Geophys.
J. lnt. (1996)125,340-354
Glacial rebound of the British Isles-III.
viscosity
Constraints on mantle
Kurt Lambeck,l Paul Johnston,l CatherineSmitherl and Masao Nakada2
lResearch School of Earth Sciences, The Austrølian National Uniuersity, Canbema, ACT 0200, Australia
zDepartment of Earth and Planetary Sciences, Kyushu Uniuersity, 33 Hakozaki, Fukuoka 812, Japan
Accepted 1995 December 13. Received1995 December 8; in original form 1995 April 7
SUMMARY
Observationsof sealevelchangesincethe time of the last glacialmaximum provide
important constraintson the responseof the Earth to changesin surfaceloading on
time-scalesof 103-104years.This responseis convenientlydescribedby an effective
elasticlithosphericthicknessand effectiveviscositiesfor one or more mantle layers.
Considerabletrade-offbetweenthe parametersdescribingtheselayerscan occur,and
differentcombinationscan give rise to comparablepredictionsof sea-levelchange.In
particular,the trade-offbetweenlithosphericthicknessand upper-mantleviscositycan
valuefor the lithosphericthicknessa correspondbe important,and for any reasonable
ing mantle viscositystructurecan be found that givesa plausiblecomparisonof seaIn particular,thinJithospheremodelswill lead to
level predictionswith observations.
low estimatesfor the upper-mantleviscosity,while thick-lithospheremodels lead to
high viscosityvalues.However,eithersolutionmay representonly a local minimum in
the model parameterspace,and may not correspondto the optimum solution. It
becomesimportant,therefore,that in the inversionof observationaldata,a comprehenspace,to ensurethat
sivesearchis conductedthroughoutthe entire model-parameter
the solution identifieddoesindeedcorrespondto the optimum solution.The sea-level
data for the British Isles lend themselveswell to such an inversionbecauseof the
relativelyhigh quality of the data,the good geographicdistributionof the data relative
observationalconstraintson the dimensionsof
to the former ice sheet,and reasonable
the formerice sheetand on its retreat.Furthermore,becauseof the contributionto the
sea-levelsignal from the distant ice sheets,as well as from the melt-waterload, the
observationaldata basefor the regionalso hassomeresolvingpowerfor the viscosity
of the deepermantle.The parameterspaceexploredis definedby up to five mantle
layers,the lithosphereof effectiveelasticthicknessDr, and a seriesof upper-mantle
eachof
layers,i:2-4, extendingdown to depthsof 200,400and 670km, respectively,
z¡¡,
to
the
coreof
viscosity
extending
down
and
a
lower-mantle
layer
viscosity4r,
x
parameters
explored
is
30<Dr<
120
km,
3
101e<4¡
The
range
of
mantleboundary.
(i:2, 3.4) <5 x 1021
Pa s, 1021
1qm31023Pas with 4z<4t<4a14¡-. Simplemodels
Pa s, and
comprising three layers with D, -70 km, Dz-670km, Ar-Ø-5) 1.020
glacial
Earth
models
q.> l}tt Pa s describethe sea-level
unloadingwell.
responseto the
with low-viscositychannelsimmediatelybeneaththe lithosphereare not required,but
if a thin lithosphere(<50km) is imposedin the inversionthen the solutionfor the
mantleviscosityleadsto a low-viscosity1< 1020Pa s) channel.Sucha model doesnot,
however,representthe overall least variancesolution that would be obtained if Dt
were also introducedas an unknown. Likewise,if a thick lithosphere(> 120km) is
highervaluefor the upper-mantle
imposed,then the solutionpointsto a considerably
only a local minimumsolution.The
viscosity(-10" Pas). But this also represents
in the viscosityof the uppermantle,
observational
data do point to somestratification
and the optimum solution is for a five-layer model with the following effective
Darameters:
340
o 1996RAS
Glacialreboundof the British Isles-
III
341.
5 5 < D t < 6 0k m
(2 <qr< 4) x 1020Pa s for (D, < D < 200)km
(4 <qr< 6) x 1020Pa s for (200< D < 400)km
4 ¿ - 2 x 1 0 2 1 Psaf o r ( 4 0 0 < D < 6 7 0k) m
4wà1022Pa s for (670<D <D"-6) km
I(ey words: British Isles,glacial rebound,mantle viscosity,sea-level.
1 INTRODUCTION
In two previouspapers (Lambeck 1993a,b)(referredto hereafter as Parts I and II, respectively)the crustal rebound of
Great Britain and the concomitantsea-levelchangeassociated
with the melting of the last greatice sheetshavebeenexamined
in detail, and models have been developedthat relate mantle
rheology and ice-sheetparametersto observationsof relative
sealevel change;i.e. the positions,in terms of both elevation
and geographiclocation, of past sea-levelwith respectto its
present position. These relations permit the viscosity of the
mantle to be estimated,and also provide a basisfor examining
the evolution of the coastal and shallow-marineenvironment
throughHoloceneand Late Pleistocene
time (Lambeck1995a).
In thesepapersa comprehensiveunderstandingof the relation
betweenthe past ice sheet over the region, sealevel change,
and shoreline evolution has been reached,although some
disagreements
betweenobservationsand predictionsthat will
require further improvementsin certain aspectsof the model
have also been noted, most notably in the details of the ice
sheetover northernmostScotlandand possiblyover lreland.
'While
the models explain many of the observationswell, the
range of plausibleearth models that describethe viscoelastic
behaviour of the planet on time-scalesof 103to 104yearshas
not been fully explored. This arises because,under some
conditions,different combinationsof the earth-modelparameterscan give rise to similar predictions,and observationsof
sealevel changefor the late- and postglacialstagesmay not
give a unique determinationof the parameters.It is this aspect
of the problem that is discussedin this paper,in which a more
comprehensive
analysisof the model-parameterspacehas been
carried out than has been the case previously. In order to
comparethe resultswith the earlier analyses,the samemathematical and physicalmodels are usedas describedin Part II
and Johnston(1993).The ice-sheet
modeland sealevelobservational data base also remain unchangedand are not discussedfurther here.
The ice sheet over the Great Britain region was small
compared with those over the former glaciated areas of
Fennoscandiaand Laurentia.But this smallice capnevertheless
produced signiûcant crustal deflection whose record is well
preservedin a rangeof now submergedor elevatedshorelines
around the coast and shallow off-shoreareas.In addition to
this local rebound, the past sea-levelsof the region have also
beeninfluencedby the rebound of the crust under the weight
of ice over Fennoscandia,as well as by the redistribution of
meltwater into the surrounding oceanswhen the global ice
sheetsdecayed.Thus, despitethe relativelysmall extent of the
former ice sheet,the sea-leveldata from the region contains
some resolving power for the lower-mantle viscosity (below
the 670km seismic discontinuity), as well- as considerable
o 1996RAS,GJr r25,340-3s4
resolving power for the viscosity of the mantle above this
boundary. This, coupled with a comprehensivedata base of
past sea-levelsand a reasonableunderstandingof the retreat
of the ice sheet since the time of the last glacial maximum,
makes this region a valuable one for the study of glacial
rebound.
