Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network Maj John Raquet [email protected] Air Force Institute of Technology (and

Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network

Maj John Raquet

[email protected]

Air Force Institute of Technology

(and The University of Calgary)

Overview• Motivation• Setting up the problem• NetAdjust solution• Implementation issues• Putting this approach in context• Covariance function description• NetAdjust test results• Covariance analysis technique• Summary/Conclusion


Reference Receiver Network Motivation (1/3)

• Single reference receiver coverage

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-80

-60

-40

-20

0

20

40

60

80

100

Ref.

Easting (km)

Nor

thin

g (k

m)

Desired Coverage Area


• One (poor) solution

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-80

-60

-40

-20

0

20

40

60

80

100

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Ref.

Easting (km)

Nor

thin

g (k

m)



• Better solution: use a network

-100 -80 -60 -40 -20 0 20 40 60 80 100-100

-80

-60

-40

-20

0

20

40

60

80

100

Ref.

Ref.

Ref.

Ref.

Easting (km)

No

rthi

ng (

km)


Phase Measurements

• Measurement with errors

• Double-differencing

NmITtctcr svu

1

ambiguity integer

multipath phase

noise meas phase

errorc ionospheri

erroric tropospher

bias clock satellite

bias clock receiver

satellite to range true

h wavelengtphase

N

m

I

T

t

t

r

sv

u

abababababab

bbaabaab

NmITr 121212121212

2121121212

1

)(

Double-Difference Phase Errors

• Highest positioning accuracy obtained by differential carrier-phase ambiguity resolution

– If “close” to reference receiver, then correlated errors are removed

ambiguity

integernoise measmultipathionotropo

NmITr

1

Nmr

errors levelcm-

1

Why Reducing Errors Helps Ambiguity Resolution

• Almost all ambiguity resolution routines use some sort of residual analysis to determine integer ambiguities

errors )( aldifferentiresiduals tmeasuremen

svpmITrN1

ˆ1

Why Reducing Errors Helps Ambiguity Resolution

• Almost all ambiguity resolution routines use some sort of residual analysis to determine integer ambiguities

errors )( aldifferentiresiduals tmeasuremen

svpmITrN1

ˆ1

Goal


Setting Up the Problem

x

ComputationPoint

Sample 5-receiver network:

Ref 1

Ref 2

Ref 3

Ref 4

Ref 5

cp

n

cp

cp

cp

cp

cp

n

n

n

n

n

nnnn l

lll

l

l

l

l

l

llll ~ ,

~

~

~

~

~ , , ,

4

3

2

1

42

32

22

12

41

31

21

11

5

4

3

2

1

321

+0p

Measurements:Measurement-minus-range observable: calcr

Setting Up the ProblemMeasurement errors:

Ambiguity

Integer

Noise) ,(MultipathErrors

edUncorrelat Position)SV Tropo, (Iono,

ErrorsCorrelated

00clock ),()( Npppl ucc d

Double-difference errors:

Nppl ucd ),( 0

Errors to be eliminated:

ucncp

nd

),( 0,

ppll

l

Measurements available:

nnnnn lBlNll :Note


NetAdjust Solution

• Use a linear minimum variance of error estimator– Generic case (to estimate x given measurements Y)

• Assumption: x and Y are jointly Gaussian

– Our case (to estimate given measurements )

• Assumes and are zero-mean

YCCx

Yx

YYCCxx

1YYx

1YYx

,

,

ˆ

0][][

][][ˆ

then,if or, EE

EE

nll

nnnlCCl lll

1 ,

ˆ

l nl

Are the Assumptions Valid?

