J ou rnal of Glaciolog)), Vol. 42, No . 142, 1996
Tidal motion, ice velocity and melt rate of Petermann Glets cher, Greenland, measured frotn radar interferotnetry
ERIC RIGNOT J et ProjJu/sion Laborat01Y, California Institute oJ T echnology, Pasadena, Califomia 91109, U. S.A.
ABSTRACT. O ver a fl oating glacier ice tongue or an ice sh elf, th e glacier motion meas ured b y a single, rep eat-pass, rad ar interferogram is difIicult to analyze, because the long-term, steady mo ti on of the ice is intermixed with its cycli c, downwa rd motion induced b y tidal forcing . Multiple interferograms a nd a quadruple-difference technique are necessary to separate the tidal signa l from th e long-term, steady motion of the ice. An example of application of this techniq ue is given here using ERS- 1 rad ar images ofPeterma nn Gletsc hel-, a m ajor outlet gla cier of northern Greenland. Tida l displacements a re m easured with < 5 mm sta ti sti cal noise . The long-term ice velocity is measured with a precision of Im a I. The inferred tidal displacem ents agree well with nl.odel predicti o ns from a fix ed elasti c beam with an elas ti c damping factor of 0.47 ± 0.01 km I. The hinge line is mapped with a precision of20- 80m. Combining th e interferometric ice velocities with ice thickn ess data, the glacier ice discha rge is calcula ted a t a nd below the hinge line. At the hinge line, the ice flu x is 12.1 ± I km3 a I. At the iee front, calf-ice production is only 0.59 km 3 a I, meaning tha t 95% of th e ice tha t crosses the grounding line melts before it reaches the calving fro nt. Ass uming stead y-sta te conditions, the melt rate of the glacier tong ue averages 12 ± I m a- I, wi th peak \'alues exceeding 20 m a- I near the hinge line. This high melt ra te cannot be accommoda ted by surface ablation alone (only about 2- 3 m a I) and is attri bu ted to pronounced basal melting of the ice tongue. Basal melting, ofte n ass umed to be negli gible in Greenla nd, is th e domina nt process of mass release from the flo a ting section of P etermann Gletsc her.
INTRODUCTION 500 m. Because both the tida l displacements a nd the ice velocity con tribu te to the observed range displacements Calving glaciers play a n essential role in the dynamics in the ERS interferog r a m , they had to use an and m ass balance of the Greenl and ice sheet, and even independen t es timate of the ice velocity to locate th e more so in the case of the Antarctic ice sh eet where ice zone of fl exing of the glacier precisely in the interfero sheh 'es and fl oating glacier ice to ngues d evelop ex ten gram. Using two interferograms, H art! and others sively (H oldsworth, 1977; Dre\\T)' a nd R obin, 1983; ( 1994) dem o nstra ted that the tidal motion of an ice Va ughan a nd Doake, 1996). Of particula r interes t for shelf could be isolated from the res t of the sig na l. The studies of the stabili ty of these glaciers is the region a t the basic premise for the method is to ass um e that over the juncti on between slower-moving inland ice where no tidal time-scal e consid ered for repeat-pass interferometric displacemen ts occur a nd faster-moving ice comprising the applicatio ns (several days) the ice motion other than ice tong ue or ice shelf where tidal forcing introduces a that due to tidal for cing is steady and continuous and cyclic, vertical moti on o f the ice surface. The grounding therefore may be eliminated by differencing of two line, where the ice decouples from the glacier bed a nd success ive radar interferograms. becom es a fl oat, is importa nt to locate precise ly, because it In this work, the same premise is used, but the provides a reference for moni to ring ch a nges in ice additional effect of surface topography is incorporated in thickness or sea level induced by clima tic cha nge the ana lysis. In addition, I propose a m e thod to (Thom as a nd Bentley, 1978). subsequently eliminate the tidal signal from individual Tida l ice-shelf fl exure m ay be measured by tiltmeters radar interferograms, so that it is also possible to es timate (Smith, 199 1) or preci e g lobal positioning system (GPS) the long-term steady motion of th e ice. I apply this kinem a tic surveys (Vaugh a n, 1995 ) . L ocating the quadruple-difference interferometry technique to ERS grou nding li ne is more difficu 1t by tra di ti onal s ta n radar images of Petermann Gletsc her, a major outlet dards, a nd there are n o systema ti c means of mapping its glacier of northern Greenla nd, which has an extensive pos ition using a rem o te-sensing instrument. Goldstein fl oating ice tongue confined within a Gord. umerous and o thers (1993) d etected the tida l m otion of the ro ck outcrops are present at the margins of the glacier to Rutford I ce Stream using a single ERS-l interferogra m , provide a reliable, fixed reference for estimating the a nd d ed uced the posi ti on of the grounding line wi thin interferome tric baselines, georeferencing the data and
