J ournalojClaciology, Vo!. 45, No. 151, 1999
Instrutnents and Methods
Modelling tnass balance using photogratntnetric and
geophysical data: a pilot study at Griesgletscher,
Swiss Alps
ANDREAS
KAAB,'
MARTIN FUNK
Versucllsanstaltflir fVa sserbau, Hy drologie and Claziologie, ETH Zentrum, CH-8092 Ziiricll, Switzerland
ABSTRACT. Th e kinematic bounda r y condition at the glacier surface ca n be used to
g ive glacier mass balance at a point as a functi on of changes in the surface elevati on, and of
the hori ZOnLal a nd vertica l vel ocities. Vcrtical velocity can in turn be estim a ted from basa l
slope, basal ice velocity and surface stra in. In a pilot stud y on the tong ue o fGri esgletscher,
Swi ss Alps, th e applicabili ty of the r e la ti on for modelling area-wide icc fl ow and massba la nce distribution is tested. Th e ke y input of the calculations, i.e. th e a rea-wide surface
velocity fi eld, is obta ined using a newl y d eveloped photogrammetric technique. Ice thickness is deri\-cd from rada r-echo soundings. Error estim ates and compa riso ns with sta ke
m eas urements show an a\'erage acc uracy of approxim a tel y ± 0.3 m a I fo r th e calcul a ted
vertical ice \'el ocity at th e surface a nd ± 0.7 m a I for the calcul ated m ass ba lance. Due to
ph o LOg ra mm etric rcs tri cti ons a nd model-inh erent se n siti\'iti es the a ppli ed model a ppcared to be most suitabl e for determining a rea-wide m ass balance a nd ice flow on f1 a tlying ablati on a reas, but is so fa r not ve ry well suited for steep abla tion a reas a nd most
pa rts of acc umul a tion areas. Nevertheless, the stud y on Gri esgletsc he r opens a new a nd
promi sing persp ec ti ve for th e monitoring of spatial a nd temporal g lac ie r mass-balance
vari atIons.
INTRODUCTION
Th e mass bala nce a t a point of th e g lacier surface represc nts
the direct elim ate influence on th e g lacier at that point. D etermining the local m ass bal ance a nd its di stribution pattern o n th e glac ier surface contributes to th e monitoring of
elim ate \'ari ati ons a nd to th e understa nding a nd m od elling
o f g lac ier reacti o n to cl i mate. Tradi tion al methods of m assba la nce determin a tio n using sta kes a nd snow pits a re timeco nsuming and ex pc nsive, even for a rudim entary network.
Just deciding wherc the "represe nta tive" points should be
located is in itself a m aj or underta king. Fin ally, it would be
des irable to obta in m ass balance for many glaciers in a geog ra phi c region, but thi s is ra rely done because of the tremendous costs involved . Our a im is therefore to devi se and
tes t a remote-se nsi ng m eth od of di rect nl,ass-bala nce determi na ti on (cr. H ae berli, 1998). In this wo rk, we use photo g ra m metricall y deri ved velocit y field s and a kinema tic icef1 0w model to determin e mass ba la nce over la rger a reas
th a n is poss ible with traditi onal sta kc m easurements.
Direct determination of mass ba lance by modelling th e
icc f10w is not ofte n done due to the need to specify a reawide boundary conditions at the g lacier surface a nd bed.
R as mussen (1988) obta ined both bed topography a nd
m ass-balance di stribution iteratively, using repeated aeroph otog ramm etric m ea surements of el eva ti on and di splace-
Present address: D epartment of G eog raphy, Uni ve rsity of
Zurich - Irchel, Winterthurerstrasse 190, CH- 8057 Zurich,
Swi tzerl and.
ments on Columbi a Gl acier, Alaska, U .S.A. Fas took a nd
others (1995) derived bedrock elcvati o ns from repeated
ae rophotogrammetry. To help a nal yze a n ice core drill ed a t
Dye r Pl a teau, Anta rctica, R aym ond and o thers (1996) ca lculated the mass balance o f the ve rtical ice column at the
drill site. For that purpose, not only the boundary co nditi ons but a lso a profil e o f vertical veloci t y from boreho le
measurem ents was used. M ore often th a n with the above
point-by-point approach es, th e di scharge fluxes for deri ving
the mass ba lance of gl ac ier sections have bee n calcul a ted .
For va ll ey glaciers, long itudina l secti o ns with known ave rage cha nge in elevati o n a nd bounded by tra nsverse profil es
of velocity measurements a rc frequently used (e.g. Rey na ud
and oth ers, 1986). Bind sc ha dler and oth ers (1996) deri ved
the mass balance of ice-stream sections by means of a reawide velocity measurem ents using a simil a r approach.
H e re we present a m e th od which requires a rea-wide surface eleva ti on, its cha nges with tim e a nd a rea-wide surface
velocities fo r input. Glac icr bed topography is known from
radar m eas urements. U sing these input d a ta a nd some simpl e ass umptions of ice m echa nics, th c three-dimensio na l
ice-velocity field at the surface, and subsequentl y, the m ass
bala nce, ca n be ca lcul a ted at every p o int of the glac ier
surface.
Owing to the size of m a ny alpine g lacie rs, the resolution
of satelli te im age ry is not a d equate to a ce u ra tely determi ne
surface topography and its temporal three-dimensiona l
va riatio ns. An alytical ae ro photogramme tr y, on the other
ha nd, is a powerful a nd pr cise remo te-se nsing tool fo r
monitor ing three-dimen sion a l geometrical changes of g la-
575
Journal qfClacio{ogy
cier surfaces. Ne\'ertheless, its applicability is limited to
areas with sufficient optical co ntrast. Thus, we appli ed th e
method desc ri bed here not on acc umul ation areas but on ly
on g lac ier tong ues. The promising possibiliti es as wel l as the
restrictions, th e results of ice-flavv and mass-ba lance computation, are shown for th e tongue of Gr iesgletscher, Val ais,
Swiss Alps, between au tumn 1990 and aut umn 1991.
