Global Positioning System Reference
In-Depth Information
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sideration. The final parameterization becomes
=
d
α
1
d
w
1
···
d
w
2
d
β
1
ds
A
(2.109)
with
a
11
a
12
a
13
a
14
a
15
a
16
=
G
1
J
1
H
−
1
1
G
2
J
2
H
−
2
=
A
:
a
21
a
22
a
23
:
a
24
a
25
a
26
(2.110)
a
31
a
32
a
33
a
34
a
35
a
36
Th
e partial derivatives are given in Table 2.5 (Wolf, 1963; Heiskanen and Moritz,
19
67; Vincenty, 1979). Some of the partial derivatives have been expressed in terms
of
the back azimuth
[49
α
2
≡ α
21
and the back vertical angle
β
2
≡ β
21
, meaning azimuth
an
d vertical angle from station 2 to station 1.
Lin
—
1
——
Sho
PgE
2.3.5.3 Implementation Considerations
It is not only easy to derive the 3D
geodetic model; it is also easy to implement it in software. Normally, the observations
will be uncorrelated and their contribution to the normal equations can be added one
by one. The following are some useful things to keep in mind when using this model:
Point of Expansion:
As in any nonlinear adjustment, the partial derivatives must
be evaluated at the current point of expansion (adjusted positions of the previous
iteration). This applies to coordinates and azimuths and angles used to express
the mathematical functions for the partial derivatives.
•
[49
Reduction to the Mark:
An advantage of the 3D geodetic model is that the
observations do not have to be reduced to the marks on the ground. When
computing
•
0
from (2.99) to (2.101), use
h
+ ∆
h
instead of
h
for the station
∆
heights. The symbol
h
denotes the height of the instrument or that of the
target above the mark on the ground.
b
always denotes the measured value,
i.e., the geodetic observable that is not further reduced. After completion of
the adjustment, the adjusted observations
a
, with respect to the marks on the
ground, can be computed from the adjusted positions using
h
in (2.99) to (2.101).
Minimal Constraints:
The
(ϕ)
or
(w)
parameterizations are particularly useful
for introducing height observations, height difference observations, or minimal
constraints by fixing or weighting individual coordinates. The set of minimal
constraints depends on the type of observations available and where the obser-
vations are located within the network. One choice for the minimal constraints
might be to fix the coordinates
(ϕ,
•
,h)
of one station (translation), the azimuth
or the longitude of another station (rotation in azimuth), and the heights of two
additional stations.
λ