The earth modelsusedin modellingglacialreboundand the
concomitantsea-levelchangeare usually describedby a series
of concentricshellsdefinedby realisticallyvarying densityand
elastic parameters and depth-averagedeffective viscosities.
Typically the shellsrepresentthe lithosphere,with an effective
elastic thickness D,, one or more upper-mantle layers of
viscosity42,4t,..., extendingfrom the baseofthe lithosphere
down to depthsD2, D3, ..., and a lower mantle of effective
viscosity 4¡o from the 670km seismicdiscontinuity down to
the core-mantleboundary. The choiceof theseparametersin
modelling the glacial rebound remains a matter for debate,
thus valuesranging from 30 km to 200km have beenusedby
different authors for the effective elastic thickness of the
lithosphere(for a summary of lithospheric-thickness
estimates
see Wolf 1993). fieldskaar & Cathles (1992), for example,
adopted the lower value throughout their modelling of the
Fennoscandianrebound,while Peltier (1984,1986)arguedfor
values of 200km from an analysis of the rebound of the
Laurentide region, without fully examiningthe trade-off that
may occr¡rbetweenthis value and the mantle-viscosityparameters.Lambeck & Nakada (1990) and Lambeck,Johnston &
Nakada (1990)found that intermediatevalues,of between70
and 150km, were more appropriate for most glacial-rebound
and sealevel modelling, and that the actual value obtained
was closely related to the estimatesof the mantle viscosity.
Nor has there been much agreementon the inferencesof the
viscosityof the mantle from the reboundand sealevelstudies.
Thus Fjeldskaar (1994) argued for a low-viscosity channel
immediatelybelow the lithospherewith an effectiveviscosity
of 2 x 101ePa s, whereasPeltier and colleaguesargued for a
nearly uniform mantle viscosity of 1021Pa s from the baseof
the lithospheredown to the corê-mantleboundary(e.g.Peltier,
Drummond & Tushingham 1986), although in their more
recentmodels the viscosity of the lower mantle has increased
somewhat(Peltier & Tushingham I99l; Mitrovica & Peltier
1993).In contrast,Nakada & Lambeck(1989)arguedthat the
lower-mantle viscosity is distinctly greater than that of the
averageupper mantle,a view recentlysupportedby Mitrovica,
Davis & Johansson(1995).
For a large part, thesedifferencesin the inferredparameters
are a consequence
of the trade-ofl that'occurs betweenthe
parametersdescribingthe mantle structure,and by an inadequate searchthrough the model-parameterspace.Only the
studiesof Nakada & Lambeck(1989)[seealso Lambeck&
Nakada ( 1990);Lambecket al. (1990);Part III haveattempted
342
K. Lqmbeck et al.
to search over a wide range of lithospheric-thicknessand
mantle-viscosityparameters,and even thesestudieshave not
covered the entire plausible range. This paper sets out to
rectify this limitation, as well as to illustrate some of the
consequences
of not conducting a full search through the
modelparameterspace.
The observation equation used to estimate the effective
viscosityof the mantle and the lithosphericthickness,as well
as other pertinent model parameters,is given by eq.(1) of
Part II, and this is written here as
A,(okp,t)
+ q(rp,t): L( "(f) + ô( " (t)
L(r-r(q,t)+ L(*
+ þL(BR(rp,t)+
:
^(prediotea,
wnere
Â(o is the observedsea-level,reducedto mean sea-level,at
location rp and time r (seePart II);
e¡ is the assumedobservationerror (seePart II);
Â(" is the eustatic sealevel function for the combined
ice sheets:
ô( " is the correctionterm to A(";
p is the scaleparameterfor the British ice sheet;
Â(BRis the predictedfirst-iterationisostaticcontributionto
sea-levelchangefrom the ice- and waterload of the British ice
sheetfor specifiedearth-modelparameters;it also includesthe
contribution arising from the changein gravity field;
Â6r-rt. thepredictedisostaticand gravitationalcontribution
from the more distant (far-field) ice sheetsof Fennoscandia,
Laurentia,Barents-Kara,and Antarctica;
A(* is the second-iterationcorrectionto the water-load
te¡msresultingfrom ( 1) the dependence
of the water-loadterm
on sea-levelchangeitself, and (2) the time dependenceof the
shorelinelocations.
Generallythe heightsof the ice sheetsare poorly constrained
by direct observational evidenceand are instead based on
indirect considerations.For this reason,the scaleparameterB
has been introduced into the observationequation (1),
although, for the British Isles, the ice heights are partly
constrainedby evidenceof trimlines near the peaks of some
of the higher mountains that stood out as nunataks,and this
information has been incorporated into the ice model used
here (Part II). As before, an iterative procedure has been
adoptedto solve for the earth-modelparameters(D¡,r¡t of the
j layer model), which are embeddeddirectly in the isostatic
functions Â( BRand Â(r-r, the ice-heightscale parameter p,
and the correctiveterm to the eustaticsea-levelfunction ä( ".
The degreeof fit of the predictions of a particular earth
model,fr,to the observationsis definedby the variancefunction
(eq.2 of Part II) as
(2)
whereM:424 is the total number of observations,eachwith
standarddeviation o'.
A svstematic model-oarameter search is conducted for
3-layer
5-layer
4-200
4-150
6)
core-mande
boundary
Figure 1. Summaryof the zonation of the visqositystructure of the mantle for the three-,four- and five-layermodels.Elasticmoduli and density
depth profiles are de¡ivedfrom seismicmodelsof the mantle,in this casethe PREM model of Dziewonski & Anderson(1981).
o 1996RAS,GJr r2s,340-3s4
Glacíalreboundof the British Islesdifferent classesof models,which are deflnedby the number
of shells in the mantle model. For each set of earth-model
parametersin this space,the p parameter(as well as, in the
higher-iterationsolutions,the eustaticsea-levelcorrectiveterm)
that givesthe best least-squaresfit to the observationaldata
for that particular earth model is estimated. The overall
minimum varianceis then found by searchingthrough a rafige
of plausibleearth-modelparameters,
k:L,...,K, that is sufficiently large to ensure that the global minimum-variance
solution, rather than simply some local minimum variance,
is found.