476 R igll ot: Tidal motion, ice l'elociO' alld melt rate oJ Peter/nalln Cletsclzer studying glacial motion. Data cO\'e rage of that part of correlation or phase coh eren ce of the cross-products, Greenland by th e ERS-I radar sys tem has been excellen t, denoted p, ta king values between 0 (no co herence) and I and S. Ekholm and R. Forsberg of KMS (Kort and (perfect temporal coherence), is hi gh (p > 0.8) over most Matrikelstyrelse n) have produced a precise topographic of the scene, yielding high-q u a li ty interferometric fringes map of the area. The objectiHs of this study were to ( Fi g. 2). Phase unwra pping was performed using utilize ERS-I radar interferometry data to meas ure the Goldstein and others' (1988) unwrapping algorithm tid al displacements of the ice tongue, m a p the grounding a ft er smoothi ng of the d a ta using a two-dimensional line of the glacier and study its ice di sc harge at and below spectral filter. th e grounding line. A companion stud y by Joughin and Upstream of the grounding line, the interferograms others ( 1995a) examined the ice velocity of Petermann ex hibit a complex pattern of closely spaced fringes (360° Gletscher hi gher up in its accumulation area. "ariations in phase) with pairs of concentri c circles " 'here phase unwrapping is diffi cult to perform. A similar fringe pattern is seen in radar interferograms of the so uth STUDY AREA western fl a nk of the Greenland ice sheet. The pairs of concentri c circles are attributed to variations in the Peterma nn G letsc her is located 60° Wand 81 ° N, on the vertical com ponent of the ice-motion vector as ice fl ows northwes tern fl ank of the Greenl and ice sheet (Higgins, past bumps a nd hollows in surface topography, several 1991 , figs I and 2) . P etermann Gletscher was first meters in height and several kilometers in diameter, documented and examined during the American Polaris created by fa ster ice-sheet now over the bedrock ex pedition under C. F. Hall in 1871 (Kollmeyer, 1980) . topography near the ice margin Uoughin and oth ers, I t is one of th e few Arctic glaciers which develops a n 1995b; Rignot a nd others, 1995) . r n those regions of more ex tensive floating ice tongue. lts terminus, only 3--4 m rapidl y varying surface slope, phase coherence is reduced a.s.l. , occasiona ll y disintegrates to yield tabular icebergs compared to that of the surrounding ice, because the 30- 50 m thick, up to 10km x 12km in area (Dun bar, 1978; Kollmeyer, 1980) . P eterm ann G letscher has the hi ghes t m easured veloci tv of a northern Greenland glacier, a bout 0.95 km a- I at th e ice front (Higgins, 1991 ). This study utilizes three consecutive passes of the ERS- I satellite acquired on 25 and 28 F ebruary and 2 M arch 1992, during orbits 3205, 3248 a nd 3291. Each rada r scene is a 100 km x lOO km fr a m e, with a 20 m pixel spacing on th e ground after averaging of 5 pixel elements in th e azimuth (or a long-track) direction. In th e rada r imagery (Fig. I), the shear ma rg ins of the glacier are pronounced, and extend far southward into the in land ice. Most of the glacier surface within th e Gord is radar-dark, indica ting a surface poorl y refl ec tive of ERS radar signa ls. The rad a r-bright region to the so uth ma rks the ed ge of th e percolation facies which is radar-brigh t because of internal refl ections in subsurface icy inclusions (Rigno t a nd others, 1993; Rignot, 1995). Fi"e glaciers descend on th e east sid e from Kane Pl a teau to merge with the ma in ice tream, the most important o ne being Porsild Gl etscher (Higgins, 1991 ) .
METHODS
InterferograDl generation
R ead ers interes ted in background info rmation on rad a r interfero m etry may consult Zebker a nd Goldstein (1986), Goldstein and others (1988 ), Gabriel and others (1989) a nd Zebker and others ( 1994). The basic principles of radar interferometry will not be repeated Fig . I. ERS-I radar amjJlilude image oJ Petemwnll here. Cletscizer , 60 km b)' 100 kill ill size, acqllired on 25 Two interferograms were formed using image 2 Febnwl)' 1.9.92. North is ujJwards, ERS-I is j7)'ing Jrolll (orbit 3248) as the r eference image. The complex ea si 10 west, Looking norlh Lo its right . The whiLe amplitude rada r images were fir t co-registered with continuolls lille locates the tidal jJrlUlle shown in Figure 6. sub-pixel accuracy, including additi o n a l pixel omets Tlte doLLed white lille rejJresents th e d),llamic cen Lerline rif over the fa t-moving p a rt of th e ice. The registered Lite glacier. Thejlow directioll is indicated scizelll(l tica!(v b)' Im ages were then cross-correlated. The normalized (IITOWJ . • \ 'orlll is indicaLed b)' all arrow . .[) ESA 1.992.
