Geophys. J . Int. (1993) 115, 960-990
Glacial rebound of the British Isles-11.
precision model
A high-resolution, high-
Kurt Lambeck
Research School of Earth Sciences, The Australian National University, Canberra ACT 0200, Australia
Accepted 1993 May 10. Received 1993 May 7; in original form 1992 September 11
SUMMARY
Observations of ice movements across the British Isles and of sea-level changes
around the shorelines during Late Devensian time (after about 25 000 yr BP) have
been used t o establish a high spatial and temporal resolution model for the rebound
of Great Britain and associated sea-level change. The sea-level observations include
sites within the margins of the former ice sheet as well as observations outside the
glaciated regions such that it has been possible to separate unknown earth model
parameters from some ice-sheet model parameters in the inversion of the glaciohydro-isostatic equations. The mantle viscosity profile is approximated by a number
of radially symmetric layers representing the lithosphere, the upper mantle as two
layers from the base of the lithosphere to the phase transition boundary at 400 km,
the transition zone down to 670 km depth, and the lower mantle. No evidence is
found to support a strong layering in viscosity above 670 km other than the
high-viscosity lithospheric layer. Models with a low-viscosity zone in the upper
mantle or models with a marked higher viscosity in the transition zone are less
satisfactory than models in which the viscosity is constant from the base of the
lithosphere t o the 670 km boundary. In contrast, a marked increase in viscosity is
required across this latter boundary. The optimum effective parameters for the
mantle beneath Great Britain are: a lithospheric thickness of about 65 km, a mantle
viscosity above 670 km of about (4-5) lo2"Pa s, and a viscosity below 670 km greater
than 4 x lo2' P a s .
Key words: British Isles, glacial rebound, mantle viscosity, sea-level.
1
INTRODUCTION
In an earlier paper (Lambeck 1993a, referred to hereafter as
Part I) it was established that the complex pattern of
sea-level indicators that formed around the British coastline
during the past 15 000 years, provides an important data set
for estimating the mantle response to changes in surface
loading produced by the melting of the Late Devonsian ice
sheet over the region, and by the concomitant rise in
sea-level produced by the melting of the much more
voluminous ice sheets of Fennoscandia and Laurentia. It
was established that both the effective lithospheric thickness
and a simple depth dependence of effective mantle viscosity
can be inferred from the observational data. Furthermore it
was shown that certain constraints on the areal extent and
thickness of the ice sheet can also be inferred from the
observations of past sea-levels. Simple models were
developed that illustrated the principal characteristics and
requirements for a model of the glacio-hydro-isostatic
rebound that is consistent with the observational evidence.
960
The recognition that Scotland had been ice covered in the
recent geological past goes back to A. Agassiz in 1840, and
A. Geikie in 1865 produced a map of the ice-movement
directions that is quite similar to the reconstruction by
Sissons (1976) more than 100 years later (Price 1983).
Despite the much greater information base now available,
knowledge of the ice sheets at the time of, and subsequent
to, the last glaciation remains limited. Denton & Hughes
(1981, p. 311), for example, conclude in a more general
context that 'our most important conclusion is that the
distribution of Late Wisconsin-Weichselian ice sheets is not
well known'. Sissons (1981) concludes that 'after 140 years
of research it is surprising how little we really know about
the extent of the last Scottish ice sheet and associated
events'. With this state of affairs it is perhaps a bold or
foolish action to propose any model for the last British ice
sheet but this has been done here in the spirit of
establishing, (1) what information is required for the
glacial-rebound calculation, (2) what information about the
ice sheet can be extracted from the observations of relative
Glacial rebound of the British Isles-I1
sea-level change, (3) what information can be extracted
from these observations about the Earth’s response to
surface loading, and (4) what information about coastal
evolution can be extracted from these rebound models. The
first two questions have been examined in Part I which
established the need for a relatively high-resolution ice
model in order to yield satisfactory predictions of the
temporal and spatial changes in sea-level observed in the
British Isles, particularly in Scotland.
In this paper a new detailed ice-sheet model for the Late
Devensian ice sheet has been developed (Section 2) from
the time of maximum glaciation to the end of the Loch
Lomond Advance ice sheet soon after 10000yrsBP,
together with an approximate model for the ice movements
during Middle Devensian time after 50 000 yr BP. In Section
3 the observational evidence for past sea-levels in Scotland,
England, Wales and the Channel coast of northern France
has been reviewed, and a series of relative sea-level curves
has been established for sites both within and without the
former glaciated region. To address the third question raised
above, this observational data base, together with the new
ice-sheet model, provide the input for the inversion in
Section 4 of the equations defining the glacio-hydroisostatic sea-level change through time. Certain ice-model
parameters are also determined in this inversion. In the
inversion the earth model is defined by realistic depthdependent elastic moduli and density as determined by
seismological methods and by a depth-dependent radially
symmetric linear viscosity structure for the lithosphere,
upper mantle, transition zone and lower mantle. Section 5
discusses the resulting earth- and ice-model parameters.
This section also includes some preliminary comparisons of
the predicted sea-level changes since the last glacial
maximum with the observed changes, but a more complete
discussion of some of the geomorphological consequences
will be given elsewhere.
2
A NEW I C E - S H E E T M O D E L
The predictions in Part I based on the ice model derived
from the Boulton et al. (1977) and Andersen (1981) results
indicated that major modifications to the ice sheet are
required. Also, since these reconstructions new evidence of
the ice movements across Great Britain and the surrounding
shallow sea-floor has accrued and this has been used to
develop a new model for the last maximum glaciation and
subsequent retreat. The principal changes from the
Andersen (1981) isochrone model and the Boulton et al.
(1977) reconstruction concern the extent of the ice cover
over the continental shelf and the North Sea, the initial rates
of retreat of the ice and the ages attributed to some of the
features defining the position of the retreating ice margin,
and the initial thickness of the ice sheet at maximum
glaciation.
Timing, areal extent and ice thickness of the Late
Devensian maximum glaciation
The maximum extent of the ice sheet over the British Isles
appears to have been reached between about 25000 and
20 000 yr BP (e.g. Huddart, Tooley & Carter 1977; Bowen
& Sykes 1988; Eyles & McCabe 1989) although the precise
961
timing of this occurrence remains uncertain and may not
have been synchronous along all fronts. Sissons (1981), for
example, suggested that the ice sheet over northern
Scotland was stationary or even retreating at a time when it
was still advancing southwards over England. In East
Yorkshire, the ice sheet may have continued to advance
until about 18 000 yr BP, the Dimlington Advance, (Penny,
Coope & Catt 1969; Rose 1985), whereas the advance from
the Irish Sea across Cheshire and Shropshire probably
occurred earlier, at perhaps 22 000 yr BP. A nominal date of
22000yrBP has been adopted here for the time of
maximum glaciation of the entire British Isles region, with
only minor changes in ice volume occurring up to about
18 000 r BP. Andersen’s (1981) estimate of the areal extent
of the [ce sheet at this time does require some modification.
Thus, the accumulating evidence that the North Sea was
largely ice free during Late Devensian time (=2600010000yrBP) is accepted and the British ice sheet is
assumed to have terminated at the Wee Bankie Moraine off
the coast of eastern Scotland (Eden, Holmes & Fannin
1978; Sutherland 1984) (Fig. 1) and to have been stationary
here between about 22000 and 18000yrBP. The Norwegian evidence also points to the Scandinavian ice sheet
not having extended across the North Sea as far as Scotland
and to the existence of an ice-free corridor between the two
ice sheets during the last glacial maximum (e.g. Larsen &
Sejrup 1990; Sejrup et al. 1987). The Shetland Islands
appear to have been largely unaffected by the lastest
Scottish and Scandinavian ice sheets and any evidence for
glaciation and ice transport across them is interpreted as
being the result of a local ice sheet or of an earlier glacial
cycle (Sissons 1981; Sutherland 1991). In the proposed
model (Fig. 2) the Orkney Islands and parts of Caithness
and Buchan were ice free during the Late Devensian
(Sutherland 1984; Hall & Connell 1991) and the nearby ice
front is assumed to have been stationary between about
22000 and 18000yr BP (Figs 1 and 2a). The Outer
Hebrides supported its own ice sheet (Peacock 1991) and
the western limit does not extend as far across the shelf as it
does in the models of Andersen (1981) or Boulton et al.
(1977) in which the Outer Hebrides ice is an integral part of
the main ice sheet and extends across the continental shelf
as far as St Kilda (Sutherland, Ballantyne & Walker 1984).
Instead the Outer Hebrides ice as assumed to have retreated
to a local ice sheet and cleared the Minches after about
18 000 yr BP (Fig. 2a). Also, the extent of the ice into the
Sea of the Hebrides and the Malin Sea is considered to have
been the result mainly of outflows from the Little Minch and
the Firth of Lorne (Davies, Dobson & Whiting 1984) and to
have been less extensive than proposed in the Andersen or
Boulton et al. models.
Furthermore, in this model the Scottish ice moved across
the north Channel of the Irish Sea but did not penetrate far
into Ireland during Late Devensian time (Stephens,
Creighton & Hannon 1975; Hoare 1991). Instead, the ice
flow was deflected southwards to form the Irish Sea Glacier
which extended as far as the southern entrance to St
George’s Channel (e.g. Garrard & Dobson 1974; Eyles &
McCabe 1989), fed also from the east by an ice sheet
centred over Wales and from the west by the Irish ice sheet.
The ice front across southern Ireland has been taken to
coincide with the South of Ireland Moraine and is given an
962
K . Lambeck
Figure 1. Location map of main geographic areas mentioned in text and the limit of the ice sheet at the last glacial maximum. The major
terminal moraines are: (1) Wee Bankie Moraine; (2) South of Ireland Moraine; ( 3 ) Galtrim Moraine; (4) Drumlin Readvance Moraine and its
extension into the Irish Sea.
age of 22000yr BP (Eyles & McCabe 1989). The initial
northwards retreat of the ice across southern Ireland and the
Irish Sea is assumed to have been rapid such that by
18000yr BP the ice margin stood north of the Galtrim
Moraine and at the Isle of Man (Stephens & McCabe 1977)
(Figs 1 and 2a), consistent with ice-free conditions having
existed in northeastern Wales by this time (Bowen 1973).
Along the western margin in Ireland the ice sheet -is
assumed not to have extended much beyond the present
shoreline such that the coastal fringe was ice free before
17000yr BP (McCabe, Haynes & MacMillan 1986; Bowen
et al. 1986).
The southern limit of the ice sheet over Wales and
England appears to be reasonably well constrained and the
limits given by Andersen (1981) have been largely adopted
here, with ice-free conditions existing locally in both
Pembroke and Glamorgan (e.g. radiocarbon dates BM 374
and Birrn-340 of 18 460 and 22 350 yr BP respectively). The
ice cover over Wales and dominated by a local ice sheet that
formed over the high ground and coalesced to the west and
Glacial rebound of the British Isles-II
T = 22000 aBP
963
T=18000aBP
T = 16ooo aBP
Figure 2. Isochrones and ice-thickness estimates for the retreat of the ice sheet over the British Isles at selected epochs (a) from 22000 to
12 750 yr BP and (b) for the Loch Lomond Advance and subsequent retreat. Growth of this new ice cap is assumed to have started at
12 500 yr BP.
north with ice that advanced down the Irish sea basin from
Scotland. The flow across Shropshire and Cheshire
terminated at Wolverhampton (e.g. Thomas 1989) but there
does appear to be a potential problem of timing here for if
the ice had retreated from the southern part of the Irish Sea;
by 18000 yr B P the maximum advance into Shropshire, fed
by ice flow from the Irish Sea, would have occurred earlier
and a nominal date of 22 000 yr B P has been adopted for this
advance. The ice front has been identified in the Peak
District at the southern end of the Pennines and the more
recent evidence is consistent with Andersen’s identification.
