Geoscience Reference
In-Depth Information
•
linear approximation in h
used very generally in first-order “Moloden-
sky corrections” such as (8-62) or (8-66) but not in (8-67) or (8-68).
In fact, to a higher approximation
∆
g
∗
=∆
g
+
g
1
+
g
2
+
g
3
+
···
(8-67)
and
ζ
=
ζ
0
+
ζ
1
+
ζ
2
+
ζ
3
+
···.
(8-68)
Generalizing (8-66), we have
g
i
S
(
ψ
)
dσ ,
R
4
πγ
0
ζ
i
=
(8-69)
σ
where
i
=1
,
2
,
3
,...
. For the deflection of the vertical we have similar
expressions, see Sect. 8.7; compare also (8-75) and (8-76).
8.6.3
Higher-order solution
The following recursion formulas are somewhat advanced and may be omit-
ted. From Moritz (1980 a: Sect. 45) we may take the recursion formula for
the correction terms
g
n
, which are evaluated recursively by
n
z
r
L
r
(
g
n−r
)
,
g
n
=
−
(8-70)
r
=1
starting from
g
0
=∆
g.
(8-71)
Here the operator
L
n
is also defined recursively:
L
n
(∆
g
)=
n
−
1
L
1
[
L
n−
1
(∆
g
)]
(8-72)
starting with
L
1
=
L
(8-73)
with the gradient operator
L
defined above, (8-60), and
z
given by (8-52).
8.6.4
Problems of analytical continuation
Analytical continuation comes from the theory of complex variables and
means extending the domain, on which the function is defined, by the use of
Taylor series. Complex functions always satisfy Laplace's equations in two
dimensions and are therefore harmonic.