477 J ournal of Glaciology .,.,--~--..., difference, ifJl2 = CP2 - ifJl' measured between antennae (orbit 3205) and 2 (3248), may then be expressed as
47r [ . Bl22 ] ifJl2 = -:\ Bl21. sm (8Bz ) - Bl211 cos (8e z) + 2R
47r + -:\ [- Vx sin (i) + Vz cos (i)]( t2 - tl )
+ ~ (Z2 - Zl) cos (i) + CP l2 ° (1)
where B l2 is the baseline or distance se pa rating antennae I and 2, Bz is th e ill umina tion angle with the hori zo ntal for a point at elevati on z, 8Bz is equal to Bz - Ba, Ba is th e ill umina ti on angle with th e horizo ntal a t the center of th e scene for a point at a reference elevati on z = 0, IYl2 is th e baseline a ngle with the horizo nta l, B 12.l. = B 12 sin (Ba + IY l2) is the component of the baseline perpendicu lar to the direc ti on of the radar ill umina tion, B 1211 = B1 2 cos(Bo + IYl 2) is the component of the baseline parallel to the direction of the rad a r ill umination, (t2 - t l) is the time lag between th e two images, i is the local in cid ence angle of the radar ill umina tion with the vertical, and CPl2 ° is an a bsolute phase offset. The first line of Equa tion (I) depends onl y on the glacier to pography and is scaled by both the baselin e and th e rad a r wavel ength . The second line is the term of ice motion a long the rada r lin e of sigh t caused by th e steady moti on of th e ice. The component Vy is abse nt from Equation ( I) because the y axis is parallel to th e fl ight Fig. 2. Flattened inteiferogram combining orbits 3205 and direction, a nd surface disp lacements are no t measured in 3248, with a color intensity modulated by phase coherence . th at direction. T he third line corres ponds to changes in Dark areas indicate low phase coherence . surface elevation along the radar line of sight caused by the downwa rd moti on of the ice under tida l forcing. With a second interferogram combining images 2 and 3, I illumina tion angle of the ice blocks changes slightly as obtain they move pas t the bumps and hollows in surface topogra phy. 47r [ . B 322 ] ifJ32 = -:\ B 321. sm (8Bz) - B32 11 cos (8Bz) + 2R The base line or distance se para ti on between the successive positions of the ERS rad ar antenna was 47r es tima ted by least-square fitting using 1400 ti e-poin ts + -:\ [-Vx sin(i) + Vz cos(i)](t 2 - t3) selected fr om a digital elevation mod el (DEM) of the 47r . glacier, 0.0050 in latitude spac ing and 0.025 0 in longitude + -:\ (Z2 - Z3) cos(i) + ifJ32° (2) spacing, provided by S. Ekholm and R. Forsberg ofKMS and referenced to sea level. The KMS DEM da ta were with a different baseline separati on, B32 , angle, 1Y32, and projected into the radar-imaging geometry, interpola ted rela tive tidal displacement, Z2 - Z3; but with the same using a bilinear interpolati on, and registered to the rad a r ice mo ti on vector, Vx a nd 11". scene within 1- 2 image pixels using on e ti e-point. If rad a r scenes I, 2 a nd 3 are acquired in seq uence and exactly 3 d apart, adding Equati ons (I ) a nd (2) elimin Interferometric products ates the term of steady ice moti on 47r I d enote Vx, Vy , 11" the components of the steady-motion ifJ12 + ifJ32 =-:\ [( Bm + B 32.l.) sin(8Bz) vector of th e ice along the x , y a nd z axes. The term "stead y" here refers to the ice motion over a time-scale B122 + B322 - (B 12 11 + B 3211) cos(8Bz) + 2R much large r than the tida l cycle. The x axis is in the cross track directi on, pointing north. The y axis is in the alo ng + [( 2Z2 - Z3 - Z1) cos(i)] + ifJ12° + ifJ32° .(3) track direc ti on, pointing wes t. The z axis is the vertical axis. U nder tidal influence , the ice tongue undergoes Using ti e-points fr om the KMS DEM o n both rock and upward and downward motion a long the z axis of ice, a t the exclusion of the floa ting sec ti on , I es timate th e a mplitude Z . I use the sign convention that the phase, baseline parameters of Equati on (3) and remove the ifJ, m easured by th e rada r is equal to -47rR / )" , where R is to pogra phy term to obtain the range distance between a point a t the surface of the 47r glacier and the center of th e sy nthetic a perture, and)" is the ifJ12,flat + ifJ23,flat = -:\ (2Z2 - Z3 - Z l) cos (i) (4) rad ar wavelength (5.66 cm for ER S- I radar). The phase
478 R ignot: Tidal motion, ice velocity and melt rate oJ Petermann Cletselzer which depends only on th e tidal displacements. The kn own. To determine ')'12, either th e tida l a mplitude or subscript " fl at" design a tes a phase value for whi ch the the ice velocity must be known at one location, otherwise effect of th e baseline and of surface to pography has been th ere is a n infinite number of solutions for ')'12. removed. The map of the relati ve tid al displacement, Here, I es timated 11 control ve locities by tracking a Z2 - O.5(Z3 - Zl), between sce ne 2 a nd the average of se t of crevasses below the grounding lin e in two ERS-l scenes 1 and 3, is shown in Fi gure 3. radar images se parated by 1 year. The rms error in th e Model predictions from th e elastic-beam theory veloc ity es timates is 50 m a- I. The leas t-square estimate of indicate that tidal displacements a t a given point along ')'12 is 1.8 ± 0.2. a n elastic beam vary linearly with the tidal amplitude. The x velocity, Vx, is d educed fr om Equation (5) Several ex perimental studies have shown tha t th e elastic ass uming' Vz: = O. In effect, the vertical motion associated beam model ma tches obsen'ations of tidal displacem ents with glacier thinning is