165: Mass and Energy Balances of Glaciers and Ice Sheets · through snow and ice that persists on the Earth’s surface for more than one year (Figure 1c). The two largest glaciers

165: Mass and Energy Balances of Glaciers andIce Sheets

J GRAHAM COGLEY

Department of Geography, Trent University, Peterborough, ON, Canada

Glaciers exchange energy and mass with the rest of the hydrosphere by snowfall, melting, vapor transfer, and thecalving of icebergs. Melting and vapor transfer are significant in both the energy balance and the mass balance,which in consequence are intimately coupled. Glacier energy balances differ from those of other natural surfacesin having small or even negative net radiation. Emission of terrestrial radiation is limited, the surface temperaturebeing no greater than the freezing point, but the surface albedo is always high. The limit on surface temperature,and the year-round tendency for net radiative cooling, means that sensible heat transfer is generally downward,while vapor transfer may be either upward or downward. Once conduction has raised a surface layer to thefreezing point, further energy surpluses are used to melt snow or ice. In winter, the energy balance is dominatedby radiative cooling. Apart from its close connection with the energy balance, the mass balance is also influencedstrongly by glacier dynamics. Glaciers and the flowlines of which they are composed exhibit vertical zonation,with net accumulation at higher and net ablation (mass loss) at lower elevations. This imbalance drives, and iscorrected by, the ice flow. The leading methods for the measurement of mass balance are the direct, geodetic,and kinematic methods. Direct measurement involves determining the accumulation and ablation in situ or byequivalent remote sensing, with separate treatment of calving where it occurs. Geodetic measurements requirethe determination of glacier thickness at two epochs; the change of thickness, approximately equal to the changein surface elevation, gives a volume balance that may be converted to a mass balance if the density of the massgained or lost can be supplied accurately. In the direct and geodetic approaches, the ice flow is assumed tointegrate to zero over any one flowline (correctly, if the entire flowline is measured). Kinematic methods are freeof this restriction. They involve measurement of all of the terms in the balance and are therefore more difficult.The need for better understanding of mass balance, at socioeconomic scales from local to global, has stimulatedintense study of ways to improve the measurements. Recent and impending methodological advances are comingfrom radar altimetry, laser altimetry, gravimetry, passive-microwave remote sensing, and interferometry usingsynthetic aperture radar. A subject requiring increased attention, as the measurements improve in precision andcoverage, is improved quantification of the measurement errors. The best current estimates of global averagemass balance are equivalent to 0.14–0.44 mm a−1 of sea-level rise, to be compared with the inferred total rate ofabout 1.9 mm a−1 . This figure is a composite of estimates for “small” glaciers (those other than the ice sheets),whose balance has been growing more negative since the 1960s; the Greenland Ice Sheet, which seems to have anegative balance; and the Antarctic Ice Sheet, for which the sign of the mass balance remains in doubt althoughits magnitude is probably within a few kg m−2 a−1 (mm a−1 water-equivalent) of zero.

INTRODUCTION

Glaciers exchange energy with the atmosphere overlyingthem and with the earth or ocean beneath. While the surfaceenergy balance of a glacier is not fundamentally different

from that of a drainage basin or other hydrological unit,the fact that glaciers have a basal energy balance and a(typically small) internal energy balance sets them apart.A more obvious distinguishing feature of glaciers is that,because they are made of frozen water which is apt to

Encyclopedia of Hydrological Sciences. Edited by M G Anderson. 2005 John Wiley & Sons, Ltd.

2556 SNOW AND GLACIER HYDROLOGY

melt, their energy and mass balances are very intimatelycoupled. Mass balance is the glaciological analog of thewater balance in hydrology. Most glaciers gain mass bysnowfall and lose mass by melting, although for someglaciers, including the largest, the calving of icebergs isan important term in the balance.

Glacier energy and mass balances are important in thefollowing unranked respects:

• Glacier meltwaters are dominant components of thewater balance of semiarid regions downstream ofglacierized mountain ranges (e.g. Su and Shi, 2002).Such regions include the Prairies of Canada, CentralAsia, and the Himalaya, and much of Andean SouthAmerica. They also constitute an important resource forhydroelectric power generation, notably in Norway; asource of revenue from tourism; and hazards (Richard-son and Reynolds, 2000).

• On the global scale, glaciers exchange mass with theocean. As it is currently understood, the water balanceof the ocean fails to add up and an accurate knowledgeof glacier mass balance is required if we are to explainthe observed contemporary rise of sea level (Churchet al., 2001; Munk, 2003). Glacier mass balance alsoaffects the salt balance of the ocean.

• Glaciers play a part both in bringing about climaticchange and in helping us to document it. They arehighly reflective and so reduce the magnitude of netradiation at the Earth’s surface, and as their extentschange so does their influence on the global energybalance and the general circulation. As independentsources of information about environmental change,they are a valuable supplement to the weather stationsfrom which we derive information about temperatureand other leading climatic variables.

• Gains and losses of glacier mass imply redistribution ofthe mass of the Earth, altering its moments of inertiawith consequences for the evolution of such geophysicalquantities as length-of-day, true polar wander and thegeoid, and with implications for understanding of theviscosity profile of the Earth’s mantle (Peltier, 1998).

DEFINITION OF TERMS

A Column of Ice

In Figure 1(a), a column extends through ice at the Earth’ssurface. Ice, a soft solid, deforms readily under stress,so we orient the column with respect to the resultingflow. We assume that net exchanges of energy and massthrough the side-walls are negligible, an idealization whichis acceptable for balance studies if not for studies ofdynamics. The lower surface makes contact with either thesolid earth or water. At its upper surface, the column will

ordinarily be exposed to the atmosphere, but there maybe a complicating mantle of rocky debris (Nakawo et al.,2000). Melting and freezing are regarded as loss and gainrespectively; that is, liquid water is “outside” the column,being assumed to run off or refreeze in a time shorter thanthe span over which we compute the mass balance (of ice).The column also has internal energy and mass balances; forexample, changes of water phase are not restricted to thesurface and the base.

Flowline

A flowline (Figure 1b) is a sequence of ice columns ofinfinitesimal cross section arranged so that each columngains mass by flow from an up-ice neighbor and losesmass to a down-ice neighbor. To a good approximation,flowlines may be identified by beginning at any point whereeither the slope changes sign – at a flow divide – or theice thickness drops to zero, and following the directionof steepest ascent or descent to another such point. Thefirst column in the sequence has zero flow through oneboundary. Most importantly, the integral of the mass fluxdivergence over the entire flowline is zero: a loss byflow from one part of the flowline must be compensatedby a gain somewhere else, which means that we canneglect the flow when estimating the mass balance ofthe flowline.

Glacier

A glacier is a collection of contiguous complete flowlinesthrough snow and ice that persists on the Earth’s surfacefor more than one year (Figure 1c). The two largest glaciersare called ice sheets: the Greenland Ice Sheet and theAntarctic Ice Sheet. An ice shelf consists of the floatingparts of two or more glaciers. There are small ice shelvesin northernmost Canada and Greenland, but otherwise iceshelves are found only in Antarctica. Ice shelves differfrom sea ice, which is a few meters thick, in being tensto thousands of meters thick.

Glacier Types and Glacier Zones

Glaciers are at or below their freezing point Tf, which inthe absence of impurities increases from 273.16 K at thesurface at a rate of about 0.67 K km−1 of ice overburden.Cold or polar glaciers are those in which temperatureT is below Tf except possibly in a surface layer, up to10–15 m thick, during summer. Temperate glaciers areat T = Tf throughout, except in the surface layer duringwinter. Polythermal glaciers have, in addition to the surfacelayer of seasonal fluctuations, a basal layer at T = Tf andan intermediate layer in which T < Tf. Cold glaciers arealso dry-based glaciers, while polythermal and temperateglaciers are, at least locally, wet-based glaciers. These types

MASS AND ENERGY BALANCES OF GLACIERS AND ICE SHEETS 2557

(a)

qin

mf

qout

pe

e

mw

w

p

m

p Precipitation; internal accumulatione Sublimation; condensationw Wind drifting, scouringm Meltwater runofff Basal freezingq Ice flow

(c)

(b)

Divide

Grounding line

Terminus(ice front)

Start of flowlineEnd of flowline

Figure 1 (a) A column of ice, showing leading mass-balance terms. Black arrows: accumulation (gain of mass); whitearrows: ablation (loss of mass); grey arrows: throughflow. (b) A flowline considered as a sequence of ice columns. Thisflowline happens to have a floating terminal section. (c) Plan view of selected flowlines (thick) on a real glacier. Thinlines: contours (100-m interval)

can be misleading, for strictly the adjectives apply only toice columns. Nevertheless, they are useful when consideringenergy and mass balances because the type determineswhether, and if so where, melting and freezing may occurbeneath the surface.

Because temperature decreases with increasing elevationat the surface, glaciers exhibit vertical zonation (Figure 2).This concept (Shumskiy, 1955; Benson, 1959) is thebasis of most remote-sensing studies of energy and massbalances. Snow is solid condensates and precipitation,including freezing rain, added to the glacier during thecurrent year. Firn is snow added in previous years. Glacierice is firn that has been compacted to a density near that ofpure ice. In the dry-snow zone, the temperature never risesto the freezing point and there is no melting. In the upperpercolation zone, melting occurs at the surface in summerbut the meltwater remains within the snow, while in thelower percolation zone some meltwater penetrates to theunderlying firn before refreezing. This constitutes internalaccumulation, which should be accounted for in the mass

balance. The slush zone (Muller, 1962) is that part of thepercolation zone in which at least some of the added snow islost from the column, either as meltwater runoff or by slushavalanching. The superimposed ice zone represents eitherexposed, refrozen percolating meltwater or the product ofslush avalanching. Ice is also found at the surface of thelowest zone, the ablation zone, but superimposed ice isnewly gained mass while exposed glacier ice implies a lossof mass.

