may need revisiting. In a few cases, the errors would warrant a careful reevaluation of the conclusions.