Global Positioning System Reference
In-Depth Information
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3 conditions for the
n
double-
difference ambiguities. Through trial and error one could attempt to identify the cor-
rect set of ambiguities. Each epoch adds another set of
n
If we consider
v
=
o
, then (7.145) represents
n
−
3 equations to (7.145). The
elements of
G
change with time as the coefficients in
A
change with the motion of
the satellites. Eventually, enough epochs will be available with different
G
matrices
to allow a unique identification of the ambiguity. Only the correct set of ambiguities
will always fulfill (7.145). In actual application where the residuals are not zero but
are small, one would be looking for ambiguity values that are close to integers. Alter-
natively, applying Equation (7.145) to several epochs can be readily used to build a
mixed-model least-squares solution to estimate
b
. The receivers could even be in mo-
tion as long as the locations are known well enough to compute the coefficients in
A
.
Hatch (1990) suggests a scheme that divides satellites into primary and sec-
on
dary ones. Consider four satellites, called the primary satellites. The respective
th
ree double-difference equations contain the station coordinates and three double-
di
fference ambiguities. When the satellite geometry changes over time, it is possible
to
estimate all of these parameters. Any satellites in addition to these four satellites,
ca
lled the secondary satellites, are strictly speaking redundant, although we know
th
at these extra satellites improve the overall solution geometry. The extra satellites
ar
e used to develop yet another procedure for rapidly identifying integer ambiguities.
The primary and secondary satellites are identified below by subscripts
p
and
s
,
re
spectively. We group the observation equations accordingly, i.e.,
−
[28
Lin
—
*
1
——
No
PgE
+
b
p
+
p
=
+
˜
p
v
p
=
A
p
x
A
p
x
(7.146)
[28
+
˜
v
s
=
A
s
x
+
(
b
s
+
s
)
=
A
s
x
s
(7.147)
Note that the 3
1 vector
x
contains only coordinate parameters. Each of the two
groups may contain observations from one or several epochs. The method assumes
that the ambiguities
b
p
are known and evaluates the effect of that assumption. The
procedure starts by computing trial sets
b
p,i
for the three primary ambiguities using
an initial position estimate
x
0
, obtained from the point positioning solution or from
the float solution if several epochs of observations are available and the receivers do
not move (see below). We can compute the change in position with respect to
x
0
for
a given set of primary trial ambiguities using the usual least-squares formulation
×
N
−
p
A
p
P
p
˜
p,i
x
p,i
=−
(7.148)
A
p
P
p
A
p
. This is a nonredundant solution because only three observation
equations are available. Each ambiguity trial set gives a different position
x
p,i
while
th
e matrices
A
p
and
P
p
do not change. The coefficients of
A
p
are evaluated for
x
0
.
Us
ing
x
p,i
the ambiguities for the secondary satellites,
b
s,i
, can be derived from
with
N
p
=
N
1
q
km,s
ϕ
1
q
km,s
1
q
km,p
(
x
p
)
=
− ρ
(7.149)