The surface volume balance of any column of the flow-line is proportional to z1 − z0 = z(t1) − z(t0) and, whenwe can neglect flux divergence and internal accumulation,the mass balance is simply this elevation difference mul-tiplied by average density. The difference, and thereforethe mass balance, is zero at the equilibrium line over thebalance year (see Figure 2) or possibly some longer span.More generally, the whole flowline or glacier is said to bein equilibrium if the sum of its column balances is zero.Figure 2 hints, unrealistically, that the flowline will growcontinually thicker in the accumulation zone and thinner in


t0

t1

t0

t1

mpd

ftoi

max

d w s

e

r

z

DSZ UPZ LPZ SIZ

AcZ AbZ

SuZ

Internalaccumulation

Figure 2 Cross section of a flowline, illustrating glacierzonation in elevation z. Broken cross-hatching indicatessnow; firn is grey (the part vulnerable to internalaccumulation being hatched), and superimposed ice isdark grey; glacier ice is unshaded. t0, t1: glacier surfaceat the start and end of the balance year; max: maximumelevation reached by transient glacier surface betweent0 and t1; ftoi: boundary between firn and ice; mpd:maximum depth to which meltwater percolates beforerefreezing or reaching the effectively impervious (but notnecessarily impermeable) barrier ftoi; d: dry-snow line(surface outcrop of mpd); w: wet snow line; r: runoff limit;s: snowline; e: equilibrium line. DSZ: dry-snow zone; UPZ,LPZ: upper and lower percolation zones; SuZ: slush zone;SIZ: superimposed ice zone; AbZ: ablation zone; AcZ:accumulation zone (the set of all zones above e)

the ablation zone. In fact, this pattern of thickness changeand differential loading is what drives the glacier flow. Thestrongest definition of equilibrium is that it is the state thatprevails when the flow is exactly that required to preservethe shape of the glacier unchanged, that is, for surface ele-vation z to remain constant everywhere. Figure 2 “works”only because of the hidden assumption that the surface z1

transforms into the surface z0 at the first instant of each newbalance year, when snow turns into firn and superimposedice into glacier ice.

Units

Glacier energy balances are usually reported in W m−2,which is the standard in climatology.

Mass balance is a rate per unit of projected horizontalarea. It is nearly always reported for balance years (begin-ning at the start of winter) or their winter and summer com-ponents, so the appropriate units are kg m−2 a−1. Severalother units are in common use. The most common are mmwater-equivalent a−1, numerically identical with kg m−2 a−1

because 1 kg of water, with a density of ρw = 1000 kg m−3,is 1 mm deep when distributed over 1 m2. Care is neededwhen mass and volume balances are discussed together,for example when the measured quantity is a change ofelevation or thickness. The thing to avoid is confusing ice

thickness h (m) and water-equivalent “thickness” (ρi/ρw)h

(kg m−2 or equivalently mm w.e.), ρi = 917 kg m−3 beingthe density of pure ice.

When discussing glacial contributions to changes of sealevel a natural unit to use is mm sea-level-equivalenta−1. A glacier mass balance of B kg m−2 a−1 is equalto −1000(S/So)B/ρw mm s.l.e. a−1, S being the area ofthe glacier and So = 362.0 Mm2 the area of the ocean andice shelves.

ENERGY BALANCE

Surface Energy Balance

The energy balance at the surface of a glacier is

Ks + Ls + H + λE + Gs + µMs = 0 (1)

where all the quantities are flux densities in W m−2. Ks =(1 − α)K↓ is net solar radiation, K↓ being the incidentsolar irradiance and α the surface albedo. Ls = L↓ + L↑is the net terrestrial radiation, the balance of gains from theupper hemisphere and the surface emittance L↑ = −εσT 4

s .The surface emissivity, ε, is the ratio of surface emittanceto that of a black body at the same temperature Ts, and σ =5.68 × 10−8 W m−2 K−4 is the Stefan–Boltzmann constant.Snow and ice are usually treated as black bodies, that is,ε = 1. H and λE are turbulent fluxes of sensible and latentheat from above respectively, and Gs is the conductiveflux of heat from below; λ = 2.835 × 106 J kg−1 is thelatent heat of sublimation. Finally, µMs is the energyused for surface melting, with µ = 0.335 × 106 J kg−1 thelatent heat of fusion and Ms the surface meltwater flux(kg m−2 s−1).

General methods for the measurement of terms in the sur-face energy balance are described by Oke (1987), and adap-tations to the peculiarities of glacier surfaces by Oerlemans(2001). Accurate microclimatological measurements aredemanding, and much of the effort in glacier climatologyis devoted to parameterizing the results of research cam-paigns so that they may be used in wider contexts. Someimportant generalizations may however be made readily.

First, the net solar radiation on glaciers is usuallysmall because the albedo is large (Table 1). At opticalwavelengths snow is the brightest of natural surfaces,although its albedo can in fact vary greatly with age,grain size, the abundance of impurities and of liquidwater, the incidence angle of the irradiant flux, and otherfactors (Warren, 1982). Exposed glacier ice is always darkerthan snow. Fresh snow may absorb three times less solarradiation than the ice that it covers, and its disappearanceis followed by a marked shift in the energy balance to amore absorbent regime in which, other things being equal,melting is accelerated.


Table 1 Typical observed energy balances of glacier surfaces

LocalityElevation

(m) Period Type α Rs H λE Gs µMs References

QML 1180 37d (s) B 0.58 50 0 −34 −16 0 Bintanja andReijmer, 2001

QML 1170 37d (s) S 0.79 2 9 −11 −1 0 Bintanja andReijmer, 2001

QML 34 2a S 0.87 −1.5 2.3 −0.8 −0.1 0 Reijmer andOerlemans, 2002

QML 1420 2a S n/a −26 26 0 0 0 Reijmer andOerlemans, 2002

QML 2892 2a S 0.84 −1.4 2.2 −0.7 −0.1 0 Reijmer andOerlemans, 2002

W Greenland 1155 60d (s) S 0.77 28 16 −6 −8 −30 Ohmura et al. 1994SW Greenland 790 512d (s) G ∼0.30 103 62 −6 n/a −161 Braithwaite and

Zhang, 1999N Greenland 540 35d (s) G ∼0.48 84 27 −24 −18 −71 Braithwaite and

Zhang, 1999Illimani, Bolivia 6340 21d (w) S 0.82 −12 12 −22 22 0 Wagnon et al. 2003Pasterze, Austria 2205 47d (s) G 0.20 180 51 11 0 −242 van den Broeke,

1997Pasterze, Austria 3225 47d (s) S 0.59 65 22 1 0 −89 van den Broeke,

1997Peyto, Alberta 2240 17d (s) G 0.36 96 51 5 0 −152 Munro, 2001Peyto, Alberta 2510 17d (s) S 0.73 28 32 5 0 −63 Munro, 2001

Fluxes, in W m−2, are positive towards the surface (Rs is net radiation, Ks + Ls; melting is negative). Error estimates vary from a few toseveral tens of W m−2. QML: Queen Maud Land, East Antarctica. Period: s, w denote winter and summer. Type: B, blue ice; G, glacierice; S, snow.

The longwave (terrestrial) balance is constrained by thefact that Ts ≤ Tf, which limits L↑ to magnitudes no greaterthan about −316 W m−2.

Because the air above glaciers is often warmer than thefreezing point in summer, and is a heat source fuelingintense radiative cooling in winter and at night, the sensibleheat flux H is generally directed downwards. The latentheat flux λE is often directed downwards also because,even when liquid water is present, the vapor pressure at thesurface will be appropriate to saturation at a temperaturenear Tf. On the lower parts of glaciers, the turbulent fluxesare enhanced by katabatic drainage of cooled air from highelevations. The katabatic wind, as well as being persistentand directionally constant, can be extremely strong.

The heat exchanged with the interior of the glacier drivesan annual variation of temperature which is confined tothe upper 10–15 m. However, in summertime, once anisothermal surface layer at the freezing point has beenestablished, the heat flux Gs must dwindle to zero, and anysurplus from the atmospheric terms in equation (1) will beused for melting. This surplus is responsible for most of theablation on most glaciers, exceeding −10 m w.e. a−1 (about−100 W m−2 over the year) at lower latitudes. It may alsobe responsible for advective heat transfer to the interior ofthe glacier if the meltwater refreezes at depth instead ofrunning off.

Some representative energy balances are summarized inTable 1. Blue ice is glacier ice exposed at the surface

because snow fails to accumulate. Apart from scouring(removal as blowing snow), the principal reason for thisin Antarctica is sublimation. Bintanja (1998) estimates bymodeling that sublimation of blowing snow may reach−15 W m−2, twice the rate of sublimation of in situ snowand ice, near the Antarctic coast.

Internal Energy Balance

The energy balance of a small volume within a glaciermay be understood (Paterson, 1994) in terms of thermaldiffusion, advection of heat by the ice flow, and energysources due to strain heating, including the compaction offirn, and the refreezing of meltwater. The strain-heatingterms are of order 10−4 W m−3 or less, and are negligiblefor balance purposes even when integrated over typicalcolumn thicknesses, but the refreezing of meltwater canbe significant. It may be expressed as µf c/Z, where c isthe surface accumulation rate, f is the fraction of c thatrefreezes in the firn, and Z the thickness over which itrefreezes. µf c is of order 0.1–1 W m −2 over the year, butZ is at most 10 m so that detectable summertime warmingof the firn is possible.

