Global Positioning System Reference
In-Depth Information
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space techniques, the respective solutions belong to the most classical of all geodetic
theories and are appropriately documented in the literature. Unfortunately, many of
the references on this subject are out of print. We therefore summarize the Gauss
midlatitude solution, the transverse Mercator mapping, and Lambert conformal map-
ping in Appendixes B and C. Supporting material from differential geometry is also
provided in order to appreciate the “roots and flavor” of the mathematics involved.
Additional derivations are available in Leick (2002) that support lectures on the sub-
ject. The following literature has been found helpful: Dozier (1980), Heck (1987),
Kneissl (1959), Grossman (1976), Hristow (1955), Lambert (1772), Lee (1976), Sny-
der (1982), and Thomas (1952). Publication of many of these “classical” references
has been discontinued.
The ellipsoidal and conformal mapping expressions are generally given in the
form of mathematical series that are a result of multiple truncations at various steps
during the development. These truncations affect the computational accuracy of the
expressions and their applicability to the size of the area. The expressions given
here are sufficiently accurate for typical applications in surveying and geodesy. Some
terms may even be negligible when applied over small areas. For unusual applications
covering large areas, one might have to use more accurate expressions found in the
specialized literature. In all cases, however, given today's powerful computers, one
should not be overly concerned about a few unnecessary algebraic operations.
Two types of observations apply to computations on a surface: azimuth (angle)
and distance. The reductions, partial derivatives, and other quantities that apply to
angles can be conveniently obtained through differencing the respective expressions
for azimuths.
[32
Lin
0.0
——
Lon
PgE
[32
9.1 THE ELLIPSOIDAL MODEL
This section contains the mathematical formulations needed to carry out computa-
tions on the ellipsoidal surface. We introduce the geodesic line and reduce the 3D
geodetic observations to geodesic azimuth and distance. The direct and inverse solu-
tions are based on the Gauss midlatitude expressions. Finally, the partial derivatives
are given that allow network adjustment on the ellipsoid.
9. 1.1 Reduction of Observations
α
The geodetic azimuth
of Section 2.3.5 is the angle between two normal planes that
have the ellipsoidal normal in common; the geodetic horizontal angle
is defined
similarly. These 3D model observations follow from the original observation upon
corrections for the deflection of the vertical. Spatial distances can be used directly
in the 3D model presented in Section 2.3.5. However, angles and distances must be
reduced further in order to obtain model observables on the ellipsoidal surface with
respect to the geodesic.
δ
9.1.1.1 Angular Reduction to Geodesic
Figure 9.1 shows the reduction of
azimuth. The geodetic azimuth,
α
, is shown in the figure as the azimuth of the normal
 
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