Geoscience Reference
In-Depth Information
with the actual gravity anomalies outside the earth, since the function
r
∆
g
is harmonic according to Sect. 2.14.
(Remark: we are consistently using the notation ∆
g
for ground level,
∆
g
harmonic
for sea level, and ∆
g
∗
for point level; see Fig. 8.5.)
It follows that the harmonic function
T
that is produced by ∆
g
harmonic
according to Pizzetti's generalization (2-302) of Stokes' formula
T
(
r, ϑ, λ
)=
R
4
π
∆
g
harmonic
S
(
r, ψ
)
dσ
(8-74)
σ
is identical with the actual disturbing potential of the earth
outside and on
its surface
.
Applications
Assume that we got in some way (e.g., by the Taylor series mentioned above
or by collocation to be treated in Chap. 10 or by a high-resolution gravita-
tional field from satellite observations) the downward continuation ∆
g
harmonic
to sea level. Then we can compute the external gravity field, its spherical
harmonics, etc., rigorously by means of the conventional formulas of Chaps. 2
and 6, provided we use ∆
g
harmonic
rather than ∆
g
in the relevant formulas.
For instance, the coe
cients of the spherical harmonics of the gravitational
potential may be obtained by expanding the function ∆
g
harmonic
according to
Sect. 1.9 together with Sect. 1.6. If we wish to compute the height anomaly
ζ
at a point
P
at ground level, we must remember that
P
lies
above
the el-
lipsoid, so that the formulas for the external gravity field are to be applied.
By Bruns' formula
ζ
=
T/γ
0
(8-50), we get
∆
g
harmonic
S
(
r, ψ
)
dσ ,
R
4
πγ
0
ζ
=
(8-75)
σ
where
r
=
R
+
h
and
h
is the topographic height of
P
in some sense of
Chap. 4. (We do not need it very accurately, but it means that
h
is
formally
added to the constant radius
R
of the mean terrestrial sphere, which has no
real-world geometric interpretation!) Cf. Eq. (6-57). The function
S
(
r, ψ
)is
expressed by (2-303), (6-22) or (6-35). Similarly,
ξ
and
η
, being deflections
of the vertical above sea level, must be computed by (6-41) and the second
and third equation of (6-30). This gives
∆
g
harmonic
∂S
(
r, ψ
)
∂ψ
t
4
πγ
0
ξ
=
cos
αdσ,
σ
∆
g
harmonic
∂S
(
r, ψ
)
∂ψ
(8-76)
t
4
πγ
0
η
=
sin
αdσ,
σ