2
THREE-LAYER
Dt=SOkm
-ro
Lower mætle Yiscosity(Pa s)
Dl=E0kn
A flrst-order zonation of the mantle is in terms of three layers
(Fig.1): an effectivelyelasticlithosphereof thicknessDt, an
upper mantle of viscosity 4, extendingfrom the base of the
lithospheredown to the major seismicdiscontinuity àt Dz:
670km depth, and a lower mantle extending from 670km
depth down to the core-mantle boundary at depth D"-u:
2891km. The parameterspacewithin which a solution for the
three-layer mantle structure is sought is deûned by
3 0 < D 1< 1 2 0 k m , l } t o < 4 r < 1 0 2 r P a s , 1 0 2<14 ¡ , < 1 0 2 3P a s ,
althoughin someof the subsequentmodel classes,
lower values
for 4, will be examined.Fig. 2 illustrates the dependenceof
(eq. 2) on the two viscosity valuesfor
the variancefactor rf.r¿
some discrete values of Dr. These results are for the firstiteration solutions, in which the meltwater load is approximated by the eustaticsealevelchange,and do not include the
eustatic correction term. For all values of Dt examined
(Table1), the most obvious result is the strong contrast
betweenthe two viscosities4t and 4¡". While the resolution
for the lower-mantleviscosity itself is not strong, the results
requirethat the lower-mantleviscosityis of the order 1022Pa s
and that it exceedsthat of the upper-mantlelayer by a factor
of as much as 50, depending on the lithospheric thickness
adopted.
As discussedbriefly above,the resolvingpower for the lower
mantle comeslargely from the far-fleld isostatic corrections,
including the waterload term. Fig. 3 illustratesthis in the form
of the ratio
5
)
5
1
t
ú
2
2
3
Lower mmtle viscosity (Pa s)
Dl=l20km.
p
2
l02l
3
4
7
5
2
tÊ2
3
4
5
7
tO21
Lower mmtle viscosity(Pa s)
Figure 2. Minimum-variancefunction Yu for the three-layermodels
for D1:50, 80, and l20km, and variable upper- and lower-mantle
viscosities.
(3)
ô( refersto the departuresof the predictedsea-levelfrom the
eustaticapproximationfor ice sheetj. The summationj is over
the totality of the ice sheets(with þ:l), and the summation
j* is over all the ice sheetsexcept the British Isles. The
summationø is,asin eq.(2), overall 424observationlocations
and times contributing to the solution of the observation
equation.The resultsfor f in Fig. 3 are for Dt:65 km, and
the contribution from the far-fleld ice sheetsof Fennoscandia,
Laurentia and Antarctica to the total sealevel signal varies,
accordingto earth model, from about l0 to 20 per cent. For
Dr:50 km this contributionis smaller,and lessmodel-dependent, being of the order of 8 to 12 per cent. For larger D,
values the contribution is both larger and more strongly
model-dependent,ranging from about 15-25 per cent for
o 1996RAS.GJI r25.340-354
343
3
MODELS
ltft','-l'l
':lrl;;ïl
III
!s-'-
à ¿
-/,"
- _.----_9_:.-./
0.10
\
'--oljs----
ì-__
\
\
0,14
Ð
0.145
012
Dl=65tcn
tût
2
3
4
s
j
rO22
2.
3
4
S
7
rO23
Lowermmtleviscosity(Pâs)
Figure 3. Ratio of the far-field to total contributions to the sealevel
change at times and locations of observations, the function f defined
by eq. (3), as a function of 4" and {1- for D, : SJ ¡¡¡.
344
K. Lqmbeck et
Table l. Summary of fi¡st-iteration three-layermodel solutions for specifrcvalues
of the lithosphericthicknessD1 and second-iterationsolutionswith and without the
corrective term ô( " for the eustatic sealevel function. For the second-iteration
solutionsonly modelswith 4rn:1.3x1022 have been considered.(Units for all
viscositiesare Pa s).
p
q2
Y
D1
4lm
50
3.3x1020
65
4.2xl}zo
z1Ú2
80
5.0x1020
21022
100
6.0x1020
21022
r20
'l.5x7}2o
2nd - iteration
0.90
4.10
1.06
2.75
2.95
(4-Ð1020
l.3xLO22
(4-5)l02o
7.3x1O22
| -44
t-t4
r.87
4.26
2.50
(excl. ô(e)
2nd - iteration
(65-70)
1.90
1.04
(incl. ô(e)
Dr:80. Theseresultsindicatethat thesefar-fleldcontributions
to the total sea-levelpredictionsare significant,ranging from
10 to 25 per cent, and that they do exhibit some dependence
on the lower-mantleviscosity,which permits someseparation
. of valuesfor the upper and lower mantle.
The resultsillustrated in Fig. 2, as well as similar resultsfor
intermediatevalues of D1, suggestthat the effectiveviscosity
of the lower mantle is of the order of I022Pas or possibly
somewhathigher but a better resolution for this parameter
from this data baseis not achievable.Instead,analysessimilar
to the presentone will be requiredfor someof the larger Late
Pleistoceneice sheets.Some trade-off betweenthe other two
parameters,D1 and 4r, is also evident. For example,if only
modelswith a 50 km thick lithosphereare consideredthen the
optimum solution, deflned as the minimum value for Yo for
Pa s. If an a priori
modelswith D1:50 km, yields4z-3 x 1020
valueof Dt:l2}km is adoptedthen 4z-7x1020 Pas, but
both solutions would only be local minima (see Fig.5(a)
below). Thus, models with a thick lithosphere lead to a
relativelyhigh estimatefor the upper-mantleviscosity,whereas
modelswith a low Dllead to a smallervaluefor 4r. This result
may at first be counter-intuitive,but it arisesfrom the fact
that often sealevel observationsare available for only the
latter part of the time span since the initiation of melting of
the ice sheet. Consider the expectedsealevel change at a
locality near the centreof the ice sheet.For a givenD1,models
with a low viscosity give a large overall relative sea-level
changewhen viewed at a time ú' sometimeafter the removal
of the ice load, and the relative sea-levelcurve will be characterizedby arapid, nearly exponentialchange[curve i, Fig.4].
In contrast,modelswith the sameD, but a high upper-mantle
viscosity give a smaller amount of relaxation by the time t',
but the sea-levelcurvewill be characterizedby a more uniform
decay (curve ii). For models with a thicker lithosphere the
relativebehaviourfor low (curveiii) and high (curveiv) mantle
viscositiesis similar,but the overallamplitudeswill be reduced.
Thus, for a given amount of observedsea-levelchangein an
(i)
Pas
30km, 5x101e
(iÐ 30km, 3x1020Pas
(iiÐ 120kn, sxl0le Pas
; z û
(iv) 120kn, 3xl02oPas
E
o
'á
--^
IJU
I
too
òo
1
6
1
4
t
2
l
0
8
ó
4
2
0
time (x1000 years BP)
F
J
U
!)
