accuracies can occur, for example, over distances of
10 km or less across former ice margins or continental
margins and the numerical solutions must be able to
represent this.
The formulation of the glacio-hydro-isostatic rebound is
well established but the definition of the rheological
parameters or aspects of the ice models themselves are
often less well known and are mostly inferred from the
analysis of the rebound evidence. Thus, for descriptive
purposes, the formulation serves as an interpolation device
between fragmentary observational evidence that also
allows for a degree of extrapolation beyond the limits of
the time–space dimensions defined by the observational
data.
The sea-level response
In the context of coral reefs, the most important consequence of the glacio-hydro-isostatic response is sea-level
change. Schematically, the equation describing sea-level
change, Δz rsl (j,t), can be written as
Dz rsl j;t
ð Þ ¼ Dz esl t
ð Þ þ Dz I j;t
ð Þ þ Dz T j;t
ð Þ þ Dz O j;t
ð Þ
with
Dz I j;t
ð Þ ¼ Dz IÀg j;t
ð Þ þ D IÀh j;t
ð Þ
Δz rsl (j, t) represents the change at location j of the sea
surface relative to land at time t compared to its present
position at time t P . The first term on the right-hand side
represents the ice-volume equivalent sea-level contribution (ESL) and is a measure of the globally averaged
sea-level change due to a change in ice volume. It is
defined as
Dz esl ðtÞ ¼ À
r i
r o
Z
t
1
A o ðtÞ
dV i dt
dt
V i is the ice volume at time t and includes the ice mass on
continents and grounded on continental shelves. A o (t) is
the ocean surface area and is defined by the shoreline
and the grounding lines of any ice on the shelves, both at
time t. r i , r ο are the average densities of ice and ocean
water. In the absence of any other factors that lead to
changes in ocean volume, the ice-volume equivalent sea
level is equal to eustatic sea level.
The second term, Dz I (j,t), is the combined isostatic
contribution. It is schematically divided here into two
contributions: the glacio-isostatic part Dz I–g (j,t), and
the hydro-isostatic part Dz I–h (j,t). Both terms include
contributions from the deformation of the crust under
the time-dependent surface load and from the change in
the gravitational potential of the earth–ice–water system
(sometimes referred to as geoid changes). Coupling
between the two isostatic terms arises from the deformation of the ocean basins and from the change in gravitational attraction by the ice. This is ignored in this
schematic description but not in the actual models
(Lambeck et al., 2003; Mitrovica and Milne, 2003).
The third term, Dz Τ (j,t), allows for any tectonic contributions to sea level. Many reefs occur in areas of tectonic
uplift or subsidence and the separation of these effects
from the isostatic signals remains a major challenge.
A frequently used criterion is to examine whether there
is evidence for a fossil Last Interglacial reef at the same
location. If this occurs within a few meters of present sea
level, then the site can be considered to have long-term
tectonic stability to better than 0.1 mm/year. Alternatively,
if the sea-level behavior can be calibrated against sites of
known tectonic stability, the comparison of model predictions and observations establishes rates of tectonic uplift
or subsidence.
The fourth term, Dz Ο (j,t), includes any oceanographic
or meteorological factors that cause sea level to depart
from a long-term equipotential surface. These are generally treated as perturbations in high-resolution analyses
of recent and Holocene sea-level change.
Despite most modern coral reefs occurring far from
the ice margins of the last glaciation – far-field sites – both
components of the isostatic response occur. The glacioisostatic response here is characterized by a broad wavelength response, reflecting global-scale mantle flow
driven by the growth and decay of the large ice sheets,
and the sea-level response is not sensitive to the details
of the ice sheets. More important here is the hydroisostatic component and this is strongly dependent on
the changes in the distribution of water around the site.
Figure 2 illustrates sea-level change at a continental
margin site. At a time t, the land surface is at a – a
0 and
the shoreline is at A(t). Between t and the present, t P , land
uplift has occurred by an amount u r and the new land surface is at b À b
0 . The ocean volume has also increased so
as to raise the ocean level relative to the center of mass
of the earth from z A to z B and the present shoreline is at
B(t P ). The original shoreline will now occur at A
0 , at
Δz(t) below the present sea level. The relative sea-level
change is therefore
Dz t
ð Þ ¼ ðz A À z B Þ À u r
and the change in elevation of any point (e.g., from C to C
0
on land or from D to D
0 offshore) measured with respect to
the sea level of the epoch is
h t
ð Þ ¼ h t P
ð Þ À z A À z B
ð
ÞÀu r
½
Š¼h t P
ð Þ À Dz t
ð Þ
This formulation is independent of the cause of land
movement or ocean volume change but for the glaciohydro-isostatic contribution only, the land movement
component will depend on the response of the earth’s surface to the changing ice–water load and the displacement
of the ocean surface will depend on how any meltwater is
distributed into the ocean basins and on the changing gravitational attraction of the solid earth and the ice on the
water surface. In the case of glacio-hydro isostasy u r and
the radial displacement of the ocean surface (z A À z B ).
GLACIO-HYDRO ISOSTASY
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