SITE DESCRIPTION
Mass-balance measurements on Griesgletsch er (Fig. I)
start ed in 1961. The annual mass balance has bee n determined using the glaciologica l method based on a set of
stakes. In add ition, the photogrammetric m et hod has been
applied at inten'als of some 5 yea rs, with res ults that compare well with the glac iological method (Siegenthaler,
1986; Aellcn a nd Funk, 1988; Funk and oth ers, 1997). Bindschadler (1981) simul ated the future evolution of Griesgletscher according to different mass-bala nce assu mptions.
H ambrey and others (1980) investigated the relation of ice
structures and the strain regime. Vi eli and oth ers (1997)
modelled th e thrce-dimensional ice fl ow and future behaviour of the whole glac ier acco rding to measured and estimated future mass balance. Since 1997, 33 stakes h m'e been
drilled over the whole glac ier a nd sun"Cyed sever al rim es a
year to obta in the surface velocity field. Hig h-resoluti on
aeria l photography from a heig ht of about 2000 m above
ground (im age scale about I : 15000) is repeatcd every
autumn by thc Swiss Federal Office of Cadastral Surveys,
cov'ering the cntire glacier on onc photo str ip of about 13
im ages (Fig. 1 ).
the en tire ice thickness, the local m ass ba lance b a t a point
(x, y) ca n be expressed as the difference between th e cha nge
in surface elevation within time, azs/ot, and the horizontal
flu x differences over ice thickness, aq.,./ ox and aqv/ ay:
oZs
ut
aq,.
x
oqy
uy
b( x,y) =~+-a +"!l
(1)
(Paterso n 1994, p. 256- 258). The hori zontal flu x is th e integ r al of the horizontal velocity components, U a nd v, over
ice thickness, and thus the flu x differences ca n be wr it te n as
oqx
ox
oqy
7) =
l
y
z
0 1 "' U d Z=Us-oZs UhOZb +
, au d
Z
ox Zb
ox
ox
"b ax
(2)
0 l
oZs
OZb
av
f} _ v dz = Vs
Vb 8- + _ 7) d z .
-= -
z,
Y
l'z,
aY -
kh
Y
Y
-h
where Zb is th e g lacier bed elevati o n, Zs is the surface elevati on, Us and Vs a re th e horizonta l co mponents o f surface
velocity, 'Ub a nd Vb are the horizonta l components o f basal
velocity, OZ5/ ax a nd OZ5/0y are the surface slope compon ents and OZb/ ax and OZ1,/ ay are the bed-slop e components. Inserting Equation (2) into Equation (1) g ives
oZs
a zs
OZ5
b(x,y) =~+us-a +vS"!l-ws,
ut
x
uy
(3)
where Ws is the vertical ice velocity at the surface
Ws
a Zb
OZb
= Ub-a
+ Vb--;:)
uy
x
1
0
'
au
-a. d z -
Zl>
12< -aav
x
zJ,
d z.
(4)
dz ,
(5)
y
Equ ation (4) can be written as
b
Ws
OZb = 'Ub aZ
ax + Vb 7)
Y
1"< '
E,·.T
dz -
1"' .
_
Eyy
.. h
Z\;
where E.T.1' and Eyy a re the horizontal st rain rates. Assuming
incompressibility of ice (E.n- + EyU + Eoz = 0), Equat ion (5)
can be expressed as
Ws
=
OZb
a Zb
Ub"!l+Vb- +
ay
uX
1
z
, .
Eoz
dz,
(6)
21>
where Ezz is the verti cal strain rate. Equations (3) a nd (6)
r eprese nt the kinem atic boundary co nditi on at th e g lacier
surface (Hutter, 1983; Paterson, 1994). This geometric
r elation is va lid a t every point of the glacier surface. Some
of the ice-flow components a nd the connection to geodetic
sta ke measurements and photogrammetric measurements
are depicted in Fig ure 2.
Ice-mechanical assumptions
Fig. 1. The tongue qf Criesgletscher between the icifall (lift)
and an artifical reservoir lake ( right). Section qf an aerial
photo taken by the Swiss Federal Office ojCadastral Surveys,
10 September 1991. Griesgletsclzer is part qf the Swiss glacier
monitoring network and the object ojnllmerolls investigations.
Inset: Location qfGriesgletscher in the Swiss Alps.
KINEMATIC BOUNDARY CONDITION AT THE
GLACIER SURFACE
Basic equations
Our calc ul ations of the local ice flow a nd m ass balance follow the kinematic boundary condition at the surface. In
this sec tion we describe the basic equations a nd the special
ice-mech anical assum ptions we used for o ur co mputation.
From the mass conservation in a vert ical co lu mn over
576
In our study, we try to determine a ll the components on the
rig hth and side of Equations (3) and (6) and, subsequent ly, to
calc ul ate the local ve rtical ice velocity at the surface, ws,
and the local mass balance, b. For that purpose, we u e rep eated standa rd ph otogrammetr y to obtain surface slopes
and su rface elevati on changes. A special photog r ammetric
technique is used to measure surface velocities. Bed topography and ice thickness are d erived from gro und-based
radio-echo soundings. The basal veloc it y (sliding and sediment deformation ) is estim ated by comparing an ex isting
ice-flow model for Gri esgletscher (Vieli and o thers, 1997)
and measured surface velocities. \lVe obtained a r atio of basal
sliding lo total surface velocity of a bo ut 75% (see be low, section on calculated ice fl ow and m ass balance).
To determ ine the \'ertical strain at the surface, Ezz" it is
necessary to know the vari ations o f vertical strain ra te with
depth. These co uld be obtained by borehole meas urements at
single points (e.g. Gudmundsson a nd others, 1997) or by ice-
Kiiiib and Funk: instruments and methods
Zs (X,t1)
Us
w,~" "q,:~. · "' ~~'
~''-'-'-'- : '-'-'-'- ....
