Geoscience Reference
In-Depth Information
Integrated form
In formula (5-114), the deflection components
ξ
and
η
refer to the geoid.
This means that the astronomical observations of Φ and Λ must be reduced
to the geoid.
It is also possible and often more convenient to apply this correction
for plumb-line curvature not to the astronomical coordinates Φ and Λ but
to the geoidal height differences computed from the unreduced deflection
components.
These
N
values, denoted by
N
, are obtained by using in (5-116) the
directly observed Φ and Λ, which define the direction of the plumb line at
the station
P
(Fig. 5.19). The notation
N
will be reserved for the correct
geoidal heights. Then we read from Fig. 5.19:
dh
=
dN
+
dH
=
dN
+
dn ,
(5-141)
where
h
is the geometric height above the ellipsoid. Thus, we see that the
difference between the unreduced and the correct element of geoidal height,
dN
−
dN
=
dH
−
dn
=
d
(OC)
,
(5-142)
is equal to the difference between the element
dH
of orthometric height and
earth's surface
dn
dH
plumb
line
ellipsoidal
normal
local horizon
W=
P
dN'
P
dN
h
geoid
normal
geoid
W=
0
dN
P
0
ds
ellipsoid
Fig. 5.19. Reduction of astronomical leveling