ó
u
time(x1000yearsBP)
Figure 4. Glacio-isostaticreboundnearthe centreof the ice load over
Scotlandfor four different earth models.If observationsare reliable
for only the last 11000a, model (i) (thin lithosphere,low upper-mantle
viscosity)gives a similar prediction to model (iv) (thick lithosphere,
higher upper-mantleviscosity).The lower panelgivesthe sameresults,
but for an expandedheight scale,
o 1996RAS,GJr r25,340-354
Glacíal rebound of the British Isles interval correspondingto the postglacialphase,about the last
10000 a in this case,it may becomediffcult to distinguish
betweena model with (1) an initially large displacement(small
Dt) and rapid relaxation(small42)(curvei), and (2) a smaller
initial displacement(large Dr) with a slower relaxation (large
4r) (curveiv). Usually few observationsare availablefrom sites
near the centre of the former ice sheet,but this argument is
also valid for sites near and beyond the former ice-sheet
margins.Thus the effectiveseparationof the two parameters
requireshigh-quality observationsthat constrainthe curvature
of the sealevel time-series,observationsthat extend as far
back in time as possible,and observationsfrom localities at
varying distancesfrom the centre of the former ice sheet.The
attraction of the British ice sheetfor glacial-reboundmodelling
is that it meetsthesecriteria well.
Fig. 5(a) illustrates the variance plots for the Dr-42 space
for modelswith a lower-mantleviscosity4¡^of 1.3x 1022Pa s.
Other valuesfor 4¡o give similar locations for the minimum
variance.This result clearly showsthe importanceof searching
through a wide domain of model parameters,and that the
III
345
prior assumptionof a value for D1 can lead to solutions that
representonly local minima.
The resultsin Fig. 5(a) are for the first-iterationsolutionsin
which the ice-heightscaleparameterhas beenincluded as an
unknown.That the B value for the minimum-variancesolution
liescloseto unity suggeststhat the useof the trim-line evidence
to constrain the ice heightsleadsto a satisfactoryestimateof
the ice thickness during the glacial maximum stage. Some
trade-off betweenthe earth-modelparametersand this scale
parametercar occur, with thick-lithospheremodels requiring
more ice in order to match a given amount of rebound, and
this could be used to place some constraint on the range of
plausible solutions. Thus, models based on a lithosphere of
120km thickness,for example,require a maximum thickness
of the ice sheet that is about 80-90 per cent greater than
inferred from the geomorphologicalevidence.The required
maximum ice height at the time of peak glaciationin this case
would be about 2600,m,comparedwith about 1500m for the
solution correspondingto the overall minimum variance,and
quite inconsistent with the trim-line evidence,as well as
a
J
0.9
7
3
5
7
1
0
2
ro2o
(b)
1
Upper mantle viscosity (Pa s)
2
3
4
s
7
1
0
2
I
Upper mantle viscosity (Pa s)
E
J
J
t
(c)
0
2
0 2 1 4 s 7 t o z
Upper¡mtle viscosity(Pa s)
|
t
(d)
0
2
0
2 j 4 ' j r O 2 1
Upper mmlte viscosity (Pa s)
Figure 5. Minimum-variancefunction Y¡ for the three-1ayer
modelswith 4n:7.3 x 1022Pa s. (a) First-iteration solution,including solving for the
p parameter.The continuouslines indicate contours of minimum varianceand the crosslocatesthe global minimum. The dashedlines indicate
contours of the p parameter(valuesin italics).(b) Y¡ for the first-iteration solution in which the B parameteris set to 1. (c) Y¡ for the seconditeration solution, including the waterload correctiveterm Â(", but excludingthe eustaticcorrectiveterm. Dashedcurvesindicate contours of
constantÉ. (d) Second-iterationsolutionsincluding the correctiveterm to the eustaticsealevelfunction. Points marked i to iu referto the models
whoseeustaticco¡rectionterms are illustratediq Fig.6(a).
o 1996RAS, GJI 125,340-354
346
K. Lambecket aL.
implausible from ice-modelling considerations(e.g. Boulton,
Smith & Morland 1984;Boulton et aI. L985).Thin-lithosphere
modelswill, in contrast,lead to an underestimationof the ice
volume (Fig. 5a),and the maximum ice-heightestimatefor the
30 km model is about 1000m. If ice thicknesseswere known
from independentobservationaldata then solutionswith B:
1 would be appropriate,and Fig.5(b) illustratesresultsbased
on this assumption.The minimum variance occurs for the
same parametersas before, but the results do illustrate the
advantageofindependentconstraintson the ice height,in that
the separationof the two parametersis considerablyimproved
becauseof the elimination of the correlationof the parameters
with p. Whetheror not p is introducedas an unknown in the
observation equation also affects the solutions carried out
within restricted ranges of earth-model parameters. For
example,by setting B to unity, implying that the ice sheetis
fully known from a priori ínformation, solutions with an
imposed thick lithosphere result in a higher upper-mantle
viscositythan if p is introduced as an unknown. Compare,for
example,the location of the minimum global variancesin Figs
5(a) and (b). Likewise,the effectof holding p constant when
imposing a thin lithosphereon the mantle model is to reduce
the estimateof the viscosityof the underlyinglayer.
Fig. 5(c) illustrates results that are comparable to those
illustratedin Fig. 5(a),exceptthat the second-iterationcorrections for the waterload term, including the time-dependent
migration of shorelines,has been included, using the formulation discussed
by Johnston(1993).The effectof the second
iteration, at leastfor thesethreeJayermodêls,is to reducethe
value of the overall minimum variancewith negligibleconsequenceon the earth-modelparametersthemselves.Hence,for
a preliminary exploration of the model space it does not
appear to be necessaryto considerthesesecond(and higher)
iterations of the sea-levelequation.