.-.-.-.-.-.- .-
.-~~.; :~ :~ tzz ~J-:
fl ow m odelling. Gudmundsson and Ba uder (1999) give fIrst
res ults of a stud y o n Unteraa rgletsc he r, Swiss Alps, whi ch
combines the approac h using th e kin em a tic bounda ry conditi o n with a three-dimensional flow m odel (Gudmund sson,
1997). On Griesgle tscher, there a re no bo rehole meas urements a\'a il abl e, a nd Vieli a nd others' (1997) fl ow model of
Griesg letscher does not include sliding, which is ass umed to
have a m a rked influence on straining due to the high sliding
ra ti o. Therefore, we dec ided to estim a te the column-a\'e raged
\'e rtica l strain using th e ice thickness, (h = Zs - ZI», th e ve rti cal stra in rate a t t he surface Ezz , and a simple ass umption fo r
the va ri ati on of stra in rate with depth:
= ~ -' f:zclz;::::: r hfe==, .
}Zh
(7)
In the case of no inte rn a l def'orm a ti o n a nd pure sliding, r
wo uld be I (i. e. \'e rtical strain rate co nsta nt with depth ). In
th e case of no sliding, the \'C rtical stra in rate at t he g lacie r
bed is 0 which, toge th e r with th e ass umpti on of lin ea r va ria tio n o f\ 'e rtica l stra in ra te with depth , g ives '"Ya \'a lue o f 0.5.
A s o ur estim ate we use '"Y ~ 0.75, co nsiderin g th e hig h
sliding rati o for Gri esgle tscher to ng ue, a nd we introdu ce a
la rge e rror estim a te (o r '"Y in our ca lcul a ti o ns.
Th e \'ertical stra in rates at th e surface in Equ a tion (7)
can be deri ved fr o m th e ho ri zon ta l o nes with -i,==, =
f..T .r , + f. yy, ' Th e ho ri zonta l strain ra tes a t the surface in turn
we re ca lcul ated from th e surface \'eloc ity fi eld, meas ured
pho tog ra mm etricall y in o ur stud y, using the ex pressio n
.
E./y
1
(av au)
=:2 ox + ay .
---...r---.....
Fig. 3. Geometl) llsedJor determining slI1foce elevation and its
changes by monotemporal photogralllmetric stereo models and
Jar determining sll1Jace displacement.> by multitemporal stereo
models. See textfor expla nation.
Fig. 2. PllOtogrammetric and geodetic measurements and their
relation to some co mponents of ice./low, which appear in the
kinematic bounda1Y condition at the sll1Jace.
E==,
Monop/otting (t2)
(8)
SURFACE TOPOGRAPHY AND SURFACE KINEMATICS
Photogrammetric methods
Aerophotogramme tric determinatio n of digita l terrain m odel s
(DTM s) and subseque nt compa rison of multi tempora l DTMs
is a n e lTccti\'e a nd wc1 l-establ ished tec h n ique for determ ining g lac ie r surface e le va ti on, surface slo p es a nd th e cha nges
in surface elevati o n. Th e method uses m o no tempora l ste reo
m od els, composed o f a t leas t two ove rl a pping photog raph s
ta ken from different places (spatia l b aseline). Th e te rra in
point A (see Fig. 3) is co m p uted by intersecting two spati al
rays, eac h fI xed by the kn o wn p roj ection centres a nd th e projecti o ns (AI ' (t J) and A / (tl )) of the selec ted terrain point.
Repeating th e procedure a t o th er points gives a DTM at time
tt. Rep ea ting th e procedure using photogra ph s ta ken at time
t2 gi\'es the eb 'ation of th e same locatio n ( point B) a nd a
DTl\I a t tim e h From bo th DTl\Is, the c ha nges in surface
ele\'ati o n throughout tll(' a rea ca n be ca leul a ted.
To d e te rmine surface di sp lace ments, a n e w photog ra mmetric m e th od to meas ure surface displ ace m ents wi th hig h
prec isio n was de\'C lop ed a nd applied, based o n th e ea rli e r
concepts o f Fl otron (1979), A rmena kis (1984·) a nd Sa kso no
(198+). Th e m ethod was impl ement ed on a co mputer-a ided
pho tog ra mme tric compil a ti o n system (a n a ly ti ca l plotte r )
(K aa b, 1996). It uses multi temporal stereo m odels composed
from ae ri a l photographs ta ke n at diffe re nt tim es (tempo ra l
baselin e ) a nd from diffe re nt pl aces (spa ti a l baselin e) (e.g.
photo I (t 2) a nd photo 2 (tIl in Fig. 3 ).
Be twee n times tt a nd t2 the surface p o int A has moved
with th e g lac ier to point G A ma rker o n th e surface or features like clo ed cre\'asses o r rough ice a re suitable ta rge ts.
Th e proj ec ti o n of such a p o int (A2' (tl )) is chosen on th e
photo a t tim e tj . Intersec tin g the spa tial ray fi xed by thi s
im age p o int a nd the kn ow n proj ect ion ce ntre w ith th e te rrain surface, represe nted by rhe DTM a t tim e t j, gi\'es th e
three-dim e nsional ground coordin ates of p o int A. This procedure is ca lled mono pl o tting. The im age p oint C 1' (t2 ),
which corres po nds to A :z' (tIl, can be fo und usi ng the stereo sco pi c o ve r lap. Th e op er a to r cance ls the te rra in moveme nt
betwee n tim es t l a nd t2 by di splac ing o nc im age whil e
simulta neo usly loo kin g a t th e multit empo r a l photographs.
Thi s simulta neous a nd sle reoscopic observa ti o n suppo rts
the ide ntificati on of co rresp o nding points, improves the accuracy o f th e meas urem e nts a nd indi cates wh e th er a loca l
displ ace m e nt measurem e nt is consistent w ith surrounding
di spl ace m e nts. In the sam e way as 1'01' th e sp a tial pos iti o n
of point A , the pos iti on o f p oint C is computed by mono pl olting. Thus, a rea-wide sp a ti a l displ ace m e nts of the glac ie r
surface ca n b e deduced .
Th e g lac ier surface fo r which the ve loc it y field is determined must show correspo nding features in every phoLOgraph,
a nd th e displ ace mems mu st be greater th a n th e acc uracy of
the me thod. Acc uracy inc reases with the time between th e
photogra phs, but the numbe r of corres po nding features may
decrease. The photogra mmetric technique d escribed here
577
J ournal ifGlaciology
works espl:c ia ll y welll or <l na lyz ing the creep of mountain permafros t (rock glaciers) and is a lso sui table for observing glacier Oow and slopc movements (e.g. Gudmundsson and others,
1997; K iiiib and others, 1997, 1998; cr. Kni zhnikov and others,
1998, for application of a simila r technique). It is ve ry difficult
to determine displ acements on snow-covered terrain, where
features are subtl e and change rapidly. Th e acc uracy of the
method is ass umed to be about ±30 /-Lm rms in im age scale
or, eonsideri ng the photog rammetric para meters of the
Gri esgletseher missions, about ± 0.4 m rms on the glacier.
