Integer ambiguity-fixed precise point positioning method and system
A method for providing and applying Precise Point Positioning-Integer Ambiguity Resolution (PPP-IAR) corrections for a Global Navigation Satellite System (GNSS). In a GNS signal correction system, PPP-IAR corrections (ambiguities plus receiver and satellite hardware delays) are calculated based on observation data (P i,a , Φ i,a ) of n satellites for individual reference stations using a functional model, and these individual PPP-IAR corrections are merged into one single and larger set of PPP-IAR corrections. In a broadcast system, the (merged) PPP-IAR corrections are encoded at a System Control Centre (SCC) and transmitted to mobiles. In a mobile system a similar functional model is used to calculate a correction Δx to an a priori position.
1. A method for providing Precise Point Positioning-Integer Ambiguity Resolution (PPP-IAR) corrections for a global navigation satellite system (GNSS), the method comprising:
calculating, by a processor, PPP-IAR corrections based on observation data P i,a , Φ i,a for n satellites received at one or more reference stations with known locations, using a functional model, the functional model being defined as:
[
P
1
,
a
P
2
,
a
Φ
1
,
a
Φ
2
,
a
]
=
[
M
a
e
n
γ
1
I
n
M
a
e
n
γ
2
I
n
M
a
e
n
-
γ
1
I
n
λ
1
I
n
M
a
e
n
-
γ
2
I
n
λ
2
I
n
]
[
T
a
t
a
I
a
A
1
,
a
,
zd
A
2
,
a
,
zd
]
,
wherein,
P i,a , Φ i,a is observation data at reference station a of the one or more reference stations,
a state vector including a zenith tropospheric delay term T a ,
t a is a clock bias term,
I a is an ionospheric delay term,
A is a set of hardware delay terms,
M a is a vector comprising tropospheric mapping function values between the reference station a and each of the n satellites,
γ i =f 1 2 /f i 2 , f i being the carrier frequency,
λ i is the carrier wavelength (λ i =c/f i , c being the speed of light),
e n is an n-unity vector with all ones as its elements,
I n is an identity matrix with 1 being the diagonal elements,
the hardware delay terms are A i,a,zd :=(a i,a 1 a i,a 2 . . . a i,a n ) T and
a
1
,
a
s
:=
N
1
,
a
s
+
1
λ
1
(
d
ϕ
1
,
a
-
d
ϕ
1
s
+
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
1
,
a
-
d
p
1
s
)
-
2
γ
1
γ
1
-
γ
2
(
d
p
2
,
a
-
d
p
2
s
)
)
a
2
,
a
s
:=
N
2
,
a
s
+
1
λ
2
(
d
ϕ
2
,
a
-
d
ϕ
2
s
+
2
γ
2
γ
1
-
γ
2
(
d
p
1
,
a
-
d
p
1
s
)
-
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
2
,
a
-
d
p
2
s
)
)
,
N i,a s are integer ambiguities at reference station a to satellite s,
d p i ,a are code hardware delays for reference station a,
d ϕ i ,a are phase hardware delays for reference station a,
d p i s are code hardware delays for satellite s, and
d ϕ i s are phase hardware delays for satellite s, and
transmitting the PPP-IAR corrections including the hardware delay terms A to a mobile station.
2. The method according to claim 1 , further comprising
deriving satellite-satellite single-differenced PPP-IAR corrections from individual reference stations;
deriving geometry-based wide-lane (WL) hardware delays and ionosphere-free combination of satellite-satellite single-differenced PPP-IAR corrections;
merging of WL hardware delays;
deriving and merging of L 1 , or L 2 or narrow-lane (NL) hardware delays;
transforming WL hardware delays and L 1 , L 2 or NL hardware delays to L 1 and L 2 hardware delays.
3. The method according to claim 2 , wherein the geometry-based WL is calculated by a wl,a s,n =a 1,a s,n −a 2,a s,n ,
the WL hardware delays are separated from the nearest integers by b wl,a s,n :=a wl,a s,n−└a w1,a s,n ┘and the WL hardware delays of all reference stations are used to compute site-independent WL hardware delays using b wl s,n := b wl,j s,n .
