Geoscience Reference
In-Depth Information
the tensor represents the gravity gradient
)
=
W
zx
W
zy
W
zz
,
(
grad
g
(21)
which describes the changes of gravity with respect to the coordinate axes.
W
zx
and
W
zy
are the two components of the horizontal gradient lying in a local horizontal
plane and
W
zz
describes the vertical gradient respectively. The vertical component is
fundamental for the interpretation of the gravity data, with Eq. (
20
) we get
2
W
δ
W
zz
=
δ
=
δ
g
2
z
=−
(
W
xx
+
W
yy
)
−
4
π
G
ρ
+
2
ω
.
(22)
z
2
δ
(
and 2
W
xy
characterize the curvature of potential surfaces, i.e. how
the shape of potential surfaces differs from the shape of a sphere,
W
xy
and
W
zy
characterize how potential surfaces are not parallel to each other (Völgyesi
2001
).
The elements of the Eötvös tensor can be measured directly in space (see Sect.
3.1.3
)
or derived fromgravitymeasurements. The torsion balance, also called torsion pendu-
lum used for geodetic application, is usually credited to Loránd Eötvös (1848-1919)
and can measure the components of the horizontal gradient
W
zx
,
W
yy
−
W
xx
)
W
zy
as well as the
curvature
(
W
yy
−
W
xx
)
and
W
xy
, but not the vertical gradient
W
zz
.
1.3 Spherical Harmonics
Given that outside the attracting masses the gravitational potential
V
is a harmonic
function, it is easier to handle if expanded into a series of
spherical harmonics
.
Spherical harmonics are a special solution of the Laplace's equation; for the full
derivation see Hofmann-Wellenhof and Moritz (
2005
).
In the exterior space
V
can be represented as
a
r
n
P
nm
(
∞
n
GM
r
V
(
r
,θ,λ)
=
cos
θ)(
C
nm
cos
m
λ
+
S
nm
sin
m
λ),
(23)
n
=
0
m
=
0
where
G
is the gravitational constant and
M
the total mass of the Earth (solid, liquid
and gaseous portions) and
a
is the mean radius of the Earth. The
associated Legendre
functions P
nm
of degree
n
and order
m
are given for the argument
t
=
cos
θ
by:
2
d
m
t
2
P
nm
(
t
)
=
(
1
−
)
d
t
n
P
n
(
t
),
(24)
with the Legendre polynomials
P
n
d
n
d
t
n
(
1
2
n
n
t
2
n
P
n
(
t
)
=
P
n
0
(
t
)
=
−
1
)
.
(25)
!