Geoscience Reference
In-Depth Information
The first step requires knowledge of the density of the topographic masses,
which is somewhat problematic.
By such a reduction procedure certain irregularities in gravity due to
differences in height of the stations are removed so that interpolation and
even extrapolation to unobserved areas become easier (Sect. 9.7).
3.2
Auxiliary formulas
Let us compute the potential
U
and the vertical attraction
A
of a homoge-
neous circular cylinder of radius
a
and height
b
at a point
P
situated on its
axis at a height
c
above its base (Fig. 3.2).
P
outside cylinder
Assume first that
P
is above the cylinder,
c>b
. Then the potential is given
by the general formula (1-12),
U
=
G
l
dv .
(3-1)
Introducing polar coordinates
s, α
in the
xy
-plane by
x
=
s
cos
α,
y
=
s
sin
α,
(3-2)
we have
l
=
s
2
+(
c
−
z
)
2
(3-3)
and
dv
=
dx dy dz
=
sdsdαdz.
(3-4)
z
z
P
P
l
c
a
a
s
z
b
b
dm
xy
Fig. 3.2. Potential and attraction of a circular cylinder on an external point