• Assumption 1: and are jointly Gaussian– Each individual error source tends to be Gaussian– Central limit theorem strengthens assumption

• Assumption 2: and are zero-mean– Reasonable for uncorrelated errors (multipath and

noise)– Reasonable for correlated errors, if systemic

biases removed by a model

nll

nll

Cleaner Statement of NetAdjust Solution

• Corrections to apply to measurements from reference receiver network

• Corrections to apply to mobile receiver measurements

• Minimizes trace --the ultimate goal!

nnnTnln

Tnllcp NlBBCBBCl

nncp

1

,ˆ

nnnTnln

Tnln NlBBCBBCl

nn

1ˆ

ncp , lC

(1)

(2)


Implementation Approach

n

n

n

n

l

l

l

l

n

2

1

n

2

1

n

n

n

n

l

l

l

l

ˆ

ˆ

ˆ

1nl̂

1nl

1nl̂cpl̂

Equation 1

Equation 2

ComputationPoint

+

+

NetAdjust

Mobile Receiver(at Computation Point)Ambiguity Resolution

and Positioning Algorithm

cpl

Mobile Receiver Position

Alternate Implementation Approach

n

n

n

n

l

l

l

l

n

2

1

n

2

1

n

n

n

n

l

l

l

l

ˆ

ˆ

ˆ

1nl̂

1nl

1nl̂

Equation 1

Equation 2

ComputationPoint

+

+

NetAdjust

Mobile Receiver(at Computation Point)Ambiguity Resolution

and Positioning Algorithm

cpl

Mobile Receiver Position

+-

Includes all corrections

How Do You Transmit for Mobile User at Any Location?

• Question that must be answered for multi-user one-way-broadcast network

• Corrections vary with location (as they should)• Variation is not easily modeled• Different approaches can be taken using grid

– Nearest point

– Interpolation (linear, quadratic)

– Update rates

• See ION AM 2000 paper by Fotopoulos

Calculation of Network Ambiguities

• Algorithm requires no initialization per se• Ambiguities between reference receivers

must be known– Best if all fixed– Will work (slightly less well) with floating ambiguity

estimates– Can account for a mix of fixed and floating

• Real-time estimation of ambiguities between network reference stations is one of the largest implementation challenges


Three “Views” of the NetAdjust Approach

• Linear Minimum of Variance Estimator– Explicitly minimizes squared error Bayes’ risk– Estimation of one variable using observables

• Least-Squares Condition Adjustment– Apply condition to measurements

• Condition is that all double-differenced measurement-minus-range observables within network are zero

• Explains the “data encapsulation” effect

• Least-Squares Collocation– Interpolation– Use of covariance kernel

Three Classes of Approaches to This Problem

• Error Mitigation Approach– Explicitly estimate individual error sources– Gao, vanderMarel, etc.

• Polynomial Fit Approach– Assume differential errors can be expressed as a particular

functional form of position– Calculate coefficients for the specified function– Varner, Wubbena, etc.

• Covariance Fit Approach (NetAdjust)– Assume error covariance can be expressed as a functional form– Use functionally generated covariance with NetAdjust


Covariance Function ConceptData from test network

Information abouterror characteristics(i.e., covariance matrix)

Express covariance in functional form:Example: function of: - Distance - SV elevation - Rcvr-specific multipath/noise levels

Use covariance function to generate predicted covariance matrix for new configuration

Predict performanceCalculate corrections

Zenith Phase Covariance Functions(Based on 55 Baselines Between 11 Receivers)

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

Zen

ith D

D E

rr V

aria

nce

(L1

cycl

es2 )

Distance Between Receivers (km)

0 100 200 300 400 500 600 7000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

Zen

ith D

D E

rr V

aria

nce

(WL

cycl

es2 )

Distance Between Receivers (km)

L1 WL

term)noisemultipath/(221

2 dcdcDD

77531.9

42048.2

2

1

Ec

Ec

85065.6

57881.1

2

1

Ec

Ec

Example of How Covariance Function Can Change

0 100 200 300 400 500 6000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Distance(km)

Var

ianc

e of

the

cor

rela

ted

erro

rs (

L1 c

ycle

s2 )

L1 correlated error function

Sept 97Sept 98

Nov 98

0 100 200 300 400 500 6000

0.005

0.01

0.015

0.02

0.025

0.03

Distance(km)

Var

ianc

e of

the

cor

rela

ted

erro

rs (

WL

cycl

es2 )

WL correlated error function

Sept 97

Sept 98

Nov 98


Norway Test Network

Region of Interest

-200 0 200 400-300

-200

-100

0

100

200

300

400

TRYRTRYM

BERG

ALES

ARERAREM

GEIM

TRON

STAV

GEIR

KRIS

Easting (km)