Basal Energy Balance

GlaciersThe energy sources at the bed of a glacier are frictionalheat and geothermal heat. If the basal temperature is below


the freezing point, the heat is conducted upwards into thebody of the glacier. The opposite situation, heat flow fromthe glacier into its bed, is possible, for example, whentemperate ice advances over permafrost, but very unusual.If the bed is at the freezing point the available energy isused to melt ice.

Pollack et al. (1993) have compiled measurements ofgeothermal heat flux. Averages are 0.09 W m−2 for Antarc-tica and 0.04 W m−2 for Greenland, with similar valuesfor other glacierized regions. Only Iceland and the RockyMountains have geothermal fluxes above 0.10 W m−2,equivalent to −10 kg m−2 a−1 of basal melting. Frictionalheating derives from the loss of potential energy in the icecolumn as it moves downslope. Its rate can be expressed asthe product of basal velocity times basal shear stress, yield-ing typical fluxes of 0.01–1 W m−2. Although the basalbalance quantities are small, they are only marginally neg-ligible given the current accuracy attainable in surfacemass-balance calculations. They are also heat sources with-out compensating sinks, so they tend to make cold glacierssteadily warmer.

Ice Shelves

Assuming that thermodynamic equilibrium prevails (Doake,1976; Holland and Jenkins, 1999), the contact betweenshelf ice and seawater must be at the freezing point of theseawater, which depends on the pressure of the overlyingshelf ice and the salinity of the water. But in general theseawater and shelf ice at some distance from the contactwill not be at Tf, so there is a heat source or sink, andtherefore melting or freezing, at the contact. We can write

Hb + Gb + µMb = 0 (2)

where Hb is the sensible heat flux from the seawater, Gb

the conductive flux from the shelf and µMb the latentheat flux; fluxes are positive towards the shelf base andmelting is negative. There are two complications. First, saltis coupled to heat because freezing increases and meltingdecreases the salinity of the seawater, altering both Tf andthe buoyancy of the water. Second, the water flow itselfis driven substantially by variations of temperature andsalinity.

Holland and Jenkins (1999) envisage a boundary layerin the water flow beneath the shelf. At some elevationbelow the base, the water is at a temperature To and salinitydetermined by the mesoscale ocean circulation, and thesensible heat flux depends on the difference between Tf

and To. Thus, the principal controls on the basal energybalance are the properties “imported” by the mesoscalewater flow. Direct measurements are very difficult, butRignot and Jacobs (2002) measured basal melting ratesindirectly at 23 shelf grounding lines (Figure 1b). Theserates are well correlated with an indirect estimate of To, and

are extremely high. They pertain to areas of only a few tensof km2 (the square of glacier width at the grounding line),but the greatest magnitude, −425 W m−2 or −44 m ice a−1,at Pine Island Glacier, is a record. The meltwater is buoyantbecause it is fresh, and flows upwards along the base ofthe rapidly thinning ice shelf to where a lesser pressureimplies a smaller sensible heat flux (higher Tf). Sometimes,it enters a regime in which it is actually colder than Tf andice begins to form, accreting as “marine ice” at the base ofthe shelf. The latent heat flux averaged over all the Antarcticice shelves, however, is believed to be negative. Jacobset al. (1996) estimate it (with an uncertainty of ±50%) as−5.4 W m−2, that is, −500 kg m−2 a−1.

If there is net freezing at the base, it is reasonable (Hol-land and Jenkins, 1999) to set the temperature gradient inthe shelf ice, and the implied heat flux, Gb = ki(∂T /∂z)|b,to zero. Where there is net melting, the shelf tempera-ture gradient is coupled to the dynamics of the shelf ice,but if we neglect this coupling a crude solution is avail-able in terms of the temperature difference between surfaceand base. Taking typical values, Ts − Tb = −30 K and ther-mal conductivity ki = 2 W m−1 K−1, we find that Gb rangesfrom −0.2 to −0.02 W m−2 for ice shelves of thickness300–3000 m.

MASS BALANCE

Methods of Measurement

The mass balance b of an ice column is

b = c + a + ci + ai + cb + ab + �q (3)

where the c are accumulation rates (black arrows inFigure 1a), the a are ablation rates (white arrows) and�q = qin − qout is the flux divergence. Subscripts i andb denote the interior and base of the column. Ablation bycalving at the terminus is a special case of equation (3) inwhich ai is equal to minus the entire mass of the column.The mass balance of a glacier or glacier flowline of areaS is

B = 1

S

∫sb ds (4)

When b is assumed to vary only with elevation, as isusual on valley glaciers, the measurements are grouped intoelevation bands and equation (4) becomes a sum of bandaverages, each average being weighted by the area of itsband.

Direct MeasurementsDirect measurements of column mass balance take the form

b = c + a = 1

�t

∫ z1

z0

ρ dz (5)


The flux divergence is ignored because the columnbalances are to be integrated over the glacier, and theother terms in equation (3) are either ignored or estimatedas corrections. If it occurs, calving must be measuredseparately. Standard methods of measurement are describedby Østrem and Brugman (1991), and Trabant and March(1999) give a detailed account of a careful protocol forfieldwork. Glaciers are dangerous places; safety in the fieldis discussed by Selters (1999).

A direct measurement of b involves emplacing a stakeand/or digging a pit. If the stake is vertical, and does nottilt, bend, or settle, measurements of stake top height abovethe surface at t0 and t1 are proportional to z0 and z1, neitherof which need be known in an independent coordinatesystem. In the ablation zone (Figure 2), the lost mass maybe assumed to have a constant density ρ = 900 kg m−3

(slightly less than ρi to allow for solid impurities, bubbles,intergranular voids, and macropores), so the mass balanceis b = ρ�z/�t . In the accumulation zone, the density ofthe mass gained must be measured in the walls of snow pits,augmented with spatially extended surveys of the variabilityof �z by probing or of b by shallow coring and weighing.

The column mass balance should be determinable witha standard error of the order of ±50 kg m−2 a−1, andusually better. Except when the measurement network isvery dense, an additional error is made by extrapolatingfrom points to the whole glacier. Cogley et al. (1996) andTrabant and March (1999) both adopt a standard error of±200 kg m−2 a−1 for elevation-band averages of b, on thebasis of the ability of single measurements to reproduceelevation-band averages determined with dense networks.Cogley (1999) showed that the uncertainty in B is notsignificantly less than this, because measurements of b atdifferent elevations are nearly perfectly correlated.

Internal accumulation is a worrisome bias on any coldglacier with a percolation zone. It is impractical to measureit, and models for estimating it are as yet quite crude.Internal ablation (Mayo, 1992) occurs because of theconversion to heat of the potential energy of meltwaterflowing down englacial channels. When the glacier isknown to be wet-based, the basal ablation can be estimatedas a function of the basal heat flux and frictional heating,although usually both it and internal ablation are neglected.Beneath polythermal glaciers, some of it is cancelled outwhen meltwater freezes on reaching cold parts of thebed, although the heat thus released helps to maintainthe temperate ice at its melting point. Extensive basalaccumulation occurs only beneath some parts of ice shelves.

The winter and summer balances, bw and bs, are mea-sured separately on some glaciers. They are defined byequation (5) with z0 and z1 taken at the endpoints of theappropriate season, and in most climates they separate thetwo main controls, winter wetness and summer warmth, ofthe annual balance b = bw + bs. This is valuable because,

for example, both controls are likely to involve more waterthan in neighboring unglacierized terrain, and relativelysmall changes in either can have substantial implicationsfor the regional water balance.

Geodetic Measurements

Geodetic measurements of mass balance have until recentlybeen used mostly as checks on the reliability of morefrequent direct measurements. They rely on pairs of datedmaps or other representations of �z/�t to give a volumebalance that may be converted to a mass balance by makingcorrect assumptions about density. Geodetic measurementsrequire the separate determination of z1 and z0 in geocentriccoordinates, which introduces a quite different set ofconcerns about accuracy. For example, the quantity whichshould be measured is actually the rate of change ofthickness, h = z − zb, and changes in bed elevation, zb,arising from glacial isostatic adjustment and other causesneed to be allowed for.

In the accumulation zone, if the density profile remainsunchanged between t0 and t1, then Sorge’s Law is saidto apply: Density is a function only of depth beneath thesurface. However, the compaction rate varies with the rateof surface loading by new snow, the temperature and,possibly, the rate of internal accumulation. Wingham (2000)modeled the compaction of dry isothermal firn, finding thatthe spatial scale of fluctuations in accumulation is of criticalimportance. Zwally and Li (2002) modeled the effectof seasonal fluctuations of temperature and accumulationloading, reproducing observed fluctuations of z with fairaccuracy. Where melting occurs and the resulting meltwaterrefreezes in the form of ice lenses, the situation is muchmore complicated, and at present it is necessary to invokeSorge’s Law arbitrarily.

Kinematic Measurements

In kinematic measurements, qin or qout or both are measuredat “gates” (cross-sections), in combination with up-ice ordown-ice measurements of accumulation and ablation. Theadvantage is that the complete-flowline assumption can berelaxed. The disadvantage is that qin and qout cannot bemeasured inexpensively. They require knowledge of icethickness across the gate and of the vertical distributionof ice velocity. In practice, only the surface velocity isknown, and either it must be assumed that the glacier movesentirely by basal sliding, or the velocity profile must bemodeled (Kostecka and Whillans, 1988). Hubbard et al.(2000) modeled glacier flow to generate a map of the fluxdivergence. This method is not likely to be applied widelybecause of the amount of boundary-condition informationrequired by the flow model. An important recent advance,discussed below, is the ability to measure q at groundinglines by radar interferometry.


