Geoscience Reference
In-Depth Information
Since these anomalies
refer to a level surface
, Vening Meinesz' formula can
now be applied directly and gives (8-77).
Relation with the ellipsoidal geodetic coordinates
The deflection components
ξ
and
η
as given by the above expressions rep-
resent the deviation of the actual plumb line from the normal plumb line at
the ground point
P
. Therefore, they are defined by
ϕ
∗
,
ξ
=Φ
−
(8-79)
λ
∗
)cos
ϕ.
η
=(Λ
−
The symbols Φ and Λ represent the astronomical coordinates of
P
referred to
the ground. The symbols
ϕ
∗
and
λ
∗
represent the
“normal coordinates”
of
P
,
defining the direction of the normal plumb line at
P
; they are not identical
with the ellipsoidal coordinates
ϕ
and
λ
of
P
, which are the coordinates of
the foot point
Q
0
of the straight perpendicular to the ellipsoid (Fig. 8.8).
ellipsoidal normal
normal
P
plumb line
earth's
surface
parallel to
equator
'
*
''
'
ellipsoid
Q
0
Q
00
Fig. 8.8. Normal latitude
ϕ
∗
and ellipsoidal latitude
ϕ
The normal coordinates of
P
,
ϕ
∗
and
λ
∗
, differ from the normal coordi-
nates of
Q
00
,
ϕ
and
λ
, by the correction for the normal curvature of the
plumb line (see Sect. 5.15). Formula (5-147) gives
h
R
ϕ
∗
=
ϕ
+
f
∗
sin 2
ϕ,
(8-80)
λ
∗
=
λ
.
Because of the rotational symmetry, we have rigorously
λ
=
λ,
(8-81)