Geoscience Reference
In-Depth Information
longitude of Greenwich itself.
These formulas (5-103) are Eqs. (7-13), (7-14), and (7-15) of Moritz and
Mueller (1987: pp. 419-420). It is interesting to note the close similarity
between the azimuth reduction (5-98) because of the “zenith variation” -
that is, the deflection of the vertical - and the longitude reduction of (5-103)
because of the polar variation. Actually, the geometry for both cases is the
same. The quantities
ξ, η,
90
◦
−
z, ϕ
correspond to
x, y, ϕ, ϕ
Gr
; the difference
in sign of sin
α
and sin
λ
is due to the fact that, when viewed from the
zenith, azimuth is reckoned clockwise and, when viewed from the pole, east
longitude is reckoned counterclockwise.
5.13
Reduction of horizontal and vertical angles
and of distances
Horizontal angles
To reduce an observed horizontal angle
ω
to the ellipsoid, we note that every
angle may be considered as the difference between two azimuths:
ω
=
α
2
−
α
1
.
(5-104)
Hence, we can apply formula (5-98). In the difference
α
2
−
α
1
,themain
term
η
tan
ϕ
drops out, so that for nearly horizontal lines of sight the whole
reduction may be neglected.
Vertical angles
The relation between the measured zenith angle
z
and the corresponding
ellipsoidal zenith angle
z
may be given as
z
=
z
+
ε
=
z
+
ξ
cos
α
+
η
sin
α,
(5-105)
where
α
is the azimuth of the target.
Spatial distances
Electronic measurement of distance yields straight spatial distances
l
be-
tween two points
A
and
B
(Fig. 5.14). These distances may either be used
directly for computations in the ellipsoidal coordinate system
ϕ, λ, h
,asin
“three-dimensional geodesy” (see Sect. 5.9), or they may be reduced to the
surface of the ellipsoid to obtain chord distances
l
0
or geodesic distances
s
0
.
We again approximate the ellipsoidal arc
A
0
B
0
by a circular arc of radius
R
that is the mean ellipsoidal radius of curvature along
A
0
B
0
. By applying
the law of cosines to the triangle
OAB
, we find
l
2
=(
R
+
h
1
)
2
+(
R
+
h
2
)
2
−
2(
R
+
h
1
)(
R
+
h
2
)cos
ψ.
(5-106)
