Geoscience Reference
In-Depth Information
Substituting these relations into (2-400) yields
∂
∆
g
∂r
=
2
γ
0
R
2
N
+
γ
0
γ
0
R
∂ξ
∂ϕ
−
γ
0
R
cos
ϕ
∂η
∂λ
.
R
ξ
tan
ϕ
−
(2-403)
Introducing local rectangular coordinates
x, y
in the tangent plane, we have
Rdϕ
=
ds
ϕ
=
dx ,
(2-404)
R
cos
ϕdλ
=
ds
λ
=
dy ,
so that (2-403) becomes
∂
∆
g
∂r
γ
0
∂ξ
.
=
2
γ
0
R
2
N
+
γ
0
∂x
+
∂η
R
ξ
tan
ϕ
−
(2-405)
∂y
The first two terms on the right-hand side can be shown to be very small in
comparison to the third term; hence, to a sucient accuracy
∂
∆
g
∂r
γ
0
∂ξ
∂x
+
∂η
=
−
(2-406)
∂y
may be used. These beautiful formulas express the vertical gradient of the
gravity anomaly in terms of the horizontal derivatives of the deflection of the
vertical. They can again be evaluated by means of numerical differentiation
if a map of
ξ
and
η
is available. They are somewhat better suited for practical
application than (2-400) because only first derivatives are required.
2.21
Practical evaluation of the integral formulas
Integral formulas such as Stokes' and Vening Meinesz' integrals must be
evaluated approximately by summations. The surface elements
dσ
are re-
placed by small but finite compartments
q
, which are obtained by suitably
subdividing the surface of the earth. Two different methods of subdivision
are used:
1. Templates (Fig. 2.20). The subdivision is achieved by concentric circles
and their radii. The template is placed on a gravity map of the same
scale so that the center of the template coincides with the computation
point
P
on the map. The natural coordinates for this purpose are
polar
coordinates ψ, α
with origin at
P
.
2. Grid lines (Fig. 2.21). The subdivision is achieved by the grid lines of
some fixed coordinate system, in particular of
ellipsoidal coordinates
ϕ, λ
. They form rectangular
blocks
- for example, of 10
×
1
◦
.
These blocks are also called squares, although they are usually not
squares as defined in plane geometry.
10
or 1
◦
×
