Global Positioning System Reference
In-Depth Information
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ϑ
Ellipsoidal (geodetic) zenith angle
θ
Total deflection of the vertical (not colatitude)
ε
Deflection of the vertical in the direction of azimuth
ξ
,
η
Deflection of the vertical components along the meridian and the
prime vertical
By applying spherical trigonometry to the various triangles in Figure 2.15, we can
eventually derive the following relations:
A CTP − α =
(
Λ CTP − λ
) sin ϕ
+
(
ξ
sin
α − η
cos
α
) cot ϑ
(2.81)
ξ = Φ CTP
ϕ
(2.82)
η =
(
Λ CTP − λ
) cos ϕ
(2.83)
[42
ϑ + ξ
ϑ
=
cos
α + η
sin
α
(2.84)
Lin
0.8
——
Sho
PgE
The derivations of these classical equations can be found in most of the geode-
tic literature, e.g., Heiskanen and Moritz (1967, p. 186). They are also given in
Leick (2002). Equation (2.81) is the Laplace equation. It relates the reduced as-
tronomic azimuth and the geodetic azimuth of the normal section containing the
target point. Equations (2.82) and (2.83) define the deflection of the vertical com-
ponents. The deflection of the vertical is simply the angle between the directions
of the plumb line and the ellipsoidal normal at the same point. By convention, the
deflection of the vertical is decomposed into two components, one lying in the merid-
ian and one lying in the prime vertical, or orthogonal to the meridian. The deflec-
tion components depend directly on the shape of the geoid in the region. Because
the deflections of the vertical are merely another manifestation of the irregularity of
[42
Figure 2.15
Deflection of the vertical components.
 
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