The first-iteration solution is basedon a eustatic sea-level
curve that has beenestimatedfrom observations,correctedfor
isostatic effects,from sites that lie far from the former ice
sheets,and where the primary contribution to sealevel is the
eustaticchange.The nominal curve usedhere is characterized
by a smoothly varying function in which all melting ceasedat
6000a BP. Earlier work on Holocenesea-levels
in the far-ûeld
hasled to the conclusionthat somemodificationof this eustatic
sealevel function is appropriate,in particular that the ocean
volumes may have increasedslightly after about 6000 a BP
(Nakada& Lambeck1988)and modiûcationsbeforethis time
may also be warranted(seePart II; Lambecket aI. 1.990).The
completesolution should also include,therefore,the corrective
term for the eustaticsea-levelfunction.The minimum-variance
functions for such solutions are illustrated in Fig. 5(d). The
principal consequence
of the inclusionof the eustaticcorrection
term is that the overall variancesare reducedthroughout the
model spaceexplored but that the minimum occurs for the
sameearth-modelparametersas before(Table 1). The correlation between this correction and the other parametersis
most pronounced with viscosity,with low and high uppermantleviscositymodelsgiving large correctionsfor late-glacial
time (Fig. 6a),but of oppositesign.Theseextremecorrections
appearto be inconsistentwith the sea-levelevidencefrom sites
far from the former ice margins,but further work on estimating
the eustatic sea-levelfunction from such far-field data is
desirable,in particular as theseestimatesfor the oldest ages
are basedon only a few data points. The cgrrectiveterms for
'5O
r^
I
tr
õ
0
o
o
o
-¡u
o
1
(b)
6
1
4
1
2
1
0
8
6
4
2
0
time (x1000 years BP)
Figure 6. (a) Estimates for the second-iteration corrective term to the
eustatic sealevel function for two models with (i, ii) a relatively thin
lithosphere (50km), (iii, iv) a relatively thick lithosphere (100km).
Models (i) and (iii) have relatively low upper-mantle viscosities and
models (ii) and (iv) have higher viscosities. (See Fig. 5d for the location
of these solutions in the Dr-4,- space.) (b) The eustatic correction for
three models near the minimum-variance solution (within the Y:2
contour of Fig. 5d).
the models near the overall minimum variance yield much
smallercorrections(Fig. 6b) and are consistentwith the earlier
conclusions
of Nakada & Lambeck(1988)that oceanvolumes
have increasedsince about 6000 a ago, such as to raise the
eustaticsea-levelby aboú 2 m.
The amplitude of the minimum-variancefunction obtained
for the most comprehensiveof the three-layermodelsis about
2, comparcdwith an expectedvalue of unity if the model is
completeand the standarddeviationestimatesadoptedfor the
observationsare true reflectionsof their accuracy.The observational accuracieshave been previously discussed(Part II)
and are consideredto be conservative,so that the minimum
value obtained for Yo suggeststhat there may be somescope
@ 1996RAS,GJr 125,340-354
Glacialreboundof the British Isles-
(a)
III
347
r@o
1020
l2
(Pa s)
7
5
Dl=50km
r¡r'=l.3x102Pas
,
(b)
Figure 7. Minimum-variance functions for fourJayer models with
D,:80 km.
Dz:200km
for further model improvement.Furthermore, the results so
far do not addressthe questionwhetherthe observationaldata
requiresthe introduction of a low-viscosityzone,as suggested
by Fjeldskaar& Cathles(1992)and Fjeldskaar(1994).Such
modelsare examinedbelow.
3 FOUR-LAYER
MODELS
Suggestions
that the glacialrebound data requirethe existence
of a low-viscositychannelgo back to the important works by
Daly (1934),van Bemmelen& Berlage(1935) and Cathles
(1975),and havebeenmost recentlysupportedby the detailed
work of Fjeldskaar(1994) (seealso Fjeldskaar& Cathles
L992\. To examinewhether the British data warrant such a
structure, the above procedureshave also been applied to a
four-layered earth model in which layer 2 (of viscosity 4r)
extendsfrom the baseof the lithosphereto a depth of 200km
below the surface(Fig.1). The third layer (of viscosity4.)
extendsfrom 200km down to the 670km discontinuity and
the fourth layer (of viscosity4tJ is the lower mantle between
o 1996RAS,GJr r25,340-354
(.)
loÐ
2
3
t
2
3
5
7
tM
ira"t"o:j
afld qtu\:7.3x1022
Pas.
(a) D1:30km;
(b) Dr:50kq
(c)
670km and the core-mantleboundary.The threeJayermodel
resultsindicated that resolvingpower for the viscosity of this
last layer is unlikely to be high, and the previouslydetermined
is adoptedin the first instance.
value of 4n:I3xI022Pas
The search through the model space for the remaining
parametersis then 30< D1< 80 km, 701e
<42< 1021 Pa s,
l0'o 116<1022Pa s,wilhqr<4.r<46. Only first-iterationsolutions will be consideredinitially. Fig. 7 illustratesthe minimumvariancefunction in tbe 4r-4, domain for three valuesof Dt.
A singlelocal minimum is found in eachinstance,and for any
value of lithosphericthicknessa well-definedsolution for both
r¡2 and 4s follows, although 42 is generallybetter determined.
For D1:30 km (Fig. 7a) the solutionpointsto a well-deflned
low-viscosity channel. Thus, if the lithospheric thicknessis
assumedto have a value of 30 km then the conclusionwould
be drawn that there is a low-viscosityzone immediatelybelow
the lithosphere.However, the minimum varianceobtained in
this caseis not the overall smallestvalue when differentvalues
for D, are considered.For 50 km, for example (Fig. 7b), the
overall minimum variance is signiflcantly reduced(Table2).
K. Lambeck eI al.
Table 2. Summaryof solutionsfor the fourJayer models.(a)Dr:266 ¡- for two differentvalues
of a¡", (b) Dz:400km, (c) Dr:150km, for specifiedvaluesof lithosphericthicknessD1 and
lower-mantleviscosity(units for all viscositiesare Pa s).
Dz ßrn)
(a) 200
(a) 2oo
(b) 400
Dl (km)
nz
q3
4rm
2.8x1O21
l.3xlo22
0.56 4.00
30
9x1019
50
1 3x1020
1.3x1021
1r3xlO22
0.76 2.65
65
2.SxL}zo
6x1020
l.3xlÙ22
1.00 2.85
80
5xl02o
5x1020
I.3xlo22
1.25 2.95
30
l.0x 1020
3xlÚr
o5xto22
0.58 4.70
50
1.1x1020
2.5x1}2r
0.5x1022
0.82 3.00
80
5x1020
5x1020
0.5x1o22
t.28 3.32
50
3x1020
2.5x102r
l.3xlÙ22
0.76 2,50
1
65
3.3x1020
2.0x1021
l.3xlO22
0.92 2.35
1
65
3;lxt02o
1.5x1021
l.3xlo22
0.91 2.21
n
80
4.Zxl}zo
l.2xl}zr
l.3x7ozz
r.t7 2.80
1
1.3x1022
0.61 3.72
(c) 150
5x1019
2xl}21
l02t
1
f
\2
3
2
2
3
5
7
n3
l02l
7
5
(c)
rl3
(d)
3
4
5
1
q3
FigureS. Minimum-variancefunctionsfor five-layermodels.4n:1.3x1022 Pas, D1:65km, and selectedvaluesfor the viscosity4o of the
t¡ansition zone.(a) q¿:5x102o, (b) 7x10'z0,(c) 1021,(d) 2xI021. Only modelsin which 4214214¿<4t^ are considered,
with Dr:2¡¡¡-,
D¡:400 km and D4:670km.
o 1996RAS,GJI 125,340-354
Glacialreboundof the British Isles-
III
349
Table 3. Summaryofsolutions for first-iterationflve-layermodelsfor specifiedvaluesofthe
lithosphericthicknessD, and the second-iterationsolution not including the correctionterm
for the eustaticsealevelfunction. Al1lower-mantleviscositiesare setto 7.3x 1022Pa s. The
last solution is for the modified British ice sheet(seetext) and includesthe correction for
the eustaticsealevelfunction.
