Geoscience Reference
In-Depth Information
2.15
Stokes' formula
The basic Eq. (2-271),
∂T
∂r
−
2
r
T,
∆
g
=
−
(2-288)
can be regarded as a boundary condition only, as long as the gravity anoma-
lies ∆
g
are known only at the surface of the earth. However, by the up-
ward continuation integral (2-282), we are now able to compute the gravity
anomalies outside the earth. Thus, our basic equation changes its meaning
radically, becoming a real differential equation that can be integrated with
respect to
r
. Note that this is made possible only because
T
, in addition to
the boundary condition, satisfies Laplace's equation ∆
T
=0.
Multiplying (2-288) by
r
2
,weget
−
r
2
∆
g
=
r
2
∂T
∂r
∂
∂r
(
r
2
T
)
.
−
+2
rT
=
(2-289)
Integrating the formula
∂
∂r
(
r
2
T
)=
r
2
∆
g
(
r
)
−
(2-290)
between the limits
∞
and
r
, we find
r
2
T
r
r
r
2
∆
g
(
r
)
dr ,
=
−
(2-291)
∞
∞
where ∆
g
(
r
) indicates that ∆
g
is now a function of
r
, computed from sur-
face gravity anomalies by means of the formula (2-282). Since this formula
automatically removes the spherical harmonics of degrees one and zero from
∆
g
(
r
), the anomalous potential
T
, as computed from ∆
g
(
r
), cannot contain
such terms. Thus, we have
R
r
n
+1
T
=
∞
T
n
=
R
3
r
3
T
2
+
R
4
r
4
T
3
+
··· .
(2-292)
n
=2
Therefore,
R
3
r
=0
,
T
2
+
R
4
r
2
(
r
2
T
) = lim
r→∞
lim
r→∞
T
3
+
···
(2-293)
so that
r
2
T
r
=
r
2
T
(
r
2
T
)=
r
2
T
−
lim
r→∞
(2-294)
∞
and
r
r
2
T
=
r
2
∆
g
(
r
)
dr .
−
(2-295)
∞
