Global Positioning System Reference
In-Depth Information
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stations. However, for short baselines, the satellites could be located approximately on
circular cones. Consider the relevant portion of the double-difference phase equation
(5.25) scaled to distances,
−
ρ
m
+···
P
pq
km
p
k
q
k
p
m
q
= ρ
− ρ
− ρ
(5.51)
The total differential
dP
pq
km
e
k
e
k
·
e
m
·
e
q
m
·
=−
·
d
x
k
+
d
x
m
+
d
x
k
−
d
x
m
(5.52)
=
e
k
−
e
k
·
d
x
k
+
e
m
−
e
q
m
·
d
x
m
expresses the change in the double-difference observable in terms of differential
changes in station coordinates. The coefficients in the brackets represent the differ-
ences in the direction cosines from one station and two satellites. For short baselines
these differences approach zero. It can readily be seen that the direction vectors
e
k
are related to the vector of directions from the center of the baseline to the satellite
e
c
as
[18
Lin
—
-
——
No
PgE
e
k
p
k
e
c
=
+
ε
(5.53)
p
k
)
. The symbol
b
denotes the length of the baseline. Referencing the other vectors also to the center
of the baseline, Equation (5.52) becomes
p
k
are of the order
O(b/
where the components of the vector
ε
ρ
=
e
c
−
k
·
+
e
c
m
·
dP
pq
km
q
k
p
[18
e
c
e
c
+
ε
p
m
q
+
ε
−
ε
d
x
k
−
−
ε
d
x
m
(5.54)
For the special case that the vertex of the circular cone is at the center of the baseline,
the condition
e
c
·
e
axis
=
cos
θ
(5.55)
is valid for all satellites on the cone. This means that the dot products
e
c
−
k
·
e
axis
=
ε
k
·
O
b/
k
q
k
p
q
k
p
p
e
c
+
ε
−
ε
−
ε
e
axis
=
ρ
(5.56)
p
k
)
. These products become smaller the shorter the
baseline. A product like (5.56) applies to every double-difference observation. There-
fore, we are dealing with a near-singular situation since the columns of the double-
difference design matrix are nearly dependent. The shorter the baseline, the more
likely it is that the near-singularity damages the baseline solution.
in (5.54) are of the order
O(b/
ρ
5.
3.5 Impact of a Priori Position Errors
A frequent concern is the need for a priori knowledge of geocentric station positions
and the effects of ephemeris errors on the relative positions. The answer to these
