Geoscience Reference
In-Depth Information
(The first term on the right-hand side of (8-58) is much smaller and can be
omitted.)
The term
∂ζ/∂r
no longer occurs in (8-57) as it did in (8-53) and (8-55),
and will not be needed.
Computational formulas; the Molodensky correction
Our computational formula is (8-57). We split it up as follows: The free-air
anomaly ∆
g
is continued (downward or upward) from ground level to the
level of point
P
, obtaining
∆
g
∗
=∆
g
+
g
1
,
(8-61)
where the Molodensky correction is
∂
∆
g
∂h
∂
∆
g
∂r
g
1
=
−
(
h
−
h
P
)=
−
(
h
−
h
P
)
(8-62)
(in spherical approximation) with
∆
g
=
R
2
2
π
∂
∆
g
∂r
−
∆
g
Q
dσ .
(8-63)
l
0
σ
Then we finally get
ζ
=
ζ
0
+
ζ
1
,
(8-64)
where
∆
gS
(
ψ
)
dσ
R
4
πγ
0
ζ
0
=
(8-65)
σ
is the simple Stokes formula applied to ground-level free-air anomalies ∆
g
,
and the Molodenski correction for
ζ
is
g
1
S
(
ψ
)
dσ .
R
4
πγ
0
ζ
1
=
(8-66)
σ
This is the first-order solution, or linear solution.
Important remark
Please note carefully that we are using “linear”, or “first-order”, in two very
different senses:
•
general linearization
, linear in quantities of the anomalous potential,
such as
N
or
ζ
, as introduced in Sect. 2.12 and Sect. 8.4 and implied
everywhere throughout the topic, and
