Geoscience Reference
In-Depth Information
where
ρ
1
(
z
) and
ρ
2
(
z
) are the density functions for crust with total thickness
t
1
and
t
2
, respectively, and
ρ
m
is the density of the mantle. When the density and seismic
velocities,
v
1
and
v
2
, are related by
ρ
1
(
z
)
=
ρ
m
−
k
v
1
(
z
)
(5.38)
k
v
2
(
z
)
where
k
is a constant, the two-way normal-incidence travel times
T
1
and
T
2
for the
two structures, given by
ρ
2
(
z
)
=
ρ
m
−
t
1
2
v
1
(
z
)
T
1
=
d
z
0
(5.39)
t
2
2
v
2
(
z
)
T
2
=
d
z
0
are equal. Although the density-velocity relationship does not exactly fit Eq. (5.38),
values of
k
between 3
×
10
6
and 4
×
10
6
kg m
−
2
s
−
1
approximate the Nafe-Drake
density-velocity curve (Fig. 4.2(d)).
Pratt-type isostatic compensation (Eq. (5.28)) with a depth-dependent density
requires that
t
1
0
ρ
1
(
z
)d
z
=
t
2
0
ρ
2
(
z
)d
z
(5.40)
In this case, the two two-way travel times (Eqs. (5.39)) are equal if the density and
seismic velocity are related by
k
v
1
(
z
)
ρ
1
(
z
)
=
ρ
k
−
(5.41)
k
v
2
(
z
)
where
k
and
ρ
k
are arbitrary constants. Thus, the Airy restriction on the
density-velocity relationship (Eq. (5.38)) is just a special case of Eq. (5.41).
It is therefore possible that observation of a nearly horizontal Moho on time
sections may just be an indication that the observed structures are isostatically
compensated. In an isostatically compensated region, if the density-velocity
relationship approximates Eq. (5.38)orEq.(5.41), a structure on the Moho would
not be seen on an unmigrated seismic section. Structures on the Moho would,
however, be seen after corrections for velocity were made.
ρ
2
(
z
)
=
ρ
k
−
5.5.5 Gravity anomalies due to some buried bodies
The gravitational attraction of some simple shapes can be calculated analytically
by using Eqs. (5.4) and (5.6). The attraction due to more complex shapes, however,
must be calculated numerically by computer. To illustrate the magnitude and type
of gravitational anomaly caused by subsurface bodies, we consider a few simple
examples.