The evidence for eastern Yorkshire suggests that a Late
Devensian ice flow occurred over the Holderness Peninsula
of northeast England, through eastern Lincolnshire, to the
northern edge of Norfolk and into the southern North Sea
south of Dogger Bank (Boulton et a f . 1985; Long et al.
1988). It is assumed here that this ice flow was a relatively
short-lived phenomenon or surge and that it had vanished
soon after 18 000 yr B P (see Fig. 2).
Two contrasting views of the last glacial maximum ice
thickness over the British Isles are exemplified by the
maximum and minimum ice sheet reconstructions of
Boulton et al. (1977, 1985) (see, Fig. 7 of Part I). In their
maximum model the ice thickness exceeds 1800 m and the
ice sheet covers the peaks of the highest mountains. Others
have argued for much thinner ice such that the higher
964
K . Lambeck
mountain tops were exposed as nunataks. Geomorphological evidence such as striae, erratics and trim lines can in
some instances place constraints on maximum ice thickness
(e.g. Ballantyne 1984) and they point to values of typically
1000-1200 m over Central Scotland. Thus Sutherland (1991)
expresses a consensus view that the maximum ice thickness
here is unlikely to have exceeded much more than 1300m.
On the Outer Hebrides, estimates of ice thickness are of the
order of 400m (Peacock 1980) while the evidence of an
absence of erratics on the higher peaks in the Southern
Uplands of Scotland suggest that here the ice thickness is
unlikely to have exceeded about 1OOOm. Ice thickness
estimates for the Central Plain of Ireland are about
400-500m according to Watts (1977) with the higher
mountains standing out as nunataks. Such reduced
ice-thickness estimates also appear to be consistent with the
evidence from flow direction indicators that point to a
polycentric ice sheet with at least two major ice foci, one
over Rannoch Moor in the north and the other over the
Southern Uplands in the south (Figs 1 and 2a), and a
number of subsidiary ice centres over Scotland, northern
England and Ireland.
Ice retreat during Late Devensian time
C
Figure 2. (Continued.)
After 18000yrBP the ice retreat is assumed to have
occurred without major standstills or readvances until about
12 500 yr BP when the ice sheets had vanished throughout
Britain and Ireland. In eastern Scotland, the retreat of ice is
marked by numerous Late Devensian shorelines formed at
the ice margin, by the deposition of marine sediments and
by glacial outflows, although the age control on these
features is limited. Thus at time of the formation of the East
Fife Shorelines, the ice margin had retreated across the low
grounds of eastern Fife, Kincardine and eastern Lothian and
locally may have retreated up the Forth and Tay valleys (see
Fig. 3 of Part I and Fig. 4 later for locations). Of this series
of shorelines, the youngest, (the East Fife 6 shoreline), has
been associated with the deposition of the arctic-faunabearing marine Errol Beds with a nominal age of
14 750 yr BP (Paterson, Armstrong & Browne 1981) and this
is adopted here. By the time of the formation of the Main
Perth Shoreline, nominally at about 13 500 yr BP (Paterson
et al. 198l), ice had retreated up the Forth and Tay River
valleys and areas such as the upper Teith Valley west of
Stirling were ice free by 12 750 yr BP (Gray & Lowe 1977).
In contrast, in Andersen’s reconstruction the ice front at
14 000 yr BP was still to the east of the Wee Bankie Moraine
and did not retreat to eastern Fife until about 13 500 yr BP
(see Fig. 9 of Part I).
The oldest Late Devensian radiocarbon age recorded in
northeastern Scotland is 16 630 f 650 yr BP for deposits
offshore from Banff (Sissons 1981) and this area is assumed
to have been ice free by this time. The Cromarty and Moray
Firths may have been occupied by ice until about
13 500 yr BP (Peacock, Graham & Wilkinson 1980) and the
ice front is considered to have stood near the upper Moray
Firth at about this time. As emphasized by Firth (1989), the
age of the ice movements across this region is poorly
constrained, but the evidence does suggest an ice margin at
14 000 yr BP that lies considerably further to the west than
assumed by Andersen (1981), consistent with other evidence
Glacial rebound of the British Isles-Il
from the Spey Valley of rapid ice retreat from the eastern
Highlands by about 13 500 yr BP (Sissons & Walker 1974).
In southwestern Scotland the ice retreated to the north of
the Solway Firth and most of the Southern Uplands are
believed to have been ice free well before 13000yrBP
(Lowe & Walker 1977; Bishop & Coope 1977). Also,
evidence from western Scotland sites such as Cam Loch
(Pennington 1977), southwest Mull (Walker & Lowe 1982)
and Lochgilphead (Peacock et al. 1977), suggest that much
of the coastal area of western Scotland was ice free before
13 000 yr BP. Taken together, the evidence suggests that a
very large part of Scotland was ice free at or soon after
about 13000yrBP (Sissons 1981; Price 1983) and by
12500yrBP, the Late Devensian ice sheet is assumed to
have vanished everywhere over Scotland. Hence the late
Glacial retreat over west and southern Scotland in this
model occurs significantly earlier than suggested by
Andersen's model in which much of the region was still ice
covered at 13 000 yr BP (Fig. 9 of Part I).
In Ireland, the early retreat of the ice sheet from south to
north is assumed to have been rapid, based on an age of
17 000 yr BP for the Main Drumlin or Kell Drumlin Moraine
(McCabe 1987; McCabe et al. 1987) rather than the
14 000 yr BP adopted by Andersen (1981). The northern
limits of the 'Drumlin event' moraines (Fig. 1) are taken
to define the ice limit at 17000yrBP along the northern
and north-western margins (McCabe et al. 1986) and by
14 000 yr BP Ireland appears to have been largely ice free
(McCabe 1987; Carter 1982). Retreat of ice from the Irish
Sea is also assumed to have been rapid, consistent with the
17 000 yr age for the Kells Drumlin Moraine extension on to
the sea-floor (Stephens & McCabe 1977), so as to have been
ice free by about 15 000 yr BP.
Few precise constraints appear to exist on the retreat of
the ice across Wales and England during the Late Glacial.
Ice-free conditions may have occurred in northwestern
Yorkshire as early as 17 000 yr BP (Gascoyne, Schwarz &
Ford 1983) and, apart from local residual ice sheets such as
that over Cumbria, the main ice sheet appears to have
retreated north of the Scottish border by 15000yrBP.
Cumbria itself is assumed to have been ice free by
13500yr BP (Gale 1986) and all glacier ice over Wales
vanished before 14 000-13 500 yr BP (Haynes et al. 1977;
Musk 1986).
Figure 2(a) illustrates the proposed isochrones for the ice
sheet over the British Isles from 22000yrBP to
12 750 yr BP. In addition to the arguments and observations
summarized above, other information used to construct
these isochrones have included the Late Devensian patterns
of frontal retreat over northern England given by Boulton,
Smith & Morland (1984) and the patterns of longitudinal
and radial features, primarily drumlins and striae, left by the
retreating ice (Boulton et al. 1985).
The Loch Lomond Readvance
The period of climatic improvement that started at about
14 000-13 500 yr BP and which led to the rapid collapse of
the Late Devensian ice sheet came to an end by about
12 500 yr BP and was followed by a gradual climatic cooling,
culminating with the Loch Lomond Stadia1 in Scotland
between about 10 800-10 300 yr RP (Price 1983; Sissons
965
1974). The extent of the resulting Loch Lomond Advance
ice sheet has been well documented for the Western
Highlands with the ice again extending down to the Clyde
estuary and into the upper valleys of the Forth and Tay
rivers (Sissons 1974, 1981). Local ice sheets developed over
the Inner Hebrides islands of Mull and Skye but no major
glaciation appears to have occurred beyond the Western
Highlands of Scotland. Ice thicknesses were never sufficient
to cover the higher peaks and the morphological indicators
suggest values of 400111, reaching 600m locally (Sissons
1974; Thorp 1986). Elsewhere in the British Isles only minor
ice sheets formed as mountain glaciers at the time of this
cold stage. Fig. 2(b) illustrates the adopted isochrones for
this short-lived advance and retreat in which all glacier ice
was finally gone by 9500 yr BP.
Ice model for 22 000-9500 yr BP
Ice heights through time have been established using eqs 4
and 5 in Part I, and the same procedure as discussed in Part
I. At the time of maximum glaciation t,,, the heights
hmax(tmax)
at the ice foci, are based on the estimates
discussed above and, with smaxestablished from the
maximum ice limits, this determines the coefficient d i )(eq.
4b of Part I) for profiles ( i = 1 . . . 9) radiating out from the
ice centres. The maximum ice heights at these foci for
subsequent epochs t are given by
h,,,(t)
1
=J
c
"(i)[s:;x(t)]"'.
i
Ice heights along the profiles then follow from eq. 5 of Part
I). Fig. 2 illustrates the resulting model at selected time
intervals. Altogether, the model of Late Devensian ice
retreat is defined at the time of maximum glaciation of
22000yrBP, then at 1OOOyr time steps from 20000 to
14000yr inclusive and thereafter at 500 year intervals until
9500 yr BP. The resulting ice model has been digitized with
a spatial resolution of 0.25" in latitude by 0.5" in longitude,
or approximately 25 km by 25 km.
Lower-middle Devensian glacial cycles
During Early Devensian time, before about 50 000 yr BP,
much of England and Ireland appears to have been ice free
(e.g. Shotton 1977; McCabe 1987). There also appears to be
little evidence for early Devensian glaciation in Scotland
before this time (e.g. Bowen et al. 1986) although some
authors have suggested that the build-up of the ice sheet
over Scotland may have started as early as 75000yrBP
(Sutherland 1984). Price (1983) has drawn a tentative
summary of environmental conditions in Scotland and
suggests that glacial/periglacial environments existed here
during 78 000-65 000 yr BP and again during 55 00050 000 yr BP, with interstadiai conditions between these two
cold intervals as well as in the interval 50000-45000BP.
Renewed ice-sheet development over Scotland possibly
occurred in the interval of 45 000-32 000 yr BP. Palaeotemperature records, particularly mean summer temperature, provides an indicator of the likelihood of ice-sheet
growth during Devensian time. Such temperature records
have been proposed by Coope, Morgan & Osborne (1971)
966
K . Lambeck
l8
..
'-
I
,
,
---
r
I
Western
'
Scotland
4
t
I
0 -
3
s'
0.1 ;
b
I
0
.-pE 0.2
-2
P
0.3
-
:
0.4 -
30
20
10
0
time ( xlo00 years BP)
Figure 3. Palaeotemperature records for central England and
western Scotland (top) and the equivalent sea-level function for the
past 50000 yr (bottom).
50
40
for lowland England, by Price (1983) for western Scotland,
and by McCabe (1987) for Ireland (Fig. 3) and, within
observational uncertainties, the results for the three regions
exhibit similar fluctuations so that a simple relationship
between temperature and ice-sheet extent can be established. For example, average summer temperatures in
central lowland England in the interval 50 000-30 000 yr BP
do not appear to have dropped below those at sites near the
edge of the ice sheet during the last glacial maximum and it
can be assumed that the ice sheet did not extend far south
into England at any time in this interval. At 35 000 yr BP, as
well as at 27 000 yr BP, temperatures were similar to those
at 15000yrBP and the extent of the ice sheet at these
earlier epochs is assumed to have been the same as that at
15 OOO yr BP. Fig. 3 illustrates the resulting equivalent
sea-level curve for the British ice sheet for the past
50 000 yr BP and it compares well with the model previously
derived from the evidence of ice movements across the
northwest shelf of Scotland and across the North Sea (Eden
et al. 1978) (Fig. 13 of Part I). Insofar as sea-levels are
predicted only for the Late- and Postglacial stages such
approximate models for the earlier glacial cycles suffice (see
Part I).
