Geoscience Reference
In-Depth Information
g
earth's surface S
z
h
g'
P
point level
U=
P
h
P
U=
0
ellipsoid
Fig. 8.4. Analytical continuation from the earth's surface to point level
surface
U
=
U
P
, which for our purpose is the same). In the spherical ap-
proximation, both surfaces
U
=
U
P
and
U
=
U
0
are concentric spheres, but
only in the precise sense of the spherical approximation as explained above.
We also use the term “harmonic continuation” because the analytically
continued function satisfies Laplace's equation. This will be explained in
detail later.
An expansion into a Taylor series gives immediately
∆
g
=∆
g
∗
+
z
∂
∆
g
∗
∂z
2!
z
2
∂
2
∆
g
∗
3!
z
3
∂
3
∆
g
∗
1
1
+
+
+
···
∂z
2
∂z
3
(8-51)
=∆
g
∗
+
∞
n
!
z
n
∂
n
∆
g
∗
1
,
∂z
n
n
=1
where
(8-52)
is the elevation difference with respect to the computation point
P
.For
the present, we assume the series (8-51) to be convergent. Here ∆
g
∗
is the
gravity anomaly at point level (Fig. 8.4). The use of a Taylor series is typical
for
analytical continuation
. For instance, Taylor series are a standard tool
for analytical continuation of functions of a complex variable.
z
=
h
−
h
P
8.6.2
First-order solution
It is particularly easy to give a solution as a first approximation. With
γ
0
from (8-50) we have
∆
g
h
S
(
ψ
)
dσ
+
∂ζ
R
4
πγ
0
∂
∆
g
∂h
ζ
=
−
∂h
h.
(8-53)
σ
This follows from the geometrical interpretation of this equation which is
evident from Fig. 8.5 a. We see that the free-air anomalies ∆
g
at ground
level are “reduced” downward to sea level to become
∂
∆
g
∂h
∆
g
harmonic
=∆
g −
h
(8-54)