The accuracy of the displ aeem ents was checked by repeating
measurem ents using semi-independent multi temporal stereo
models a nd by comparing th e results with geodetic stake
measurem ents on several glaciers and rock glaciers. The comparison for Griesgletscher indicated an accuracy (rms) of the
displacement measurements of about ± 0.6 m a- I in the case of
a I yea r interva l betwee n the two photo missio ns.
±0.4 m, so the elevation differences haye a sta nda rd devia tion of about ± 0.6 m a I rms. O ver the period 1990- 91 the
tongue thinned by up to 4 m a l in the pl anar m id dle part
(Fig. 5), a consider able thickness loss amounting to about
3% of total ice thickness. Du r ing the same year, m ost other
g laciers in th e Sw iss Alps also exp eri enced significa nt thinning due to th e ex tremely wann summer. The r aw elevation
differences outside the glacier m a rg in a rc at the error level,
with some la rger a mounts caused by (moraine) er os ion and
melting of ice rem ains. In Fig ure 5, depicting the filtered
d ata, the rem a in ing non-zero e levation differences adj acent
to th e glacier a re mos tl y an artifac t of the low-pass filtering,
which smoo ths sha rp edges. Thi s can be seen from the fact
that the elevati o n differences d ecr ease to small a m o unts at
g reater dista nces fro m th e g lacier. The filterin g effect is
especiall y evident at the lake, which despite being a t different levels in 1990 a nd 1991, does not show a uniform cha nge
in elevation (Fi g. 5).
Sur face elevatio n
Surface dis placem.ents
For the photogrammetrie measurements two stereo pairs of
photog raphs, ta ken on 22 August 1990 and 10 September
1991, were used. The orientation was computed from artificial
ground m arkers painted on stable bedrock beside the glacier
for which coordinates were known from a geodetic netwo rk.
DTMs at both times were composed of a reg ula r grid of elevation points with a spacing of 50111. Each DTM includes
about 1000 points on the glacier surface. Con to ur lines we re
interpolated from these DTM points and the m argin line of
the glacier (a lso determined ph otogrammetrically) using th e
DTM interpolation softwa re HIFI (height interpolation
using finite elements; Ebn er and Reinhardt, 1984). Below the
steep icefa ll, the surface slope of the tongue decrease to a
nearly plana r middl e part a t about 2600 m a .s. !. , before increasing again towards the terminus (Fig. 4).
C ha nges in el evati on for th e period 1990- 91 were computed as th e d ifference between the two DTM grids. A
Gaussian low-pass fil te r ave raging nine weighted differences to obta in a new sm oo thed value at th e centre point of
the 3 x 3 cell fil ter m ask was appli ed to red uce noise caused
by terrain roughness and r a ndom measurem ent errors (Fig.
5). Th e acc uracy of one height measurement is about
W
+E
N
lOOm S
500 m
•
Glacier margin
Uncertain glacier margin
Mass-balance stake
Topography 1991
Fig. 4. Surface contour lines if the Griesgletsclzer tongue (in
m a.s.l.), derivedfrom a 1990 DTM, and positions qftlze stakes
Jor determination of mass balance and ice velocity. Coordinates
( Swiss system) are in metres.
578
W+E
N
S
.000
•• 0
'
I\)
o • (J1
o·
000
0
0
000000
••
0
0
0
+3ma- 1
+ 2 ma- 1
-2ma- 1
-3 m a- 1
-4ma-1
Fig. 5. Cha nges in sUljace elevation on Ihe tongue, 20 August
1990 la 10 September 1991.
The photogrammetrie monoplo tting technique for d etermining surface di splacements, outli ned earlier in this section, was
used to deri ve surface velocities and, subsequentl y, to calcul ate surface strain rates and estimate surface strain and
basa l velocity. Two separate sets of di splacements were measured for the p eriod 1990- 91, each consisting of a bout 700
points on the glacier surface. Two multitempora l stereo
models we re constructed for tha t purpose, each co mposed of
different photographs of the photo strips of 1990 and 1991and
each newly oriented (i.e. one m od el of photo I (t2 ) a nd photo
2 (td, and the o th er of photo 1 (t l ) a nd photo 2 (t2); cr. Fig. 3).
The semi-indep endent sets of di splacemenLs help reduce distortion effects in single photogra phs, errors in the orientation
procedures a nd, most importa nt, errors in single measurements. Genera l error inOuences, such as bad ground-control
points or cam e ra errors, are detected to a lesser extent. Figure
6 shows the raw velocity field obta ined from one of the displacement meas uremenL sets. It g ives an impression of th e basic
data quality, whil e the Oow field itsel f is discussed below (Fig.
7). The velocity vectors in Fig ure 6 have a density similar to
that of the DTMs, i. e. 50 m. Sufficient targets a t both times
were found in m ost regions, a nd measurement was fairly
straightforward. I n some regio ns missing ta rgets, low surface
Kddb and Funk: Instruments and methods
-+
20ma- 1
15
10
Horizontal
t'
ice velocity
Q)
E15 [ma"l
E
~10
0
.r:.
0..
7
10
9
5
contras t or la rge surface cha nges prevented determ ination of
surface displacements. And a t a few places, simi la r but not
tru ly correspondent ta rge ts were identified , leading to
obvious data blunder.
10 compa re the two vel ocity fields, the scattered data
were interpolated to a gr id with a reg ula r spacing of 50 III
using a m odu le of the IMSL m athematical softwa re library
(least-squa re fit to neighbo urs using quadra tic p olynomi al;
Akim a, 1978). Extrapolati ons greater tha n 50 m from original input data were deleted . Each vec tor in the first
gridded d a tase t was compa red with the co rresp o nding vector in the seco nd se t, a nd a fina l value was comp uted as the
a\'erage of both veero rs. Velocities d iffering > 15% in onc or
more of the two \'ector components (u. v) in both sets were
eli minated , as were velocities measured only in onc dataset.