D1
lst iteration
30(1)
50
n2
Y
Tl4
n3
1020
2xl02r
2xl02r
0.58 4.04
1.4x1020
7xl02o
2.5x102r
0.77 2.40
57.5 2.8x1020 3 . 1 x 1 0 2 0 2.5xI02rQ) 0 . 8 4 2 . 1 6
2nditeration
65
3.3x1020 3.3x1020
80
4.2xl}2o
0.92 2.35
1.2x702r
t.t7 2.84
5o(2) t.4x1o2o
ro21
2.5x1}2r
0.74 2.t5
5i.5Ø 2.2xto2o
5x1020
2.5x102r
0.81 2.10
2x1úr
0.89 2.27
l02r
1.11 2.80
65
3.5x1020 3.5x1020
8oG) 4.5xto2o
2nditeration
4.2x1020
2x102r
57.5
2x1}2o
4.5x1020
6x1020
2.5xt}2r
0.80 1.35
(1)The sea¡chthrough
42 and 43 was carried out for Dr : 30km with 4. set to 2 x 1021Pa s,
based on the values obtained for the other models for which the searchthrough the 4a
dimensionwas more comprehensive.
(2)Thesearchat thesevaluesof D1 was conductedthrough
nz-43 spaceonly, with 44 set to
2.5x 1021Pa s.
(3)Modelsrilith
44:1021Pa s only considered.
5x1020
1021
(a)
tl¡
(b)
'n3
Figure 9. Minimum-variance function for the frrst- and second-iterations,(excludingthe correctiveterm for the eustatic seaJevel) for 4¡-:
I.3 x 1022,
4¿:102r Pa s and Dr: 57.5km.
to
As Dt is increasedfurther, 41 increasesand q2 decreases,
thepoint wherc41:4r1or D1:80 km (Fig.7c)andthe solution
becomesidentical to the three-layeredmodels. This latter
solutioûis similar to that discussedin Part II (seealsoLambeck
1993c),in which the searchthrough the rangeoflithosphericthicknessvalues was only partially complete and which, in
consequence,
led to the conclusionthat there was no strong
evidencefor viscositylayeringin the mantle abovethe 670km
discontinuity. The correspondingminimum variance for this
solution is larger than that for smaller D1 and the global
minimum-variancesolution for this classof modelsoccursfor
50>Dl >65 km, with rtz-G-3) x 1020, and 4r=l02rPas
(Tab\e2). The second-iterationsolutions give essentiallythe
sameresultsfor the earth-modelparameters,but with a further
350 K. Lambeck et al.
ã
0)
F
C)
()
(.)
.ñ
Ø
I
q
G
1
5
1
0
5
0
time (x1000 yearsBP)
Figurel0. Rangeof estimatesof the eustaticsea-level
correctiontermsfor the five-iayered,
second-iteration
solutionswith D1:57.5km
corresponding
to earthmodelsin thevicinityof theminimumvariancemodelcompared
with simila¡resultsfor thethree-layer
model.
0.5
'40
ã
Q
^ 1
o
6
9)
l l z o
d
o
o
o
'tt
é
A
v.z
!¿
z
o
.g -20
I
0.1
_40
(a)
-
4
-
2
0
2
-60 L
-60
4
(observed-predicted/standarddeviation
(b)
40
-20
0
20
40
60
Observedrelativesealevel (m)
Figure 11. Statistical summary of the optimum frve-layer model. (a) Histogram of residuals normalized by the standard deviations of the
observations compared with the expected distribution of normally distributed data with a normalized variance of 1.4 times the assumed values for
the individual observations. (b) Linear regression plot of the observed and predicted sea-leveis. The open circles refer to the data points from the
Beauly-Inverness area.
reduction in the overall minimum variance.The conclusion
now would be that the observationaldata are consistentwith
some depth-dependentstructure in the upper mantle. Similar
results are obtained for a lower-mantle viscositv of
0.5x 1022Pa s, except that the overall variances are about
10-20 per cent greater than for the previous case (Table2),
confirming the earlier conclusionthat the lower mantle has a
considerablyhigher viscosity than the averagevalue of the
o 1996RAS,GJI tzs,340-354
Glacial reboundof the British Isles-
Glencrse & Bridge of Eam
Firth ofTay
Anprior
Firth of Fofl¡
III
Beauly Finh
Invemess
60
\
- J U
6
I
Í40
ú
=
w
I
{
v
\I
20
I
I g
rle
\''
{1'
:
2
Lochgilphead
Argyll
t
s
t0
Morecmbe Bay
-2.4
-10
-10
-15
-õ-rs
-20
ts
Bridgwater
Bristol Bay
l0
8
4
6
time(xl00oyeæ BP)
2
Exmoutlr
Devon
-35
0
-40
10
1.5
5
2.s
tine (x1000yem BP)
Fenlands
East Anglia
0
8
6
4
2
time(x1000yeæ BP)
0
Figure 12. Comparisonof observations(opencircleswith error bars) with predictionsfor selectedsitesusing the optimum five-layermodel (solid
lines),and, for comparison,the optimum threeJayermodel (dashedlines).Both solutionsinclude the seconditeration,and eustaticcorrectiveterms.
overlyingsub-lithosphericmantle.The thin lithospheremodels
requirequite low valuesfor the B parameter,and the predicted
maximum ice height is only of the order of 800m.
The overall minimum varianceobtainedfor thesefourJayer
modelsis lessthan that for the ûrst-iterationthree-layermodels
(compareFigs 7 and 5(a);seealso TablesI and2), indicating
that there may be somestructurein the depth dependenceof
the viscosity of the upper mantle, although there appearsto
be no caseto support the existenceof a pronouncednarrow,
low-viscositychannel beneatha thin lithosphere.Four-layer
o 1996RAS,GJr r25,340-354
modelsin which the boundary of the secondlayer is placedat
Dz:400 km (Fig. 1) give similar resultsto that of rhe Dt:
200km models,exceptthat the viscositiesare slightly modiûed
and the overallminimum varianceis further reduced(Table 2),
arguing againstmuch structure in the mantle viscosityabove
400km depth. Likewise,four-layermodelsin which D, is less
than 200km lead to smaller valuesfor 42 (Table2), but the
minimum varianceis significantlygreaterthan for any of the
mantle models without the pronouncedlow-mantle-viscosity
channelimmediatelybelow the lithosphere.
