Geoscience Reference
In-Depth Information
These are the
formulas of Vening Meinesz
. Differentiating Stokes' function
S
(
ψ
), Eq. (2-305), with respect to
ψ
,weobtain
Vening Meinesz' function
dS
(
ψ
)
dψ
cos(
ψ/
2)
2sin
2
(
ψ/
2)
+8sin
ψ
3
1
−
sin(
ψ/
2)
sin
ψ
=
−
−
6cos(
ψ/
2)
−
(2-387)
+3sin
ψ
ln
sin(
ψ/
2) + sin
2
(
ψ/
2)
.
This can be readily verified by using the elementary trigonometric identities.
The azimuth
α
is given by the formula
cos
ϕ
sin(
λ
− λ
)
cos
ϕ
sin
ϕ
−
sin
ϕ
cos
ϕ
cos(
λ
− λ
)
,
tan
α
=
(2-388)
which is an immediate consequence of (2-382).
The form (2-385) is an expression of (2-386) in terms of ellipsoidal co-
ordinates
ϕ
and
λ
. As with Stokes' formula (Sect. 2.15), we may also use an
expression in terms of spherical polar coordinates
ψ
and
α
:
2
π
π
1
4
πγ
0
∆
g
(
ψ, α
)cos
α
dS
(
ψ
)
dψ
ξ
=
sin
ψdψdα,
α
=0
ψ
=0
(2-389)
2
π
π
1
4
πγ
0
∆
g
(
ψ, α
)sin
α
dS
(
ψ
)
dψ
η
=
sin
ψdψdα.
α
=0
ψ
=0
The reader can easily verify that these equations give the deflection compo-
nents
ξ
and
η
with the correct sign corresponding to the definition (2-230);
see also Fig. 2.13. This is the reason why we introduced the minus sign in
(2-373).
We note that the formula of Vening Meinesz is valid as it stands for an
arbitrary reference ellipsoid, whereas Stokes' formula had to be modified by
adding a constant
N
0
. If we differentiate the modified Stokes formula with
respect to
ϕ
and
λ
, to get Vening Meinesz' formula, then this constant
N
0
drops out and we get Eqs. (2-386).
For the practical application of Stokes' and Vening Meinesz' formulas
and problems, the reader is referred to Sect. 2.21 and to Chap. 3.
2.20
The vertical gradient of gravity
Bruns' formula (2-40), with
=0,
∂g
∂H
=
−
2
gJ−
2
ω
2
,
(2-390)
