Geoscience Reference
In-Depth Information
For each compartment
q
k
, the gravity anomalies are replaced by their
average value ∆
g
k
in this compartment. Hence, the above equation becomes
∆
g
k
S
(
ψ
)
dσ
=
S
(
ψ
)
dσ
4
πγ
0
k
4
πγ
0
k
R
R
N
=
∆
g
k
(2-408)
q
k
q
k
or
N
=
k
c
k
∆
g
k
,
(2-409)
where the coe
cients
S
(
ψ
)
dσ
R
4
πγ
0
c
k
=
(2-410)
q
k
are obtained by integration over the compartment
q
k
; they do not depend
on ∆
g
.
If the integrand - in our case, Stokes' function
S
(
ψ
) - is reasonably
constant over the compartment
q
k
, it may be replaced by its value
S
(
ψ
k
)at
the center of
q
k
.Thenwehave
4
πγ
0
S
(
ψ
)
q
k
dσ
=
R
2
dσ .
R
S
(
ψ
k
)
4
πγ
0
R
c
k
=
(2-411)
q
k
The final integral is simply the area
A
k
of the compartment and we obtain
c
k
=
A
k
S
(
ψ
k
)
4
πγ
0
R
.
(2-412)
The advantage of the template method is its great flexibility. The influ-
ence of the compartments near the computation point
P
is greater than that
of the distant ones, and the integrand changes faster in the neighborhood
of
P
. Therefore, a finer subdivision is necessary around
P
. This can easily
be provided by templates. Yet, the method is completely old-fashioned and
thus obsolete.
The advantage of the fixed system of blocks formed by a grid of ellipsoidal
coordinates lies in the fact that their mean gravity anomalies are needed for
many different purposes. These mean anomalies of standard-sized blocks,
once they have been determined, can be easily stored and processed by a
computer. Also, the same subdivision is used for all computation points,
whereas the compartments defined by a template change when the template
is moved to the next computation point. The flexibility of the method of
standard blocks is limited; however, one may use smaller blocks (5
×
5
,for
example) in the neighborhood of
P
and larger ones (1
◦
×
1
◦
, for example)
farther away. With current electronic computation, this method is the only
one used in practice. The theoretical usefulness of polar coordinates will be
shown now.