3 A SUMMARY OF OBSERVATIONS OF
SEA-LEVEL CHANGE
Quantitative evidence for sea-level change along the coast of
Great Britain comes from a variety of different sources, the
particular problems of interpretation of which have been
discussed in the volumes edited by van de Plassche (1986),
Smith & Dawson (1983), Tooley & Shennan (1987) and
Pirazzoli (1991). An important source of information,
particularly for change during the latter part of the
Holocene, is from the aquatic and herb flora of peats
defining marine transgressions or regressions. Often these
peats are intercalated with marine or freshwater sediments
so that it becomes possible to deduce both the position of
past shorelines and indicators of whether sea-levels were
rising or falling at the time of deposition. In other examples,
the past shorelines are inferred from the position of fossil
intertidal or shallow-water marine molluscs, or from
erosional features left when the sea retreated, such as the
rock platforms and shingle beaches found in western
Scotland. Generally, the height formation will not
correspond to mean sea-level, the reference level used in all
model calculations, and the successful interpretation of the
observations in terms of sea-level change requires a
knowledge of the relation of the stratigraphic boundaries or
shoreline features to sea-level. Thus the peat sequences
corresponding to the freshwater-to-marine transgressions,
will usually correspond to a level near mean high-water
spring (MHWS) tide although the actual formation range
about this level may be quite large (e.g. Shennan 1986) as well
as a function of tidal amplitude (e.g. Heyworth & Kidson
1982). Additional uncertainties result if the peats formed in
a backswamp rather than in an open coastal environment,
because in the former case the formation height may be near
the middle of the tidal range rather than near the MHWS
level. Compaction of the peats and consolidation of the
underlying sediments add still further uncertainty to the
inferred sea-levels. Other shoreline markers present a
similar range of problems. For example, shingle beaches
define a maximum level that generally lies above MHWS
tide because of storm activity and the highest astronomical
tide level may provide a more appropriate reference height.
In contrast in situ molluscs define a lower part of the tidal
range or a limiting depth below the tidal range, depending
on species. Because of these different formation levels it is
important that all shoreline markers are reduced to the same
level, particularly in a region such as the Solway Firth where
the tidal range presently exceeds 10m, or in the Bristol
Channel where it exceeds 14 m. Mean tide level is the most
appropriate level for this and all shoreline information will
be reduced to this level, with the questionable assumption
made that the tidal range has remained constant throughout
a time interval in which major changes in sea-level have
occurred in environments of indented shorelines and
variable topography. Because the tidal range can vary
significantly within a region, corrections have been applied
individually to the data points at each location using the
tidal amplitude information compiled by Shennan (1992).
The height measurements of the shorelines are usually
expressed relative to the British Ordinance Datum (OD) at
Newlyn in the south of England. Zero-height O D does not
generally coincide with mean sea-level (MSL) which lies at
up to 0.35 m O D in northern England and up to 0.5m O D in
Scotland and all heights have been reduced to MSL using
the corrections given by Thompson (1980) except in a few
instances where heights refer to a local datum in which case
the reference is assumed to approximate MSL.
Glacial rebound of the British Isles-II
The quoted precision of the height determinations based
on the peat evidence is generally small, of the order of
0.1 m, but the accuracies of the inferred sea-levels will be
considerably lower, of the order of f l m or more when
some of the above-mentioned uncertainties are taken into
consideration. These error estimates are comparable to the
amplitudes of the minor oscillations in sea-level inferred, for
example, by Tooley (1978) or Shennan (1986) and the latter
are considered as local ‘noise’ superimposed on the more
regional sea-level trends. The resulting sea-level trend is,
therefore, represented by a smooth curve through the MSL
estimates with upper and lower limits indicated by the
spread of depths of the individual observations and by the
accuracy estimates of the observations.
All ages used for the shorelines and sea-levels correspond
to the conventional radiocarbon time scale because the
history of the movements of the ice sheets are almost wholly
defined within the same time scale. Fractionation and
reservoir corrections will generally have been made or
estimated. Age constraints for some of the major shorelines
recorded in Scotland are poor and inferred rather than
based on direct radiocarbon ages and the inferences of ages
by correlating shoreline fragments from one area to another
introduces further uncertainty, particularly because the
formation of a shoreline need not be synchronous;
diachronous formation being suggested by both the
observational evidence (Morrison et al. 1981) and by
rebound modelling (Lambeck 1991).
Southeast Scotland
This area encompasses the Forth and Tay valleys and
adjacent coastal areas from the English border (near
Berwick) to about Aberdeen (Fig. 4). Well-developed
elevated shorelines occur up to elevations of about 40 m and
these are believed to post-date the last ice retreat across the
area from east to west. The main raised shorelines are the
East Fife group, the Perth sequence and the postglacial
features, some of which have been correlated across the
regions using their morphological, stratigraphic and
sedimentary characters, as well as their heights and
gradients. Along the river valleys the evidence for change in
sea-level is primarily in the form of marine deposits, peats,
or erosional features, but only in a few instances has it been
possible to establish time series for the sea-level change.
The East Fife Shorelines were initially mapped by
Cullingford & Smith (1966). Six separate beaches have been
identified in the sequence, each of which terminates in
outwash plains with the break in slope between the beach or
terrace and the back slope marking the shoreline. The
lowermost feature East Fife 6 (EF-6) Shoreline extends
furthest westward and it has been suggested that it
correlates with the Errol Beds in the Tay estuary (Paterson
et al. 1981) where they occur to the west of Perth, and the St
Abbs Beds in the Forth Estuary where they have been
recognized as far west as Inverkeithing (Paterson et al. 1981;
Sissons & Rhind 1970). Based on age estimates of these
clays, Browne (1980) tentatively associates an age of
14 750 yr BP with the EF-6 Shoreline, a suggestion that is
adopted here. Further north, Cullingford & Smith (1980)
have identified the same shoreline sequence on the basis of
the correlation of their heights and gradients with the
967
sequences of East Fife. Fig. 4(a) illustrates the shoreline
isobases, contours of constant shoreline elevation above the
present OD height datum, based primarily on the
Cullingford & Smith analysis and on the elevations of the
top of the Errol and St Abbs Beds in the Tay and Forth
valleys. No reductions to MSL have been made at this stage.
Another conspicuous elevated beach and shoreline
throughout southeast Scotland is the Main Perth Beach
(Sissons & Smith 1967; Sissons, Smith & Cullingford 1967)
and associated shoreline (MPS). Again, quantitative age
information for this shoreline is lacking and its age is
inferred to be about 13 500 yr BP from evidence of the ice
front which stood near the shoreline during a temporary halt
in the ice retreat (Paterson 1974; Sissons 1974). In East Fife
the MPS has been identified at elevations below the EF-6
beaches (Smith, Sissons & Cullingford 1969) but the
correlation is based on elevations and gradients rather than
on quantitative evidence. The corresponding isobases of the
MPS shorelines are illustrated in Fig. 4(b).
The best defined shoreline of all in the Forth valley is the
Main Postglacial Shoreline defined by the abutment of the
present flat valley floor, the ‘carse’ surface, against the
higher adjacent ground. These carse clays were deposited in
an estuarine environment during a time of relative sea-level
rise and are overlain in some localities by peats carbon
dated at 6500 yr BP. This determines the time of the end of
the transgressive phase and the formation of the Main
Postglacial (MPg) Shoreline (Sissons 1967a). It is also well
defined in the Tay and Earn River valleys and along the Tay
Firth (Cullingford, Caseldine & Gotts 1980; Morrison et al.
1981), in East Fife and along coastal areas as far north as
the Ythan valley in Aberdeenshire (Smith et al. 1980, 1985).
The precise age of the shoreline remains uncertain and the
culmination is variously reported as between 6800 and
5700 yr BP although its formation may not be synchronous.
Smith et al. (1985), for example, suggest that the shoreline
decreases in age by as much as 500yr from the upper Tay
and Earn River valleys to East Fife. A nominal age of
6000yr BP is adopted and Fig. 4(c) illustrates the
corresponding isobases.
While the cane clays form a remarkably uniform deposit
some layering and lamination does occur. One persistent
feature near the top of the sequence is a thin silty-tofine sand layer constrained in age by peats above and below
it dated at about 7000 f 200 yr BP. It is found in the upper
Forth valley (Sissons & Smith 1965), in East Fife (Morrison
et al. 1981), near Montrose (Smith et al. 1980), and in the
Ythan valley (Smith, Cullingford & Brooks 1983). Sissons
(1983) and Smith et al. (1985) have suggested that this
feature may have been produced during a major storm surge
while Dawson, Long & Smith (1988a) and Long, Smith &
Dawson (1989) have suggested that it may have been
deposited by a tsunami generated by a major submarine
landslide in the Norwegian Sea. Irrespective of its cause, the
uniqueness of this layer makes it an ideal shoreline marker
for a well-defined instant in time. Fig. 4(d) illustrates the
‘Tsunami’ isobase from localities where it has been
identified and from elevations given by Long et al. (1989).
The reference height for this shoreline is unknown but the
MHWS tide level is adopted.
Cores taken through the carse clays in the Forth valley
have revealed a number of buried beaches, the deepest and
968
K . Lambeck
I
50
I
40
30
20
Figure 4. Isobases of the major shorelines identified in eastern Scotland. (a) The East-Fife 6 Shoreline, (b) the Main Perth Shoreline, (c) the
Main Postglacial Shoreline, (d) the 7000 yr BP ‘tsunami’ Shoreline, (e) the Main Late Glacial Shoreline.
most extensive of which is a gravel layer backed by cliffs
formed by marine erosion of older tills. This is the Buried
Gravel Beach of Sissons (1967a) or the Bothekennar Gravel
(Browne et al. 1984). The shoreline, defined by the base of
these buried cliffs and referred to as the Main Late Glacial
(MLg) Shoreline, has been identified from near Grangemouth in the west where it occurs near present sea-level to
near Berwick in the east where it occurs at - 1 8 m O D
(Sissons 1974). It post-dates the Main Perth Shoreline and
pre-dates the overlying sediment deposited at about
10 300 yr BP (Sissons 1967b). The extent of erosion suggests
that sea-level may have stood at this height for a
considerable duration.
Figure 5 illustrates the time series of the change in
shoreline heights for three locations with all heights reduced
to MSL. The Arnprior results of Sissons & Brooks (1971)
have been extended to late glacial time by a linear extrapolation of the MLg and MP shorelines of Sissons (1974) about
10 km and 20 km respectively beyond the western limits at
which they have been observed. The upper Tay Forth result
for the localities of Bridge of Earn and Glencarse is from
Cullingford et al. (1980) and from the data points from
Glacial rebound of the British Isles-II
I
-10
'I
MLg
I
10
15
5
0
(b)60
(Earn, Abernethy, Glen Carse)
-20
15
5
10
0
969
because no direct age measurements exist for the actual
shorelines before about 9000 yr BP and the older ages are all
inferred from relationships between the retreating ice front
across the region and shoreline development. The results for
Arnprior and Errol both indicate the occurrence of small
oscillations in sea-level between about 10 000 and 8000 yr BP
corresponding to the three Buried Shorelines of Sissons
(1967a and b) (see also Paterson et al. 1981). The age of the
first of these, the High Buried Shoreline, is constrained by
pollen ages as having formed at about 10 200 f 100 yr BP,
soon after the formation of the Main Late Glacial Shoreline.