negligible compa red to the "vell (H old wo rth, 1969). rf we assume that tidal forcing hori zonta l mo ti on, and ice fl ows nearl y in the horizontal is th e same everywhere along the beam as in plane since the glacier slope is less than I % . The x Holdsworth's (1969) study and tha t the elas ti c damping veloci ty was subseq uen tl y transformed in to a two factor of the ice does not cha nge wi th tidal amplitude dimensional velocity by assuming a fl ow direction '(Holdsworth, 1977), a differen t reali zation of tidal para ll el to the dynamic center line o f the glacier forcing should exhibit th e sam e pa ttern of tidal (dotted line in Figure I ). The dyna mi c cen ter line was displacement as that given in Equatio n (4), scaled b y a drawn based on intense surface crevassing at the center different relative tidal amplitude. Under these circum of th e fl oating part of the glacier and using the line of stances, I rew rite Equa tion (1) a ft er removal of the maximum x velocity on grounded ice. The res ult is topography term as shown in Figure 4. Once the tidal amplitude is kn ow n, it is a lso poss ible to 47r Fig. 3. T idal disjJlacements (color-coded between - 50 and 180 mm and modulated by the radar brightness Jor display Fig. 4. Ice velocity oJ Petermann Cletseher, calor-coded purposes), and hinge line (con tinuolls wllite line) oJ between 0 and 1200 m a- /, and modulated b)1 the radar Petermamz Cletscller . D ark pa tches indicate areas with no brightness. Dark patches indicate areas with no intelJero intelJeromet rie data. metric data. 479 Joumal of Glaciology probably not due to baseline uncertainties, since its magnitude does not increase with surface elevation. It i probably due to a combina ti on of atmospheri c and ionos pheri c propagati on d elays (Goldstein, 1995), and surface effects in cluding for insta nce a cha nge in snow thi ckn ess of th e ice caps. T o average out these errors, additional interferograms a re necessary. RESULTS Tidal dis placements The pattern of relati ve tida l displ acements (Fig. 3) delin eates the part of the glacier th at is afloa t. Nea rl y the entire section of th e glacier below the grounding line undergoes tida l moti on, with a sharp discontinuity between the rock ma rgin a nd th e ice tongue. This obse rvati on suggests th ere is little mechanical coupling between the ice tongue a nd the rock margin. The tid al displacements increase rapidly from zero to a maximum valu e about 6 km downstream , a nd subsequently decrease slowly toward an asymptotic valu e. On the eastern sid e of the glacier, where Porsild Gl etscher merges with the m a in stream of Peterm ann Gl etscher, the pattern of tida l displ acement is more co mplex . Phase unwra pping fail ed a t the junction between the two glaciers, bu t Porsild Gletsch er is likely to undergo tidal moti on as well. The pa ttern of tid al moti on proba bl y refl ec ts the interplay of the grounding Fig. 5. Sll1Jace topograj)hy of Petemzann Gletsclzer color zones from both glaciers. coded between 0 and 900 m. Dark patches indicate areas T o explain the pattern of tidal motion d erived from with no interferometric data . th e interferometric da ta a nd to characterize the fl ex ura l ri gidity of the ice, I compa red a tid al profile ex tracted along the wes tern half of the ice tongue (Fig. I) with Measurement uncertainties model predictions from an elasti c beam of infinite length, with one end ri gidly cl a mped on bedrock (Holdsworth, The fins error o[ the phase values calculated during the 1977 ). The predicted tida l a mplitude at time t and baseline estima ti on process was 3 rad fo r pair 3248- 3205 abscissa x along th e beam is (Equa tion (1)) , 0.6 for pair 3248- 3291 (Equati on (2)) and 5 [or the two pairs combined (Equa ti on (3)) . These Zt,x = Zt{l - e-,BX [cos (/3x) + sin (f3x)]} (6) phase errors transla te in to uncertainties in surface topogra phy 0[, respecti vely, 75 , 400 a nd 100 m. The where Zt is the asympto tic value of th e tida l displacement errors a re large because the perpendicul a r baselines a re at time t referenced to mean sea level , and f3 is the elas ti c short (respecti vely 58, 2 and 60 m). The velocity errors damping factor of th e ice given by a re conversely small because th ey do no t depend on the baseline separation a nd are, res pec ti vely, equal to 4 a nd (7) I mm d- I [or the individua l pairs, which means about 1 m a I uncertainty in ice velocity; and 10 mm for the two where E is Young's elasti c modulus of ice, Pw = pairs combined, which means 5 mm uncertainty in 103 0 kg m 3 is the densi ty of sea water, 9 = 9.81 m S- 2 is re lative tid al displacement. These errors are, res pec th e acceleration of gravity, 1/ = 0.3 is the Poisso n tive ly, three orders of magnitude less than the velocity of coeffi cient fo r ice, and h is the glacier thickness. The the glacier (1000 m a- I) a nd two orders o[ magnitude less bes tfitisobtainedfor /3 = (4 .7 ± 0.1 ) x 10-4 m- I, with tha n the largest rela tive tida l displacement which can be a rms fit error ofO.8mm (Fig. 6). recorded over a complete cycle (800 mm; see below). Judging from the low rms error and the high number Errors in tid al displacement are visible in several a reas of points used in th e co mpa rison, th e model predicti ons fit ou tside of th e main ice stream of Peterma nn Gletsc her. the measurements very well a nd expl ain the pattern of For instance, in the center top of the