Hydrological MeasurementsIn the hydrological approach, precipitation and evapora-tion over the glacier are estimated along with the runoffof meltwater, and the water balance is solved. This is onlydone routinely for Aletschgletscher, Switzerland. Bhutiyani(1999) has published hydrological estimates for SiachenGlacier in the Karakoram. The uncertainty in the hydro-logical method is in practice much greater than in a typicaldirect measurement. For example, glacier runoff must beseparated from that contributed by unglacierized parts of thecatchment tributary to the discharge measurement station.

Surrogate ObservationsGlacier mass balance B is well correlated with the elevatione of the equilibrium line at the end of the balance year(Braithwaite, 1984). This offers a means of inferring B

from less expensive observations of e, possibly fromspace, but, apart from glaciers on which B is alreadymeasured, the only regular reports of e are those ofChinn (1999) for New Zealand glaciers. B is equally wellcorrelated with the accumulation area ratio, which is thearea of the accumulation zone divided by the area ofthe glacier (Slupetzky, 1989). These economical surrogatesare valuable for extending knowledge of mass balancevariability, but they are necessarily quite uncertain.

It is easier to measure the position of a glacier’s terminusthan to measure its mass balance. Terminus fluctuations arereported annually for several hundred glaciers, as againstfewer than 100 reports of mass balance (Haeberli et al.,1998). Unfortunately, the link between mass balance andglacier length is indirect. An observed annual change ofterminus position is a response to balance forcing integratedover the glacier and over some indefinite span much longerthan one year.

Developing and Emerging TechnologiesConventional measurement methods are labor-intensive anduncertain, but the need for better and more comprehensiveestimates of mass balance has stimulated vigorous researchinto alternatives. Some of these are discussed briefly here.

Radar Altimetry Radar altimeters, mounted on air-craft or satellites, emit pulses towards the nadir (the pointon the surface vertically beneath the sensor) and “track” thewaveforms of the return pulses. Tracking involves predic-tion of future ranges (distances to the surface, convertibleto elevations z) from ranges detected in the immediate past.Poor predictions, as when the surface is steep or undulating,result in loss of “lock” on the surface. A cross-track interfer-ometer on the future CryoSat radar-altimetry mission willaddress this problem, which at present compromises theglaciological use of radar altimetry outside the interiors ofthe two ice sheets. In those regions, however, repeated radaraltimetry has transformed our knowledge of accumulation.

Accurate identification of orbit crossover points is essential,so orbital errors, discussed in detail by Davis et al. (2000),become important. Wingham et al. (1998) give a long listof corrections needed before the range differences may beinterpreted as elevation changes. It is also assumed that theradar interacts with the glacier by surface backscatter, withextinction and subsurface volume backscatter being unvary-ing. This appears to be a good description of the interactionin dry-snow zones, but melting and internal accumulationcomplicate matters. Bamber et al. (2001) showed that theroot mean square (rms) error of radar altimetry with respectto collocated estimates from airborne laser altimetry was∼ 7 m. The laser-altimeter measurements have decimeter-level accuracy (Krabill et al., 1995).

Laser Altimetry Laser altimetry is the measurementof surface elevation by measurement of the two-way traveltime of a pulse of optical radiation. Repeated measurementsyield the change of elevation, leaving the density of theice column to be supplied, for example, by Sorge’s Law.The position of the laser must be known precisely, andit must be possible to reoccupy horizontal positions withan accuracy comparable to the radius of the “footprint” ofthe laser pulse. These problems have been solved by theintegration of GPS measurements into altimeter systems.Neither laser altimeters nor radar altimeters can estimate theflux divergence, so it is necessary to accumulate a glacier-wide coverage of column measurements.

The first satellite laser altimeter, GLAS (the GeoscienceLaser Altimeter System), was launched on ICESat inJanuary 2003. Airborne laser altimetry was used by Krabillet al. (2000), who found significant thinning at lowerelevations and overall balance at higher elevations onthe Greenland Ice Sheet over a five-year span. Arendtet al. (2002) presented balance estimates for 67 glaciersin Alaska, relying on maps drawn from aerial photographyflown in the 1950s and on laser-altimetric surveys in the1990s. A notable methodological conclusion is that errorsin the geodetic balance are dominated by the errors in theold maps. The old maps offer the chance of extending thehistorical span of the measurements, but their inaccuracylimits significantly what can be achieved.

Gravimetry Velicogna and Wahr (2002) have studiedthe joint resolving power of GLAS, GPS measurements, andthe Gravity Recovery And Climate Experiment (GRACE).Launched in March 2002, GRACE is a pair of satellitesflying about 200–250 km apart along-track. Each satelliteranges to the other using microwave phase measurements.When nongravitational accelerations are accounted for, theresidual fluctuations in range may be used to map thegravity field with a horizontal resolution of a few hundredkilometers once every 30 days. With the aid of GLASand GPS estimates of change in elevation, GRACE geoidchanges are interpreted as a result of glacial isostatic


adjustment and of changes in ice mass. Errors in the latterof the order of ±16 kg m−2 a−1 may be expected over250-km spatial and 5-year temporal scales. This is quitelarge by comparison with the minimum accumulation in theinterior of East Antarctica, but GRACE offers a substantialimprovement over current coverage.

Ice-penetrating Radar and Ice Cores Radarswith wavelengths in the meter range (frequencies of10–500 MHz) can yield information on the depth to reflec-tive horizons within the ice. The vertical separation betweenany two such horizons, if both horizons are isochronousand of known age, is a measure of volume balance. Thesame reasoning applies to horizons identified in ice cores.Only the accumulation at the core site can be measured,but the resolution in time is likely to be much better, atleast in the upper core where annual layers are recogniz-able. When a well-dated core record can be tied to extensiveradar surveys, a great increase in coverage results. It wouldbe valuable if this capacity, at present largely restricted tocentury and longer timescales (e.g. Siegert, 2003), couldbe extended to shorter and more recent time spans andto smaller glaciers. Here, the aim is to find summer sur-faces – the crusts that separate balance years. Palli et al.(2002) estimated errors for a traverse in Svalbard with a50-MHz radar. Most of the uncertainty was due to reflector-tracking error. For 13-year-old (Chernobyl) and 36-year-old(nuclear test) layers, with accumulation rates of 670 and580 kg m−2 a−1, errors were ±134 and ±49 kg m−2 a−1.

Microwave Emission and Backscatter Passive-microwave radiometers sense the emission of surfaces atwhich they are pointed. Microwave emission from glaciers(Matzler, 1987) is always reported in terms of the brightnesstemperature TB = εT − Ta , where ε is the emissivity ofthe medium (a mixture of ice, air, and possibly water), T

is its physical temperature and Ta , often neglected, is thebrightness temperature of the downwelling sky radiation. Indry snow, the emissivity is determined by volume scatteringat interfaces such as grain boundaries and larger structuressuch as buried hoar layers. Grain growth rate depends onboth temperature and accumulation rate, and it is possible toexploit this (Zwally, 1977) to model accumulation rate as afunction of TB and T . Estimates of c from the Zwally modelcompare well with estimates from surface measurements(Vaughan et al., 1999).

Scattering at air-water interfaces is much more effectivethan at air-ice interfaces, and, when melting occurs at thesurface, volume scattering ceases to be significant and theemissivity approaches unity. Abrupt changes are observedin TB as meltwater comes and goes, and have been usedto estimate the extent and duration of melting over the icesheets (Abdalati and Steffen, 2001; Mote, 2003). Ramageand Isacks (2003) have exploited the same behavior to

determine the start and end of the melt season on Alaskanglaciers.

Active-microwave sensors (scatterometers and imagingradars) measure the backscatter from pulses whichthey themselves emit. Like radiometers, scatterometershave poor spatial resolution (tens of kilometers orworse), although it can be improved by temporalaveraging. Drinkwater et al. (2001) reported standard errorsof 50–70 kg m−2 a−1 for a linear regression betweenclimatological estimates of accumulation and NSCATscatterometer data in Greenland. In the percolation zone ofwestern Greenland, Wismann (2000) showed an excellentcorrelation between seasonally integrated reduction ofbackscatter, with respect to a reference wintertime average,and positive degree-days. Because of its buried ice lenses,resulting from internal accumulation, the percolation zoneis one of the most radar-bright surface types on Earth whencold and dry, and one of the darkest when wet.

Synthetic aperture radars (SARs), unlike scatterometers,can resolve surface features as small as 5–100 m. Theycover much less ground per megabyte of data, so that com-plete one-time coverage of an ice sheet is a major under-taking (e.g. Jezek, 2002). In SAR studies of small glaciers,the emphasis, to date, has been on delineating zones and onsearching for the equilibrium line (e.g. Demuth and Pietron-iro, 1999). Cogley et al. (2001) demonstrated a differentapproach to SAR observations, relying on browse imagesto identify the seasonal course of melting as a function ofelevation. There was a clear dependence of change in imagebrightness upon positive degree-days accumulated betweensame-day image pairs from summer days. In a similar way,Nghiem et al. (2001) were able to map the extent of meltand refreezing in Greenland, relying on diurnal variationsmeasured by the Quikscat scatterometer.

Microwave estimates of accumulation in the dry-snowzone are already among the best available, and estimates ofmelting in the percolation zone have shown great promise.However, the ablation zone lacks a distinctive seasonalmicrowave signature.

