Geoscience Reference
In-Depth Information
Effect of the neighborhood
This issue is interesting and instructive. In the innermost zone, even the
template method may pose diculties if the integrand becomes infinite as
ψ
→
0. This happens with Stokes' formula, since
2
ψ
=
S
(
ψ
)
(2-413)
for small
ψ
. This can be seen from the definition (2-305), because the first
term is predominant and is, for small
ψ
,givenby
1
sin(
ψ/
2)
1
(
ψ/
2)
2
ψ
.
=
=
(2-414)
Vening Meinesz' function becomes infinite as well, since to the same approx-
imation,
dS
(
ψ
)
dψ
2
ψ
2
.
=
−
(2-415)
In the gradient formula (2-394), the integrand
1
l
0
1
R
3
ψ
3
=
(2-416)
behaves in a similar way.
Therefore, it may be convenient to split off the effect of this innermost
zone, which will be assumed to be a circle of radius
ψ
0
around the compu-
tation point. For instance, Stokes' integral becomes in this way
N
=
N
i
+
N
e
,
(2-417)
where
2
π
ψ
0
R
4
πγ
0
N
i
=
∆
gS
(
ψ
)
dσ ,
α
=0
ψ
=0
(2-418)
2
π
π
R
4
πγ
0
N
e
=
∆
gS
(
ψ
)
dσ .
α
=0
ψ
=
ψ
0
The radius
ψ
0
of the inner zone corresponds to a linear distance of a few
kilometers. Within this distance, we may treat the sphere as a plane, using
polar coordinates
s, α
,where
.
=2
R
sin
ψ
=
Rψ
=
R
sin
ψ
s
2
,
(2-419)
so that the element of area becomes
R
2
dσ
=
sdsdα.
(2-420)