scene, running tid al displacements measured by radar interferometry. a lm ost eas t- west, a 10 km wid e segment shows a rela ti ve The inferred value of /3 is in reasonabl e agreement with tidal displ acement of - 10 to - 50 mm (colored white in the curve of Vaughan ( 1995) relating /3 to th e ice Fi gure 3) outside of the glacier a rea. This anoma ly thi ckness a t the grounding line. T o obtain a measurement coi ncides wi th the ice-covered areas of VVashington La nd point lying exactl y on his curve, th e glacier thickn ess and K a ne Plateau (see Higgins, 199 1, fi g. 2). This error is wo uld have to be 863 m, or 288 m thicker than th e ice 480 Rigllot: Tidal 1Il0tiOIl, ice velocil)' alld meli rate oj Petermallll Cletseher 200 the hinge lin e can be detected w ithin 1 pi xel or 20 m. 1 ear the center of the glacier, tb e precision is less, because -s the !tinge line shifts in th e cross-track direc ti on by several S 150 pixels over a n across -O o\\' distance of about 500 m , and .,.J the tidal profil e no longer ex hibits a sharp minimum. At ~ the glacier ma rgin, the hinge line cannot be detected Q) 100 S accurately, sin ce the fringe ra te is too hi gh, phase Q) C,) coh e re nce is much lower and the phase \'alu es cannot .....(tl be unwrapped . ~ 50 ...en The achi eved mapping precision still remains more "C than one order of magnitude superior to that quoted by -ro 0 Golclstein and others ( 1993 ) \\'ho utilized a single radar "C -- inte rfe rogram. The reason for the lo\ver precision of the ...E-< sing le interferogram technique is tha t it includes the - 50 ~~~~~~~~~~~~ longitudinal gradients in ice velocity which tend to -4 -2 0 2 4 6 8 10 smooth out the local minimum in tidal displacement Distance from hinge line (km) and bias its locati o n , H ere, the bi as in a bso lute location is of the order of se \'era l hundred meters. In order to m a p Fig , 6, TidaL disjJlaeements ( dots ) along a 160111 wide the hinge line of a Ooating glacier precisely, it is therefore jHofile shown in Fig1l1'f 1, cOl71jJored to model jJredietiol1s essentia l to utili ze multiple interic rograms and elimina te (solid line) frOIll an elastir-beam theO/)'. The arrow jJoints the lo ngitudinal g radients in ice velocity. to the area oj ma\imlll11 bendillg stress, or hinge line. Ice v elocities thickness measured by an ice-sounding radar (see below). The ice \'e loc it y varies from 400 m a 1 at 900 m elevati on In an eq ui\'alent fashion, the value of E calcula ted from to I 100 m a I a t the grounding line, d ecreasin g th ereafter Equation (7) is 3 ± 0.2 GPa, or three times larger th an to a bout 900 m a 1 toward th e eclge of the scene (Fig, 7), the value d erived by Vaugha n (1995 ) using da ta from R emo\'a l of th e tidal signal clearly reduces the vari atio ns several glaciers, H ence, the method of using Eq uation 7) in ice \'elocity across the grounding line, yielding a m o re is probably not a se nsiti w predictor of th e elasti c modulus reasonable \'e loci ty profile. of the ice. R es idua l errors in model fitting are present down stream from the point of m aximum tidal deflection (Fig, 6), where wc ea rlier no ti ced th e presence o f res idual phase errors running across the scene, These errors ~ 7 1000 ,; remain small compared to the tidal displacement, cd ,. .-... ~ ..... recorded on the ice tongue. "'... El ,. " "-.. 800 \\ Hinge line and grounding line $ po .. ;.o ..... I I", e.> 0 0 .: 00 In the clasti c-beam theory (Holdsworth, 1969), th e Q) 2 -~ 600 bending stress of the ice, 1;'.('.1' = E zE.r.r (1- v ) - I , where Q) Z is the \'e rtical distance to the ne utra l axis of the beam, e.> and E.r,r is th e bending strain ra te, is maximum a t th e ice - surface for Z = ±h/2 a nd a t the hinge lin t" for T = 0, Locatin g the maximum of the bending stress, howewr, ilwoh 'es second-order deri\'ati\'es of th e phases, which -40 -20 o 20 40 increase the noise lewl of the d a ta. In stead , I propose to Distance from hinge line (km) d efin e th e interferometri c grounding lin e as the location of th e minimum relati\'e tidal displacement m easured Fig. 7. Ire velorities ill the flow direction along the glacier a long a tid a l pro fil e extracted in the glacier-Oow direc ti on cen ter line before (dotted, grey line ) al/d aJiet (solid line) ( Fi g , 6). I n the elas ti c-beam theo ry, th e p oint of tidal (ormtiolls. The width rif the jJrojile iJ 80 Ill. minimum deflection also coincides with the hinge line, The hinge line may not necessarily coin cide with th e g rounding line or with th e line of hydros tatic equilibrium I n se\'eral parts of the Ooa ting sec ti on of th e glacier, of the glacie r (Smith, 199 1) , The grounding line is la rge discontinuities in ice ve locity a re cl etected (Fig. 4 ) . typicall y downstream from the hinge lin e and upstream Al o ng the ce nter line, the eastern sla b of the Ooating ice fr om th e lin e of hydrostatic equ il ihrium. On Pete rma nn tongue mo\'es a bou t 30 50 m a 1 fa ster th a n th e wes tern Gletscher these three zo nes a re separated by 1- 2 km, as slab. About 20 km d ownstream fr om the grounding line, disc ussed later, where the \'elocit)' difference be tween the t\\'o sla bs The hinge line of Petermann Gletsc her is shown in reaches 50 m a I , the velocity of thc eastern slab a bruptly Figure 3, overl aid on th e tidal displacements. Based on d ecreases by 40 m a I , The discontinuity in \'e loc ity and the phase noi se