Geology Reference
In-Depth Information
Consequently,
V
is harmonic in any region
R
where ¡
D
0. According to the theorems proved in
on the basis of a given set of boundary conditions.
However, in geophysics it is also quite com-
mon calculating the potential or the field through
direct integration, starting from the continuous
mechanics version of Newton's law (see Eqs.
both these approaches to study the Earth's gravity
field.
Z
Z
2
d¥
S
nm
.™;¥/S
sr
.™;¥/sin ™d™
0
0
2
2n
C
1
.n
C
m/Š
.n
m/Š
•
ns
•
mr
I
m
¤
0 (14.7)
D
Furthermore,
Z
Z
2
d¥
R
nm
.™;¥/S
sr
.™;¥/ sin ™d™
0
0
D
0 for any n;m;r;s
(14.8)
14.2
Spherical Harmonic
Expansion of the
Geopotential: The Geoid
Z
Z
2
4
2n
C
1
ŒR
n0
.™/
2
sin ™d™
D
d¥
(14.9)
0
0
A solution of Laplace's equation for the geopo-
tential can be obtained using the techniques de-
harmonic expansion has the form:
The determination of the coefficients
a
nm
and
b
nm
in (
14.5
) can be accomplished multiplying
both sides of this equation by some polynomial
R
rs
(™,¥) or, respectively,
S
rs
(™,¥), and integrating
over the unit sphere. Using the orthogonality
relations (
14.6
), (
14.7
), (
14.8
), and (
14.9
)we
obtain:
R
r
n
X
GM
r
V.r;™;
¥
/
D
n
D
0
X
n
Z
Z
2
Œa
nm
cosm
¥
C
b
nm
sin m
¥
2n
C
1
2
.n
m/Š
.n
C
m/Š
a
nm
D
d¥
V.™;¥/
mD0
P
nm
.cos ™/
I
r
R
0
0
(14.5)
R
nm
.™;¥/sin ™d™
I
m
¤
0
where
M
is the Earth's mass,
R
is the Equatorial
radius, and the coefficients
a
nm
and
b
nm
are called
Stokes
'
coefficients
. Just like in the case of the
geomagnetic potential, specific orthogonality and
normalization constraints are used for the sur-
face spherical harmonics
R
nm
(™,¥)
P
nm
cos
m
¥
and
S
nm
(™,¥)
P
nm
sin
m
¥ (e.g., Heiskanen and
Moritz
1993
):
(14.10)
Z
Z
2
2n
C
1
4
a
n0
D
d¥
V.™;¥/P
n
.cos™/d™
0
0
(14.11)
Z
2
Z
2n
C
1
2
.n
m/Š
.n
C
m/Š
b
nm
D
d¥
V.™;¥/
0
0
Z
Z
2
S
nm
.™;¥/ sin ™d™
I
m
¤
0
d¥
R
nm
.™;¥/R
sr
.™;¥/ sin ™d™
(14.12)
0
0
2
2n
C
1
.n
C
m/Š
.n
m/Š
•
ns
•
mr
I
m
¤
0 (14.6)
These
are
the
basic
equations
used
to
D
determine approximate values of
the
Stokes'