The res ulting ave rage velocity fi eld was smoo th ed by th e
same G aussia n low-pass fi lter desc ribed earl ier. After that fi ltering, da ta gaps of not more tha n one gridcell (50 m) were
elosed by interpolati on. The velocit y fi eld resulting from thi s
post-process ing procedure (Fig. 7) was used for th e foll owing
calcu lations. The hori zonta l velocity acc uracy a t a single
point, deduced by compar ison with stake meas urements (inset in Fig. 7), was about ± 0.6 m a 1 rms; the acc uracy of th e
average velocit y values was a bout ± 0.4 m a 1 rms.
Th e hig hes t annu al surface velocities durin g 1990- 91 on
the Griesgletscher tong ue occur in th e icefa ll (up to
40 m a \ Th e surface vel ocities dec rease to 9 m a 1 in th e
p la na r midd le pa rt. T he photog rammetric res u lts reveal
most ice in th e a blation a rea coming from the so uth ern (S)
p a rt of the acc umu lation a rea. Thus, th e Oowl i nes of th e
northwestern (NW ) part o f the tong ue a rc sh ifted against
the NW va ll ey nanks. Th e la rge ice velocities directl y at
the NW m a rgin sugges ts g la cier sli ding. Below 2550 m
a.s. l. , ice Oow increases to > 10 m a 1 a nd th en decreases
again to 6- 7 m a 1 near th e terminus.
5
15
20
10
S1ake measurement
p lace ment meas urements (cr. F ig. 6), so wc apply the
smoothed surface velocities (cr. Fig. 7). \Ve use the s urface
velocity fielel desc ribed in the previous subsection to ca lcu late
ho ri zontal stra in ra tes following Nye (1959), in which deformation at a gridpo int is computed fro m velocities at th a t point
and its four nearest neighbours (see a lso Bind schad ler and
o thers, 1996). Fi g ure 8 shows the horizontal stra in ra tes at
th e glacier surface transformed to th eir principa l axes. The
average strain-ra te acc uracy, deduced by redunda ncy in
Nye's algorithm , is a bout ± 0.0002 a \ only a few p er cent of
the calcu lated defo rm alions. Th e vert ical strain rate a t the
surface (Fig. 9) is d eri ved from the incompressibility co ndition, f..r,y + Eyy + Ezz = O.
The increas i ng velocities in the upper part of the icefa ll
cause marked ex te nding Oow. Strong horizonta l compression, and hence vertical extensio n b elow the icefall , is d ue
to decelerating Oow. As expec ted , the hori zonta l stra in-rate
p attern at th e m a rg ins (Fig. 8) shows a form and ori e nta tion
ty pi cal for shear d e form ation. A lso, below 2600 m a .s.l. the
ce ntral glacier ex p eri ences \"er y low strain rates as o ften as
longitud inal compress ion. At 2600-2580 m a.s.l. the ice Oow
1
\
To model local ice now a nd mass balance, we estimate the
\"Crti cal stra in from the vertical strain rate at the surface (c[
Equation (7)). For th at purpose, we derive the hori zontal
strain rates from the surface velocit y fi eld using Equation
(8). The res ulting f.. tU a re se nsitive to noise in the raw di s-
10
8
Fig. 1 SwJace velocityfield qfthe Criesgletscher tongue during
1990- 91, derived from two measurement sets like Figure 6 .
The results can be compared with sll1face velocit)lJimn stake
measurements (inset).
0.05 a -1
Surfac e s train rates
15
Stake 6
~
Fig. 6. R aw sll1Jace displacement vectors measllred by simultaneous mOlloplolling using a multitempoml plLOtogmmmetric
stereo model.
20
,
Horizontal compression
Horizontal extension
/ " Crevasses 1990
'")
Fig. 8. Horizontal p rincipal surface strain rales during 199091 calculated fm m the velocity field. The crevasses qf 1990
were mapped pllOtogrammetrically.
579
J ournal cifGlaciolofJ'
••
Crevasses 1990
extension
0.04 a·'
0.02 a·'
0
0.02 a·'
0
0.04 a ·'
Vertical compression
Fig. 9. Vertical sll1jace strain rates during 1990- 91 calculated
from the horizontal strain rates using the incompressibility
condition.
against th e N vV vall ey fl ank causes transverse h ori zontal
compress ion due to converging flow a nd, additionally, longitudina l horizonta l co mpression due to decreasing ice
velocit y. Both togeth er res ult in strong compressio n a nd a
rota tion of th e principal strain-ra te axes of about 45° with
respect to fl ow directi on. Di verging flow and inc reasing
velocity produce hori zontal extensio n in th e middl e of th e
g lac ier tong ue a t a bout 2560- 2580 m a.s. l. In m os t oth er
pa rts of the to ng ue, horizonta l compression a nd a ssociated
vertical extensio n predomin ate, as ge nerall y exp ected for
abl ati on a reas.
Fig ures 8 a nd 9 a lso inelude th e 1990 crevasse p a ttern
m apped photogra mm etrically. Crevasses are expected
where hori zonta l extension preva il s. Th e orientati o n of the
crevasses should theoretically be perpendicul a r to the
directi on of th e la rgest horizonta l extension (principa l axis)
(Vaughan, 1993; H a rper and others, 1998). This ori entation
is consistent for m ost mapped cr evasses on Griesgle tscher
(Fig. 8; cf. H ambrey and oth ers, 1980). Excepti o ns m ay
occur, for exa mpl e, wh ere crevasses advect from r egions of
hori zontal extensio n into regions o f horizontal compression,
without having yet totally elosed. Sm all zones of ho rizontal
extension, indicated by crevasse fo rm ation but witho ut co rresponding evidence in the stra in-rate pattern , m ay have
been missed in the strain-rate calcu lations, possibly due to
inadequ ate resolution of th e velocity grid pacing, or to
fa ult y interpola tion or excessive sm oothing of the velocity
field. Fin all y, the observed pa ttern of crevasses is th e result
of some cumulative strain ex pe rience prior to 199 1, a nd i
not necessaril y the resultofth e stra in-rate pattern m easured
fo r the peri od 1990- 91.