The second, the Main Buried Shoreline, is constrained in
age by both pollen and radiocarbon age as about 9600 yr BP
and the youngest, the Low Buried Shoreline, is constrained
in age as 8700 yr BP (Sissons & Brooks 1971). Within the
firths of eastern Scotland, at least, sea-level fluctuations
exhibit a complex sequence of oscillations in earliest
Holocene time. Fig. 5 also illustrates the envelopes of the
adopted sea-level curves for the three localities, all referred
to MSL, and the associated accuracy estimates. The
standard deviations for the older shorelines (EF-6, MPS) are
5 m while for the MLg and Buried Shorelines accuracies of
3 m and for the MPg and Tsunami Shorelines values of
between 1 to 2 m have been adopted depending on the
quality of the original observations. Figs 4 and 5, with the
former shoreline heights reduced to present MSL,
summarize the eastern Scotland sea-level observation data
that will be used below for estimating glacial-rebound
parameters.
Northeast Scotland
(c) 50
Errol
(Firth of Tay)
. L I )
&
-20' . . .
15
.
. .
10
5
time (~1000years BP)
.
'
.
Most evidence for past sea-levels in this area comes from the
shores of the Cromarty, Moray, Inverness and Beauly Firths
(Fig. 6) and this has been reviewed by Firth (1989) for Late
Devensian time and by Firth & Haggart (1989) for Holocene
time. Within the inner Moray Firth a sequence of beaches of
1
0
58.0"
Figure 5. Time series of height-age relationships of former
shorelines (relative to MSL) at (a) Amprior, (b) Bridge of Earn and
Glencarse, (c) Errol. Also illustrated are the envelopes of the
inferred sea-level change for these localities. In (a) and (c) HBS,
MBS and LBS refer to the high-, main- and low-buried shorelines
respectively.
Paterson et af. (1981) for the MP and the top of the marine
clays identified with the EF-6 Shorelines. The Errol curve,
to the east of Glencarse, is from Paterson et al. (1981) and
includes the EF-6 Shoreline height taken from their
height-distance diagram for the Tay Firth shorelines. It
needs to be emphasized that these curves are less accurate
than may be implied by the generally consistent pattern
j
L
1
I
I
40
50
Figure 6. Location map for northeast Scotland and (inset) the
heights of the ILg-4A Shoreline. (1) Beauly Firth, (2) Firth of
Inverness, (3) Cromarty Firth, (4) Dornoch Firth, (5) Munlochy
Bay, (6) Beauly, (7) Inverness, (8) Nairn, (9) Banff, (10) Fort
George, (11) Wick.
970
K. Lambeck
Late Devensian age reach altitudes of over 30m, and
possibly as high as 42m (Synge 1977), associated with the
westward retreat of the ice from Nairn to Inverness. Firth
(1989) has identified at least 10 Late Glacial sequences,
referred to collectively as the Inverness Late Glacial (ILg)
Shorelines of which the best defined are the Ilg-4A and
Ilg-5A surfaces with gradients of about 0.4 m km-' in an
approximately northeasterly direction (Fig. 6). Direct age
information for these shorelines is not available. They must
post-date the retreat of the last ice cover and this places an
age constraint on ILg-4A of about 13 500 yr BP corresponding to the deglaciation of Cromarty (Firth 1989), although
Peacock (1974) suggests that this deglaciation may have
occurred later. A nominal age of 13500yrBP is adopted
here, equivalent to the Main Perth Shoreline of the Forth
region (Firth 1989). Firth & Haggart (1989) have also
identified a buried erosional surface covered by sediments
dated at 9610f 130yrBP. This surface is terminated at
some locations by the base of a cliff line and they suggest
that it corresponds to the Main Late Glacial Shoreline of
southeastern Scotland. This association is also adopted here.
A detailed examination of the Beauly evidence by
Haggart (1986) and Firth & Haggart (1989) has indicated
that after about 10 000 yr BP small sea-level oscillations
occurred that are similar to those reported for the upper
Forth valley (Fig. 7). The 'Tsunami Shoreline' has also been
identified here as well as in Munlochy Bay, Dornoch Firth
and Wick (Long et al. 1989) and its age is constrained to lie
between 7430 f 120 and 7270 f 90 yr BP (Haggart 1986).
The culmination of the early Holocene transgression forms
the Inverness Flandrian Shoreline (IF1) and is defined by
estuarine flats, terraces and shingle ridges along the margins
of the firths of the region, attaining a height of about
9-10 m OD near Beauly and sloping in a north-northeast
direction with a gradient of about 0.07mkm-'. The
formation time is between 7100 and 5500 yr BP and Haggart
(1986) suggests an age of about 6400 yr BP and correlates it
with the Main Postglacial Shoreline of the Forth-Tay
region. Further north, elevated shorelines of Late Holocene
age have not been identified in Caithness or in the Orkney
Islands while in Shetland, shorelines formed at 6000-
Beauly Firth
', '
h
E
v
3
e
20-
.
t
'
,'
8
'
, '
\
'
Figure 7. Time series of height-age relationship in the upper
Beauly Firth region and the inferred sea-level change envelope.
7000 yr BP, now appear to be well below present sea-level
(Hoppe 1965; Flinn 1964). The observational data base used
for the Moray region consists of the sea-level curve for the
upper Beauly Firth (Fig. 7) and the elevations of the
LIg-4A, MLg, Tsunami, and the IF-1 shorelines along the
Inverness and Moray Firths.
Western Scotland
This area includes the Inner and Outer Hebrides and the
mainland coast from Wester Ross in the north to the Firth
of Clyde, Kintyre and Arran in the south. Evidence for
sea-level change in this region is distinctly different from
that in the southeastern region, reflecting the differences in
geology and sediment supply and the fact that the west coast
has been subjected to a greater degree of glacial erosion and
scouring than the east coast. Raised shorelines and elevated
marine sediment deposits have been recognized throughout
the region at elevations up to 100m but no systematic
studies for their origins and ages have yet been published.
Some of the elevated deposits are believed to be of
pre-Devensian age while others may hav been emplaced as a
result of glacier action (Synge 1977; Dawson 1984).
Well-developed rock platforms at up to 50 m OD also attest
to higher shorelines at some time in the past but these also
are of uncertain age (Sissons 1982, 1983; Dawson 1984).
One region where a variety of fossil-bearing Late Glacial
sediments have been preserved is in the Clyde Firth and
Clyde River. A useful data set is from the Ardyne locality
(see Fig. 9b for location) where Peacock, Graham &
Wilkinson (1978) have examined the heights and ages of in
situ marine bivalves and their relationship with the
maximum elevations of the same sedimentary units in
nearby localities. They succeeded in estimating the sea-level
curve from about 12000 to 10000yrBP and while
uncertainties are large (Fig. 8a), this result indicates a rapid
fall followed by a nearly constant level for about 1500yr.
Other data points for the region include in situ marine
molluscs dated at 13 780 f 124 yr BP (Browne, McMillan &
Graham 1983) and a series of peat and mollusc dates
between about 13 000 and 8000 yr BP. The in situ n~ollusc
(points 1-3 in Fig. 8a) represent lower limits and, by
analogy with the Ardyne results, could be up to 20 m below
sea-level (see radiocarbon dates GU-12, Birm-122, SRR479). Point 4 is also based on marine shells but the height
has been reduced to sea-level using nearby morphological
indicators (Baxter, Ergin & Walton 1969). Points 5 and 6
(Bishop & Dickson 1970) represent lower limits whereas
points 7 and 8 (Bishop & Coope 1977) define upper limits.
The level at 6000yrBP is based on a number of
observations throughout the firth and associated lochs and
sounds, although nowhere is the shoreline defined with
precision (e.g. Jardine 1986; Boyd 1986; Gemmell 1973).
For a wide range of plausible ice and earth models, the
predicted regional variability for the sites within the Firth of
Clyde region are generally smaller than the uncertainties
associated with the observations, and the data points can be
combined into a single sea-level curve (Fig. 8a) that is
representative of the region between Dumbarton and
Ardyne. (See also Fig. 21 of Part I).
Other quantitative data for establishing part of a sea-level
curve, come from the Lochgilphead region of Argyll
Glacial rebound of the British Isles-II
971
55
(a) 40
(weslern Scotland)
30
E
-2
P
3e
regional
.
7
5
20
.
10
0 .
15
13
11
9
(b) 60
\Oban-Lome
50
"
15
10
(western Scotland)
5
0
time (~1000years BP)
Figure 8. (a) Height-age relationships for sea-level indicators in the
Clyde Firth and the lower Clyde River. The data from Ardyne are
illustrated by the solid circles with error bars and the numbered
points are discussed in the text. All elevations are with respect to
MSL. The estimated sea-level curve is also shown. (b) Sea-level
evidence for the Firth of Lome and Loch Fyne regions. The Lower
Fyne curve is given by Dawson er al. (1988b), the Oban-Lorne
curve is from Synge (1977), the Lochgilphead observations (with
error bars) are from Peacock et al. (1977) and the open circle
corresponds to the rock barnacles at Oban (Gray 1974).
(Peacock et al. 1977) (see Fig. 9b for location), where the
evidence is similar to that obtained by Peacock et al. (1978)
for Ardyne in the Clyde Firth. Fig. 8(b) illustrates this result
and here also the evidence points to a rapid fall in sea-level
up to about 12 000 yr BP followed by a period of at least
lo00 yr during which sea-level remained at an approximately
constant level. Dawson et al. (198%) give a sea-level curve
for the lower Fyne region that is presumed to incorporate
the Lochlgilphead data and only the Holocene part of this
curve is shown in Fig. 8(b). Near Oban, barnacles of a
species that normally lives in the intertidal zone have been
found at 2 0 m O D (Fig. 8b) with an age of about
12 000 yr BP (Gray 1974).
Elsewhere, the only information is from the rock
platforms and the shingle beach ridges. The latter are
believed to correspond to the Main Postglacial shoreline of
eastern Scotland and reach a maximum elevation of about
I
70
I
6O
I
so
I
1
Figure 9. (a) Elevations (OD) of the Main Postglacial Shoreline for
the Inner Hebrides and Western Highlands. Numbers in italics refer
to locations as follow(1) Scarba, (2) Oronsay, (3) Oban, (4) Loch
Fyne. (b) Same as (a) but for the Main Rock Platform. (1) Ardyne,
(2) Dumbarton, (3) Lochgilphead, (4) Firth of Lome, (5)
Ardnamurchan, (6) Moidart.
14 m OD near the heads of the inner sea lochs (Gray 1974).
The appropriate reference level for these features is the
height of modern deposits which in some instances may be
up to 3 m above the MHWS tide level (e.g. Sissons &
Dawson 1981). Fig. 9(a) illustrates the isobases for this
972
K . Lambeck
shoreline based on the work of Gray (1974) for the Firth of
Lorn and Mull; of Dawson (1984) for Jura, Islay and
Scarba; of Jardine (1978) for Oronsay; Synge & Stephens
(1966) for Iona, Kintyre and Loch Etive; and the mean
Clyde Firth result discussed above. In Wester Ross this
shoreline feature is found at about 9 m near Applecross and
at about 7 m OD between Loch Gairloch and Little Loch
Brook (Sissons & Dawson 1981). All shingle beach deposits
have been reduced to mean sea-level assuming a formation
height at MHWS + 2 m for the exposed outer coast sites and
MHWS 1 m for the protected inner coast sites.