of th e tidal signal (Fig, 6), I es tima te tha t a ppa rent surface rupture re\'eals a n overriding of the 48 1 Journal of GLacioLogy northern sla b by the faster-moving so uthern sla b. A la rger ice thickness th an calcula ted from hydrosta ti c further 20 km downstream , a similar discontinuity in equili brium of the ice. CO I1\'ersely, between km 5 and 25, velocity occurs on the wes tern side of the glacier. The two ice thickness is slig h tly overes tima ted by the DE~I d a ta . If sla bs 'ubsequently mO\"C a t comparable speed s. the KMS DEM is accurate, this res ult suggests tha t either These discontinuities in ice \"C locity occur in th e a la rge part of the ice tongue is not in hyd rosta tic turning sec ti on of the glacier. Surface rupturing could be equilibrium, or the ice-so unding rad a r data include due to th e differenti al veloci ty betwee n th e two sid es of sp a ti a l irregula ri ties in ice thic kness th a t d o not the glacier associa ted with fl ow turning. O\'erriding of th e corres pond to stead y-state conditions. Multiple ice iee indicates tha t the resista nce to ice fl ow is larger sounding radar pro fil es are need ed to interpret these downstream fr om the grounding line, during the glacier differences more com pl etely. turn, whi ch is consistent with additional fri c tion at th e About I ± 0. 5 km from the hinge lin e, the ice side margi ns during fl ow turning. sounding radar d a ta transition to a regimr of la rger v a ria ti ons in ice thickn ess (Fig . 8 ) and th e b asal Ice thickness refl ecti ons produce h yperboli c rad io-echo records U ezek and oth ers, 1995) . This fea ture is likely caused by bo ttom 1 es timated the glacier thi ckn ess using the KMS DEM data crevassing of the glacier and indicates the approxima te by ass uming tha t th e fl oating glacier ice tong ue is in position of the groulldillg line Uezck a nd others, 1995). hydrosta ti c equili brium. T o perform this calcula ti on, 1 The grounding line, the hinge line and th e line of used an ice density of9 17 kg m 3 a nd a density of sea water hy drostatic equilibrium of Peterma nn Gl etscher a re of 1030 kg m 3 The corres ponding valu es of the ice therefo re not coincident and a re pro bably separa ted by thickn ess averaged ac ross th e glacier ",id th , h, a re shown a bo ut 1- 2 km. in Figure 8, with error bars corresponding to one stand ard devia ti on in ice thickn ess across the glacier width. Ice discharge 800 The ice flu x, Q, fi'om Petermann Gletscher at and below the hinge line is calcula ted using (8) 8 600 m where 1fr is th e ice velocity along the x direction averaged m Q) across the glacier wid th , and W IJ is th e glacier wid th ~ 400 m easured in th e y direc ti on (Paterson, 1994). The glacier Co) thickness used in the calculation is tha t deri ved fr om the :a..., K NIS data. Where the glacier is a fl oat, basal \'elocities Q) sh ould eq ual the surface velocities, so the surface \'elocities .8 200 m easured by rad ar interferome try a re equivalent to verticall y integra ted ice \·e locities . 3 Ice di sc harge is 12.3 ± 1 km a I at th e hinge line a nd 3 12 .1 ± I km a I a bout I km d ownstream where the o 10 20 30 40 50 glacier IS more likely to be in hydros tatic equilibrium Distance from hinge line (km) 13 Fig . 8. Ice thickness measured b)! Ihe University oJ 12 Kansas's airborne ice-sounding radar ( fSR, guy Line ) ..-. 11 i aLo ng Ihe gLacier cenler line com/](lred with Ihe ice thickness t\l 10 dedllcedJrom Ihe KMS data ( DEAf. solid lille) assuming .. S 9 /I)!dros latic equilibrium of tlte ice alld averaged across Ihe ~ "-' 8 glacier widlh . Q) b.O 7 ..c:'"'t\l 6 Ice thickness was coincidenta ll y measured by an Co) 5 ,'"'m airborne ice-sounding rad a r, d esigned and opera ted by ~ 4 the Unive rsity of K ansas (Chuah and others, 1996) and Q) 3 Co) fl ying on board a NAS A P-3 a ircra ft , along the center line .... 2 of Peterma nn Gletscher, on 26 ~Ia y 1995. The ice 1 0 sounding rad a r operates a coherent rada r system at a 0 0 center fr equency of 150 MHz. 0 20 40 60 80 At the hinge line, ice thickn ess is sli g htly over Distance from hinge line (km) es tim ated by the DEM d a ta (Fig. 8). The two thickn ess es tima tes become equal a t a bo ut 2.7 ± 0. 5 km from th e Fig. 9. Ice disch{l1ge ( da is) oJ Pelermanll eletselzer at and hinge line, which indicates the approxima te locati on of below tlte hinge Line, alld average disc /i (l1ge (solid Line ) th e lin e of hydros tatic equilibrium of th e glacier tongue. averaged over 1.5 km segments. Diamond symbols indicale Betwee n km 25 and 45, the ice-sounding rada r indica tes a the ice-flux estimates by Higgills ( 1991) . 482 Rigllo/: Tidal mo/ion, ice l'e/ocif)' alld lIIe/t rate q/ Petermallll Cle/scher ( Fi g . 9 ). The m easurement error is assoc ia ted with DISCUSSION uncerta inties in ice velocity a t th e ice m a rgin , a nd with uncerta inties in ice thickn ess. About 30 km upstream fro m The hinge line of Peterma nn Gletscher m ay mo\'e back the hinge line, J oug hin a nd oth ers ( 1995b) es tim a ted a n and forth with the ocean tide, depending o n the geometry 3 ice flu x of 12.7km a I , consis tent with our res u lts. of the hinge zone (H old sworth, 19 77 ). The K~ r S DE1\ l D o wnstream [i"om the hinge li ne, the ice discharge indicates tha t th e glacier slope at the cente r of th e glacier d ecreases rapid ly. At km 52, th e ice disc ha rge is only is a bout 0.8% a t the hinge line, 1 % 4 km abo\'e, and 3 2. 