PSEUDORANGE AND CARRIER PHASE OBSERVABLES - GPS Satellite Surveying - page 181

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

p and station 2 to satellite p. However, a slip in the latter phase sequences affects only
the double differences containing satellite p . Other double-difference sequences are
not affected.
For a session network, the double-difference observation is

1 m = ϕ 1 −

ϕ m − ϕ 1 ϕ m

ϕ 1 p − (5.47)

The superscript p goes from 2 to S , and the subscript m runs from 2 to R .Itis
readily seen that a cycle slip in ϕ 1 affects all double-difference observations, an error
in ϕ m affects all double differences pertaining to the baseline 1 to m , an error in
ϕ 1 affects all double differences containing satellite p , and an error in ϕ m affects
only one series of double differences, namely, the one that contains station m and
satellite p . Thus, by analyzing the distribution of a blunder in all double differences
at the same epoch, we can identify the undifferenced phase observation sequence
that contains the blunder. This identification gets more complicated if several slips
occur at the same epoch. In session network processing, it is always necessary to
carry out cross-checks. The same cycle slip must be verified in all relevant double
differences before it can be declared an actual cycle slip. Whenever a cycle slip
occurs in the undifferenced phase observations from the base station or to the base
satellite, the cycle slip enters several double-difference sequences. Actually it is not
necessary that the undifferenced phase observations be corrected; it is sufficient to
limit the correction to the double-difference phase observations if the final position
computation is based on double differences.
It is also possible to use the geometry-free functions of the observables to detect
cycle slips. The geometry-free functions are discussed in Chapter 7.

[18

Lin
—
6.2
——
No
PgE

[18

5.3.4 Singularities

A case of a critical configuration for terrestrial observations is discussed in Chapter 4.
Fo r example, Figure 4.12 shows how ellipses of standard deviation display the change
in the geometry as the critical configuration (singularity) is reached for the plane
re sections. The dilution of precision (DOP) introduced in Section 7.4.1 is a one-
nu mber indicator for the geometry of the point positioning solutions. At the critical
co nfigurations the columns of the design matrix become linearly dependent. When
th e satellite constellation approaches a critical configuration, the resulting positioning
so lution can be ill conditioned.
Linearizing the pseudorange equation (5.7) around the receiver location x k gives

x p dx k

dy k
dz k

y p z p

− x k − y k − z k

=−

dP k e k =− · d x k

p
k

p
k

p
k

ρ ρ ρ

(5.48)

1
ρ

p
k

=−

k ρ

· d x k

p