Global Positioning System Reference
In-Depth Information
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the gravity field or the geoid, they are mathematically related to the geoid undula-
tion. Equation (2.84) relates the ellipsoidal and observed zenith angle (refraction not
considered).
Equations (2.81) to (2.83) can be used to correct the reduced astronomic latitude,
longitude, and azimuth and thus to obtain the ellipsoidal latitude, longitude, and az-
imuth. It is important to note that the reduction of a horizontal angle due to deflection
of the vertical is obtained from the difference of (2.81) as applied to both legs of the
angle. If the zenith angle to the endpoints of both legs is close to 90°, then the correc-
tions are small and can possibly be neglected. Historically, Equation (2.81) was used
as a condition between the reduced astronomic azimuth and the computed geodetic
azimuth to control systematic errors. This can best be accomplished now with GPS.
However, if surveyors were to check the orientation of a GPS vector with the as-
tronomic azimuth from the sun or polaris, they must expect a discrepancy indicated
by (2.81).
Equations (2.81) to (2.83) also show how to specify a local ellipsoid that is tangent
to the geoid at some centrally located station called the initial point, and whose
semiminor axis is still parallel to the CTP. If we specify that at the initial point the
reduced astronomic latitude, longitude, and azimuth equal the ellipsoidal latitude,
longitude, and azimuth, respectively, then we ensure parallelism of the semimajor axis
and the direction of the CTP; the geoid normal and the ellipsoidal normal coincide
at that initial point. If, in addition, we set the undulation to zero, then the ellipsoid
touches the geoid tangentially at the initial point. Thus the local ellipsoid will have
at the initial point:
[43
Lin
—
1.0
——
Sho
PgE
ϕ
= Φ
CTP
(2.85)
[43
λ = Λ
CTP
(2.86)
α =
A
CTP
(2.87)
=
N
0
(2.88)
Other possibilities for specifying a local ellipsoid exist.
The local ellipsoid can serve as a convenient computation reference for least-
squares adjustments of networks typically encountered in local and regional surveys.
In these cases, it is not at all necessary to determine the size and shape of a best-fitting
local ellipsoid. It is sufficient to adopt the size and shape of any of the currently valid
geocentric ellipsoids. Because the deflections of the vertical will be small in the region
around the initial point, they can often be neglected completely. This is especially true
for the reduction of angles. The local ellipsoid is even more useful than it appears
at first sight. So long as typical observations, such as horizontal directions, angles,
and slant distances, are adjusted, the accurate position of the initial point in (2.85)
and (2.86) is not needed. In fact, if the (local) undulation variation is negligible, the
coordinate values for the position of the initial point are arbitrary. The same is true
for the azimuth condition (2.87). These simplifications make it attractive to use an