1 km a I. These results a re con sistent with those 0.2% 4 km belo\\' . The tide a mplitudes of Thank God o bta ined by Higgins (199 1) near the ice front (Fig. 9 ). Ha rbor (81 °36' N, 61 °40' \\') \\'ere measured by C. F. H a ll At the glacier fr o n t, Higgi ns ( 199 1) es ti mated a calf-ice during the Polaris exped ition around 187 1 (perso na l production of only 0.59 km 3 ai, or 20 times small er than communicati on from R. Fo rsberg, 1996) a nd indica te the ice discharge a t th e hinge line. Abla ti on processes th a t the m aximum tid al displacement recorded over a 3 d therefore melt more tha n 95% of the ice th a t crosses the repeat-pass cycle should no t exceed 800 mm. Ass uming hinge line. Calf-i ce production plays a minor role in the that the b edrock slope is 0.8%, short-term \'ari a ti ons in mass discharge to the ocean from Petermann G letscher. sea le\'el sho uld not displace th e hinge line by more th a n 100 m o r 5 pi xels. Larger displ acements o f the hinge lin e Melt rates would indicate a cha nge in glacier thic kness of I m per 125 m of ho ri zontal disp lacement, assuming th a t other Assuming th at the ice d ensity is e\'e ry\\'here th e sa me, the effects, suc h as th e isosta ti c uprise of the seabed, a re equa ti o n of mass consen 'a ti on integrated \'ertically a nd negli gible during tha t time period. Using the interfero across th e glacier \V id th is metric hinge lin e as a refe rence, it shou ld be possible to detec t fin e c ha nges in glacier conditions. (9) The pattern of ice discharge fro m Peterma nn Gletscher is unex pec ted . C lose to the ice fr ont, Higgins (199 1) m easured a ra te o f g lacier th inning of 2.7 m a 1 in where b is th e glacier net bala nce, positive if the glacier th e last 17 km of glacia l Oow, wh ich he a ttributed to accum ulates mass (Pa terso n, 1994) . I now ass ume that surface ablation. If we assume a surface ablation ra te of the glacier is in stead y-state conditio ns, m eaning fJ h/fJt = about 2 3 m a 1 for the ice to ngue, glacie r thinning near 0, a nd calcul ate the glacier net ba la nce, b, ri"o m the th e grounding line canno t be attributed to surface g radient in ice flu x di\'ided by the g lacier width. abl a tion a lo ne. A sig nificant amount of ice must be The largest source of error is the unce rtainty in removed thro ugh basal m elting. The sig lla ture of the thic kness gradi ent. T o reduce th a t error, I calcula te the radar echoes fr om th e ice-sounding rada r (not shown glacier net balance at a discre te num ber of locati ons, over here) su ppo n s th at concl usio n Ueze k a nd o thers, 1995). 5 10 km long segments. The res ults a re shO\\'Il in Fig ure The basal melt rate o r Pe terma nn Glctscher should 10 . The la rgest \'a lue, a bout 2+ ± 5 m ai, is record ed a\'erage a bo ut 10 ± 2 m ai, \\'ith peak \'alues exceed ing cl ose to th e grounding lin c. Near the ice front, the glacier 20 m a 1 near th e grounding lin e. ne t bala nce is only 2- 3 ma 1 (Higgins, 1991 ). The ice flux The m e lt ra tes ofOoating ice tongues or ic e shelves a rc d ec reases from 12. 1 to 1, 5 km :l a lover an a bla ti on a rea of poorly kno \\'n near the g ro unding line, a nd it is not well 2 886 km , correspond ing to a n average melt ra te of kn own ho w soon hi gh ra tes of melting develop (jacobs 12±2 m a 1 and o the rs, 1992 ). Here, basal melting is m os t acti\'e in lh e first 4- 5 km dO\\'nstream from th e hinge line. The 0 pa ttern o f m elt ra te shown in Figure 10 is consistent with earli er o bservations cond ucted by J enkins and Doake ,~ " ...... ~ . ~ (199 1, fi g. 10 ) [or th e Filchner- R onne lee Shelf o r -5 Thom as ( 19 76, fi g. 5) fo r the R oss Jce Shel f. i (0;1 The sam e mechanisms which control basal melting on El -10 an ice sh elf must be acting on the ic e tong ue ofPeterma nn '-" Gletscher. These mech a nisms are we ll known a nd Cl) C) correspo nd to a large-scale ice pump (L ewis and Perkin, !=: -15 (0;1 1986). In a n ice pump, d eep thermoha li ne convecti on is -('0 induced b y melling of ice in th e deepes t part of th e oD.... -20 fl oating ice and dri"en b y th e press ure d e pe ndence or th e Cl) fr eez ing po int. Ma rine ice accumula tes a t the base of th e Z -25 g'lacier, as ice pla telets ri sing in th e water column accrete to th e bo tto m of th e ice she lf (O en er a nd o the rs, 1992 ). rn -30 th e case o rPe terma nn Gletscher, th e ice-pump effec lmust be amplified by strong tida l pumping a nd mixin g of th e 0 10 20 30 40 50 60 70 80 water col um n. O ceanograph ic obse rva ti o ns a re needed to Distance from hinge line (km) determine the cha racteri sti cs of th e wa te r column a nd co nfirlll the existence of stro ng basal melting. Fig. 10. ,\felt rale 0)' the floating ire tOllgue I'S the One o ther possibl e inte rp reta ti on of the net balance dowllstream distance .from /he hinge line with 20% error da ta is tha t the glacier is no t in steady-sta te conditions bars. a nd is actua ll y thi ckening . The corres pond ing thickening 483 Journal of Claciology rate is large and would suggest a major change in mass ACKNOWLEDGEMENTS balance of Petermann Gletscher. J oug hin and others (1995a), however, found tha t the ice disc ha rge from the This work was performed at the J et Propulsion equilibrium line of Petermann Gletscher was nearl y in Laboratory, California Institute of T ec hnology, under a bala nce with its