Geology Reference
In-Depth Information
coefficients from observed values of gravity.
The conventional normalization (
14.6
), (
14.7
),
(
14.8
), and (
14.9
) is often substituted by the so-
called
full normalization
, which allows a better
manipulation of the spherical harmonics. In this
instance, the surface harmonics
R
nm
(™,¥)and
S
nm
(™,¥) are replaced by the
fully normalized
harmonics
defined as follows:
R
nm
.™;¥/
S
nm
.™;¥/
of artificial satellites and updated regularly using
terrestrial data. Examples of recent geopoten-
tial models are GEM-T3 (Lerch et al.
1994
),
GRIM4 (Schwintzer et al.
1997
), GRIM5 (Bian-
cale et al.
2000
), GGM02 (Tapley et al.
2005
),
and EGM2008 (Pavlis et al.
2012
). These models
can be used to build reliable representations of
the Earth's gravity field in a variety of techni-
cal and scientific applications. Starting from a
geopotential model, it is possible to build equipo-
tential surfaces that locally define the vertical
direction. Among the infinitely many equipo-
tential surfaces of the Earth's gravity field, the
one passing through the mean surface of the
oceans (removing the effect of tides) is used as
a reference surface for measuring elevation. This
surface is the
geoid
and represents the math-
ematical figure of the Earth as determined by
the density distribution in the Earth's interior
and by rotation. The (usually) curved lines that
intersect orthogonally any equipotential surface
are called
plumb lines
. As illustrated in Fig.
14.1
,
given two values
V
1
and
V
2
for the potential, the
corresponding equipotential surfaces
V
(
r
)
D
V
1
and
V
(
r
)
D
V
2
are not generally parallel each
other. This fact leads to a curious paradox. Let us
consider two near points at the Earth's surface,
A
and
B
. Now let us assume that a level placed at
an intermediate location
C
indicates that the two
points lie on the same horizontal surface. In this
instance, they should have the same altitude.
However, the effective altitude of the two
points relative to the geoid will be in most cases
slightly different, depending on the shape of the
plumb lines. In fact the altitude, more precisely
the
orthometric height H
, of a point
P
is defined
as the length of the plumb line between the actual
location of
P
and the geoid. For this reason,
a precise determination of orthometric altitudes
requires in general a combination of gravity mea-
surements and optical leveling.
s
2.2n
C
1/
.n
m/Š
.n
C
m/Š
D
R
nm
.™;¥/
S
nm
.™;¥/
I
m
¤
0
(14.13)
p
2n
C
1R
n0
.™/
D
p
2n
C
1P
n
.cos™/
(14.14)
R
n0
.™/
D
With these definitions, the orthogonality rela-
tions (
14.6
), (
14.7
), (
14.8
), and (
14.9
)arethen
substituted by more canonical orthogonality re-
lations:
Z
Z
2
d¥
R
nm
.™;¥/R
sr
.™;¥/ sin ™d™
D
4 •
ns
•
mr
0
0
(14.15)
Z
Z
2
d¥
S
nm
.™;¥/S
sr
.™;¥/sin ™d™
D
4 •
ns
•
mr
0
0
(14.16)
Therefore, the average square of fully nor-
malized harmonics over the unit sphere is unity,
whether or not
m
is zero. Also the coefficients of
the spherical harmonic expansion have now more
simple expressions:
Z
Z
2
1
4
a
nm
D
d¥
V.™;¥/R
nm
.™;¥/sin ™d™
0
0
(14.17)
Z
Z
2
1
4
b
nm
D
d¥
V.™;¥/S
nm
.™;¥/ sin ™d™
14.3 Geoid and Ellipsoid
0
0
(14.18)
The shape of the geoid is determined by the dis-
tribution of masses in the Earth, especially from
lateral density variations in the Earth's mantle.
Stokes' coefficients are determined empiri-
cally from the analysis of orbital perturbations
