Geoscience Reference
In-Depth Information
cannot be directly applied to determine the gradient
∂g/∂H
because the
mean curvature
J
of the level surfaces is unknown. Therefore, we proceed in
the usual way by splitting
∂g/∂H
into a normal and an anomalous part:
∂g
∂H
∂H
+
∂
∆
g
∂γ
=
.
(2-391)
∂H
The normal gradient
∂γ/∂H
is given by (2-147) and (2-148). The anomalous
part,
∂
∆
g/∂H
=
∂
∆
g/∂r
, will be considered now.
Expression in terms of
∆
g
Equation (2-272) may be written as (note that
r
∆
g
is harmonic and the
factor must be 1 for
r
=
R
)
R
r
n
+2
∆
g
(
r, ϑ, λ
)=
∞
∆
g
n
(
ϑ, λ
)
.
(2-392)
n
=0
By differentiating with respect to
r
and setting
r
=
R
, we obtain at sea level:
∞
∞
∂
∆
g
∂r
1
R
1
R
2
R
∆
g.
−
(
n
+2)∆
g
n
=
−
n
∆
g
n
−
=
(2-393)
n
=0
n
=0
Now we can apply (1-149), setting
V
=∆
g
and
Y
n
=∆
g
n
. The result is
∆
g
=
R
2
2
π
∂
∆
g
∂r
−
∆
g
P
2
R
∆
g
P
.
dσ
−
(2-394)
l
0
σ
In this equation, ∆
g
P
is referred to the fixed point
P
at which
∂
∆
g/∂r
is
to be computed;
l
0
is the spatial distance between the fixed point
P
and the
variable surface element
R
2
dσ
, expressed in terms of the angular distance
ψ
by
l
0
=2
R
sin
ψ
2
.
(2-395)
Compare Fig. 1.9 of Sect. 1.14; the element
R
2
dσ
is at the point
P
.
The important integral formula (2-394) expresses the vertical gradient
of the gravity anomaly in terms of the gravity anomaly itself. Since the
integrand decreases very rapidly with increasing distance
l
0
, it is sucient in
this formula to extend the integration only over the immediate neighborhood
of the point
P
, as opposed to Stokes' and Vening Meinesz' formulas, where
the integration must include the whole earth, if a sucient accuracy is to be
obtained.