Geology Reference
In-Depth Information
GM
r
C
G
2r
3
Œ.A
C
B
C
C/
3I .
or small bodies falling in a vacuum. In the
latter case, a test mass is dropped inside a
vacuum chamber for some centimeters and
the trajectory is monitored accurately using
a laser interferometer and an atomic clock.
The precision of these instruments is of a
few Gal (10
8
ms
2
) in static conditions. In
general, the observed gravity results from the
superposition of several factors, which must
be accurately separated to obtain a data set
that can be interpreted geologically. In fact,
the final objective of most gravity surveys
is to determine
gravity anomalies
associated
with short-wavelength lateral density variations
close to the Earth's surface and to reconstruct
the density distribution in the underground.
Therefore, the contribution of these small-scale
density anomalies must be isolated in the gravity
measurements from other factors that influence
the variability of the observed data. These factors
include:
The variations of latitude, which affect the
formula of Somigliana (
14.35
);
The effect of elevation above the sea level;
The average crustal mass above the sea level;
The lateral variations of altitude about the
average mass above the sea level;
Periodic tidal changes in the shape of the
Earth;
Variations of the centrifugal potential caused
by the motion of the gravimeter, for example
when the measurements are made by aircrafts
or ships. This is called
Eötvös effect
;
The reduction of raw gravity data to a format
that can be interpreted in terms of density anoma-
lies is performed through a series of corrections.
In small-scale studies, the difference between
geoid and reference ellipsoid is usually neglected,
thereby the orthometric altitude
H
relative to the
geoid is assumed to be coincident with the
geo-
metric altitude h
relative to the ellipsoid. While
shipboard gravity data can be compared directly
with the normal gravity ”
0
, measurements per-
formed at elevation
h
above the ellipsoid must
be adjusted to take into account of the variation
of gravity with altitude. The normal gravity at a
small altitude
h
above the reference surface can
be expressed as a Taylor's series expansion about
V.r/
Š
r/
(14.46)
This formula reveals the role of the moments
of inertia for the determination of the Earth's
gravity field. Assuming that the Earth can be
approximated by an oblate spheroid and that the
z
axis coincides with the rotation axis, we have
that
A
D
B
and the inertial tensor is diagonal,
r/ can
be expressed as follows:
r/
D
A
r
y
r
x
Cb
r
z
I.
b
C
C
b
D
A
1
b
r
z
r
z
D
A
C
.C
A/
r
z
C
C
b
b
D
A
C
.C
A/ cos
2
™
(14.47)
where ™ is the colatitude of
r
. Therefore, assum-
ing axial symmetry we have that MacCullagh's
formula (
14.46
) reduces to:
GM
r
G
r
3
.C
A/P
2
.cos ™/
(14.48)
V.r/
Š
Comparing this expression with the spherical
harmonic expansion (
14.23
), we obtain an ex-
pression for the first coefficients
J
n
:
C
A
MR
2
J
0
D
1
I
J
1
D
0
I
J
2
D
(14.49)
Therefore, MacCullagh's formula allows to
determine the excess moment of inertia about the
spin axis relative to the moment of inertia about
an equatorial axis, associated with the Earth's
flattening.
14.5 Gravity Measurements
and Reduction of Gravity
Data
Gravity measurements are made through
gravimeters
close to the Earth's surface, by
aircraft, ships, or land surveys. Gravimeters may
be based on precise spring balances, pendulums
