isolating and estimating undifferenced gps … and estimating undifferenced gps integer ambiguities...
TRANSCRIPT
Isolating and Estimating UndifferencedIsolating and Estimating UndifferencedGPS Integer AmbiguitiesGPS Integer Ambiguities
Paul CollinsPaul CollinsGeodetic Survey Division, Natural Resources CanadaGeodetic Survey Division, Natural Resources Canada
Institute of Navigation, National Technical MeetingJanuary 28-30, 2008; San Diego, California
2
IntroductionIntroduction
Undifferenced Ambiguity Resolution has been anelusive goal in GPS processing.
Recent techniques have been introduced that appear to“show the way”. (All aspects addressed?)
Concurrently, there have been on-going investigationsinto the so-called “code-biases”.
The goal of the presentation is to show how: the “standard model” of undifferenced ionosphere-free
observables is sub-optimal; and rigorous modelling of code biases facilitates estimation of
integer ambiguities from undifferenced observables.
3
Standard Observation EquationsStandard Observation Equations
( )
( ) 222
2
2222
1111111
222
2
2
1111
1111
)(
)(
)(
)(
)(
L
s
L
r
L
sr
L
s
L
r
L
sr
P
s
P
r
P
sr
P
s
P
r
P
sr
C
s
C
r
C
sr
bbdtdtcIqTLN
bbdtdtcITLN
bbdtdtcIqTP
bbdtdtcITP
bbdtdtcITC
!"#
!"#
!"
!"
!"
+$+$+$+==+%
+$+$+$+==+%
+$+$+++=
+$+$+++=
+$+$+++=
distinguish between geometric and non-geometric (timing)parameters
b** represent synchronisation errors betweenmeasurements – codes and phases measured separately
understanding their role is crucial to isolating integerambiguities from undifferenced carrier phases
4
Standard Observable ModelStandard Observable ModelRe-assessedRe-assessed
singular due to functionally identical clocks & biases by combining code clock and bias parameters and
retaining common oscillators:
hence, even if bL3* known, ambiguities are not isolated
333333
3333
)(
)(
L
s
L
r
L
sr
P
s
P
r
P
sr
NbbdtdtcTL
bbdtdtcTP
!"#
!#
+$$+$++=
+$+$++=
33333
3333
)(
)(
LP
s
P
r
P
P
s
P
r
P
AdtdtcTL
dtdtcTP
!"
!"
++#++$
+#++$
3333333NbbbbA
s
P
s
L
r
P
r
LP!"+""=where
5
PseudorangePseudorange Clock Datum Clock Datum
Because each carrier phase is uniquely ambiguous,the pseudoranges provide the datum for the clocksolutions.
Implication: A change in dual-frequency pseudoranges manifests
itself in estimated clocks and ambiguities. Example:
Compute P1-C1 bias from 2 standard model solutions:( )
( ) ( )r
CPPP
s
CP
s
P
s
P bAAbdtdt
PPCPwherePCfP
PPfP
11'3311'33
121'2'21'3
213
,
,
!! !!==!
!+==
=
6
Deriving satellite P1-C1 biasesDeriving satellite P1-C1 biases
standard model still optimally parameterised if b**
are constant, but…
RMS of fit = 0.3ns/0.1m
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
G01
G02
G03
G04
G05
G06
G07
G08
G09
G10
G11
G12
G13
G14
G15
G16
G17
G18
G19
G20
G21
G22
G23
G24
G25
G26
G27
G28
G29
G30
G31
P1
-C1
Est
ima
te (
ns)
CODE
satclks
amb
7
Day-Boundary Clock JumpsDay-Boundary Clock Jumps
YELL-AMC2 clock error
(Jan07-Jan13, 2007; common linear fit removed)
-6
-4
-2
0
2
4
6
0 1 2 3 4 5 6 7
Day of Week (GPS 1409)
Clo
ck
Error (
ns)
code clock
standard model
IGS Rapid
9
Decoupled Clock ModelDecoupled Clock Model
Problem: Apparent code and phase oscillator measures are
significantly different. Solution:
Decouple the code and phase clocks:
No assumptions about bias ‘stability’ required.Implication:
Pseudorange datum removed from carrier phase. Replace with Ambiguity Datums.
