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

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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

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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

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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

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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

8

YELL Residuals YELL Residuals –– P3 Obs. & Avg. P3 Obs. & Avg.

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.

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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.

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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

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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

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0.0E+00

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2.0E-13 ex

t mo

del P

3 a

vg

res (m

)

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0.0E+00

5.0E-07

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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

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2.0E-16 ext m

od

el L3

av

g res (m

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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

16

WidelaneWidelane Ambiguities Ambiguities –– Float & Fixed Float & Fixed

17

NarrowlaneNarrowlane Ambiguities Ambiguities –– Float & Fixed Float & Fixed

18

L3 Residuals L3 Residuals –– Float & Fixed Float & Fixed

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.