4. The method according to claim 2 , wherein the L 1 and L 2 hardware delay is corrected using the difference between the float double-difference WL float ambiguity and its integer, according to
a
^
1
,
a
s
,
n
:=
a
1
,
a
s
,
n
+
f
1
f
1
-
f
2
c
wl
,
a
s
,
n
and
a
^
2
,
a
s
,
n
:=
a
2
,
a
s
,
n
+
f
1
f
1
-
f
2
c
wl
,
a
s
,
n
,
wherein
c
wl
,
a
s
,
n
:=
a
wl
,
a
s
,
n
-
(
b
wl
s
-
b
wl
n
)
-
N
wl
,
a
s
,
n
and
N
wl
,
a
s
,
n
:=
[
a
wl
,
a
s
,
n
-
(
b
wl
s
-
b
wl
n
)
]
.
5. The method according to claim 2 , wherein the NL hardware delay is derived using the ionosphere-free combinations and the WL integer ambiguities:
a
nl
,
a
s
,
n
:=
a
if
,
a
s
,
n
-
f
2
f
1
-
f
2
N
wl
,
a
s
,
n
wherein
a
if
,
a
s
,
n
=
f
1
·
a
1
,
a
s
,
n
-
f
2
·
a
2
,
a
s
,
n
f
1
-
f
2
.
6. The method according to claim 2 , wherein the PPP-IAR corrections for broadcast to the mobile station comprise:
the L 1 and L 2 hardware delays; or
the WL and L 1 hardware delays; or
the WL and NL hardware delays.
7. A method for applying Precise Point Positioning - Integer Ambiguity Resolution (PPP-IAR) corrections and position determination of a mobile station r, the method comprising:
receiving, by one or more processors, PPP-IAR corrections from a global navigation satellite system (GNSS) signal correction system, the PPP-IAR corrections including hardware delay terms A;
acquiring, at the one or more processors, GNSS data including pseudo range mobile observations and carrier phase mobile observations from a plurality of GNSS satellites;
applying, by the one or more processors, the PPP-IAR corrections to a functional model, the functional model being defined as:
[
P
1
,
r
P
2
,
r
Φ
1
,
r
Φ
2
,
r
]
=
[
μ
r
M
r
e
n
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
γ
2
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
1
E
n
×
(
n
-
1
)
λ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
2
E
n
×
(
n
-
1
)
λ
2
E
n
×
(
n
-
1
)
]
[
Δ
x
r
T
r
t
p
1
,
r
t
p
2
,
r
t
ϕ
1
,
r
t
ϕ
2
,
r
I
r
,
sd
A
1
,
r
,
sd
A
2
,
r
,
sd
]
wherein,
P i,r , Φ 1,r is observation data at the mobile station r,
a state vector comprising an a priori position correction Δx,
T r is a zenith tropospheric delay term,
t p1,r , t p2,r , t Φ1,r , t Φ2,r are clock bias terms,
I r is ionospheric delay terms,
M r is a vector comprising tropospheric mapping function values between the mobile station r and each of the n satellites,
γ i =f 1 2 /f i 2 , f i being the carrier frequency,
λ i is the carrier wavelength (λ i =c/f i , c being the speed of light),
e n is an n-unity vector with all ones as its elements,
μ
r
:=
(
μ
r
1
μ
r
2
…
μ
r
n
)
T
t
p
1
,
r
:=
t
r
+
γ
1
I
~
r
n
t
p
2
,
r
:=
t
r
+
γ
2
I
~
r
n
t
ϕ
1
,
r
:=
t
r
-
γ
1
I
~
r
n
+
a
1
,
r
n
t
ϕ
2
,
r
:=
t
r
-
γ
2
I
~
r
n
+
a
2
,
r
n
I
r
,
sd
:=
(
I
~
r
1
,
n
I
~
r
2
,
n
…
I
~
r
n
-
1
,
n
)
T
A
i
,
r
,
sd
:=
(
a
i
,
r
1
,
n
a
i
,
r
2
,
n
…
a
i
,
r
n
-
1
,
n
)
T
E
n
×
(
n
-
1
)
:=
(
I
n
-
1
0
n
-
1
T
)
=
(
1
⋱
1
0
…
0
)
,