333333
3333
)(
)(
L
s
L
r
L
P
s
P
r
P
NdtdtcTL
dtdtcTP
!"#
!#
+$$++=
+$++=
10
Ambiguity Datum FixingAmbiguity Datum Fixing
One ambiguity per phase clock fixed, less one One phase clock fixed as the network datum
Identical concept to fixing the ‘reference clock’ instandard model network processing
One code clock fixed also Ambiguity can be fixed to arbitrary integer value
acts as Partial Integer Constraint remaining ambiguities are integer!
Phase clock estimates are integer ambiguouswith respect to the code clock estimates.
11
Relationship to Goad ModelRelationship to Goad Model
Goad, C.C. (1985). “Precise Relative Position Determination UsingGlobal Positioning System Carrier Phase Measurements in aNondifference Mode.” Proceedings of the First InternationalSymposium on Precise Positioning with the Global Positioning System.US Dept. of Commerce. Rockville, Maryland, April 15-19. Vol. 1, pp. 347-356.
)()()(/)()(
)(/)()(
)(/)()(
)(/)()(
1
1
2
1
1
2
12
12
2
2
2
2
1
2
1
2
1
2
2
1
2
1
2
1
1
1
1
1
1
1
tBtBtBNtGt
tBtGt
tBtGt
tBtGt
!+++="
+="
+="
+="
#
#
#
#
)()(/)()(
)()(/)()(
)(/)()(
)(/)()(
2
1
12
1221
2
2
2
2
1
121
1
2
1
2
2
1
2
1
2
1
1
1
1
1
1
1
tBNtBtGt
tBtBtGt
tBtGt
tBtGt
+++=!
++=!
+=!
+=!
"
"
"
"
base-station–base-satellite datum fixing4 observations : 4 clk/amb unknowns when G(t) knownG(t): a-priori values or pseudorange estimates
}network datum
12
Extended ModelExtended Model
Problem: λIF(L1,L2) ≈6mm. (Note: λIF(L2,L5) ≈12cm!)
Implication: Intermediate step required for L1,L2 processing
Solution: Melbourne-Wübbena combination for WL
are not constant – ‘delta-clocks’ Ambiguity Datum fixing Processed simultaneously with P3 and L3
With WL fixed, λIF(L1,L2) = λNL≈11cm
44444544 A
s
A
r
ANbbPLA !" +##=#=
*
4Ab
13
YELL P3 & L3 Average ResidualsYELL P3 & L3 Average Residuals
-2.0E+00
-1.5E+00
-1.0E+00
-5.0E-01
0.0E+00
5.0E-01
1.0E+00
1.5E+00
2.0E+00
0 12 24 36 48 60 72
Elapsed Time (hour)
std
mo
del
P3
av
g r
es (
m)
-2.0E-13
-1.5E-13
-1.0E-13
-5.0E-14
0.0E+00
5.0E-14
1.0E-13
1.5E-13
2.0E-13 ex
t mo
del P
3 a
vg
res (m
)
-2.0E-06
-1.5E-06
-1.0E-06
-5.0E-07
0.0E+00
5.0E-07
1.0E-06
1.5E-06
2.0E-06
0 12 24 36 48 60 72
Elapsed Time (hour)
std
mo
del
L3
av
g r
es (
m)
-2.0E-16
-1.5E-16
-1.0E-16
-5.0E-17
0.0E+00
5.0E-17
1.0E-16
1.5E-16
2.0E-16 ext m
od
el L3
av
g res (m
)
14
Station Clock Parameter EstimatesStation Clock Parameter Estimates
Rapid/phase clock de-trended RMS = 0.08ns/0.02m
15
Satellite Clock Parameter EstimatesSatellite Clock Parameter Estimates
Rapid/phase clock de-trended RMS = 0.17ns/0.05m
19
Implications for PPP Implications for PPP —— Summary Summary
Each observable requires a satellite ‘clock’ parameter: dtP3, dtL3, bA4
as well as satellite X, Y, Z coordinates. In practice (dtP3−dtL3) and bA4 variations may allow
transmission as ‘slow’ corrections. Standard Ambiguity Resolution Techniques (e.g.
LAMBDA) become applicable to PPP. PPP-AR becomes possible in principle. ALL predicated on good orbits! (IGS Rapid here)
20
ConclusionsConclusions
Synchronisation of code and phase measurementsis significantly different.
The Standard Model allows the pseudorange biasesto directly interfere with the carrier phase biases.
The Decoupled Clock Model provides: unambiguous, but imprecise code clock estimates precise, but ambiguous phase clock estimates integer ambiguities.
Extended Model required for L1, L2 processing. Provides a path for PPP-AR in a very generic way
extension of generic LS, no a-priori bias assumptions.