Global Positioning System Reference
In-Depth Information
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R
=
(2.112)
according to (2.98). The law of variance-covariance propagation provides the
3
×
3 covariance submatrices:
Σ (x) R T
Σ (w) =
R
(2.113)
(x) J 1 T
J 1
Σ (ϕ,
=
Σ
(2.114)
λ
,h)
Leveled Height Differences: If geoid undulation differences are available, then
leveled height differences can be corrected for the undulation differences to yield
ellipsoidal height differences. The respective elements of the design matrix are
1 and
1. The accuracy of incorporating leveling data in this manner is limited
by our ability to compute accurate undulation differences.
[51
Refraction: If vertical angles are observed for providing an accurate vertical
dimension, it may be necessary to introduce and estimate vertical refraction
parameters. If this is done, we must be careful to avoid overparameterization by
introducing too many refraction parameters that could potentially absorb other
systematic effects not caused by refraction and/or result in an ill-conditioned
solution. However, it may be sufficient to correct the observations for refraction
using a standard model for the atmosphere.
In view of GPS capability, the importance of high-precision vertical angle
measurement is diminishing. The primary purpose of vertical angles is to give
sufficient height information to process the slant distances. Therefore, the types
of observations most likely to be used by the modern surveyors are horizontal
angles, slant distances, and GPS vectors.
Lin
0.5
——
Sho
PgE
[51
Horizontal Angles: Horizontal angles, of course, are simply the difference of
azimuths. Using the 2-1-3 subscript notation to identify an angle measured at
station 1 from station 2 to station 3 in a clockwise sense the mathematical model
for the geodetic angle
δ 213 is
tan 1
sin
λ 1
x 12 +
cos
λ 1
y 12
δ 213 =
sin ϕ 1 cos
λ 1
x 12
sin ϕ 1 sin
λ 1
y 12 +
cos ϕ 1
z 12
(2.115)
tan 1
sin
λ 1
x 13 +
cos
λ 1
y 13
sin ϕ 1 cos
λ 1
x 13
sin ϕ 1 sin
λ 1
y 13 +
cos ϕ 1
z 13
The partial derivatives can be readily obtained from the coefficients a 2 i listed in
Table 2.5 by applying them to both legs of the angles and then subtracting.
Height-Controlled 3D Adjustment: If the observations contain little or no
vertical information, i.e., if zenith angles and leveling data are not available, it is
still possible to adjust the network in three dimensions. The height parameters
h can be weighted using reasonable estimates for their a priori variances. This
 
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