Interferometric Synthetic Aperture Radar In asuitable configuration (e.g. Madsen and Zebker, 1998), twoSAR images of a ground target can be used to estimateits elevation z by interferometry. More important payoffsof InSAR can be realized once the effect of topographyis removed: the measurement of surface velocity and themapping of grounding lines. When the images are separatedby one day, as in the ERS tandem mission, 1995–1996,errors in surface velocity are of the order of meters/year.Vertical motions can be resolved as well, and near togrounding lines the contribution of tidal flexure to thevertical motion can be evaluated. The downstream changein flexure is largest just downstream of the grounding line,so the number of interference fringes is greatest there.Accurately located grounding lines are valuable because


at the grounding line the ice thickness is a function of z

by hydrostatic equilibrium, and the discharge q may beestimated by integrating thickness times velocity acrossthe gate. Speckle tracking (Gray et al., 2001) is a lessaccurate but more robust way of measuring velocity. It isanalogous to feature tracking in optical imagery, althoughthe conditions for interferometry must still be satisfied.

Undersampling

Sparse information is a problem common to nearly allmass-balance methods. We wish to know the balance atM points in space during N spans of time, but havemeasurements only at a smaller number m of points for asmaller number n of spans. In whatever way m, n and M , N

are related, interpolation or extrapolation will be required.How do spatial and temporal variability affect our abilityto interpolate or extrapolate? This question can appearin different guises. Altimeter measurements of elevationchange over a few years, and interferometric estimates ofice discharge over a few days, need to be compared withestimates of accumulation over much longer spans. In icecores, measurements can be well resolved in time, butthey may measure spatial as well as temporal variability

if sastrugi and snow dunes have been migrating across thecore site. Global estimates of small-glacier mass balancerely on observations spanning 1–50 years from at most afew hundred glaciers, and some of the unmeasured glaciersare thousands of kilometers from the nearest measuredglacier. Information on time and space scales of variabilityis essential in each of these contexts if the measurementsare to be interpreted meaningfully, that is, if they are tobe given accurate error bars. This statistical problem iscomparable in magnitude with the observational challengesbeing addressed by new technologies.

Results

Small GlaciersMeasurements of mass balance are reported to the WorldGlacier Monitoring Service (WGMS), Zurich, which pub-lishes biennial bulletins (Haeberli et al., 2001) and quin-quennial summary volumes (Haeberli et al., 1998). Not allmeasurements find their way into the WGMS database.Cogley and Adams (1998) and Dyurgerov (2002) areamong those who have published more complete com-pilations. Dyurgerov’s is the most comprehensive, and

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is available in spreadsheet form on CD-ROM. That ofCogley and Adams (1998; “CA” hereafter), coveringonly annual mass balance, may be obtained from http:

//www.trentu.ca/geography/glaciology.htm. Mostreported measurements are direct measurements. Conven-tional geodetic measurements are unlikely to alter the pic-ture greatly, although recent laser-altimetric measurements(Arendt et al., 2002) have established more confidently thatAlaska is a substantial contributor to sea-level rise.

Small glaciers occupy between 0.6 and 0.7 Mm2, consist-ing of 0.539 Mm2 plus about 0.070 Mm2 in Greenland andan undetermined extent in Antarctica not belonging to theice sheet. At present (2004) the CA dataset contains mea-surements from 310 glaciers dating back to 1885, althoughcontinuous measurements began only in the 1940s and aworldwide picture is only available after 1960. The spa-tial distribution of measured glaciers is uneven (Figure 3a),with high northern latitudes, Patagonia, and Tibet underrep-resented at the expense of less remote regions (Europe andwestern North America). Glaciers with calving terminuses(19 of the 310) are not well represented. Most records areshort. The modal length is 1 year, only 51 are 20 years orlonger, and in no one year have as many as 100 glaciersbeen measured (Figure 4c). To set against this evidence ofsparse coverage, there are about 2500 km2 of ice per mea-sured glacier, to be compared with about 30 000 km2 ofland per station for temperature climatologies. Moreover,CA note that three quarters of the small-glacier ice has atleast one annual balance measurement within 400 km, andthat the decorrelation distance for balance time series isabout 600 km.

Spatial variations in mass balance are difficult to iden-tify, at least at zonal resolution (Figure 3b). Allowing foruncertainties, most measurement series have averages indis-tinguishable from zero, but together they give a globalarithmetic average of −165 ± 34 kg m−2 a−1 for the refer-ence period 1961–1990. (Here, and below, uncertaintiesare twice the standard error.) The error estimate is some-what optimistic, and the balance estimate is biased by theneglect of internal accumulation in cold glaciers and ofinternal ablation in temperate glaciers, and other factorsincluding possibly the spatial unevenness of the measure-ment network.

As records grow longer, however, the evolution ofmass balance presents an increasingly coherent picture(Figure 4b). The world’s small glaciers were close toequilibrium in the 1960s and have been losing mass sincethen at a growing rate. When spatial bias is correctedwith an interpolation algorithm, the global average for1961–1990 increases to −123 kg m−2 a−1, which is a bestestimate.

Mass balance is well correlated with temperature(Figure 4a; r = −0.79 for the spatially corrected balance).The association would no doubt be closer if regional

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Figure 4 (a) Northern Hemisphere surface temperatureanomalies; (b) small-glacier mass balance from the CAdataset, with shaded confidence region; crosses showeffect of correcting for spatial bias; and (c) annualmeasurements contributing to each pentadal averagebalance (Jones and Moberg, 2003. American Meteoro-logical Society. http://www.cru.uea.ac.uk/cru/data/temperature/)

balances were matched to regional anomalies. Figures 4(a)and 4(b) help to justify the modeling of mass balance asa function of temperature (Wild et al., 2003), and moreimportantly show that two independent measures agreein identifying the late twentieth century as a period ofsignificant global change.

Greenland Ice SheetThe ice sheets are too big for an integrated measurement ofmass balance to be practical. Instead the aim is to compilethe results of separate evaluations of each component.

For Greenland, accumulation is the best-known com-ponent. There are up to 400 column measurements, andthree recent analyses interpolate to unmeasured parts ofthe ice sheet in different ways, hand contouring (Ohmuraet al., 1999) and kriging (Calanca et al., 2000; Bales et al.,2001). Figure 5 (Cogley, 2004) is constructed with another


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Figure 5 Accumulation on the Greenland Ice Sheet. The rate is below 100 kg m−2 a−1 in the northern interior andapproaches 1400 kg m−2 a−1 in the southeast (Reproduced from Cogley (2004) by permission of American GeophysicalUnion)


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Figure 6 Standard error of accumulation on the Greenland Ice Sheet. Errors (for an assumed 30-year span) are as lowas 8 kg m−2 a−1 in the interior and reach several hundred kg m−2 a−1 in places near the margin, where the interpolationalgorithm falters for lack of information (Reproduced from Cogley (2004) by permission of American Geophysical Union)


interpolation algorithm. It is broadly similar to the othermaps, but the algorithm has the advantage of generat-ing formal estimates of the error at each interpolationpoint (Figure 6). The result is an accumulation estimate of299 ± 23 kg m−2 a−1, very close to the Ohmura, Calanca,and Bales estimates, 297, 290, and 305 kg m−2 a−1 respec-tively. Uncertainty is small in the interior of the ice sheet,where measurements are relatively abundant, and growsrapidly towards the edge where measurements are few (ornonexistent because the balance is negative and no snowsurvives).

The spatial and temporal variability of accumulation hasof late received considerable attention (e.g. McConnellet al., 2001), stimulated by advances in geodetic measure-ment technology which have led to improved estimates ofsurface elevation change (e.g. Krabill et al., 2000; Daviset al., 2000). The variability is such that the patterns of ele-vation change determined by altimetry can be understoodmostly in terms of short-term fluctuations of the compactionrate and the accumulation rate (Braithwaite and Zhang,1999) and spatial “glaciological noise”.

According to the altimetry, average elevation change inthe interior of the ice sheet is close to zero, implyingb = c + �q � 0 because ablation a is near to zero. Thekinematic measurements of Thomas et al. (2001) supportthis conclusion. The altimetry shows, however, that partsof the ablation zone are thinning rapidly.

The ablation zone occupies 10–15% of the GreenlandIce Sheet. Surface measurements are too few for a coherentpicture to be drawn from them, and the most comprehensivecurrent understanding of surface ablation derives fromobservations of melt extent and duration (Abdalati andSteffen, 2001) and from modeling. Mote’s model (2003),which builds on passive-microwave observations of meltduration, yields a 12-year average of −155 kg m−2 a−1 formeltwater runoff from the whole ice sheet. Wild et al.(2003) parameterized ablation as a function of temperatureand surface elevation to obtain an equivalent estimate of−152 kg m−2 a−1.

The estimates given so far imply a net surface massbalance of about 150 kg m−2 a−1. Zwally and Giovinetto(2000), using passive-microwave and thermal infraredsatellite observations and a parameterization of meltwa-ter runoff, estimated this quantity as 128 kg m−2 a−1, andsummarized earlier estimates ranging between 97 and211 kg m2 a−1.

