Geoscience Reference
In-Depth Information
We next see from the derivation of Stokes' formula by means of the
upward continuation integral (2-282) that it automatically suppresses the
harmonic terms of degrees one and zero in
T
and
N
. The implications of this
will be discussed later. We will see that Stokes' formula in its original form
(2-304) and (2-307) only applies for a reference ellipsoid that (1) has the
same potential
U
0
=
W
0
as the geoid, (2) encloses a mass that is numerically
equal to the earth's mass, and (3) has its center at the center of gravity of
the earth. Since the first two conditions are not accurately satisfied by the
reference ellipsoids that are in current practical use, and can hardly ever be
rigorously fulfilled, Stokes' formula will later be modified for the case of an
arbitrary reference ellipsoid.
Finally,
T
is assumed to be harmonic outside the geoid. This means that
the effect of the masses above the geoid must be removed by suitable gravity
reductions. This will be discussed in Chaps. 3 and 8.
A bonus application to satellite geodesy
As a somewhat unexpected application, not related to Stokes' formula, we
note that Eq. (2-280) can be used to compute gravity anomalies ∆
g
from
a satellite-determined spherical-harmonic series of the external gravitational
potential
V
!
2.16
Explicit form of Stokes' integral and Stokes'
function in spherical harmonics
We now write Stokes' formula (2-307) more explicitly by introducing suitable
coordinate systems on the sphere.
The use of spherical
polar coordinates
with origin at
P
offers the ad-
vantage that the angle
ψ
, which is the argument of Stokes' function, is one
coordinate, the
spherical distance
. The other coordinate is the
azimuth α
,
reckoned from north. Their definitions are seen in Fig. 2.16. Denoting by
P
both a fixed point on the sphere
r
=
R
(or in space) and its projection on
the unit sphere is common practice and will not cause any trouble.
If
P
coincides with the north pole, then
ψ
and
α
are identical with
ϑ
and
λ
. According to Sect. 1.9, the surface element
dσ
is then given by
dσ
=sin
ψdψdα.
(2-308)
Since all points of the sphere are equivalent, this relation applies for an
arbitrary origin
P
. In the same way, we have
=
2
π
α
=0
π
.
(2-309)
ψ
=0
σ
