Geoscience Reference
In-Depth Information
Astronomical field observations for latitude, longitude, and azimuth have
an accuracy around 0
.
3
, which is sucient for classical trigonometric net
computation and astrogeodetic observation of the geoid (Sect. 5.14).
8.8
Gravity disturbances: the GPS case
The basic fact is that for gravity disturbances the derivation of “Molodensky
corrections”
g
n
is identical to the ∆
g
case. The reason is that the gravity dis-
turbance
δg
has exactly the same analytical behavior as the gravity anomaly
∆
g
since
rδg
, as a function in space, is harmonic together with
r
∆
g
.Thus,
the arguments are literally the same, only ∆
g
has to be replaced by
δg
,and
Stokes' formula must be replaced by the Neumann-Koch formula (8-47) and
similarly for Vening Meinesz' formula.
Therefore, we obtain
δg K
(
ψ
)
dσ
+
∞
g
n
K
(
ψ
)
dσ ,
R
4
πγ
0
R
4
πγ
0
ζ
=
(8-87)
n
=1
σ
σ
δg
dK
dψ
g
n
cos
αdσ
+
∞
1
4
πγ
0
1
4
πγ
0
dK
dψ
ξ
=
cos
αdσ,
n
=1
σ
σ
(8-88)
δg
dK
dψ
g
n
sin
αdσ
+
∞
1
4
πγ
0
1
4
πγ
0
dK
dψ
η
=
sin
αdσ.
n
=1
σ
σ
For the “Vening Meinesz GPS formula” (8-88), we find by differentiation of
(8-49):
dK
dψ
1
2
cos(
ψ/
2)
sin
2
(
ψ/
2)
1
1+sin(
ψ/
2)
.
−
=
(8-89)
The correction terms
g
n
are evaluated recursively by
n
z
r
L
r
(
g
n−r
)
,
g
n
=
−
(8-90)
r
=1
but now we start from
g
0
=
δg .
(8-91)
We only have to replace ∆
g
by
δg
and
S
(
ψ
)by
K
(
ψ
). The operators
L
remain the same.
Let us summarize again our trick for solving the modern boundary-value
problems (Molodensky and Koch). It is dicult to directly work with the
complicated earth's surface
S
. Therefore, by analytical continuation of ∆
g
or