Of the rock platforms only the low rock platforms, the
Main Rock Platform of Gray (1974), ranging in elevation
from 0-10mOD are used. They are particularly well
defined around the Firth of Lorne and the west coast of Jura
but they have also been identified in Arran, Kintyre and the
Firth of Clyde and as far north as Ardnamurchan and
Moidart (Dawson 1984, 1988; Gray 1978). The spatial
variation of the platform elevations forms a relatively simple
pattern (Fig. 9b) with the highest values occurring along the
inner sea lochs of the Highlands and decreasing with
distance from the centre of maximum glaciation, and it is
usually assumed that they correspond to the same erosion
event(s). Different ages have been attributed to this feature
at various times but the present consensus seems to be that
it formed or was reshaped during the Loch Lomond Stadia1
by periglacial rock erosion (Dawson 1984, 1989). Within the
Clyde Firth the platforms occur at between 5 and 10 m O D
(Gray 1978), consistent with the minimum sea-level
recorded at Ardyne in the time interval 11500-10 000 yr BP.
The association of the platform with the relative lowstand
during the Loch Lomond Interstadial at 11000-10 500 yr BP
is therefore plausible and accepted here, either as a time of
formation or as a time of re-occupation and re-shaping of an
older feature. The High Rock Platforms form a less
coherent pattern and the consensus is that they are a result
of shaping by successive glacial events, possibly of
pre-Devensian age, with the latest modifications having
occurred at 13 500-13 000 yr BP when the ice front stood at
the western edge of the Highlands. Because of the
considerable uncertainty in their interpretation these
shorelines are not considered here as appropriate for
constraining models of Late Devensian rebound.
+
Southwest Scotland
This region includes Southern Ayr and the coastal regions of
Wigtown, Kirkcudbright and Dumfries along the Solway
Firth (see Fig. 5.3 for locations). Elevated shorelines have
not been identified in the region (Jardine 1971) and as the
coastal area was generally ice free soon after 14000yrBP
(Fig. 2a), the absence of such features, if confirmed, places
sea-level as being near or below the present level at that
time. The oldest indicators of sea-level occur in the upper
reaches of the firth and span a time interval of 12 290 f 250
to 10300f 185yrBP (Bishop & Coope 1977) and suggest
that mean sea-level in this period was at -3 to -4m below
the present level. The Holocene evidence has been
examined by Jardine (1975) for the eastern Solway near
Annan, and for Wigtown Bay and Luce Bay in the western
Solway Firth. Fig. 10 illustrates the results, including the
pre-Holocene result for the eastern Solway region. Four
1
1
(Redkirk, Annan)
-8
1 4 1 2 1 0 8
6
4
2
I
0
West Solway Firth
(Wigtown Bay & Luce Bay)
-8
1 4 1 2 1 0
8
6
4
time (~1000years BP)
2
0
Figure 10. Age-height relations of sea-level indicators and
estimated sea-level curves, all reduced to MSL, for southwest
Scotland. (a) East Solway Firth near Annan and Redkirk point, (b)
the west Solway Firth near Luce Bay and Wigtown.
data points are given by Jardine (1975) for the Girvan area
on the south coast of Ayr and these indicate that mean
sea-level here was at about 2 m between 9400 and
8400 yr BP.
England and Wales
The palaeo sea-level evidence for England and Wales is
confined mainly to river estuaries, shallow bays and
low-lying coastal plains where sediments and organic
materials deposited in tidal flat or lagoonal environments
have been preserved. The most complete examination of the
evidence for northwestern England is by Tooley (1978). The
preliminary model predictions in Fig. 21 of Part I, indicate
that considerable spatial variation in sea-level response is
predicted for this area and that it may not be appropriate to
combine all data into a single sea-level curve. Sufficient
information exists to construct sea-level curves for three
localities within the region: Morecambe, the Ribble
Estuary, and Formby. Fig. 11 illustrates the results.
Following Tooley, all heights are assumed to correspond to
the MHWS level and have been reduced to MSL in this
figure. Height errors ranging from f 1 m to f 2 m have been
adopted and the minor oscillations noted by Tooley have
been ignored. A partial sea-level curve for a fourth region,
the Duddon Estuary near Millom, has been inferred from a
few data points in Tooley (1978) and from the mainly
Glaciai reborrnd of the British Isles-I1
973
0
-5
0
, I
,, #I
b
4
cd
2 -10
-
, I
, , II
e,
.>,
*
1'1
, , II
cd
4
-10
, I
, , I1
-15
8
'
'
-15 .
Morecambe Bay
-20 .
-25
10
8
(northwest England)
I
6
4
2
0
t
-20
-25'
10
Ribble Estuary
(northwest England)
'
.
'
8
.
'
'
6
4
'
.
2
'
0
e,
.L
S -15 .
4
t
1
-20 .
-25
'
10
,
' *
Formby
Millom
(northwest England)
(northwest England)
.
1
8
6
4
time (x1OOO years BP)
2
0
Figure 11. Sea-levels for northwest England. (a) Morecambe, (b) Ribble Estuary, (c) Formby, (d) Duddon Estuary, Millom. In (d) rn refers to
molluscs which are considered not to be in their growth position but to define an upper limit to sea level. P refers to peat samples at a level
near the MHWS level with P-T referring to results from Tooley. W a n d C are wood and charcoal samples which also define an upper limit to
MSL.
mollusc and wood fragment results by Andrews, King Lk
Stuiver (1973). The shell data have come from dune and
shingle ridge deposits and because they do not appear to be
in their growth position they are assumed to define upper
limits to sea-level. Fig. ll(d) illustrates these results but
generally they do not provide useful constraints on past
sea-levels. [The discovery of a spoon in the same strata as
these shells (Tooley, private communication) highlights the
uncertainty of this data.]
Tooley (1978) (see also Gaunt & Tooley 1974) discusses
evidence from northeastern England, mainly from the
Humber Estuary in an area between Spurn Head, Hessle
and Brigg. Four points from Chapel Point in this
compilation are also found in Shennan's (1986) data set for
the Fenlands and are included below. The evidence is of
several types; peats, wood fragments within estuarine and
saltmarsh clays, and molluscs, and they generally constrain
only the limits to sea-level height. Fig. 12(a) summarizes the
observed height-age relationships, with all heights reduced
to MSL. Further south, in the Fenlands of East Anglia,
unconsolidated sediments have been deposited in lagoonal,
perimarine and tidal flat environments that extend over an
area of nearly 5000 km'. The sea-level information, in the
form of peats intercalated between lagoonal and tidal flat
sands and clays, has been examined in detail most recently
by Shennan (1986). It includes data from Devoy's (1982)
analysis of sea-level in southern England as well as the
Chapel Point data included in Tooley (1978). Shennan's
estimates of MHWS have been reduced to MSL and his
error bounds have been adopted (Fig. 12b). Predicted
sea-levels through the region based on the preliminary
model (Fig. 21 of Part I) exhibit some spatial variability
across the region, but this generally lies within the range of
the adopted error bounds and the single curve is assumed to
be representative of a location near the geographic centre of
the various data points.
974
K . Lambeck
.
.
-2
.
.
I
.
4 h
$
4
I
0
.e
3
e:
-6
.
-8 .
-10 .
r
'
-12 .
Humber Estuary
-14 .
t
-16
8
'
"'
(northeastern England)
.
I
2
4
6
,
,
.
.
~
0
~
.
-
(East Anglia)
6
2
4
0
4
Yare Valley
Waveney Valley
-16
Norfolk
-20
8
6
4
2
time (~1000years BP)
0
Figure 12. Sea-levels for (a) the Humber Estuary in northeast
England including the Chapel Point data, (b) the MSL curve for the
Fenlands from Shennan (1986), (c) the Norfolk coast from three
different locations.
The Norfolk coast has provided more localized environments for the preservation of Holocene sea-level indicators.
Evidence from the northern coast between Brancaster and
Cley is discussed by Funnel1 & Pearson (1989), observations
from the Yare estuary included in Devoy's (1982)
compilation, and the data from the Waveney Valley has
been examined by A. M. Alderton (Shennan 1987). Fig.
12(c) illustrates the results for the three localities, all
reduced to MSL. The observations are consistent with each
other within their errors and the range of spatial variability
predicted across the region for the past 7000yr (cf. Fig. 21
of Part I).
An important source of information on sea-level change
in southeastern England is provided by Devoy (1979, 1982).
His data includes observations from the Fenlands and
Norfolk and these have been included in the abovediscussed sea-level curves. The remaining data points fall
into three regions within each of which the predicted spatial
variability is smaller than the accuracy estimates for the
reduced mean sea-levels. These are the Essex coast, mainly
from the Blackwater and Colne River estuaries, the Thames
Estuary as far upstream as London, and the eastern section
of the Channel coast between Margate and Weymouth. The
evidence is from a variety of peats that are indicative of
changes from freshwater to marine-brackish environments.
Fig. 13 illustrates the mean sea-level curves for the three
localities. All heights have been reduced to mean sea-level
and the corrections given by Devoy for consolidation and
compaction of the peats and underlying sediments have
been applied. The data from Devoy is in two parts. For the
first (Devoy 1982, Table 3) the evidence is from peats that
are assumed to have formed near the MHWS tide level. For
the second (Devoy's Table 4) the dated horizons cannot be
reliably related to this reference and they provide mainly
upper limit estimates of the sea-level curve but they are
useful for the interval 10000-8000yrBP for which no
information would otherwise be available from this region.
Sea-level information for the western sector of the
Channel coast is given by Heyworth & Kidson (1982) and is
based on freshwater-estuarine transition peats and in situ
tree roots and stumps. Fig. 13(d) illustrates the resulting
mean sea-level curve for this part of the coastline. Another
important collection of data points is given by Heyworth &
Kidson (1982) for the Bristol Channel and Wales where the
information is similar to that from the western Channel
coast. From Fig. 21 of Part I it would appear inappropriate
to group all Bristol Channel observations into a single
sea-level curve and instead curves for three localities have
been constructed, for Bridgwater Bay, Swansea Bay (Port
Talbot, Fig. 3 of Part I), and the upper Bristol Channel
(Fig. 14). For Wales, much of the evidence comes from the
submerged forest beds near Borth (Heyworth & Kidson
1982) and this is the only locality in Wales for which it is
possible to construct a sea-level curve for middle and late
Holocene time (Fig. 14d). The tree-stump data is based on
the assumption that the trees have been killed by rising
groundwater associated with a rise of sea-level, although the
relationship is not straightforward, particularly if there has
also been significant coastal erosion (e.g. Tooley 1986).
Despite this reservation, these observations are assumed
here to be indicative of an upper limit to the sea-level curve.
French Atlantic Coast
To complement the observations from southern England,
sites where sea-level is relatively insensitive to the changing
British ice sheet but more strongly dependent on the
changes in the global ice sheets through the equivalent
sea-level function, observations from the French Atlantic
coast are also considered. Ters (1986) has compiled
observational evidence from locations along the entire
Atlantic coast of France and estimated a single sea-level
curve, but analogous to the predictions in Fig. 21 of Part I,
some spatial variability in sea-level response can be
expected through the spatial dependence of the water-load
term and through the variable distances of the sites from the
Glacial rebound of the British Isles-II
-.-----$--$
--- -,,.I-:.f.
-
0 h
.
p
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I
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I
3 ;
:;
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-20
975
t
.
.
-9n I
10
.
.
*
*
Table 4
4,
Mersea
6
8
Thames Estuary
(Essex)
.
-50
I
2
4
10
0
8
-30
10
6
4
2
time (x1OOO years BP)
8
'
Channel Coast
-30
10
.
