Geoscience Reference
In-Depth Information
This observation is explained by the notion of the strong lithosphere, which is
able to support small mass anomalies but bends or flexes under very-large-scale
mass anomalies, causing flow and readjustment in the weaker underlying astheno-
sphere. Erosion also occurs. Isostatic balance is thus achieved.
The geoid height anomaly resulting from an isostatic density distribution is
not zero and can be calculated. It can be shown (see Turcotte and Schubert 2002,
p. 217) that the geoid height anomaly at any point P is given by
D
0
ρ
2
π
G
g
h
=−
(
z
)
z
d
z
(5.49)
where
y
is the reference gravity value,
(
z
) the anomalous density at depth
z
beneath point P and
D
the compensation depth. Depth
z
is measured positively
downwards with
z
ρ
0 corresponding to the spheroid. Equation (5.49) therefore
gives the geoid height anomaly due to long-wavelength isostatic density anoma-
lies. Geoid anomalies can be used to estimate the variation of density with depth.
In practice, we need to work in reverse, by first calculating the geoid height
anomaly for an isostatic density model and then comparing the calculations with
the observed measurements.
As
=
an
example,
consider
the
Airy
compensation
model
illustrated
in
Fig. 5.6(a). The reference structure is an upper layer of density
ρ
u
and thick-
ness
t
and a substratum of density
ρ
s
. All density anomalies are
with respect
to this reference structure
. The geoid height anomaly over a mountain range of
height
h
1
, calculated by using Eq. (5.49), is
0
−
h
1
ρ
u
z
d
z
+
(
ρ
u
−
ρ
s
)
z
d
z
2
π
G
g
t
+
r
1
h
=−
t
−
h
1
ρ
u
+
(
ρ
u
−
ρ
s
)
2
tr
1
+
r
1
=−
π
G
g
(5.50)
After substituting for
r
1
from Eq. (5.24) and rearranging terms, we finally
obtain
g
ρ
u
h
1
2
t
π
G
ρ
s
h
1
ρ
s
−
ρ
u
h
=
+
(5.51)
10
3
kg m
−
3
,
respectively, and a reference crust 35 km thick, the geoid height anomaly is
10
3
Therefore, for crustal and mantle densities of 2.8
×
and 3.3
×
h
6
h
1
(0
.
7
+
0
.
066
h
1
)m
(5.52)
where
h
1
is in kilometres. Thus, a compensated mountain range 3 km high would
result in a positive geoid height anomaly of about 16 m.
