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density below the points of observation. On
scales smaller than 100 km or so the strength
(or very high viscosity) of the lithosphere can
support departures from isostatic balance that
are apparent as free-air anomalies, notably at
the margins of continents. This is an indication
of what is known as the depth of compensation,
that is the depth below which viscosity is low
enough to equalize pressures (or below which
homogeneity can be assumed). As we see from
the global analysis in Section 9.3, on a larger
scale isostasy prevails and this means equality
of the masses in all vert ical columns, so that
free-air anomalies are weak. However, on a
scale of several thousand kilometres free-air
anomalies larger than those at intermediate
scales are apparent from the geoid plot in
Fig. 9.1. They are attributed to heterogeneity of
the lower mantle. This is possible because,
although the relatively low viscosity of the
asthenosphere explains the isostatic balance
at intermediate scales (200–2000 km) it does
not nullify the effect of irregular masses in
the more viscous lower mantle. These masses
must be deep as they are not evident at the
intermediate scale.
For the interpretation of local geol ogical
structures it is often more effective to calcu-
late gravity on the geoid assuming complete
removal of all material above it, instead of col-
lapse to the geoid. This gives Bouguer anoma-
lies. In the simplest cases, with no allowance
for topography or heterogeneity, the removed
material is assumed to be an extensive slab of
uniform thickness equal to the height at the
point of measurement. The gravity due to a
slab of thickness h,density and infinite hori-
zontal extent, is (see Problem 9.2)
g ¼ 2pGh; (9:19)
and is independent of distance from it. Thus,
calculation of gravity on the geoid by the
Bouguer method means downward extrapola-
tion from elevation h by a Bouguer gradient
which is equal to the free-air g radient minus
g/h ¼2pG. Commonly a standard density,
2670 kg m
3
, is used for this purpose and
then the Bouguer gradient is 0.1967 mGal m
1
(1.967 10
6
s
2
), about 2/3 of the free-air gra-
dient. If the density of the surface layers is
knownthenitisobviouslybettertousethat
rather than the standard value and, as with
free-air calculations, topographic corrections are
usually necessary. Bouguer anomalies are of inter-
est in studies of local crustal structure because
they reflect density variations immediately
below the geoid. On a continental scale Bouguer
anomaly maps show systematic lows over conti-
nents because, by Eq. (9.19), they differ from free-
air anomalies by 2pGh at height h and on this
scale isostatic balance prevails, making free-air
anomalies small.
The purpose of free-air and Bouguer anom-
aly maps is to remove the effect of ground ele-
vation, which obscures the underlying density
variations. They are complementary, giving dif-
ferent information, and it is instructive to have
both.Thethirdmethodistousegeoidanoma-
lies, that is variations in the height of the geoid,
taking advantage of satellite observations of
large-scale features, as in Fig. 9.1, and giving a
different perspective on isostasy. The general
idea can be understood from a simple example.
If, in a broad, topographically featureless area
of average d ensity r,wehaveapatchwithden-
sity ( þD) underlying a layer of equal thick-
ness but density ( D) then, by the argument
in Section 9.3 the patch would be isostatically
balanced. However, it would nevertheless
appear as a geoid anomaly. The reason is that
the deeper, denser layer gives enhanced gravity
within and above it and, since this is the gra-
dient of gravitational potential, in integrating
upwards from the depth of compensation
(below the patch) t he potential corresponding
to the geoid is reached at a lower level than in
areas outside the patch. A density dipole of this
form gives a geoid low. Conversely, if a denser
layer overlies a less dense one it gives a geoid
high. Thus a geoid anomaly gives information
about the depth distribution of mass and not
just its total. In this simple, plane layered
model the geo id anomaly, DN (metres), is
related to the depth dependence of the density
anomaly by integrating Eq. (9.19) through it
from the depth of co mpensation to the geoid
potential,
126 THE GEOID, ISOSTASY AND REBOUND