(western sector)
s
0
0
1 ,
1 ,
:&
,,
-25
(eastern sector)
2
,+'
-20 :
Channel Coast
4
6
8
6
1
2
4
0
time ( ~ 1 0 0 0years BP)
Figure 13. Sea-levels for (a) Mersea on the Essex coast, (b) Thames Estuary, (c) the eastern sector of the English Channel coast from Margate
to Weymouth and (d) the western sector of the Channel coast from west of Weymouth to Cornwall.
Upper Bristol .Channel
fb)
-2
.
'
.
.
.
(southwest England)
.
-4.
-6 .
-30
-8 .
Swansea Bay .
(Wales)
-10
10
8
2
4
6
(c)
. . 0I
0
"
'
-
8
10
I
6
4
1
2
0
2
0
J
a*--5
-A
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-10
-
'
,'% '
.
-5 .
,&$
-10 .
@
-15 .
:
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'I
-20 .
-25
-30
'
10
-15
::
-20 .
;;
I
'
v
.
-25 .
Bridgwater
(southwest England)
8
6
4
time (xlOc0 years BP)
2
1
0
-30'
10
'
.
.
'
8
I
6
4
time (x1ooO years BP)
Figure 14. Sea-levels for (a) Swansea Bay, south Wales, (b) the upper Britsol Channel near Avonmouth, (c) Bridgwater (Somerset), (d)
Cardigan Bay, mainly from Borth Bog and the Dovey Estuary but including single data points from Newport and Newquay.
976
K . Lambeck
(4 5
'I
I
0 -
-5
h
-5 1
>
3
-
g -10
OY
ACo(qpI A, t ) + 4%A, t )
= Ag'(t)
Sg'(t)
+
0
--
E -10
v
3
I
0
.2
-20
t..""
+ B"" ACBR(q, A, t )
+ ACFEN(q,I , t ) + ACf-'(q, I , t ) + SC"
W
-30
= ACpredicted
(1)
where
-50
Manche 81 Calvados
m
-60
-'
10
-15
8
6
2
4
.
Cotes-du-Nord
-20'
10
.
'
8
.
0
1
1
6
2
4
time (~1000
years
0
BP)
Figure 15. Sea-level curves for the France Atlantic coast at (a) Pas
de Calais, (b) Manche & Calvados and (c) CBtes-du-Nord.
Fennoscandian ice sheet. The data has, therefore, been
grouped to construct regional curves for three locations:
Pas-de-Calais, Manche and Calvados, and CGtes-du-Nord.
The observations in each area of a diverse nature and the
relation of the sample positions to the tidal range are not
always known with precision so that many of the
observations represent lower limits only. Fig. 15 illustrates
the three mean sea-level curves. The high-frequency
oscillations in sea-level proposed by Ters have been ignored.
4
assumed to be a function of depth) and certain ice model
parameters. The local ice model is assumed known except
for the possibility of a constant scale parameter 0"" to be
applied in space and time to the height of the British ice
sheet. The predicted sea-levels for most sites are not
strongly dependent on the Fennoscandian ice sheet, and
without including sea-level data from northern European
sites it is not possible to solve for a similar parameter BFEN.
An earlier study using such sites indicated that B""" = 1
(Lambeck, Johnston & Nakada 1990) and this value is
adopted here. Also, a time-dependent equivalent sea-level
correction S g ' ( t ) is introduced to allow for any inadequacies
in the models of the distant ice sheets. The observation
equation then becomes (cf. eq. 6 of Part I)
AN IMPROVED REBOUND MODEL
A strategy for parameter estimation
The parameters to be determined from the observations of
sea-level include both the effective earth model parameters
(the lithospheric thickness He and the mantle viscosity r]
A&, = observed sea-level reduced to MSL, at location
(q,A) and at time t.
E ( ) = estimate of the observation error.
Ag' = equivalent sea-level function for the totality of the
ice sheets.
Sg' = correction term to Ag'.
p"" = scale parameter for the British ice sheet.
ACnR = predicted first-order contribution to sea-level
change from the ice- and water-load. Terms of the
British ice sheet for specified earth model
parameters.
ACFEN= predicted contribution to sea-level change from
the ice- and water-load terms of the
Fennoscandian ice sheet.
A CCf = predicted contribution to sea-level change from
the water- and ice-load terms of the distant ice
sheets of Laurentia, Barents Sea and Antarctica
S t w = second-order corrections to the total water-load
terms arising from (1) the dependence of the
water-load term on sea-level change itself, and (2)
the time dependence of the ocean function (see
Part I). The full formulation for this term as
developed by Johnson (1993) is used here.
An iterative procedure has been adopted to solve for the
various parameters, the H e , r] defining the A < functions,
B"" and S g ' ( t ) . In the first iteration predictions are made
for the three sea-level terms AC"", AtFEN,AC'-' for earth
model parameters covering a wide range of values.
Predictions for each earth model k(k = 1 * * . K) within this
range are made for each instant of time at every site where
sea-level has been observed for a total of M ( m = 1 . . . M )
observations. The corrective terms Sg' and SC" are initially
set to zero and for each model k the corresponding
value is estimated that minimises the quantity
a,""
Glacial rebound of the British Isles-I1
where d m is
) the standard deviation of the mth observation.
For each earth model k the @"" value is established that
gives the minimum solution variance and a search is then
conducted through the earth model space to determine
those parameters that give an overall minimum variance.
Once this minimum variance solution is established, the
second-iteration correction terms Sc" are computed only for
those earth models in the neighbourhood of this
first-iteration minimum because of the substantial computing
requirements for these terms and because these corrections
are relatively small. The solution of (1) for H,, q , p"" and
the correction vector 6r(t)
is then repeated for the subset
of earth models and the new minimum variance solution is
sought.
977
.I
E
Earth model parameters
In the first series of calculations the earth model comprises a
lithosphere (layer l ) , a combined upper mantle and
transition zone down to a depth of 670km (layer 2) of
viscosity q 2 and a lower mantle (layer 3) of viscosity q,,.
(This is the three-layered model of Table 1.) The
viscoelastic lithosphere is assumed to have a high viscosity
(lo2' Pa s) and its effective thickness He is to be determined
from the comparison of the observations with the predicted
sea-levels. The preliminary calculations (Part I) indicated
that the predicted sea-levels are relatively insensitive to the
choice of qImand the initial search will be for He and q2 for
qIm= 1.3 x
Pa s (see below) with the search conducted
within the range
(50 I He 5 100) km,
(10'"
5
q2 5 5 X lo2') Pa s.
(3)
Figure 16 illustrates a typical result for the first-iteration
solutions for a subset of the K earth models and they indicate
that while some trade off occurs between the earthand ice-model parameters a separation of q2 and @""
appears feasible. This is further illustrated in Fig. 17 which
summarizes the minimum variances and associated p"" for
the parameters within the earth model space defined by (3).
The results are for three observational data sets: (1) the
observations from sites outside the former ice sheet margins
in England, Wales and France; (2) the observations from
sites within the formerly glaciated areas of Scotland and
northwest England; and (3) the combined data set. The first
data set (Fig. 17a) has no resolving power for @"" (see Part
I) and this parameter has been set to unity in the first
Table 1. Definition of the multilayered earth models. CMB refers
to the core-mantle boundary.
'three-layer
Five-layer
Layer 1
0-HQ
HQ = 8 0 krn
Layer 2
Hp -670
Hp -200
Layer 3
670-CMB
200-400
Layer 4
400-670
Layer 5
670-CMB
0.5
0.7
0.9
1.1
1.3
1.5
PBR
Figure 16. The variance function u: for three three-layer earth
models with different q2 values and H, = 80 km as a function of the
ice-height scale parameter, B"".
instance. Two local minima are identified in the q2 - He
parameter space, one near q2 = 2-3 X lo2' Pa s , and the
second near 4 X 102"Pas. This example provides a good
illustration of the frequently encountered existence of two
solutions in viscous relaxation problems with one solution
corresponding to an early stage of relaxation of a process
with a long time-constant and the other corresponding to the
late stage of relaxation of a short time-constant viscous
model (e.g. O'Connell 1971). In the absence of any other
information the higher viscosity solution would be the
preferred one in this case. However, for the Scotland data,
where the dominant contribution to sea-level change at the
sites near the centre of loading comes from the local
rebound, this higher viscosity mantle model does not predict
well the considerable spatial and temporal variability
observed and this solution is now excluded (Fig. 17b). The
combined data set (Fig. 17c) also excludes the higher value
and a well-defined optimum solution space is defined by the
= 3 contour and indicates earth parameters that lie in the
range (60 5 He 5 75) km and (3 x lo2" 5 q2 5 5 x lo2")Pa s.
The overall minimum variance is (1.58)2 whereas the
expected value from a perfect model and realistic estimates
of observation variances would be unity and it appears that
there is some room for model improvement unless the
observational occurrences have been systematically
underestimated.
The statistics of the minimum variance solution
(He = 65 km, q2 = 4 x 102"Pa s) are summarized in Fig. 18
in terms of the degree of correlation between the M
observed and predicted sea-levels (Fig. 18a) and in terms of
the histogram of the residuals E ~ )(eq. 1) (Fig. 18b) and
although some of the individual observational errors are
large, there does appear to be a high degree of consistency
between the model predictions and the observations of
sea-level change. At this stage no attempt has been made to
solve for the correction 85;e(t) to the equivalent sea-level
function and a first estimate of it is given as the weighted
mean of the residuals E ~ for
,
each epoch t. This function is
978
K . Lambeck
England &Wales
earth model dependent although this dependence is slight
for solutions about the minimum variance solution.
In the second iteration the correction term
is evaluated
and included in the solution of (1) for parameters in the
neighbourhood of the first-iteration solution. Also, the
first-iteration estimated of bf'"(r)is now included, and the
solution is illustrated in Figs 17(d) and 18(c, d). The
resulting minimum variance occurs for
cw
He = 65-70 km
(4a)
and
r ] , = (4-4.5)102"
I
l(Pl
En+nd,
I
I
,
2
3
1
I
IOU
I
1
I
2
3
5
Wales & Sro(l.nd
I
1
I
I
I
l P
2
3
5
I l(P1
,
I
2
I
3
5
rli(P8 s)
Figure 17. The variance function u: as a function of earth
parameters qz (the combined upper mantle and transition zone
viscosity) and H , for the first iteration solution (a-c) and the
three-layered model. (a) The England, Wales and northern France
data set only, (b) the Scotland data set only, and (c) the combined
data set. In (c) the B dependence on v2 and H , is shown by the
dashed lines. In (d) the results for the second-iteration solution are
shown for the combined observational data set. All solutions
correspond to the fully non-adiabatic models.
Pa s.
(4b)
Neither the inclusion of the second-order terms nor the
introduction of the equivalent sea-level correction term
significantly affects the solution for the earth model
parameters, at least for the British rebound problem, and it
appears reasonable to ignore these second-order terms in
any further preliminary exploration of the full range of
possible earth model parameters although these terms have
been included in the five-layer solutions discussed below.
Further iterations of the solution of (1) also are not
required.
In the solutions of the sea-level equations discussed here
the response of the mantle to loading has been assumed to
be fully non-adiabatic in the sense that any particle
displaced to a new depth in a medium of depth-dependent
density retains its original density and thereby experiences a
buoyancy force. It has been pointed out by Cathles (1975)
and Fjeldskaar & Cathles (1984) that adiabatic models may
be more appropriate when the mantle is chemically
homogeneous or at phase transition boundaries. In these
models the displaced particles blend with the material at
their new depth and experience no buoyancy. Recently
adiabatic solutions have been developed by P. Johnston so
that it has been possible to estimate the effect of this
assumption on the model parameters. Generally the
differences between the two models as applied to the British
ice load are relatively small, essentially because the surface
load is composed largely of high wavenumber terms whose
corresponding viscoelastic Love numbers differ little for
either the adiabatic or non-adiabatic assumption. Furthermore, the two models lead to results that are nearly linearly
proportional to each other and, for this region at least, a
direct trade off occurs between the degree of adiabaticity
assumed and the scale factor /3 for the ice sheet. Both the
adiabatic and non-adiabatic solutions lead to the same
viscosities but not the same 0, with the latter leading to an
underestimation of the ice height by about 10 per cent
(Lambeck 1993b). Thus, in the following solutions, only the
non-adiabatic models are considered in the discussion of the
earth parameters.