It remains to evaluate the losses due to calving, or prefer-ably the discharge qout across the grounding line. Manyoutlet glaciers, particularly in north Greenland, have float-ing terminal sections. We would prefer the grounding-lineflux because the floating ice has already made its contribu-tion to sea-level rise, and because the basal mass balanceof the floating terminuses is difficult to evaluate. Bigg(1999) estimated ablation due to calving using empirical

relationships to predict calving velocity as a function ofice thickness or water depth. The latter were taken mostlyfrom bathymetric charts. The estimates range from −73 to−132 kg m−2 a−1 with an uncertainty estimated as ±70%.Rignot et al. (1997) used InSAR to locate the groundinglines of 14 outlet glaciers of the northern Greenland IceSheet and to measure qout as 136 kg m−2 a−1 from an areaof 0.332 Mm2. The surface mass balance, estimated froma map of accumulation and a simple degree-day modelof melting, was 113 kg m−2 a−1, so the total balance wasB = −25 kg m−2 a−1. The calving flux at the ice front wasseveral times smaller than the grounding-line flux, requir-ing that basal melting of the floating ice be of the orderof thousands of kg m−2 a−1. Rignot et al. (2001) foundB = −2 kg m−2 a−1 for a different but overlapping set ofnorth Greenland glaciers. On 8 of 12 outlet glaciers, thegrounding line retreated inland over 1992–1996. This isconsistent with the laser-altimetric observations of near-coastal thinning, which also imply that the margin shouldbe retreating where it is on land. There is some limitedevidence for this (Sohn et al., 1998).

Not all of the balance components are accompanied bydetailed error estimates, but it seems likely that the massbalance of the Greenland Ice Sheet still cannot be distin-guished reliably from zero. Krabill et al. (2000) estimated itas −27 kg m−2 a−1, but this figure awaits analysis of errorsand more complete documentation. Nevertheless prioritiesfor future study are clear. Radar interferometry and laseraltimetry both suggest that the closest attention should begiven to lower altitudes of the ice sheet, where surface mea-surements are fewest and the energetics of melting and thedynamics of thinning need to be better understood.

Antarctic Ice Sheet

The Antarctic Ice Sheet is seven times the size of theGreenland Ice Sheet: 12.3 Mm2 of conterminous groundedice as against 1.7 Mm2, plus about 1.6 Mm2 of ice shelf andice rises.

Accumulation reaches less than 25 kg m−2 a−1 in the inte-rior of East Antarctica and exceeds 2500 kg m−2 a−1 inthe mountains of the Antarctic Peninsula (Turner et al.,2002). Vaughan et al. (1999) compiled surface observa-tions of accumulation and extrapolated them using a modelbased on microwave brightness temperature (Zwally, 1977)as a background field. They estimated accumulation to be149 kg m−2 a−1 over the grounded ice and 166 kg m−2 a−1

over the entire ice sheet. Giovinetto and Zwally (2000) con-toured the surface observations by hand and constructedfrom the contour map a grid of interpolates by eye. Theresulting estimate of accumulation over the entire ice sheetwas 159 kg m−2 a−1, to which they applied somewhat con-jectural bulk adjustments totalling −10 kg m−2 a−1 for lossby melting and deflation (wind scouring and sublimation)in coastal areas.


Although ablation by meltwater runoff is small in Antarc-tica, not all of the ice sheet has a positive surface bal-ance. Bintanja (1999) reviewed the widely distributedareas of blue ice, where there is no snow at the sur-face. Mass balance minima range from −350 kg m−2 a−1

in northern coastal areas where melting is significant tosmaller magnitudes (∼ −50 kg m−2 a−1) at higher eleva-tions where sublimation is low because temperatures arelow. Winther et al. (2001) estimated the extent of blueice to be 0.12–0.24 Mm2. Over about half of this extentof the negative balance is due to melting and over theother half it is due to scouring and sublimation. Theseresults suggest that blue-ice areas reduce the mass balanceof the grounded ice sheet by perhaps 0.5–4 kg m−2 a−1,which is negligible given the present accuracy of accu-mulation estimates at the ice sheet scale. Surface ablationmay not be negligible, however, in some of the smallerbasins.

Rignot and Thomas (2002) reported ice fluxes obtained atgrounding lines by InSAR and coupled these estimates withthe Giovinetto–Zwally estimates of accumulation to yieldthe nearest approach to date to a whole-glacier estimate ofB for catchments in Antarctica. The 33 catchments, labelledin Figure 7, cover 7.2 Mm2. Their ice discharges range froma negligible 3 kg m−2 a−1 (from the inactive Ice Stream C)to more than 1000 kg m−2 a−1 (Smith and DeVicq Glaciersin the Amundsen sector). Their balances (accumulationminus discharge) range from 131 kg m−2 a−1 (Ice Stream C)to well below −500 kg m−2 a−1 (the small outlet glaciers inthe Amundsen sector). The spatially variable picture thus

presented is not the least significant contribution made bythis work.

When pooled, the Rignot–Thomas measurements give abalance of −3 ± 4 kg m−2 a−1. If the Vaughan accumulationfield is used in place of the Giovinetto–Zwally field, thebalance becomes 6 ± 4 kg m−2 a−1. It is not clear how muchweight should be given to the reservations that led Rignotand Thomas to prefer the Giovinetto–Zwally field, and, byanalogy with the error analysis for Greenland (Figure 6),the standard error of accumulation may not be as small astheir chosen 5%. If it is larger, or if equal weight is givento the two accumulation fields, then the Rignot–Thomasresults resemble all earlier estimates of Antarctic massbalance in showing no significant difference from zero.A further source of uncertainty, not considered by Rignotand Thomas, is the difference in time span between theaccumulation maps, based on measurements covering up toseveral decades, and the quasi-instantaneous 1996 InSARmeasurements of discharge. Too little is known about thetemporal variability of the discharge of ice streams andoutlet glaciers for us to be confident that the InSARsnapshots are good estimators of multidecadal averages; thisis a question requiring more systematic attention.

Radar-altimetric surveys of thickness change (Wing-ham et al., 1998; Davis et al., 2001) agree with theInSAR/accumulation estimates in finding no significantchange in the interior of East Antarctica. (This is a puz-zle awaiting an explanation, for in a warmer world, as inFigure 4(a), the atmosphere should deliver more snow toAntarctica.) In the Amundsen sector of West Antarctica, the

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Figure 7 Major basins of Antarctica. The underlying field is the balance velocity (the column-averaged velocity qout/ρiwhich makes b + qout zero in equation 3) (Reprinted from Rignot and Thomas, 2002. Mass balance of polar ice sheets.Science, 297, 1502–1506. 2002 American Association for the Advancement of Science). A color version of this imageis available at http://www.mrw.interscience.wiley.com/ehs


two methods also agree in finding a substantial negativebalance. Shepherd et al. (2002) measured inland migra-tion of the grounding lines and rapid thinning of PineIsland, Thwaites, and Smith Glaciers (Figure 7). Theyestimated the balances of Thwaites and Smith Glaciersas −22 ± 8 kg m−2 a−1 and – 233 ± 34 kg m−2 a−1 respec-tively for 1991–2001, while Rignot and Thomas (2002)gave −123 ± 92 kg m−2 a−1 and −698 ± 222 kg m−2 a−1.The discrepancies obviously require investigation, but maybe consistent with other evidence suggesting that the thin-ning began rather abruptly at some time during the 1990s.An abrupt onset would invite the speculation that the thin-ning is due to changes in ocean circulation leading toincreased basal melting near the grounding line (Rignot andJacobs, 2002).

The mass balance of the ice shelves is difficult to assess,mainly for lack of complete estimates of its components.The grounding-line flux estimates of Rignot and Thomas(2002) sum to 754 Gt a−1. If they account for the samefraction of shelf nourishment as the fraction of groundedice from which they come, 59.3%, the total grounding-line flux would be 1271 Gt a−1. Vaughan et al. (1999)estimated the surface accumulation on the ice shelves as478 Gt a−1. Several of the most northerly shelves havedisintegrated in recent years, and surface melting has beenimplicated in these events, but quantitatively it makes littlecontribution to the balance; Jacobs et al. (1992) gave acrude estimate of −36 Gt a−1. They also gave −2016 Gt a−1

for the rate of calving, based mainly on iceberg censuses.From oceanographic observations, Jacobs et al. (1996)estimated the basal balance to be −756 Gt a−1. The twoinputs and the three outputs sum to −1059 Gt a−1, or−708 kg m−2 a−1.

Although there are scattered in situ and remotely sensedmeasurements, at present the only realistic way to get abroad view of the basal mass balance of an ice shelfis to model it using an ocean circulation model. Forexample, Williams et al. (2001) simulated the basal balanceof Amery Ice Shelf to be −254 kg m−2 a−1, which issubstantially less negative than the Jacobs estimate of−596 kg m−2 a−1. Nevertheless, the appropriate conclusionfrom the incomplete evidence appears to be that the iceshelves of Antarctica are losing mass at a rate that is veryuncertain.

SUMMARY

Imprecise measurements, with insufficient spatial densityand coverage, are a significant constraint upon what can besaid about glacier mass balance in its global context. Thecost of improving the methods and extending the scope ofthe measurements is high, but so are the probable costs offailure to understand mass and energy exchange betweenglaciers and the rest of the hydrosphere. For this reason,

measurement technology is the subject of vigorous researchand is improving rapidly. In the near future, the massbalance of the two ice sheets will become known with anuncertainty small enough to state with confidence whetherthey are growing or shrinking. At present the weight ofevidence, some of it circumstantial, implies that both havenegative mass balance, but the errors are not adequatelyquantified and for the Antarctic Ice Sheet, in particular,a conclusion about the sign of the mass balance wouldbe premature. For the small glaciers, the picture is moreclear. Their balances are negative on average and have beengrowing more negative since the 1960s, but here also thereis scope for a more thorough assessment of errors.