μ r s is the unit vector of line-of-sight between the mobile station r and satellite s
a
1
,
r
s
:=
N
1
,
r
s
+
1
λ
1
(
d
ϕ
1
,
r
-
d
ϕ
1
s
+
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
1
,
r
-
d
p
1
s
)
-
2
γ
1
γ
1
-
γ
2
(
d
p
2
,
r
-
d
p
2
s
)
)
a
2
,
r
s
:=
N
2
,
r
s
+
1
λ
2
(
d
ϕ
2
,
r
-
d
ϕ
2
s
+
2
γ
1
γ
1
-
γ
2
(
d
p
1
,
r
-
d
p
1
s
)
-
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
2
,
r
-
d
p
2
s
)
)
I
~
r
s
,
n
:=
I
~
r
s
-
I
~
r
n
=
I
r
s
-
I
r
n
+
γ
1
γ
1
-
γ
2
(
-
d
p
1
s
+
d
p
1
n
+
d
p
2
s
-
d
p
2
n
)
a
1
,
r
s
,
n
:=
a
1
,
r
s
-
a
1
,
r
n
=
N
1
,
r
s
,
n
-
1
λ
1
(
d
ϕ
1
s
,
n
+
γ
1
+
γ
2
γ
1
-
γ
2
d
p
1
s
,
n
-
2
γ
1
γ
1
-
γ
2
d
p
2
s
,
n
)
a
2
,
r
s
,
n
:=
a
2
,
r
s
-
a
2
,
r
n
=
N
2
,
r
s
,
n
-
1
λ
2
(
d
ϕ
2
s
,
n
+
2
γ
2
γ
1
-
γ
2
d
p
1
s
,
n
-
γ
1
+
γ
2
γ
1
-
γ
2
d
p
2
s
,
n
)
,
and
N i,r s are integer ambiguities for mobile station r to satellite s,
N i,r s,n are satellite-differenced integer ambiguities for mobile station r and between satellite s and reference satellite n,
d p i ,r are code hardware delays for mobile station r,
d ϕ i ,r are phase hardware delays for mobile station r,
d p i s are code hardware delays for satellite s,
d ϕ i s are phase hardware delays for satellite,
d p i s,n are satellite-differenced code hardware delays between satellite s and reference satellite n, and
d ϕ i s,n are satellite-differenced phase hardware delays between satellite s and reference satellite n and
wherein the PPP-IAR corrections are applied to the functional model.
8. The method according to claim 7 , wherein the functional model is augmented with a stochastic model according to
E
{
y
}
=
(
A
B
)
(
a
b
)
D
{
y
}
=
Q
y
wherein Q y is the covariance matrix of y, E{•} and D{•}denote expectation and dispersion, respectively, and
y
:=
[
P
1
,
r
P
2
,
r
Φ
1
,
r
Φ
2
,
r
]
,
A
:=
[
λ
1
E
n
×
(
n
-
1
)
λ
2
E
n
×
(
n
-
1
)
]
,
B
:=
[
μ
r
M
r
e
n
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
γ
2
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
2
E
n
×
(
n
-
1
)
]
,
a
:=
[
A
1
,
r
,
sd
A
2
,
r
,
sd
]
,
b
:=
[
Δ
x
r
T
r
t
p
1
,
r
t
p
2
,
r
t
ϕ
1
,
r
t
ϕ
2
,
r
I
r
,
sd
]
and a least squares solution is calculated accordingly.
9. The method according to claim 7 , wherein the received PPP-IAR corrections are applied to the phase observations:
y
:=
[
P
1
,
r
P
2
,
r
Φ
1
,
r
-
λ
1
E
n
×
(
n
-
1
)
A
1
,
sd
Φ
2
,
r
-
λ
2
E
n
×
(
n
-
1
)
A
2
,
sd
]
,
and the ambiguity parameters are:
a
:=
[
N
1
,
r
N
2
,
r
]
wherein N i,r =A i,r,sd −A i,sd =(N i,r 1,n N i,r 2,n . . . . N i,r n−1,n ) T
and a solution is calculated using an integer-fixing scheme.
10. The method according to claim 7 , wherein the received PPP-IAR corrections are alternatively applied to estimated float ambiguities, resulting in
â′=â−A,
wherein A is the vector of PPP-IAR corrections,
a fixing scheme is applied to â′ and Q ââ , resulting in integer ambiguities ă′, and
an ambiguity fixed solution of the rest parameters is then given by
ă=â−Q {circumflex over (b)}â Q ââ −1 (â′−ă′).