352
4
K. Lambeck et al.
FIVE-LAYER
MODELS
The resultsfor the four-layer models suggestthat the observational data may have someresolvingpower for the viscosity
structureof the upper mantle, and that, becauseof the tradeoff that occursbetweenthe upper-mantleparameters,further
exploration of the model spaceis warranted,particularly as
the leastvarianceis still significantlygreaterthan unity. In the
following seriesof models the mantle above 670km depth is
definedby the lithosphereas before,and by layersbottoming
(Fig. 1). Basedon the
at 200,400,and 670km, respectively
three- and four-layer-model results, the lower mantle is
assumedto have an effectiveviscosityof 4*:1.3 x 1022Pa s.
Comparisons
with earlierresultsbasedon 4h:0.5 x 1022Pa s
(Part II) conûrm that the higher-viscosityvalue leads to a
smalleroverallminimum-variancefactor without modifying in
any significant manner the estimatesfor the other mantle
parameters.
Only modelswifh 4t<4r<4q<qt ^ are considered
in the presentcalculations.Fig.8 illustratesthe first-iteration
resultsfor the caseDt:65 km, 4*: I.3xL022Pa s and for
discrete values of r¡a, the viscosity of the transition layer
between400 and 670km depth.For eachvalueof4a illustrated
in the four panels of Fig.8 the local minimum occurs for
1z-Ut, and,at leastfor this choiceof Dr, thereappearsto be
no evidencein support of viscosity layering in the mantle
betweenthe baseof the lithosphereand 400km depth. But, as
alreadyintimated by the fourJayer model results,the viscosity
of the transition zone may be considerablylarger than for the
sub-lithospheric mantle above 400km. Results for thicklithospheremodels are similar (Table3) exceptthat the minimum varianceis increased.For smallervaluesof the lithospheric thicknessa low-viscosity zoîe immediatelybeneath the
lithosphere is again implied (Table3), but such a model
representsonly a local minimum in the wider five-layer
modelspace.
The second-iterationsolutions result in a further reduction
in the overall minimum variance, although for Dr: fJ ¿rr¿
80 km the earth-model parameters remain essentially
unchanged(Table3). For the lower Dr values(Dr:57.5 and
50 km), however,some shift in the 4t-4, parametersis noted
(Fig.9; Table3). This apparentlycomesabout becausethe
characteristicspatialwavelengthof the second-iterationcorrections, determinedmainly by the gravitationalattraction of the
ice sheetsand by the surfacedeformationand the dimensions
of the North Sea,is of such a magnitudethat theseload terms
stress the upper mantle and contribute, therefore, to the
separationof the mantle parameters.The introduction of the
correctiveequivalentsealevel term further reducesthe minimum variance,to about 1.5,without any further shift in the
earth-modelparameters.The solution for the correctiveterm
itself (Fig. 10) is similar to that for the three-layermodels,
except that the magnitude of the correctionsfor the earlier
epochs is substantially reduced, pointing to the five-layer
models'providinga betterfit to the late-glacialobservational
data.
5
DISCUSSION
Within the range of models exploredhere, the model in best
overall agreement\ryiththe observationsis the f,ve-layermodel
with the followins effectiveDarameters:
55<Dl <60 km;
(2<ttr< ) x 1020
Pa s for (Dr<D<200)km;
(4<qr<6) x 1020Pa s for (200<D<400)km;
Pas for (400<D<670)km;
4¿-2x 1021
4*-à1022Pa s for (670<D<D"-6) km.
This is basedon the assumptionsthat 4z<42<4¿<I¡, and
that no other minima develop with smaller overall variance
beyond the exploredparameterrangeof approximately:
3 x l01e> 41,42,4t, 4¿,> 5 x 1021Pa s;
l}t'>rt^> 1023Pa s;
30>D1> 120km.
The above estimates(4) are effectiveparametersin the sense
that they describethe responseof the planetto surfaceloading
well on time-scalesof the order of 10oa. with stressdifferences
of the order of 10 MPa, and based on the assumptionof a
Maxwell viscoelasticrheology.Models with a viscosityinversion, other than at the base of the lithosphere,have not at
presentbeenexamined.Models with, for example,a viscosity
varying more smoothly with depth may also flt the observational sea-leveldata equally wel1,but suchmodelshave not
beenexamined.The resultsindicatethat thereis no compelling
casefor a strong increasein the mantle viscositybetweenthe
baseof the lithosþhereand the 400 km seismicdiscontinuity
and that a narrow low-viscosity channel immediatelybelow
the lithosphereis not required.But the solutionsdo point to
a substantialincreasein the viscosity,by a factor of three or
four betweenthis upper-mantleregionand the transition zone.
A further increase of viscosity by an order of magnitude
appears to be required across the 670km boundary. The
resolving power of this data for the viscosity of the lower
mantle is poor, and no attempt has been made to establish
whether depth dependenceof viscosityoccursbelow 670km.
The surfaceresponseto the loading of the larger ice sheets
may,however,helpto resolvethe questionof viscositystructure
in this region,as well as improving upon the resolution of the
structureabove670km.
With the exception of the possiblezonation betweenthe
baseof the lithosphereand 400km depth, the abovefeatures
of the mantle rheology appear to be robust and independent
of detailsof the solutions,suchaswhetherthe second-iteration
termsare includedor not, although the inclusionof theseterms
doesimprove the overall fit of the model to the data. Also, as
discussedin Part II and in Lambeck (I993c),the solutionsfor
the earth-modelparameters,basedon surfaceloadsof relatively
small extent, appear to be independentof the assumptions
made about the nature of the responseto loading of the phase
transformation boundaries at 400 and 670km. The present
modelsassumethat the kinetics of the phasetransformations
are slow enough such that they do not occur on the timescalesconsideredhere,and that densitycontrastsacrossthese
boundariesare advectedwith the flow, but modelswith rapid
phase-transformation
kineticsdo not yield very differentresults
for the Great Britain loading problem, other than a modification of the ice-scalingparameter,B, which is reduced by
about 10 per cent. Thus, for this scaleof ice load at least,and
provided that the ice-scaling parameter is introduced, the
assumptionof no exchangeof mass acrossthe boundariesis
adequate(seealso Johnston,Lambeck& rWolf1996).