It has previously been established (Part I) that the
sea-level predictions for the British Isles are not strongly
dependent on the properties of the lower mantle because of
the relatively small dimensions of the ice load. However, the
response to the water load, occurring with dimensions equal
to those of the Atlantic Ocean, does stress the lower mantle
to a certain degree and some dependence of the sea-level
response to qI,,, can be anticipated. Fig. 19 illustrates the
variance factor u', as a function of both r]* and qIm for
He = 80 km and the result confirms the weak dependence of
Glacial rebound of the British Isles-11
observed sea level (m)
-p
v
40
-
2 20 .
U
8
B
4g
0 :
-20
observed-predicted sea level (m)
.-'.
r.
.
',,.' .
/
80
-
[
B
60
40
0'
,a.
-40.'
're
#7?-
6
.
979
p =0.949
A$red = - 0.16 + 1.013 A{ob,
20
-60
observed-predicted sea level (m)
observed sea level (m)
Figure 18. Statistical evaluation of the minimum variance solutions for (a, b) the first iteration solution and (c, d) the second iteration solution
for the three-layered earth model. In (a) and (c) the observed sea-levels are plotted as functions of the predicted values and the (unweighted)
best fit linear trend is shown. In (b) and (d) the histograms are of the residuals E,, of eq. (1).
the rebound on qlm, with the smallest value occurring for
qIm= 1.3 x
3 -
-
10
I
I
1
I
I
Pas.
(4c)
Although the resolving power of this data set for the lower
mantle viscosity is poor, models with qlm5 4 x lo2' P a s
appear to be excluded and there is strong support for
models in which the mantle viscosity increases substantially
across the 670 km boundary, consistent with previous studies
of the crustal rebound in other areas (Lambeck & Nakada
1990; Lambeck el al. 1990). How this increase in viscosity is
distributed with depth below 670 km cannot, however, be
resolved with the present data.
The estimates (4) for the mantle response are effective
parameters that describe the response of the Earth's surfaceto-glacial unloading on a time scale and surface-load scale
that is characteristic of the British ice sheet, but it remains
to be demonstrated that this simple three-layered rheological model does in fact give the best representation of the
upper mantle response. Therefore, an additional class of
model is considered comprising five mantle layers (Table 1).
Fig. 20 illustrates the first-iteration results for this model
980
K . Lambeck
7
7
?'p7Xl@O
q4=3x1020 Pa s
Pa s
5 -
7(3'7(4
I
1
8
2
3
5
1
7
2
3
5
7
10
0
4
x
W
2
10
r
1
10
q4=10x1#*
7(4=5x1020 Pas
7 -
Pas
7 -
n3w4
1
2
3
5
7
10
q3 ( ~ 1 0 2 Pa
0 s)
Figure 20. Minimum variance solutions for the second-iteration five-layered earth model as a function of the viscosities ( q 2 ,11) for the two
upper mantle layers and the viscosity q4 for the transition zone from 400 to 670 km equal to (a) 3 X 10'" Pa s. (b) 5 X 102"P a s , (c) 7 X 10'" Pa S ,
(d) 10" Pas. The other earth model parameters are He = 80 km, qlm= 5 x 10" Pa s.
with H, = 80 km and in which the upper mantle comprises
three layers; the lithosphere, a second layer which extends
from the base of the lithosphere to 200 km with viscosity r]',
and a third from 200 km to 400 km with viscosity q 3 . The
transition zone down to 670 km forms the fourth layer of
viscosity q4. The lower mantle viscosity in this example is
equal t o 5 x 10" P a s for all models, near the lower limit
permitted by the earlier solution (Fig. 19). Only the range of
models in which the viscosity in any layer (other than the
lithosphere) is less than or equal to that of the layer below it
(Table 1) are considered at this stage. The variances are
plotted as functions of r]' and q3 for discrete values of v4
corresponding to the viscosity of the transition zone between
400 and 670 km depth, and for each q4 value the minimum
variance occurs only when r ] , = q3. The overall minimum
variance for the H , = 80 km solutions occurs for r ] , = q3 =
4.5 x 10'" Pa s and q4 = 7 x 102"Pa s suggesting that there
may be a small increase in viscosity between the upper
mantle and transition zone although solutions with
r ] , = q3 = q4 = 5 x lo2"Pa s yield only a slightly larger
minimum variance. Fig. 2 l f a ) illustrates the solution
variance for those models for which r ] , = q3 and it confirms
that any increase in viscosity from the upper mantle to
transition zone is either zero or very small. Similar results
are illustrated in Fig. 21(b) for Q , =~ 102'Pas and
H, = 80 km and the conclusion is the same as before, that
there is no compelling evidence f m substantial layering in
the mantle viscosity above 670km. Nor d o the results
support mantle models with a significant inversion of
viscosity across the 400 km boundary (Fig. 21b).
The pattern of the isovariance curves does not suggest that
models with low-viscosity channels (small r]' values) below
the base of the lithosphere down t o a depth of 200km,
except possibly if both q3 and q4 are substantially increased.
However, even in this case, the results in Fig. 20(d) suggest
that the smallest variance for such models will be greater
than that found for solutions without such a channel. This is
further illustrated in Fig. 21(c) for models in which q3 = v4
981
Glacial rebound of the British Isles-II
100
/
70
50
30
h
m
20
a"
2
sx 10
W
N
F
7
5
3
2
I
I
I
I
I
I
I
2
3
5
7
10
20
30
q3=(q4)(x1020
Figure 21. Minimum variance solutions for the second-iteration multilayered earth model with HI
(b) q2 = q 3 , q,,,,= 10'' P a s and (c) with q3 = q4, qlm= 5 x 10" Pa s.
and in which the viscosity q2 of the uppermost mantle layer
from the base of the lithosphere to 200 km depth is allowed
to be as low as 10" Pa s and the observations do not support
mantle models with very low viscosities for the uppermost
mantle.
Ice model parameters
While the solutions for earth model parameters, at least in
the case of the British rebound, are not significantly affected
by the inclusion of the second-iteration corrective term in
eq. (l), the estimate of the scale factor p"" for the British
ice sheet is, and the inclusion of these terms results in a
lowering of the total ice thickness required (compare Figs
17c and d). The solution for the minimum variance
three-layer earth model points to p"" = 1.02 f 0.04 (Fig.
17d) and indicates that the maximum ice thickness is
unlikely to have exceeded about 1500m over central
Scotland, wholly consistent with the geomorphological
evidence (Section 2). Some trade off between this parameter
and the earth-model parameters does occur (Fig. 17d), with
increased lithospheric thickness estimates leading to greater
p"" values being required in order to produce a comparable
= 80 km
3
Pa s)
and (a) q2 = q 3 , qlm= 5 x lo2' P a s ,
rebound amplitude, although the minimum variance is
increased. These @ estimates are for the so-called fully
non-adiabatic models. Fully adiabatic models yield estimates
that are about 5 per cent smaller and largely within the
uncertainty estimate of this parameter.
The estimated correction 6c(t)to the equivalent
sea-level function (eq. 1) from the second-iteration solution
is illustrated in Fig. 22 and indicates that after about
6000yrBP, a rise of about 2 m in equivalent sea-level
occurred; that sufficient glacial melting continued after this
time over the next 1500 to 1000 years to raise sea-level
globally by about 2 m . This result is consistent with earlier
analyses of Late Holocene sea-levels in both the Australian
region and the northwestern European region (Nakada &
Lambeck 1988; Lambeck et af. 1990) and lends support to
glacial models in which the Antarctic ice sheet continued to
add meltwater to the oceans after the complete melting of
the northern ice sheets. Before 6000 yr BP the equivalent
sea-level corrections change sign, indicating a more complex
melting model than the relatively uniform melting rates
adopted for the ARC and ANT ice models (see Part I).
While the accuracy estimates of these corrective terms for
earlier times are relatively low, the Sr(t)
function points to
982
K. Lambeck
in the second regional solution the earth parameters (4)
have been adopted and local p values are estimated
independently for each of the four subregions. Of these
solutions only the Moray and Solway yield p"" estimates
that are significantly different from the combined solution
(Table 2). In particular, the Moray solution suggests a need
to increase the ice load by about 17 per cent over northern
Scotland, while the Solway solution points to a decrease in
the ice heights by about 7 per cent over northwest England.
These regional solutions are discussed further below.
15
10
5
5
DISCUSSION
0
time (x1OOO years BP)
Figure 22. The corrective term 6T'(t) to the nominal equivalent
sea-level function A r ( f ) . Intervals in which the gradient of this
function is positive indicate times of more rapid global melting than
assumed in the nominal melting model.
a faster global melting rate in the interval before about
11 000 yr BP than assumed (that part of the function for
which the gradient of
is positive) and to a slower melting
rate at the time of the Loch Lomond Stadial, suggesting that
the residual global ice sheets expanded in the interval 11 000
to 10500yrBP so as to lower sea-level by about 3-4m.
However, the correction at 10500yrBP is based almost
entirely on the nominal date given for the Late Interglacial
Buried shoreline and Main Rock Platform of Scotland, and
without a firmer basis for this age it would be premature to
interpret this anomaly in the equivalent sea-level function in
terms of a global sea-level response. A similar examination
of the sea-level evidence along the coasts of Norway and
southern Sweden is warranted.
Regional models
Generally, agreement between observations and predictions
based on the above optimum models (4) with p"" = 1.02 is
satisfactory, but some significant discrepancies occur, most
notably for sites in the Beauly Firth of northeastern
Scotland (Fig. 23c), which point to either a regional
variation in the earth response or to local departures of the
ice load from that assumed in the model. To examine these
alternative interpretations, the observational data base for
Scotland and northwestern England has been divided into
four subregions: (1) southeastern Scotland, comprising
primarily the Forth and Tay estuaries and valleys, (2)
northeastern Scotland with the sites occurring in the Moray,
Inverness and Beauly firths, (3) the Argyll region of western
Scotland including the Clyde Estuary and Inner Hebrides,
and (4) the Solway Firth of southwestern Scotland and the
Morecambe Bay area of northwest England. In a first
calculation, effective earth and ice parameters for the
three-layer model are estimated independently for each
subregion.
However, the resulting q2 and He all lie within the range
of values defined by the overall solution (4) and a strong
case for lateral variation in earth response cannot be made
particularly as the solutions for both the Moray and Solway
regions are not well constrained by the relatively small
numbers of observations available for these regions. Hence,
A comparison of observations and predictions of sea-level
change
The comparison of the predicted sea-levels with the
observed levels can be affected in several ways, including
the time series of sea-level change at characteristic sites
(Fig. 23), as isobases of the mean sea-level for specific
epochs (Fig. 24), and as height-distance diagrams of past
mean sea-level surfaces relative to their present levels (Fig.
25). For the Forth and Tay firths of eastern Scotland the
predicted isobases for mean sea-level at the time of
formation of the principal shorelines (Fig. 24) can be
compared with the observed isobases in Fig. 4 with the
proviso that the latter correspond to the shoreline-formation
heights (mainly the MHWS level which is typically at 2-3 m
above MSL throughout the region). The generally
satisfactory agreement between the two is also evident in the
sea-level gradient plots for the major shorelines along the
southern shore of the Firth of Forth (Fig. 25a), except that
the predicted gradients for the older East Fife 6 and Main
Perth Shorelines are somewhat steeper than observed.