DETERMINATION OF GLACIER BED TOPOGRAPHY
To estim ate th e vertical ice vel ocity at the surface fo ll owing
Equ ati on (6), bed slope a nd ice thic kness have to b e known.
Both a re deri ved from radi o-echo so undings perfo rmed in
spring 1988 a nd 1990 along II profiles using a m on o pulserada r dev ice (Funk and oth ers, 1995). The method used to
interpret the d a ta is ex plained in Fa bri (1991). The acc uracy
of the measurem ent method as deduced from compa risons
of rada r-deri ved ice thickn ess with deep ice drillin gs on
580
seve ra l other glaciers is esti mated to be a bo ut 5- 10 % of th e
ice thickness (H ae berli a nd Fi sch, 1984). Additi ona l radi oecho soundings for Griesgletscher a re pla nn ed, especiall y
in the stee p midd le part of the glacier where no meas urements h ave been possibl e so far.
A regul ar 50 m g ridded digital elevati on model (DEM )
and contour lines (Fig. 10) were interpolated from the ice-thicknes profil es Llsing the soft wa re HIFI (Ebner and Reinh a rdt,
1984). Th e most notable features of th e bed topography are the
overdeepening between profiles R6 and R 9 a nd the associa ted tra nsverse bedrock riegel between profil es R 9 a nd
RIO. Thi s riegel cannot be deduced from the radar measurements a lone, and has been inferred from the bedrock topography outside of the glacier. The ex istence of the interpola ted
ri egel seems to be confirmed by the comparison between
modelled ice fl ow a nd sta ke measurement (see following secti on). Ice thickness of th e tongue is up to 150 m in th e ove rdeep ening, and about 100 m over the riegel, both relative to
th e 1991 surface elevation.
W+E
N
100m S
'"
o
o
500m
•
Stakes
Radar profiles
100 m surface contou r lines
Fig. 10. Glacier bed topograpil)' (in m a.s.l) cif the Griesgletscher tongue interj)olatedfrom prcifiles cifradio -ecllO soundings ( R5- Rll) and bedrock topography cif/he surrounding iceJreearea.
CALCULATED ICE FLOW AND MASS BALANCE
So fa r we have de termined a DTM of th e gl acier surface,
surface velocities a nd surface strain rates between 1990 a nd
199 1, a nd a DEM of th e glacier bed. To model vertical ice
vel ocity at the surface (ws ) foll owing Equ ation (6), ass umption s for the basal veloc ity and th e relation between the
measured surface strain rates and those at depth (er. Equ atio n (7)) are necessar y.
Vieli and oth ers (1997) simul ated th e ice fl ow of Gri esgle tscher using surface a nd bed elevatio n data meas ured by
pho togramm etr y a nd r adi o-echo so undings, respectively.
Applying a flow m odel of Blatter (1995), th ey calc ula ted ice
defo rm ation neglecting basa l sli ding. Comparing the
m od elled ice deform a ti on at the surface with the velocities
meas ured photog ra mm e trieally gives a ra tio of basal sli ding
to to ta l surface vc10cit y of about 75% . At most places on the
g lacier tongue the sliding ratio differs by only a few p er ce nt
from thi s value. D evia ti o ns in the iccfa ll a nd at the terminus
a re a rti facts of the fl ow m odel's grid sp acing (150 m ) a nd of
insufficient bed da ta . Thus, we decided to use the average
heidb and Funk: i nstruments and methods
Table I. Approximate average, maximum and error if terms if
the kinematic boundmy condition at the gLacier surface Jor
Griesgletscher tongue (values in flow direction )
.
...,/" .
//•
ilverage amollnt
Term
Dzs/
at
It,
8ZjO.L
sliding/tota l \'e locit ),
DZh/ OX
IE". I
h
assumpti on for "r
(sce Equati o n (7))
calcu lated 10,
calcu lated b
3ma 1
I
9ma
0.13
0.75
0.10
I
0.01 a
80 m
0.75
.\/arimuJ1I amolllll
'4ma I
20 III a 1
- 0.+
I
0.3
0.03 a I
150 m
Average (ll'tllrtll)'
±0.6ma I
±O..'m a I
±0.01
± 0.15
± 0.05
± 0.0002 a
± 10m
± 0.25
- -im£l
I
I
(er. Fig. II I
6 ma I
±03 m a
± 0.7 m a
•• . ••• . • 0 0 - · · · ' . ' .0
••• 0 •
.
•
••
_ 0000 '
. ... /; : : : . : : : : : : : : ~
.•.
• • 0
~ ~ ~ g~ ~ : :.
o:::::: :::::: 7: :: : , : :~~~. >
/D
. : : . :~::: : ~ : : ::;: : : : : . •
•• ••• 0 •
j
....
•
•
00, • •
•
;,~~6:::~::::~~~:: ': :
o. • • • •
0·
...
~;7--
I
"
000
0
•••
.
(
.
..., ..... /"-
r/~
: : :::::::::::;: : :::.
-5 -4 -3 -2 +2 +3 +4 ma- 1
- 0.2 m a
. • •• 00000
• • . • • • • .
. . • 00
• • • · . 0 0 · · · · · . · · 00
/"
Vertical
icevelocily[ma- 11
..-'
\
\
·1,5·1.0·0.5
9
0.5
·1.0
-1.5
I
I
sliding ra tio o f 75 % O\'er th e whole tongue. Th e basal
velocity is assumed to have th e sa m e directio n as th c surface
velocity. For th e abO\'e approach, we introduce a n error of
10- 20%. The vertical stra in at th e glacier surface was es tim a ted fr om th e ice thickness and the \'ertica l stra in rate at
the glac ier surface using Equation (7) and a , of 0.75, a nd
introducing a la rge error for thi s ass umpti on (Ta ble I).
Using the meas ured photogrammetric and geophysica l
d a ta, and the ass umptions made abo\'e, it is possibl e to compute the vert ical ice velocity at the surface and th e local
m ass ba la nce a t eve ry point from Equ ations (3), (6) a nd (7).