11. A global navigation satellite system (GNSS) signal correction system for estimating and transmitting Precise Point Positioning-integer Ambiguity Resolution (PPP-IAR) corrections to a mobile station, the GNSS signal correction system comprising:
one or more reference stations including a GNSS receiver configured to acquire GNSS data including pseudo range system observations P a and carrier phase system observations Φ a from a plurality of GNSS satellites transmitted over multiple epochs;
one or more processors, configured to receive the GNSS data from the multiple reference stations in real time, and calculate the PPP-IAR corrections based on observation data P i,a , Φ i,a for n satellites received at one or more reference stations with known locations, using a functional model, the functional model being defined as:
[
P
1
,
a
P
2
,
a
Φ
1
,
a
Φ
2
,
a
]
=
[
M
a
e
n
γ
1
I
n
M
a
e
n
γ
2
I
n
M
a
e
n
-
γ
1
I
n
λ
1
I
n
M
a
e
n
-
γ
2
I
n
λ
2
I
n
]
[
T
a
t
a
I
a
A
1
,
a
,
zd
A
2
,
a
,
zd
]
,
wherein
P i,a , Φ i,a observation data at reference station a of the one or more reference stations,
a state vector including a zenith tropospheric delay term T a ,
t a , is a clock bias term,
I a is an ionospheric delay term,
A is a set of hardware delay terms,
M a is a vector comprising tropospheric mapping function values between the reference station a and each of the n satellites,
γ i =f 1 2 /f i 2 ,f i being the carrier frequency,
λ i is the carrier wavelength (λ i =c/f i , c being the speed of light),
e n is an n-unity vector with all ones as its elements,
I n is an identity matrix with 1 being the diagonal elements,
the hardware delay terms are A i,a,zd :=(a i,a 1 a i,a 2 . . . a i,a n ) T and
a
1
,
a
s
:=
N
1
,
a
s
+
1
λ
1
(
d
ϕ
1
,
a
-
d
ϕ
1
s
+
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
1
,
a
-
d
p
1
s
)
-
2
γ
1
γ
1
-
γ
2
(
d
p
2
,
a
-
d
p
2
s
)
)
a
2
,
a
s
:=
N
2
,
a
s
+
1
λ
2
(
d
ϕ
2
,
a
-
d
ϕ
2
s
+
2
γ
2
γ
1
-
γ
2
(
d
p
1
,
a
-
d
p
1
s
)
-
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
2
,
a
-
d
p
2
s
)
)
,
N i,a s are integer ambiguities at reference station a to satellite s,
d p i ,a are code hardware delays for reference station a,
d ϕ i ,a are phase hardware delays for reference station a,
d p i s are code hardware delays for satellite s, and
d ϕ i s are phase hardware delays for satellites;
a signal transmitter connected to the one or more processors configured to transmit the PPP-IAR corrections to the mobile station.
12. A mobile station, comprising:
a mobile global navigation satellite system (GNSS) receiver configured to acquire GNSS data including pseudo range mobile observations P r and carrier phase mobile observations Φ r from a plurality of GNSS satellites transmitted over multiple epochs,
a mobile signal receiver configured to receive GNSS corrections,
a mobile processing unit connected to the mobile GNSS receiver and the mobile signal receiver, the mobile station is configured to:
receive Precise Point Positioning-Integer Ambiguity Resolution (PPP-IAR) corrections from a global navigation satellite system (GNSS) signal correction system, the PPP-IAR corrections including hardware delay terms A;
acquire GNSS data including pseudo range mobile observations and carrier phase mobile observations from the plurality of GNSS satellites;
apply the PPP-IAR, corrections to a functional model, the functional model being defined as:
[
P
1
,
r
P
2
,
r
Φ
1
,
r
Φ
2
,
r
]
=
[
μ
r
M
r
e
n
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
γ
2
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
1
E
n
×
(
n
-
1
)
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
2
E
n
×
(
n
-
1
)
γ
2
E
n
×
(
n
-
1
)
]
[
Δ
x
r
T
r
t
p
1
,
r
t
p
2
,
r
t
ϕ
1
,
r
t
ϕ
2
,
r
I
r
,
sd
A
1
,
r
,
sd
A
2
,
r
,
sd
]
wherein,
P i,r , Φ i,r is observation data at the mobile station r,
a state vector comprising an a priori position correction Δx,
T r is a zenith tropospheric delay term,
t p1,r , t p2,r , t ϕ1,r , t ϕ2,r are dock bias terms,
I r is ionospheric delay terms,
M r is a vector comprising tropospheric mapping function values between the mobile station r and each of the n satellites,
γ i =f 1 2 /f i 2 , f i being the carrier frequency,
λ i is the carrier wavelength (λ i =c/f i , c being the speed of light),
e n is an n-unity vector with all ones as its elements,
μ
r
:=
(
μ
r
1
μ
r
2
…
μ
r
n
)
T
t
p
1
,
r
:=
t
r
+
γ
1
I
~
r
n
t
p
2
,
r
:=
t
r
+
γ
2
I
~
r
n
t
ϕ
1
,
r
:=
t
r
-
γ
1
I
~
r
n
+
a
1
,
r
n
t
ϕ
2
,
r
:=
t
r
-
γ
2
I
~
r
n
+
a
2
,
r
n
I
r
,
sd
:=
(
I
~
r
1
,
n
I
~
r
2
,
n
…
I
~
r
n
-
1
,
n
)
T
A
i
,
r
,
sd
:=
(
a
i
,
r
1
,
n
a
i
,
r
2
,
n
…
a
i
,
r
n
-
1
,
n
)
T
E
n
×
(
n
-
1
)
:=
(
I
n
-
1
0
n
-
1
T
)
=
(
1
⋱
1
0
…
0
)
,
μ r s the unit vector of line-of-sight between the mobile station r and satellite s
a
1
,
r
s
:=
N
1
,
r
s
+
1
λ
1
(
d
ϕ
1
,
r
-
d
ϕ
1
s
+
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
1
,
r
-
d
p
1
s
)
-
2
γ
1
γ
1
-
γ
2
(
d
p
2
,
r
-
d
p
2
s
)
)
a
2
,
r
s
:=
N
2
,
r
s
+
1
λ
2
(
d
ϕ
2
,
r
-
d
ϕ
2
s
+
2
γ
1
γ
1
-
γ
2
(
d
p
1
,
r
-
d
p
1
s
)
-
γ
1
+
γ
2
γ
1
-
γ
2
(
d
p
2
,
r
-
d
p
2
s
)
)
I
~
r
s
,
n
:=
I
~
r
s
-
I
~
r
n
=
I
r
s
-
I
r
n
+
γ
1
γ
1
-
γ
2
(
-
d
p
1
s
+
d
p
1
n
+
d
p
2
s
-
d
p
2
n
)
a
1
,
r
s
,
n
:=
a
1
,
r
s
-
a
1
,
r
n
=
N
1
,
r
s
,
n
-
1
λ
1
(
d
ϕ
1
s
,
n
+
γ
1
+
γ
2
γ
1
-
γ
2
d
p
1
s
,
n
-
2
γ
1
γ
1
-
γ
2
d
p
2
s
,
n
)
a
2
,
r
s
,
n
:=
a
2
,
r
s
-
a
2
,
r
n
=
N
2
,
r
s
,
n
-
1
λ
2
(
d
ϕ
2
s
,
n
+
2
γ
2
γ
1
-
γ
2
d
p
1
s
,
n
-
γ
1
+
γ
2
γ
1
-
γ
2
d
p
2
s
,
n
)
,
and
N i,r s are integer ambiguities for mobile station r to satellite s,
N i,r s,n are satellite-differenced integer ambiguities for mobile station r and between satellite s and reference satellite n,
d p i ,r are code hardware delays for mobile station r,
d ϕ i ,r are phase hardware delays for mobile station r,
d p i ,r s are code hardware delays for satellite s,
d ϕ i s are phase hardware delays for satellite,
d p i s,n are satellite-differenced code hardware delays between satellite s and reference satellite n, and
d ϕ i s,n are satellite-differenced phase hardware delays between satellite s and reference satellite n; and
wherein the PPP-IAR corrections are applied to the functional model.
13. The GNSS signal correction system to claim 11 , wherein the one or more processors are further configured to:
derive satellite-satellite single-differenced PPP-IAR corrections from individual reference stations;
derive geometry-based wide-lane (WL) hardware delays and ionosphere-free combination of satellite-satellite single-differenced PPP-IAR corrections;
merge of WL hardware delays;
derive and merge of L 1 , or L 2 or narrow-lane (NL) hardware delays;
transform WL hardware delays and L 1 , L 2 or NL hardware delays to L 1 and L 2 hardware delays.
