Geoscience Reference
In-Depth Information
P
l
H
s
dx
dy
xy
F
R
2
d
¾
R
R
0
Fig. 6.2. Plane approximation
6.4
Gravity disturbances by upward continuation
We apply Poisson's integral formula (1-123) to the harmonic function
T
:
T
l
3
=
R
(
r
2
R
2
)
−
T
P
dσ .
(6-44)
4
π
σ
In the neighborhood of
P
(Fig. 6.2), the sphere practically coincides with
its tangent plane at
F
. Since the value of the integrand is very small at
greater distances from
P
, we may extend the integration over the tangent
plane instead of over the sphere. Then, according to Fig. 6.2,
l
=
s
2
+
H
2
.
(6-45)
We introduce a rectangular coordinate system
x, y, z
,the
x
-axis pointing
north and the
y
-axis pointing east in the tangent plane. Then we may also
write
l
=
x
2
+
y
2
+
H
2
,
(6-46)