The best balance estimates, in sea-level equivalents, are:for small glaciers, as calculated above, 0.21 mm s.l.e. a−1

with a poorly quantified uncertainty of the order of 0.06 mms.l.e. a−1; for the Greenland Ice Sheet (Krabill et al., 2000),0.13 mm s.l.e. a−1 with no estimate of uncertainty; and forthe Antarctic Ice Sheet (Rignot and Thomas, 2002; Vaughanet al., 1999), between 0.10 and −0.20 mm s.l.e. a−1 with anuncertainty of at least 0.13 mm s.l.e. a−1. The sum of theseestimates, 0.14 to 0.44 mm s.l.e. a−1, is a small proportionof the contemporary rate of sea-level rise, 1.84–1.91 mms.l.e. a−1 (Peltier, 2001; but see also Miller and Douglas,2004, and references cited therein).

REFERENCES

Abdalati W. and Steffen K. (2001) Greenland ice sheetmelt extent: 1979–1999. Journal of Geophysical Research,106(D24), 33983–33988.

Arendt A.A., Echelmeyer K.A., Harrison W.D., Lingle C.S. andValentine V.B. (2002) Rapid wastage of Alaska glaciers andtheir contribution to rising sea level. Science, 297, 382–386.

Bales R.C., McConnell J.R., Mosley-Thompson E. and Csatho B.(2001) Accumulation over the Greenland ice sheet fromhistorical and recent records. Journal of Geophysical Research,106(D24), 33813–33825.

Bamber J.L., Ekholm S. and Krabill W.B. (2001) A new, high-resolution digital elevation model of Greenland fully validatedwith airborne laser-altimeter data. Journal of GeophysicalResearch, 106(B4), 6733–6745.

Benson C.S. (1959) Physical Investigations on the Snow and Firnof Northwest Greenland 1952, 1953, and 1954 , Research Report26, U.S. Army Corps of Engineers Snow, Ice and PermafrostResearch Establishment: Wilmette, p. 62.

Bhutiyani M.R. (1999) Mass-balance studies on Siachen glacierin the Nubra valley, Karakoram Himalaya, India. Journal ofGlaciology, 45(149), 112–118.

Bigg G. (1999) An estimate of the flux of iceberg calvingfrom Greenland. Arctic, Antarctic and Alpine Research, 31(2),174–178.

Bintanja R. (1999) On the glaciological, meteorological andclimatological significance of Antarctic blue ice areas. Reviewsof Geophysics, 37(3), 337–359.


Bintanja R. (1998) The contribution of snowdrift sublimation tothe surface mass balance of Antarctica. Annals of Glaciology,27, 251–259.

Bintanja R. and Reijmer C.H. (2001) Meteorological conditionsover Antarctic blue-ice areas and their influence on the localsurface mass balance. Journal of Glaciology, 47(156), 37–50.

Braithwaite R.J. (1984) Can the mass balance of a glacierbe estimated from its equilibrium-line altitude? Journal ofGlaciology, 30(106), 364–368.

Braithwaite R.J. and Zhang Y. (1999) Relationships betweeninterannual variability of glacier mass balance and climate.Journal of Glaciology, 45(151), 456–462.

Calanca P., Gilgen H., Ekholm S. and Ohmura A. (2000) Griddedtemperature and accumulation distributions for Greenland foruse in cryospheric models. Annals of Glaciology, 31, 118–120.

Chinn T.J. (1999) New Zealand glacier response to climate changeof the past 2 decades. Global and Planetary Change, 22,155–168.

Church J.A., Gregory J.M., Huybrechts P., Kuhn M., Lambeck K.,Nhuan M.T., Qin D. and Woodworth P. (2001) Changesin sea level. In Climate Change 2001: The Scientific Basis,Houghton J.R., Ding Y., Griggs D.J., Noguer M., van derLinden P.J., Dai X., Maskell K. and Johnson C.A. (Eds.),Cambridge University Press: New York, pp. 639–693.

Cogley J.G. (1999) Effective sample size for glacier mass balance.Geografiska Annaler, 81A(4), 497–507.

Cogley J.G. (2004) Greenland accumulation: an error model. Jour-nal of Geophysical Research, 109(D18), D18101, doi:10.1029/2003JD004449.

Cogley J.G., Ecclestone M.A. and Andersen D.T. (2001) Meltingon glaciers: environmental controls examined with orbitingradar. Hydrological Processes, 15, 3541–3558.

Cogley J.G. and Adams W.P. (1998) Mass balance of glaciersother than the ice sheets. Journal of Glaciology, 44(147),315–325.

Cogley J.G., Adams W.P., Ecclestone M.A., Jung-Rothen-hausler F. and Ommanney C.S.L. (1996) Mass balance ofWhite Glacier, Axel Heiberg Island, N.W.T., Canada, 1960–91.Journal of Glaciology, 42, 548–563.

Davis C.H., Belu R.G. and Feng G. (2001) Elevation-changemeasurement of the East Antarctic Ice Sheet, 1978 to 1988,from satellite radar altimetry. IEEE Transactions on Geoscienceand Remote Sensing, 39(3), 635–644.

Davis C.H., Kluever C.A., Haines B.J., Perez C. and YoonY.T. (2000) Improved elevation-change measurement of thesouthern Greenland Ice Sheet from satellite radar altimetry.IEEE Transactions on Geoscience and Remote Sensing, 38(3),1367–1378.

Demuth M.N. and Pietroniro A. (1999) Inferring glacier massbalance using RADARSAT: results from Peyto glacier, Canada.Geografiska Annaler, 81A(4), 521–540.

Doake C.S.M. (1976) Thermodynamics of the interaction betweenice shelves and the sea. Polar Record, 18(112), 37–41.

Drinkwater M.R., Long D.G. and Bingham A.W. (2001)Greenland snow accumulation estimates from satellite radarscatterometer data. Journal of Geophysical Research, 106(D24),33935–33950.

Dyurgerov M.B. (2002) Glacier Mass Balance and Regime: Dataof Measurements and Analysis, Occasional Paper 55, Institute of

Arctic and Alpine Research, University of Colorado: Boulder,p. 87, 4 Appendices and CD-ROM.

Giovinetto M.B. and Zwally H.J. (2000) Spatial distribution ofnet surface accumulation on the Antarctic ice sheet. Annals ofGlaciology, 31, 171–178.

Gray A.L., Short N., Mattar K.E. and Jezek K.C. (2001) Velocitiesand ice flux of the Filchner Ice Shelf and its tributariesdetermined from speckle tracking interferometry. CanadianJournal of Remote Sensing, 27(3), 193–206.

Haeberli W., Hoelzle M., Suter S. and Frauenfelder R. (1998)Fluctuations of Glaciers 1990–1995, Vol. VII, InternationalCommission on Snow and Ice of International Association ofHydrological Sciences/UNESCO: Paris.

Haeberli W., Frauenfelder R. and Hoelzle M. (2001) Glacier MassBalance Bulletin No. 6 (1998–1999), International Commissionon Snow and Ice of International Association of HydrologicalSciences/UNESCO: Paris.

Holland D.M. and Jenkins A. (1999) Modeling thermodynamicice-ocean interactions at the base of an ice shelf. Journal ofPhysical Oceanography, 29(8), 1787–1800.

Hubbard A., Willis I., Sharp M., Mair D., Nienow P., Hubbard B.and Blatter H. (2000) Glacier mass balance determined byremote sensing and high-resolution modelling. Journal ofGlaciology, 46(154), 491–498.

Jacobs S.S., Hellmer H.H. and Jenkins A. (1996) Antarctic icesheet melting in the southeast pacific. Geophysical ResearchLetters, 23(9), 957–960.

Jacobs S.S., Hellmer H.H., Doake C.S.M., Jenkins A. and FrolichR.M. (1992) Melting of ice shelves and the mass balance ofAntarctica. Journal of Glaciology, 38(130), 375–387.

Jezek K.C. (2002) RADARSAT-1 Antarctic Mapping Project:change-detection and surface velocity campaign. Annals ofGlaciology, 34, 263–268.

Jones P.D. and Moberg A. (2003) Hemispheric and large-scalesurface air temperature variations: extensive revisions and anupdate to 2001. Journal of Climate, 16(2), 206–223.

Kostecka J.M. and Whillans I.M. (1988) Mass balance along twotransects of the west side of the Greenland Ice Sheet. Journalof Glaciology, 34(116), 31–39.

Krabill W., Abdalati W., Frederick E., Manizade S., Martin C.,Sonntag J., Swift R., Thomas R., Wright W. and Yungel J.(2000) Greenland Ice Sheet: high-elevation balance andperipheral thinning. Science, 289, 428–430.

Krabill W.B., Thomas R.H., Martin C.F., Swift R.N. and FrederickE.B. (1995) Accuracy of airborne laser altimetry over theGreenland Ice Sheet. International Journal of Remote Sensing,16(7), 1211–1222.

Madsen S.N. and Zebker H.A. (1998) Imaging radarinterferometry. In Principles and Applications of ImagingRadar, Manual of Remote Sensing, Third Edition , Vol. 2,Henderson F.M. and Lewis A.J. (Eds.), John Wiley: New York,pp. 359–380.

Matzler C. (1987) Applications of the interaction of microwaveswith the natural snow cover. Remote Sensing Reviews, 2,259–387.

Mayo L.R. (1992) Internal ablation – an overlooked componentof glacier mass balance. Eos, 73(43), 180, (abstract).

McConnell J.R., Lamorey G., Hanna E., Mosley-Thompson E.,Bales R.C., Belle-Oudry D. and Kyne J.D. (2001) Annual


net snow accumulation over southern Greenland from1975 to 1998. Journal of Geophysical Research, 106(D24),33827–33837.

Miller L. and Douglas B.C. (2004) Mass and volume contributionsto twentieth-century global sea-level rise. Nature, 428,406–409.

