Geoscience Reference
In-Depth Information
g
i
S
(
ψ
)
dσ ,
R
4
πγ
ζ
i
=
(11-15)
σ
∆
g
∗
=∆
g
+
g
1
+
g
2
+
g
3
+
···
.
(11-16)
The correction terms
g
n
are evaluated recursively by
n
z
r
L
r
(
g
n−r
)
,
g
n
=
−
(11-17)
r
=1
starting from
g
0
=∆
g.
(11-18)
Here the operator
L
n
is also defined recursively:
L
n
(∆
g
)=
n
−
1
L
1
[
L
n−
1
(∆
g
)]
(11-19)
starting with
L
1
=
L
(11-20)
with the gradient operator
L
defined by the integral (8-60), that is,
f
L
(
f
)=
R
2
2
π
−
f
Q
dσ .
(11-21)
l
0
σ
This means: take
g
0
=∆
g
,where∆
g
is the free-air anomaly at ground level
in the sense of Molodensky, then compute
g
1
by (11-17) with
n
=1,then
compute
g
2
by (11-17) with
n
=2and
L
2
by (11-19), then
g
3
by (11-17)
with
n
=3and
L
3
by (11-19), etc.
The operator
L
behaves like differentiation (
L
(
f
)=
∂
∆
g
) and, there-
fore, “roughens” the function
f
; this means that each successive
L
becomes
rougher and rougher. This is not conducive to the convergence of Molodens-
ky's series unless the original ∆
g
is very smooth, which cannot be assumed
in mountainous areas.
In such cases, some smoothing of ∆
g
is inevitable. Numerical analysis
is constantly confronted with problems of smoothing, so many techniques
of smoothing have been developed such as the sliding average. For evalu-
ating the integral
L
, fast Fourier methods are available. The problem is to
find an appropriate degree of smoothing which makes consecutive correc-
tions
g
1
,g
2
,g
3
,...
decrease in order to achieve practical convergence with-
out “oversmoothing”. At any rate, smoothing must ensure that
g
5
,g
6
, ...
are practically negligible since they cannot be meaningfully computed be-
cause of the inevitable accumulation of round-off errors, which finally tends
to producing pure noise.
∂r
