Geology Reference
In-Depth Information
angle between the normal to the ellipsoid and the
equatorial plane, while ”
a
and ”
b
are respectively
the equatorial and polar normal gravity. In the
WGS84, these parameters are given by:
The
formula
of
Somigliana
(
14.35
)pre-
dicts
an
average
latitudinal
variation
of
”(§)of
0.8
mGal/km
in
N-S
direction
(1 Gal
D
10
-2
m/s
2
).
Now we are ready to give an intuitive repre-
sentation of the Earth's figure through a com-
parison of the geoid with the reference ellip-
soid. In fact, any departure from the geometrical
simplicity of the ellipsoid of revolution must be
associated with localized variations of density in
the Earth's interior. In particular, a local excess of
density in the mantle or in the lithosphere tends to
increase the gravity in that area, possibly leading
to an upward displacement of the geoid relative
to the ellipsoid. Conversely, a mass deficit could
cause a decrease of gravity and a consequent
downward bending of the geoid. The method
for calculating these
geoid undulations
(or
geoid
anomalies
) is rather complex and goes beyond
the scope of this topic. The interested reader can
find an exhaustive description of the technique in
the classic book of Heiskanen and Moritz (
1993
),
while a modern approach to the problem can be
found in Rapp (
1997
). The undulations of the
recent geoid model EGM2008 (Pavlis et al.
2012
)
are illustrated in Fig.
14.3
. They range from
107 m in the central Indian Basin to
C
86 m
”
a
D
9:7803253359 ms
2
”
b
D
9:8321849378 ms
2
(14.36)
while
b
can be calculated using (
14.24
)andthe
reference values of Table
14.1
:
b
D
a.1
f/
D
6;356;752:3 m
(14.37)
Table 14.1
Basic reference values in the WGS84 refer-
ence frame
Quantity
Symbol Value
Units
Semi-major axis
a
6,378,137.0
m
m
3
/s
2
Geocentric
gravitational constant
(mass of Earth's
atmosphere included)
GM
3.986004418
10
14
Reciprocal of
flattening
1/
f
298.257223563
s
1
Mean angular velocity
7,292,115
10
11
m
2
s
2
Theoretical normal
potential of the
ellipsoid (derived)
U
0
62,636,860.8497
Fig. 14.3
Geoid undulations relative to the WGS84 ellipsoid in the EGM2008 model (Pavlis et al.
2012
)