Mote, T.L. (2003) Estimation of runoff rates, mass balance, andelevation changes on the Greenland Ice Sheet from passive-microwave observations. Journal of Geophysical Research,108(D2), 4056, doi:10.1029/2001JD002032.

Muller F. (1962) Zonation in the accumulation area of the glaciersof Axel Heiberg Island, N.W.T., Canada. Journal of Glaciology,4(33), 302–313.

Munk W. (2003) Ocean freshening, sea level rising. Science, 300,2041–2043.

Munro D.S. (2001) Linking the Weather to Glacier Hydrology andMass Balance at Peyto Glacier. In Science Report 8, NationalHydrology Research Institute, Environment Canada: Saskatoon,pp. 135–175.

Nakawo M., Raymond C.F. and Fountain A. (Eds.) (2000) Debris-covered glaciers. International Association of HydrologicalSciences Publications, Vol. 264, IAHS Press: Wallingford, p.288.

Nghiem S.V., Steffen K., Kwok R. and Tsai W.-Y. (2001)Detection of snowmelt regions on the Greenland Ice Sheet usingdiurnal backscatter change. Journal of Glaciology, 47(159),539–547.

Oerlemans J. (2001) Glaciers and Climate Change, Balkema:Lisse, p. 148.

Ohmura A., Calanca P., Wild M. and Anklin M. (1999)Precipitation, accumulation and mass balance of the GreenlandIce Sheet. Zeitschrift fur Gletscherkunde und Glazialgeologie,35, 1–20.

Ohmura A., Konzelmann T., Rotach M., Forrer J., Wild M.,Abe-Ouchi A. and Toritani H. (1994) Energy balance for theGreenland Ice Sheet by observation and model computation.International Association of Hydrological Sciences Publica-tions, 223, 85–94.

Oke T.R. (1987) Boundary Layer Climates, Second Edition ,Routledge: New York, p. 435.

Østrem G. and Brugman M.M. (1991) Glacier Mass-BalanceMeasurements: A Manual for Field and Office Work , ScienceReport 4, National Hydrology Research Institute, EnvironmentCanada: Saskatoon, p. 224.

Palli A., Kohler J.C., Isaksson E., Moore J.C., Pinglot J.F., PohjolaV.A. and Samuelsson H. (2002) Spatial and temporal variabilityof snow accumulation using ground-penetrating radar and icecores on a Svalbard glacier. Journal of Glaciology, 48(162),417–424.

Paterson W.S.B. (1994) The Physics of Glaciers, Third Edition ,Elsevier Science: Tarrytown, p. 480.

Peltier W.R. (2001) Global glacial isostatic adjustment andmodern instrumental records of relative sea level history. InSea-Level Rise, Douglas B.C., Kearney M.S. and LeathermanS.P. (Eds.), Academic Press: San Diego, pp. 65–95.

Peltier W.R. (1998) Postglacial variations in the level of the sea:implications for climate dynamics and solid-Earth geophysics.Reviews of Geophysics, 36(4), 603–689.

Pollack H.N., Hurter S.J. and Johnson J.R. (1993) Heat flowfrom the earth’s interior: analysis of the global data set.Reviews of Geophysics, 31(3), 267–280, [Data set availablefrom http://www.heatflow.und.edu/data.html.].

Ramage J.M. and Isacks B.L. (2003) Interannual variations ofsnowmelt and refreeze timing on southeast-Alaskan icefields,U.S.A. Journal of Glaciology, 49(164), 102–116.

Reijmer C.H. and Oerlemans J. (2002) Temporal and spatialvariability of the surface energy balance in Dronning MaudLand, East Antarctica. Journal of Geophysical Research,107(D24), 4759, doi:10.1029/2000JD000110.

Richardson S.D. and Reynolds J.M. (2000) An overview ofglacial hazards in the Himalayas. Quaternary International, 65,31–47.

Rignot E. and Jacobs S.S. (2002) Rapid bottom meltingwidespread near Antarctic ice sheet grounding lines. Science,296, 2020–2023.

Rignot E. and Thomas R.H. (2002) Mass balance of polar icesheets. Science, 297, 1502–1506.

Rignot E., Gogineni S., Joughin I. and Krabill W. (2001)Contribution to the glaciology of northern Greenland fromsatellite radar interferometry. Journal of Geophysical Research,106(D24), 34007–34019.

Rignot E.J., Gogineni S.P., Krabill W.B. and Ekholm S. (1997)North and north-east Greenland ice discharge from satelliteradar interferometry. Science, 276(5314), 934–937.

Selters A. (1999) Glacier Travel and Crevasse Rescue, SecondEdition , The Mountaineers: Seattle, p. 143.

Shepherd A., Wingham D.J. and Mansley J.A.D. (2002) Inlandthinning of the Amundsen Sea sector, West Antarctica. Geo-physical Research Letters, 29(10), doi:10.1029/2001GL014183.

Shumskiy P.A. (1955) Osnovy Strukturnogo Ledovedeniya,Izdatel’stvo Akademiy Nauk SSSR: Moscow, p. 492;Translated by Kraus D. (Ed.) (1964) as Principles of StructuralGlaciology, Dover: New York, p. 497.

Siegert M.J. (2003) Glacial-interglacial variations in central EastAntarctic ice accumulation rates. Quaternary Science Reviews,22(5–7), 741–750, doi:10.1016/S0277-3791(02)00191-9.

Slupetzky H. (1989) Die massenbilanzmessreihe vom StubacherSonnblickkees 1958/59 bis 1987/88. Zeitschrift fur Gletscher-kunde und Glazialgeologie, 25, 69–89.

Sohn H.-G., Jezek K.C. and van der Veen C.J. (1998) JakobshavnGlacier, west Greenland: 30 years of spaceborne observations.Geophysical Research Letters, 25(14), 2699–2702.

Su Z. and Shi Y. (2002) Response of monsoonal temperateglaciers to global warming since the Little Ice Age. QuaternaryInternational, 97, 123–131.

Thomas R.H., Csatho B., Davis C., Kim C., Krabill W.,Manizade S., McConnell J. and Sonntag J. (2001) Mass balanceof higher-elevation parts of the Greenland Ice Sheet. Journalof Geophysical Research, 106(D24), 33707–33716.

Trabant D.C. and March R.S. (1999) Mass-balance measurementsin Alaska and suggestions for simplified observation programs.Geografiska Annaler, 81A(4), 777–789.

Turner J., Lachlan-Cope T.A., Marshall G.J., Morris E.M., Mul-vaney R. and Winter W. (2002) Spatial variability of AntarcticPeninsula net surface mass balance. Journal of GeophysicalResearch, 107(D13), 4173, doi:10.1029/2001JD000755.


van den Broeke M.R. (1997) Structure and diurnal variation of theatmospheric boundary layer over a mid-latitude glacier duringsummer. Boundary-Layer Meteorology, 83(2), 183–205.

Vaughan D.G., Bamber J.L., Giovinetto M., Russell J. and CooperA.P.R. (1999) Reassessment of net surface mass balance inAntarctica. Journal of Climate, 12(4), 933–946.

Velicogna I. and Wahr J. (2002) A method for separating Antarcticpostglacial rebound and ice mass balance using future ICESatGeoscience Laser Altimeter, Gravity Recovery and ClimateExperiment, and GPS satellite data. Journal of GeophysicalResearch, 107(B10), 2263, doi:10.1029/2001JB000708.

Wagnon P., Sicart J.-E., Berthier E. and Chazarin J.-P.(2003) Wintertime high-altitude surface energy balance of aBolivian glacier, Illimani, 6340 m above sea level. Journal ofGeophysical Research, 108(D6), doi:10.1029/2002JD002088.

Warren S.G. (1982) Optical properties of snow. Reviews ofGeophysics and Space Physics, 20, 67–89.

Wild M., Calanca P., Scherrer S.C. and Ohmura A. (2003)Effects of polar ice sheets on global sea level in high-resolution greenhouse scenarios. Journal of GeophysicalResearch, 108(D5), doi:10.1029/2002JD002451.

Williams M.J.M., Grosfeld K., Warner R.C., Gerdes R.and Determann J. (2001) Ocean circulation and ice-ocean

interaction beneath the Amery Ice Shelf, Antarctica. Journalof Geophysical Research, 106(C10), 22383–22399.

Wingham D.J. (2000) Small fluctuations in the density andthickness of a dry firn column. Journal of Glaciology, 46(154),399–411.

Wingham D.J., Ridout A.J., Scharroo R., Arthern R.J. and ShumC.K. (1998) Antarctic elevation change from 1992 to 1996.Science, 282, 456–458.

Winther J.-G., Jespersen M.N. and Liston G.E. (2001) Blue-iceareas in Antarctica derived from NOAA AVHRR satellite data.Journal of Glaciology, 47(157), 325–333.

Wismann V. (2000) Monitoring of seasonal snowmelt onGreenland with ERS scatterometer data. IEEE Transactions onGeoscience and Remote Sensing, 38(4), 1821–1826.

Zwally H.J. (1977) Microwave emissivity and accumulation rateof polar firn. Journal of Glaciology, 18(79), 195–215.

Zwally H.J. and Giovinetto M.B. (2000) Spatial distribution ofnet surface mass balance on Greenland. Annals of Glaciology,31, 126–132.

Zwally H.J. and Li J. (2002) Seasonal and interannualvariations of firn densification and ice-sheet surface elevationat the Greenland summit. Journal of Glaciology, 48(161),199–207.

Documents

165: Mass and Energy Balances of Glaciers and Ice Sheets · through snow and ice that persists on the Earth’s surface for more than one year (Figure 1c). The two largest glaciers