Geoscience Reference
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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 .
(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 , 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 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.
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