Geoscience Reference
In-Depth Information
(a)
(b)
(c)
n=3
n=3, m=2
n=m=2
Fig. 3
Spherical harmonics:
a
zonal,
b
tesseral,
c
sectoral
For the gravity field determination,
fully normalized spherical harmonics
are usually
used, with the condition that the average square of any fully normalized harmonic is
unity:
k
(
2
n
+
1
)(
n
−
m
)
!
P
nm
(
t
)
=
P
nm
(
t
),
(26)
(
n
+
m
)
!
with
k
=
1for
m
=
0 and
k
=
2for
m
=
0. Legendre functions multiplied by
cos
m
are called surface spherical harmonics and are used as in Eq. (
23
)
to describe the spatial characteristics of
V
on a spherical surface.
The associated Legendre functions change their sign (
n
λ
or sin
m
λ
−
m
) times within the interval
0
≤
θ
≤
π
and the functions cos
m
λ
and sin
m
λ
do change the sign 2
m
times in the
interval 0
, dividing the surface at geodetic parallels and meridians into
cells which are alternatingly positive and negative. For
m
≤
λ
≤
2
π
0 they divide the sphere
into a chequered pattern and are called tesseral harmonics (Fig.
3
b). For special cases
when
m
=
0 the spherical harmonics do not depend on the longitude and divide the
sphere into zones parallel to the equator and are called zonal (Fig.
3
a), or if
n
=
m
then the spherical harmonics degenerate into functions that divide the sphere into
sectors following the meridians and are therefore named sectoral (Fig.
3
c).
Comparing Eq. (
23
) with Eq. (
6
) it becomes clear that the spherical harmonic
coefficients
C
nm
and
S
nm
are describing mass integrals of the Earth gravity field with
C
nm
S
nm
=
cos
m
d
M
1
λ
r
n
P
nm
(
=
cos
θ)
.
(27)
sin
m
λ
(
2
n
+
1
)
Ma
n
Earth
The first degree coefficients are related to the rectangular coordinates of the center
of gravity, i.e. the geocenter. Following Hofmann-Wellenhof and Moritz (
2005
) and
Torge (
1989
) we get:
1
a
2
M
1
a
2
M
z
d
M
x
d
M
C
10
=
,
C
11
=
,
Earth
Earth
1
a
2
M
y
d
M
S
10
=
.
(28)
Earth