Before desc ribing the res ults, we di sc uss th e acc urac y of th e
calcula ti o ns. To a nal yze the e rror propagation a nd the fin al
acc uracy o f th e calcul ated ice now a nd mass bal a nce, we
estim ate erro rs for the input data, the ice-m ec hanical
ass umptions a nd the final res ults (1a bl e 1). To obtain th e
ave rage acc uracy of o ur model ( rm s), on th e o ne ha nd, a nd
to estim a te a m ax imum erro r r a nge, on th e o th er ha nd, we
ta ke both a verage a nd maximum \'a lues for th e input data,
a nd analyze e rror propagation fo r both cases, Th e res ulting
m ax imum m odel errors are g ive n below in pare ntheses, Fo r
th e error a na lysis wc neglect th e steep part of th e icefall.
Sl opes a nd velociti es of the esti mates are tra n sform ed to
th e now direction, Th e errors in bed slope a nd ice thickness
a rc rough but m ax imum estimates, Consequent ly, the error
of I' has a n efrect of up to ± O.2 m a 1 (max. 1.1 ) on th e \'ertica l ice \'elocity, while the error of basal \'eloc it y has a n effec t
o f o nl y ± 0.04 m a 1 (mal'. 0.9). Th esc est imates sh ow that in
th e specia l case of the Griesgle tscher tongue, th e rather
r o ugh assumptions fo r sliding \'C locity and to ta l vertical
strain have a minor effect on W s due to the low bed slope
and low stra in rates at the surface (cr. Equ a tion (6)). All
e rro rs take n togeth er cause an uncerta inty in the calcul a ted
\'e rtica l ice velocity of about ± 0.3 m a 1 (max. 1.5), which is
do minated by the uncertainty in f'. Errors in elevati on
change, surface velocity a nd surface slope contribute a n
e rror in th e calcul ated m ass ba la nce of ±0.5 m a 1 (max,
0.6). Thus, th e to tal error of th e mass bala nce m ode ll ed for
a point is about ± 0.7 m a 1 (m al'. 1.7).
The calc ul a ted \'e rtical ice velocity shows o m e interesting features (Fi g. Il ). The la rge negati\'e \'ertica l velocity in
th e icefall is due to the steep b ed slope a nd is therefo re to be
ex pected . Th e region along the NW margin where th e glac ier Oows towa rd s the N\ V vall ey n a nks is a co nspicuous
feature, in whic h the model calcula tion yields positive vertical \'eloeities o f up to +3 m a l . This m ay be due to the \'er-
Fig. 11. Calculated vertical ice lIelocitJlJo r 1.990- 91, including
com/Jarisons with stake measurements ( inset ).
tica l strain (cC Fi g. 9). In add ition , the now direction in thi s
a r ea is rotated towa rd the margin , causing the a long-now
bed slop e to increase, a nd hence co uld result in positive vertica l velocities a nd dec reasing ho ri zo nt al \'elocities, i. e.
compressive now. Positi\'e ve rtical \ 'C lociti es at the s urface
have been ca lculated upstream of the tra nsverse b edrock
riege l, mainly as a result of the increas ing bed slop e a t the
riege l. In the remaining parts of th e planar middle a rea the
ic e nows approxim a tel y horizontally, as expec ted because of
the low strain rates a nd fl at bed. At the steeper terminus,
b elow a surface a ltitude of abou t 2550 m, it has been calc ul a ted th at th e ice novvs dow nwa rd s in two stages which
a r e th e result of bed-slope \'ariations.
Sume calc ulated values of vertica l ice \'elocity ca n be
compa red with sta ke measureme nts (in set in Fig. 11). Th e
difTc rences show a n ag reement withi n th e range of th e e rror
estim ate of ±O.3 m a I. At sta ke 7, abo \'C the trans\-crse ri eg el, the sta ke m eas ure ment of the vertica l \'eloc it y co nfirms
th e interpolation o f thi s riegel.
Using the verti cal surface velocity, th e surface slope, th e
elevation cha nges a nd hori zontal surface \'elocit)', th e distribution of the m ass b a lance for 1990- 9 1 can be calc ul ated
(Equ a tions (3), (6) and (7)). The resu lts, pl otted on a 100 m
g rid, a re shown in Fi g ure 12, The m ode ll ed mass-balance
di stributi on mainl y res ults from the d istributi ons of\'e rtica l
ice \'e locit y a nd c h a n ge in cb'ati o n, The glaciologica lm assb a la nce meas ure m e nts at stakes 8- 10 ag ree with ca lcul a ted
values with in th e e rror estim ate of ±0.7 m a I, At sta ke 7 no
mass bala nce was available, At sta ke 6 in the lowe r pa rt of
th e icefall , the res ulting mass balance does not ag ree well
w ith the sta ke m eas urement , e\'en tho ugh the calc ul a ted
vertical \'elocit y a pp eared reasonable. Additional photog ra m metric measu reme nts were m ade in thi s area w ith out
detec ting any gross e rro rs. The horizo nta l \'eloc ity co n 'espond s well with th e sta ke measurement. Our fa\'oured hypoth es is is tha t this represents a g ross error in th e stake
m eas urement ; howe\ 'e r, a difTcrence between modelled a nd
m eas ured mass balance, in principle, may also be ca used by
the d ifTerent pe ri o d s of photogrammetry and stake m easure m ent. The aer ial ph otos used were ta ken 0 11 22 August
1990 a nd 10 September 1991, while the sta kes were measured
o n 25 September 1990 a nd 2'~ August 1991. Different melting
rates in the non-overl a pping periods would influence th e
mass-bala nce m eas u rem ent a nd therefore a ffect all co mparison s between m odel a nd fi eld data.