14. The GNSS signal correction system to claim 13 , wherein the geometry-based WL is calculated by a wl,a s,n =a 1,a s,n −a 2,a s,n ,
the WL hardware delays are separated from the nearest integers by b wl,a s,n :=a wl,a s,n −└a wl,a s,n ┘, and the WL hardware delays of all reference stations are used to compute site-independent WL hardware delays using b wl s,n := b wl,j s,n .
15. The GNSS signal correction system to claim 13 , wherein the L 1 and L 2 hardware delay is corrected using the difference between the float double-difference WL float ambiguity and its integer, according to
a
^
1
,
a
s
,
n
:=
a
1
,
a
s
,
n
+
f
2
f
1
-
f
2
c
wl
,
a
s
,
n
and
a
^
2
,
a
s
,
n
:=
a
2
,
a
s
,
n
+
f
1
f
1
-
f
2
c
wl
,
a
s
,
n
,
wherein
c
wl
,
a
s
,
n
:=
a
wl
,
a
s
,
n
-
(
b
wl
s
-
b
wl
n
)
-
N
wl
,
a
s
,
n
and
N
wl
,
a
s
,
n
:=
⌊
a
wl
,
a
s
,
n
-
(
b
wl
s
-
b
wl
n
)
⌋
.
16. The GNSS signal correction system to claim 13 , wherein the NL hardware delay is derived using the ionosphere-free combinations and the WL integer ambiguities:
a
nl
,
a
s
,
n
:=
a
if
,
a
s
,
n
-
f
2
f
1
-
f
2
N
wl
,
a
s
,
n
wherein
a
if
,
a
s
,
n
=
f
1
·
a
1
,
a
s
,
n
-
f
2
·
a
2
,
a
s
,
n
f
1
-
f
2
.
17. The GNSS signal correction system to claim 13 , wherein the PPP-IAR corrections for broadcast to the mobile station comprise:
the L 1 and L 2 hardware delays; or
the WL and L 1 hardware delays; or
the WL and NL hardware delays.
18. The mobile station according to claim 12 , wherein the functional model is augmented with a stochastic model according to
E
{
y
}
=
(
A
B
)
(
a
b
)
D
{
y
}
=
Q
y
wherein Q y is the covariance matrix of y, E{•} and D{•} denote expectation and dispersion, respectively, and
y
:=
[
P
1
,
r
P
2
,
r
Φ
1
,
r
Φ
2
,
r
]
,
A
:=
[
λ
1
E
n
×
(
n
-
1
)
λ
2
E
n
×
(
n
-
1
)
]
,
B
:=
[
μ
r
M
r
e
n
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
γ
2
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
1
E
n
×
(
n
-
1
)
μ
r
M
r
e
n
-
γ
2
E
n
×
(
n
-
1
)
]
,
a
:=
[
A
1
,
r
,
sd
A
2
,
r
,
sd
]
,
b
:=
[
Δ
x
r
T
r
t
p
1
,
r
t
p
2
,
r
t
ϕ
1
,
r
t
ϕ
2
,
r
I
r
,
sd
]
and a least squares solution is calculated accordingly.
19. The mobile stations according to claim 12 , wherein the received PPP-IAR corrections are applied to the phase observations:
y
:=
[
P
1
,
r
P
2
,
r
Φ
1
,
r
-
λ
1
E
n
×
(
n
-
1
)
A
1
,
sd
Φ
2
,
r
-
λ
2
E
n
×
(
n
-
1
)
A
2
,
sd
]
,
and the ambiguity parameters are:
a
:=
[
N
1
,
r
N
2
,
r
]
wherein N i,r =A i,r,sd −A i,sd =(N i,r 1,n N i,r 2,n . . . N i,r n−1,n ) T
and a solution is calculated using an integer-fixing scheme.
20. The mobile stations according to claim 12 , wherein the received PPP-IAR corrections are alternatively applied to estimated float ambiguities, resulting in
â′=â−A,
wherein A is the vector of PPP-IAR corrections,
a fixing scheme is applied to â′ and Q ââ , resulting in integer ambiguities ň′, and an ambiguity fixed solution of the rest parameters is then given by
{hacek over (b)}={circumflex over (b)}−Q {circumflex over (b)}â Q ââ −1 (â′−{hacek over (a)}′).