The resultsfor the multi-laveredmodelsillustratethe tradeo 1996RAS,GJr l2s,340-354
Glacialreboundof the British Islesoffs that can occur betweenthe various parametersthat define
the Earth's response.They also illustrate the consequences
of
not searchingthrough a complete range of plausible model
space,although the expandedsearch does incur the cost of
very substantial computing time. If, as done by Fjeldskaar
(1994), the lithospheric thicknessis txed at a relatively low
value ()50km) then the solution points to a low-viscosity
channelbeneaththe lithosphere,a result that is reinforcedif
the thicknessof this channelis also reduced,such as the fourlayer model with D2:150km (Table2). But thesesolutions
representlocal minima only, and more solutions covering a
wider parameter spacedo not support such results, at least
not for the Great Britain region. Likewise,the imposition of a
thick lithosphereon the viscositystructure,without conducting
a full searchof the model space,leads to a local minimum
solution that does not correspond to the global minimum
variance.Thus, an adoption of a relatively thick lithosphere,
of say 100-120km, as usedin the recentmodelsof Peltierand
colleagues(Mitrovica & Peltier 1993;Tushingham& Peltíer
1992),leadsto an overestimation
of the upper-mantle
viscosity
and to models in which the contrast between upper- and
lower-mantleviscosityis much lessthan that found here (see
e.g. Fig. 2c). However, such solutions representonly local
minima within a larger earth-modelparameterdomain.
If independentinformation on the appropriate choice for
lithospheric thicknesswere availablethen this would significantly improve the solutions for the viscosityproûle because
it would eliminate some of the correlations that otherwise
occur betweenthe various parameters.Through the representation of the lithosphereas a simple elasticlayer, its effective
thicknesswill be dependenton the duration of loading because
of the tendencyfor the load stresses
within this layer to migrate
with time from the weakerzonesinto the strongerzones.Thus,
seismicdata will generallygive relatively large values for Dr,
whereasestimatesbasedon tectonicstudieswith characteristic
loading time-scalesof 106to 107a will usually lead to lower
values. The glacial-rebound estimates,appropriate for the
responseto loading on time-scales
of the order of 103to 10s
a, are expectedto fall betweentheselimits, but exactly where
processes
dependson the stress-relaxation
in the layer.Realistic
a priori assumptionsabout the lithosphericthicknesscannot
be madefor this problem, and the estimationof this parameter
from the glacial-rebound studies is, in fact, an important
element in reaching an understanding of the lithospheric
responseto loading over a wide rangeof load cycles.
One consequenceof the trade-off betweenthe earth-model
parametersis that it is not permissibleto mix models,as has
sometimesbeen done. For example, Fjeldskaar & Cathles
(1992) combined a set of mantle-viscosityparametersof
Lambeck et aI. (1990), correspondingto a thick-lithosphere
(D1:200) model(whichthe latter rejectedon the groundsthat
it producedan excessively
large correctionto the eustaticsealevel),with their own thin-lithospheremodel to arguethat the
former viscosity parameters do not lead to a satísfactory
estimate of present changesin seaJevelfor the Scandinavia
area.But theseviscosityparametersare only valid if usedwith
the correspondingD, value,and, if the requirementof internal
consistencyof model parametersis maintained,the model does
lead to reasonablepredictionsof the glacio-hydroisostatic
contribution to presentsea-levelchange[see,for example,the
predictedcontributionsto presentsea-level
for Great Britain
and the North Sea in Lambeck(1993d,e) for earth models
III
353
with D1:65 km and no low-viscositychannel].The trade-off
between ice- and earth-model parameters may also have
important consequences.
Clearly, any ice-model parameters
inferredf¡om relative sea-levelobservationswill be a function
of the adoptedearth-modelparameters.A model with a thick
lithospherewill, for example,lead to an increasein ice volume
in such an inference,but the solution will not representa
global minimum unlessa completesearchthrough both iceand earth-modelparameterspaceis carriedout.
As alreadynoted in Part II the optimum three-layermodel
describesthe characteristicpatterns of the sea-levelchange
observedaround the British Isles well and gives results that
are very similar to the optimum five-layermodel for most of
the sites. Thus the former model is largely adequate for
exercisessuch as predicting sho¡elineevolution or estimating
ice heights.Fig. 11 illustratessome of the statisticsof the
comparisonof the optimum five-layersolution with the observational data. The histogram of the residuals,the observed
minus the predictedsealevelsnormalizedby the observational
standarddeviations,is illustratedin Fig.11(a).The residuals
are consistentwith normally distributed observationalerrors
with standarddeviationsthat may be about 20 per centgreater
than the assumedvalues.The plot of observedversuspredicted
sealevelsconfirmsthe generallygood agreement,with a linear
regressioncoefficientof 1.00.The small group of outliers (open
circlesin Fig.11b), falling below the linear regressionline,
correspondsmainly to observationsfrom the Beauly and
InvernessFirths wherethe modelsystematicallyunderestimates
the rebound.This is typified by the Beaulyresult illustrated in
Fig.12, and it has previously been concluded,from observations in this region,as well as from data from north-western
Scotland,that this points to a deficiencyin the ice load over
northern Scotland,north of the Great Glen, at the time of the
last glacialmaximum (Part II; Lambeck l995a,b).Comparisons
of predictionsand observationsfor someof the other sitesare
also illustrated in Fig. 12, and these,also,point to a generally
good agreementbetweenthe two.
Testswith the presentice model, as well as with a modified
ice model, in which the ice thicknessove¡ northernmost
Scotlandhas been increased(seePart II, Section5), have
shown that the limitation of the ice model usedin the above
analysesdoesnot modify the mantle parametersin any significant way, provided that the overall scalefactor BGBis introducedas an unknown. The modified ice model,with increased
ice over northern Scotland,does lead to a further variance
reduction, and for the optimum fiveJayermodel parameters,
including the far-field correction, this is reduced to 1.35
(Table3). Important new observationaldata for this region
are now being collectedby I. Shennanand colleagues,and this
should provide important constraintsin future modelling of
the ice sheet.
ACKNOWLEDGMENTS
This researchhas been partly funded by the Ministerie van
Verkeer en Waterstaat,Directoraat-GeneraalRijkwaterstaat,
the Netherlands.
REFERENCES
Boulton,
G.S.,Smith,G.D.& Morland,
L.W.,1984.
Thereconstruction
of former ice sheets and their mass balance characteristics using a
nonlinearly viscous flow model, J. Glac., 30, 140-1,52.
354
K. Lambecket a7.
Boulton, G.S.,Smith, G.D., Jones,A.S. & Newsome,J., 1985.Glacial
geologyand glaciology of the last midlatitude ice sheets,J. geol.
Soc.Lond., 142, 447-47 4.
Cathles,L.M., 1975. The Viscosityof the Eørth's Mantle, Pnnceton
University Press,Princeton,NJ.
Daly, R.4., 1934.The ChangingWorld of the lce Age, Yale University
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