D espit e th e mass ba lance at sta ke 7, all difTerences
581
J ournal qfClaciology
1°
00
00
9
000000
00000000
o
~
/ <50000000 OpO
/ooooooooo
°&5000 dooo 0
0
0
00 0 0 00000
,_ / 0000000000 ,-/
00000000000 ,/
6 0 0000 0
0 :
•
o -2
-3
-4
0
0
0
/-'
-4
Mass balance
[m a- 1]
-3
-2
6
-3
0
. o/;?--
o
o
o -5 ma-1
Stake
-4
8
<L-----c
_4.----;
_3.-~
_2 -5
measured
Fig. 12. Calculated mass-balance pattern qfthe Criesgletscher
tongue fo r 1990-91, including comjJarisons with stake measurements ( inset).
between the calcul ations a nd the m easured ve rtical ice
velocity and mass balance ar e within the error es tim ates.
Therewith, th ey a rc the expected res ult of the acc uracy of
the photogrammetric and geophysical method s used , the
ice-mechanical ass umptions a nd the error propagation
through the kin em atic bound a ry co nditi on.
The modelled m ass-balance distribution (Fig. 12 ) shows
a high spati a l vari ability. The values range up to nearly - 6 m
and var y by a factor of up to 2 at the same altitude. Even
taking into acco unt the acc uracy of the applied m ethod,
these differences remain significant. Several influences may
play a rol e, anal ysis of whi ch wil l require furth er investigations. The obta ined mass-bala nce p a ttern may be a n expression of the solar radiati on pattern, th e snow di stribution and
a height gradient of air temperature. The spatial variations
of th e solar radiati on, affecting the snow- and ice m elt, depend on the to pog raphy of th e glacier a nd of th e s urrounding terrain. So me low mass-bala nce values at the SE m a rgin
of the glacier m ay be attributed to the shading of the adj acent mountain ridge, while hig he r m ass-balance va lues at
the NW ma rg in m ay refl ect t he absence of such shading.
The ice mass-ba la nce pattern is influenced by the snowmelt
pattern, a nd therefore also by the distributi on of sn ow depth
over th e glacier, which we do not know. Th e general
decrease of m ass-balance values to the SVV is exp ected due
to the height g radient of air temperature. Enl a rged area of
the ice surface, a nd thus high m elting rates, may furth er influ ence m ass ba lance at zones with crevasses or o ther high
acti vity such as deform ati on (e.g. crevasses a round stake 7,
or strong deform ation at the NvV m a rgin; cr. Figs 8 a nd 9).
CONCLUSIONS AND OUTLOOK
The kinematic boundary conditi on at the glacier surface
provides a va lu abl e tool for using a rea-wide pho togrammetrically derived velocity fi elds and elevatio n d ata to
m odel ice flow a nd mass ba la nce. Ice-mecha nical ass umptions a nd d ata noise can have sig nificant influence on calcul ated ve rtical ice velocity a nd m ass-balance va lues. The
effect of errors and assumptions will increase with increasing basal velocity, increasing bed slope and incr easing surface slope (cr. Equati ons (3) a nd (6)). Therefore, the m ethod
is most suita ble for planar g lacier sections. Additio na lly, surface strain rates a nd surface strain can be expected to be low
582
under such conditi o ns. Applying the m od el to ice fl ow over
steep er glacier secti o ns like icefalls will b e difficult, but, on
the o ther hand, may provide a useful a lte rnati\'e means of
determining mass ba la nce for such diffic ult-to-reach ar eas.
In addition to these m odel-inherent restricti ons, there a re
limits imposed by ph otogrammetric techniques. Photogrammetric determination of elevation and di splacement requires
sufficient and persistent surface contrast a nd corresponding
features, and is therefo re restricted to a bla tion zones, or to
crevassed or rock-covered areas in th e acc umulation zone.
Contras t-giving features on the snow, such as sastrugi or dust,
a llow for ele\'ation m easurement, but often a rc not persistent
enoug h for displ acem ent measurem ent. Photogramm f' try
works much better o n contras t-rich debri s-covered ice tha n
on th e debris-free ice which predomin ates on Griesgletscher.
Th er efore, the metho d described here is especiall y suita bl e
for d e termining the mass balance of debri s-covered glaciers,
which is diflicult with o ther approaches.
To summari se, th e m odelling of ve rtical ice fl ow a nd
m a ss-ba lance distribution using area-wide pholOgrammetric
a nd geophysical data is most promi sing for ablation a reas
with low slope (cr. Ba uder, 1996; Kaab, 1996). Under the above
restrictions, this approa ch has the potenti a l to reduce the expenditure involved in traditional sta ke-based mass-ba la nce
prog ra mmes, for exa mple by helping in siting representa tive
sta kes. Fin ally, thi s a pproach provides the spatial va ri ability
of the mass balance, which might contribute to improving
three-dimensional m odels of glacier fl ow, energy-ba la nce
m od els and interpretation of mass-bala nce processes.
Th e future goal o f m odelling glacier m ass balance using
photog rammetric a nd geophysical d a ta is to co ntribute to a
remote-sensing stra tegy for monitoring mass-balance va ria tions. For that purpose, it is necessar y to apply repeated
a nnu a l photogramm e tr y to obtain a time series of m assba la nce distribution. In the case of such time seri es, the
effect of some time-i ndep endent errors will be markedly reduced by analyzing o nly relati ve m ass-ba lance variations.
In fac t, first results of a corresponding 20 year series (K aab,
1996) gave a reasona ble mass-bala nce cur ve and suggest
th a t even strong error s in glacier bed d etermination and in
the ice-mechanical assumptions wer e m o t1 y elimin ated by
calcul ating differences between annu a l m ass balances.
ACKNOWLEDGEMENTS
\lVe g ratefull y ackno wledge the car eful a nd constr uctive review s of two anony m o us referees. Tha nks are due to H.
Gudmunclsson a nd W. H aeberli for m a ny useful di sc ussio ns
a nd to numerous co lleag ues for their help with fi eldwo rk.
The study was pa rt o f th e Swiss Na tiona l Research Prog r a mme No. 31on "C lim ate Change a nd Natural Disasters".
Th e photogramm etric investigations wo uld not have been
p ossibl e without th e va lu able ae ri a l photographs ta ken by
th e Swiss Federal Offi ce of Cadas tra l Su r veys.
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A kim a, H. 1978. A me thod of bi\'aria tc interpola ti on a nd smooth sur face
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Fastook,.J. L. , H. H. Brech erandTJ. Hughes. 1995. Derived bedrockclevations.
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