Geoscience Reference
In-Depth Information
refers to sea level and ∆
g
to the earth's surface. With GPS we have gravity
disturbances
γ
(
P
)
.
(8-129)
Regarding plumb line definition, we must distinguish three lines (Fig. 8.13):
1. the straight ellipsoidal normal
Q
0
P
,
2. the actual plumb line
P
0
P
,
3. the normal plumb line
P
0
P
.
Geometrically, the ellipsoidal normal is defined as the straight line through
P
perpendicular to the ellipsoid. The (actual) plumb line is defined by the
condition that, at each point of the line, the tangent coincides with the
gravity vector
g
at that point; the plumb line is very slightly curved, but its
curvature is irregular, being determined by the irregularities of topographic
masses. The normal plumb line, at each of its points, is tangent to the normal
gravity vector
∆
g
=
g
(
P
)
−
γ
; it possesses a curvature that is even smaller and completely
regular.
The points
P
0
,
P
0
,and
P
0
coincide within a few decimeters, and we will
not distinguish them in what follows. The reason is that the distance, in arc
seconds, between
P
0
and
P
0
is much smaller than the effect of plumb line
curvature (Sect. 5.15). The same applies for
Q
0
,Q
0
,and
Q
0
.
The direction of the gravity vector
g
is the direction of (the tangent to)
the plumb line. It is determined by two angles, the astronomical latitude Φ
and the astronomical longitude Λ. Let Φ
,
Λ be referred to the earth's surface
(to point
P
)andΦ
0
,
Λ
0
to the geoid (strictly speaking, to point
P
0
). The
differences
Λ (8-130)
express the effect of plumb line curvature (Fig. 8.14). You may also wish to
refer back to Fig. 5.18. Hence, we have
δϕ
=Φ
0
−
Φ
,
λ
=Λ
0
−
Φ
0
=Φ+
δϕ ,
Λ
0
=Λ+
δλ .
(8-131)
Knowing the plumb line curvature
δ
Φ
,δ
Λ, we could use these simple formulas
to compute the sea-level values Φ
0
,
Λ
0
from the observed surface values Φ
,
Λ.
In the same way as Φ
,
Λ are related to the actual plumb line, the ellip-
soidal latitude
ϕ
and the ellipsoidal longitude
λ
refer to the straight ellip-
soidal normal. The quantities
ξ
=Φ
− ϕ,
η
=(Λ
− λ
)cos
ϕ
(8-132)
are the components of the deflection of the vertical in a north-south and an
east-west direction. For an arbitrary azimuth
α
, the vertical deflection
ε
is
given by
ε
=
ξ
cos
α
+
η
sin
α.
(8-133)