Along the northern shore of the Tay Firth the observed and
predicted gradients are consistent (Fig. 25b) although the
predicted heights for the East Fife 6 Shoreline lie below the
observed heights. This, together with the Forth result,
suggests that the ice load should be increased to the north or
northwest of the Firth of Tay.
The regional solution for the Moray-Beauly region
indicated a need to increase the ice load in this region (Fig.
23c) and Fig. 24(b) illustrates the sea-level isobases over
northern Scotland based on the best-fitting local /3 value of
j3 = 1.20. The predicted isobases in the region trend in a
nearly easterly direction, whereas the observed trend is east
of northeast (compare Fig. 24b with 6) and this points to a
need to place the additional ice volume over the region of
Ross & Cromarty to the northwest of Beauly (see Fig. 3 of
I for location map). This conclusion is reinforced by the
limited evidence from Wester Ross (in the area of Malvaig,
Fig. 3 of Part I) where the model predicts Late Glacial
shorelines that are near or below present mean sea-level
(Fig. 24b), whereas the rock platforms attributed to a Late
Glacial origin, Sissons & Dawson (1981), occurs at heights
of 20-30 m. Thus, a considerably greater ice load is required
in the northern region of Scotland unless the Wester Ross
rock platform pre-dates the last glacial event. The
predictions for the Main Postglacial event in this region are
generally consistent with the observed levels reported by
Sissons & Dawson (1981). Further north, in Orkney and the
Glacial rebound of the British Isles-11
Glencarse & Bridge of Earn
Firth of Tay
- 10
Beauly Firth
Arnprior
Forth River
40
983
lnverness
30
..
16
12
14
10
8
6
4
15
S
10
0
16
12
14
10
8
6
4
2
s
Lochgilphead
Loch Fyne
1
-10 1
15
5
10
-20
-25
-
16
.-
Thames Estuary
M
8
10
0
Morecambe Bay
Lancashire
6
2
4
0
10
8
6
4
2
0
(9)
01
-20
Fenlands
East Anglia
-15
8
6
4
2
time (x1ooO years BP)
-
1
-25
1
-30
1
0
-
1
T/
f
0
I
Bridgwater Bay
Bristal Channel
8
6
4
2
time (x1ooO years BP)
0
time (x1ooO years BP)
Figure 23. Predicted and observed sea-levels at selected sites in Scotland and England. The predictions are based on the earth model
parameters defined by (4) and o n the local /3 factors (Table 2). For the Beauly Firth the prediction based on the global /3 value is also shown
(broken line).
Outer Hebrides, the model predicts sea-levels that are
below the present level throughout Late Devensian and
Holocene time, consistent with an absence of observed
highstands in the region as well as with sea-levels at
8000-5000 yr BP in the Outer Hebrides being several metres
below present levels (Binns 1972; Ritchie 1985).
For the west coast of Scotland the agreement between
observed and predicted sea-level curves for Lochgilphead
and the Clyde Firth is also satisfactory (Fig. 23d) and the
association of the Main Rock Platform with sea-level at
10500yrBP leads to a consistent model (Fig. 25d). The
older and higher rock platforms of western Scotland at
heights of up to 50 m OD are predicted to occur only during
the earliest stages of deglaciation, at about 16 000 yr BP
984
K . Lambeck
Table 2. Regional solutions for pBRand corresponding minimum
variances for the earth model defined by (4). For the southern
England, Wales and northern France solution, the PBR value has
been set to unity.
Region
PR
Variance
Au data
1.03
2.14
Forth-Tay
1
.oo
0.96
Moray Beady
1.20
0.74
Western Scotland
1.03
1.52
Solway and NW England
0.96
1.78
Southern England, Wales, France
1 .00
1.36
(e.g. Fig. 23d), when the region is believed to have still been
ice covered and by the time the Inner Hebrides became ice
free the predicted isobases are significantly lower than the
observed heights. The model, therefore, supports the
interpretation that these high-level platforms pre-date the
last glacial cycle.
In neither eastern nor northeastern Scotland do the
models predict the short-duration oscillations in sea-level as
defined by the buried shorelines between about 10200 and
8700yrBP. If the fluctuations are the results of the Loch
Lomond Advance then the ice volumes involved must have
been greater than adopted and its duration and timing must
be different from what is usually assumed. During the
advance, renewed crustal loading will produce an apparent
rise in sea-level at a site such as Arnprior near the advance's
maximum limit (Fig. 17) but this would be expected to have
occurred somewhat earlier than the age of 10200yrBP
attributed to the highest of the buried beaches. Also, if the
other buried beaches are attributed to the Loch Lomond
Advance then the ice retreat would have occurred
non-uniformly and more slowly until after 8700 yr BP, the
age of the Low Buried Beach. Generally, neither the
observational constraints nor the numerical resolution of the
present rebound model are adequate to examine in detail
the relative sea-level response to the Loch Lomond
Advance.
Agreement between observations and predictions for the
northern England sites is typified by the comparison for
Morecambe (Fig. 23e). In this region the Main Postglacial
zero contour isobase for mean sea-level is predicted to pass
through Morecambe Bay and small highstands at about
6000 yr BP are predicted for sites such as the Esk or Duddon
Estuaries to the north (Fig. 24a). The predictions for these
localities suggest that sea-level should have reached its
present level earlier than observed although the observational data base (Andrews et al. 1973) is unsatisfactory for
this region (see Fig. 1ld). To the south, at Southport and
Formby, agreement between observations and predictions is
again satisfactory. Also, the gradient of the Main Postglacial
shoreline from the Solway Firth to north of Morecambe Bay
is predicted to be about 0.04 m km-l which agrees well with
the observed gradient of 0.03 m km-' for raised beaches
reported by Walker (1966).
For the sites in Wales, southern England and the French
Channel coast (Fig. 25), the predicted sea-levels are
generally consistent with observations, although at a number
of sites the observations lie above the predicted level. This is
the case, for example, for the result at Borth Bog in
Cardigan Bay (Fig. 2 5 ) where the tree stumps may define a
level above MHWS tide as suggested by Tooley (1979). For
the Thames Estuary the discrepancy is greatest for Devoy's
(1982) Table 4 results which define mainly upper limits to
the sea-level curve. Thus the discrepancies noted for
southern England may reflect the different nature of the
observational data as much as limitations of the model
predictions, and no systematic pattern of the differences for
the region as a whole occurs.
Mantle rheology
The effective parameters for the mantle below the Great
Britain region are defined by the results (4) for the
three-layer earth model as
He = 65-70 km
r ] , = (4-4.5)102"
Pa s
4 x lo2' Pa s < qIm< 2 x lo2' Pa s
where r ] , represents the average viscosity for the mantle
between the base of the lithosphere and the 670 km seismic
discontinuity. These values are effective parameters that are
representative of the mantle response to load cycles
operating on the order of 104yr. The associated stress and
strain-rate fields have not been evaluated at this stage, but
simple order-of-magnitude calculations for the mantle above
670 km suggest deviatoric stresses and strain rates of the
order of 1 MPa and (2-5)10-16 s-' respectively and
considerably smaller values at greater depths.
Tests with multilayered models have shown that: (1) there
is no significant change in viscosity from the upper mantle to
the transition zone (Figs 20 and 21) and, (2) there is no
strong evidence for viscosity layering in the upper mantle,
including a low-viscosity layer immediately below the
lithosphere (Fig. 20). The absence of a marked viscosity
gradient above 670 km is perhaps not an obvious conclusion
in view of the many variables that can be expected to affect
the creep strength (i.e. viscosity) of the mantle materials. In
particular, because garnet appears to have a considerable
higher creep strength than either olivine or spinel (Karat0
1989), it has sometimes been argued that the transition zone
could be expected to possess a significantly higher viscosity
(Spada et al. 1992), but the result of these rebound
inversions suggests instead that the creep strength of this
layer as a whole is controlled by the weaker spinel phase
such that the effective viscosity at ambient temperatures and
pressures is not significantly different from that of the
predominantly olivine mantle above 400 km depth. Another
factor that could produce a strong depth dependence of
viscosity is the mantle temperature, but the inferred near
constancy of viscosity with depth suggests that, if this is an
important mechanism, then the ratio of temperature to
melting point temperature does not change significantly with
depth in the mantle beneath the British Isles.
The solution for the earth model parameters based on the
British Isles data agrees well with that previously
Glacial rebound of the British Isles-II
985
15 000 years BP
6 000 years BP
Figure 24. Contours of the predicted constant relative mean sea-level over northern Great Britain based on the earth model defined by (4) and
(a) the global p"" value, and (b) for northern Scotland based on the p value from the northeastern Scotland regional solution.
determined from the Fennoscandian rebound using the same
approach although in the latter case only three-layered
mantle models were considered (Lambeck et al. 1990). In
that study relative sea-level observations were used from
sites either near the centre of rebound or well outside the
former ice margin so that the model results were not
strongly dependent on limitations of the ice model used (the
ARC model discussed in Part I). Because the Fennoscandian ice sheet is considerably larger than the British ice
sheet the depth dependence of the mantle response to the
loading is different in the two cases so that the similar result
for q2 lends further support to models of nearly constant
viscosity profiles between the base of the lithosphere and the
670 km boundary.
986
13 500 years BP
15 OOO years BP
\
\
\
-100
10 500 years BP
6 OOO years BP
'I
Figure 24. (Continued.)
-
-
' I
\
\
(b)
Amprior
Kincardie
bbar
Leith
Stkling
Aberlady
t
Firth of Forth (south side)
20
0
c
/
-20
40
60
80
distance (km)
120
100
lo Tay Firth (north side)
n
I , , .
" [
-20
0
20
40
distance (h)
..
1
80
60
(d115 I
I
T
Western Scotland
0 -.
-_
Moray & Beady firths
_-5
-10
0
5
10
15
distance (km)
20
25
-20
k3g
cyuy
,
0
20
(Jmf
40
60
distance (km)
T
upper Sound
O h
of Jura
80
100
120
Figure 25. Predicted'(continu0us lines) and observed (broken lines and error bars) for some of the major shorelines (reduced to MSL)
observed in Scotland.
Glacial rebound of the British Isles-I1
The effective viscosity q2 obtained for both the
Fennoscandian and British regions is higher by a factor of
about 2 than that obtained from analyses of sea-level change
along the Australian margin. In the last example, the
departures from the eustatic sea-level change is primarily a
result of the response of the mantle to the water loading and
to a transport of material between oceanic mantle and
continental mantle. The value of (2-3)102"Pa s obtained
(Lambeck & Nakada 1990) can, therefore, be interpreted as
some average of the two mantle regions and the difference
between this and the European value can be attributed to
lateral variation in mantle structure (Nakada & Lambeck
1991). Such variations in upper mantle viscosity are
generally consistent with observations of lateral variations in
seismic shear wave velocities (e.g. Woodhouse & Dziewonski 1984; Tanimoto 1990a,b; Romanowitz 1990) and seismic
wave attenuation.
Pa s for the lower mantle
While the above estimate of
viscosity is constrained primarily by the mantle response to
the water loading which occurs on length scales up to that of
the dimension of the Atlantic Ocean, the resolution is not
high (Fig. 19), but models with q l r n 5 4X lo2' P a s are
excluded. The result is consistent with previous studies
(Lambeck & Nakada 1990; Nakada & Lambeck 1991) and
the conclusion that the lower mantle has a significantly
higher viscosity than the mantle above 670 km appears to be
a robust one.
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