accumulation . Their res ult argues contract with the National Aeronautics and Space trong ly in favor of assuming steady-state conditions for Administration. I wou ld like to thank S. Ekholm and th e ice tongue. R. forsberg for graciously providing a high-quality In Greenland, it is generall y ass umed that ice removal topographic map of Petermann Gl etsc hcr, C. ""ern er proceeds through surface ablation and calf-ice production for use of his SAR processor, K . J ezek for enriching (Reeh, 1985). This is not true of Petermann Gletscher, disc ussions on the glaciology and d ynamics of Petermann where calf-ice production a nd surface a blation are sma ll G letsc her and for pointing ou t the importance of basa l com pa red to basal melting. Ice tongues like th at of melting, and P. Gogin eni for sharing in ad\'ance of Peterma nn Gletsc her do not de\'elop extensively in the publication his ice-so unding radar observations. Arcti c, but a re present in other parts of northern Greenla nd. for mass-balance studies, and in order to avoid the difficu lty of m easuring the basal melt rates of REFERENCES the ice tongues, it seems essential to estimate ice discharge Chuah, T. S. , S . P. Gogincni, C. Alien and B. \\·ohl e tz. 1996. Radar a t th e grounding line, instead of com bini ng estimates of Ihickness measurelllenls over Ihe 11.0 1'1 hem jJarl oJ Ihe Greenland ice sheel. surface ablation and calf-ice production. Lawrence, KS, C nil'ersit I' of K a nsas. Radar Systems a nd Rcmote In Antarctica, where far more glaciers develop a Se nsing Labora ton ". Tcchnical R eport 10+70-3. ) floating ice tongue or an ice shelf, basal m elting is alread y Drcwr)', D.J. a nd G. de Q Robin. 1983. Form and fl ow of the Antarcti c ice sheet during the last million years. In Robin, G. d e Q , ed. The known to play an important rol e in the overall mass clilllalic record ill polar ire sheels. Cambridge. etc. , Cambridge U nil'e rsity bala nce of the ice sheet 0 acobs and others, 1992). Recen t Prcss, 28~38 . studies of Pine Island G lacier U acobs and others, 1996) Dunbar, :\1. 1978. Correspondence. Petermann Glctscher: possible and Rutford fee Stream Uenkins and Doake, 1991; so urce of a tabular ice bcrg o ff the coast of :\'ell'foundland. ]. Glaciol., 20(84), 595597. Sm ith, in press ) also show that basal melt rates of the Gabriel, A. K ., R . :\1. Goldstein and H. A. Zcbkcr. 1989. :\rapping sma ll magnitude of those detected on Petermann Gletscher a re e1 cl'a ti on changes Ol'er large a reas: dilTcrential radar interferometry. not unlikely. These results ta ken together make it more ]. Ceophys. Res ., 94(B7 ), 9183 9 19 1. imperative to measure ice discharge of polar ice shee ts at Goldstcin, R.I\1. 1995. Atmospheric limitations to repeat-track radar interferometry. Geoj)I!)'J. Re>. Lelt., 22( 18 ), 25 1 7 ~ 2520. th e grounding line rather tha n at the ice front. Goldstein , R . :\f.. H . A. Zcbkcr a nd C. L. \\'erner. 1988. Satellite radar R adar interferometry appears to be a powerful intcrfcrome try: two-dimensiona l phase unwrapping. Radio Sci .. 23 4\' tec hnique for locating the grounding line with precision 713 720. and providing essen tial information to calculate ice Golcl stei n, R . :\1., H. Engelhardt, B. K amb and R.:\L F rolich. 1993. Sa tellite rad a r interferomctry [or moni to ring ice sheet motion: di sc harge. Ice thickness could be es timated from inter a pplica ti on to a n Alllarctic ice stream. Science. 262 (5 139), 1525 1530. ferometricall y derived topog ra phic data in places where Hartl. P .. 1'- . -H . Thicl, X. \l'u. C. S.I\f. Doakc and j. Sievers. 1994. the ice is in hydrostatic equilibrium, but more direct Applicati o n of SAR intcrfcrometry with ERS-I in the AlllarClic. means of measuring ice thickness are probably desirable Ellrlh Observalion QjI""rler0', 43, 1-4·. Higgi ns, A. K . 1991. North Greenland glacier I"elociti es a nd calf icc to increase co nfidence in the res ul ts. production. Polmforsc/lUllg, 60( I ), 1990, 1 23. Holdswo rth, G. 1969. Flcx urc of a fl oating icc tongue. ]. Glaciol., 8(54 ), 385~397. CONCLUSIONS Holdswo rth. G. 19 77 . Tidal intcraetion with ice shell-cs. A IIII. Clopl!)'s., 33 1 ~ 2 ) , 1 33~ 146. J acobs, S. S. , H . H. Helmer, C. S.I\1. Doakc, A. j cnkins and R.I\ f. Mul tiple repeat-pass ERS- l rad a r observations of Froli ch. 1992. 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K wok, i\1. Fa hnestock, S. Gogineni a nd C. Alien. 1995a. or isos tatic uprise of the seabed. Interferom e trica ll y deril'ed topography. I"elocity, a nd ice-nux Mc! t rates of the ice tongue of Petermann Gletsche r estim ates fo r the Peterm ann G lacier. [Abstracl.] EOS, 76(46 ), Fa ll deduced from the interferometric veloci ties and ice i\fceting Supplement, 1'1 84. J oughin. 1. R. , D. P. \\'in ebrenner and \1. A. Fa hneslOck. 1995b. thickness data appear to be very hi gh, especiall y near the Obse f'l'ations of icc-sheet motio n in Greenland using satellite radar grounding line. These high melt rates are a ttributed to intcrfcrom e try. Ceoj)h)'5. Res . Lell., 22(5 ). 57 1 ~ 57+. pronouneed basal melting of the ice tongue at about Kollmc),cr, R. C. 1980. \\'est Greenla nd outlet glacicrs: an it1l'entory of the major iceberg producers. 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Re; .• 99(BI O, 19.617- 19.63+. meas ured from seismic obseryations. lGR Ocealls. JlIS recei1'l'd 21 "larch 1996 and acee/lled ill rel,ised Jorm 15 JUll e 1996 485