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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)
!
 
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