Global Positioning System Reference
In-Depth Information
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The design matrix looks like one for a leveling network. The coefficients are either
1,
1, or 0. Each vector contributes three rows. Because vector observations contain
information about the orientation and scale, one only needs to fix the translational lo-
cation of the polyhedron. Minimal constraints for fixing the origin can be imposed by
simply deleting the three coordinate parameters of one station, holding that particular
station effectively fixed.
Inner constraints must fulfill the condition
−
Ex
=
o
(8.4)
according to (4.201), or, what amounts to the same condition,
E
T
A
=
O
(8.5)
[30
It can be readily verified that
Lin
—
0.0
——
Lon
PgE
E
=
[
3
I
3
I
3
I
3
···
]
(8.6)
fulfills these conditions. The matrix
E
consists of a row of 3
3 identity matrices.
There are as many identity matrices as there are stations in the network. The inner
constraint solution uses the pseudoinverse (4.203)
N
+
=
A
T
PA
×
E
T
E
−
1
E
T
EE
T
EE
T
−
1
E
+
−
(8.7)
of the normal matrix. If one sets the approximate coordinates to zero, which can be
done since the mathematical model is linear, then the origin of the coordinate system
is at the centroid of the cluster of stations. For nonzero approximate coordinates,
the coordinates of the centroid remain invariant; i.e., the values are the same whether
computed from the approximate coordinates or the adjusted coordinates. The standard
ellipsoid reflects the true geometry of the network and the satellite constellation. See
Chapter 4 for a discussion on which quantities are variant or invariant with respect to
different choices of minimal constraints.
The GPS-determined coordinates refer to the coordinate system of the satellite po-
sitions (ephemeris). The broadcast ephemeris coordinate system is given in WGS84,
and the precise ephemeris is in ITRF. Both coordinate systems agree at the couple-
of-centimeters level.
The primary result of a typical GPS survey is best viewed as a polyhedron of sta-
tions whose relative positions have been accurately determined (to the centimeter or
even the millimeter level), but the translational position of the polyhedron is typically
known only at the meter level (point positioning with pseudoranges). The orientation
of the polyhedron is implied by the vector observations. The Cartesian coordinates
(or coordinate differences) of the GPS survey can, of course, be converted to geode-
tic latitude, longitude, and height. If geoid undulations are available, the orthometric
heights (height differences) can be readily computed. The variance-covariance com-
ponents of the adjusted parameters can be transformed to the local geodetic system
for ease of interpretation using (2.113).
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