No

rth

ing

(km

)

Norway Network

• 24 hours of data at 2 second intervals• Ambiguities calculated

– Between every pair of reference receivers– Over 24 hour period

• Receiver positions calculated– Based on ionospheric-free carrier-phase

observable (requires L1 and L2 ambiguities)– Network adjustment procedure– Relative positioning accuracy: 2-3mm horizontal,

5-7mm vertical

Seven Test Networks

-200 0 200 400-300

-200

-100

0

100

200

300

400

TRYRTRYM

BERG

ALES

ARERAREM

GEIM

TRON

STAV

GEIR

KRIS

Easting (km)

No

rthi

ng

(km

)

Test NetworksARER-0GEIR-29ARER-67STAV-143GEIR-164GEIR-223-sparseALES-242

Testing NetAdjust on Norway Network

• Improvement in double-difference measurement error

• Improvement in differential positioning accuracy (using correct integer ambiguities)

• Improvement in carrier-phase ambiguity resolution

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

GE

IR-2

9

GE

IR-1

64

ST

AV

-143

AR

ER

-67

ALE

S-2

42

AR

ER

-0

GE

IR-2

23-s

pars

e

Dou

ble

Diff

eren

ce M

eas

Err

or R

MS

(L1

cycl

es)

Distance To Nearest Reference Receiver (km)

RawNetAdjust

Improvement in DD Measurement Error

L1 Phase

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

GE

IR-2

9

GE

IR-1

64

ST

AV

-143

AR

ER

-67

ALE

S-2

42

AR

ER

-0

GE

IR-2

23-s

pars

e

Dou

ble

Diff

eren

ce M

eas

Err

or R

MS

(WL

cycl

es)


RawNetAdjust

Improvement in DD Measurement Error

WL Phase

Improvement in Positioning Accuracy L1 Phase (Fixed Integer Ambiguities)

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


3-D

RM

S P

ositi

on E

rror

(m

)

RawNetAdjust

Improvement in Positioning Accuracy WL Phase (Fixed Integer Ambiguities)

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


3-D

RM

S P

ositi

on E

rror

(m

)

RawNetAdjust

Improvement in Ambiguity Resolution

• University of Calgary’s FLYKINTM software– Run iteratively, start times staggered by 10

minutes (138 runs over 24 hours)– Stopped immediately if integer ambiguities

determined

• Three performance criteria– Percentage of correct fixes– Percentage of incorrect fixes– Average time to resolve ambiguities

Improvement in Ambiguity ResolutionPercentage of Correct Fixes - L1 Phase

0 50 100 150 200 250 0%

20%

40%

60%

80%

100%

AR

ER

-0

AR

ER

-67

GE

IR-2

23-s

pars

e

GE

IR-1

64

ST

AV

-143

GE

IR-2

9

ALE

S-2

42

Per

cent

age

Cor

rect

Fix

es

Distance to Nearest Reference Receiver (km)

Raw Code, Raw L1 PhaseRaw Code, NetAdjust L1 PhaseNetAdjust Code, NetAdjust L1 Phase

Improvement in Ambiguity ResolutionPercentage of Correct Fixes - WL Phase

0 50 100 150 200 250 0%

20%

40%

60%

80%

100%

AR

ER

-0

AR

ER

-67

GE

IR-2

23-s

pars

e

GE

IR-1

64

ST

AV

-143

GE

IR-2

9

ALE

S-2

42Per

cent

age

Cor

rect

Fix

es


Raw Code, Raw WL PhaseRaw Code, NetAdjust WL PhaseNetAdjust Code, NetAdjust WL Phase

Improvement in Ambiguity ResolutionMean Time to Fix - WL Phase

0 50 100 150 200 2500

1

2

3

4

5

6

7

GE

IR-2

9

GE

IR-1

64

ST

AV

-143

AR

ER

-67

ALE

S-2

42

AR

ER

-0

GE

IR-2

23-s

pars

e

Mea

n T

ime

to R

esol

ve A

mbi

guiti

es (

min

utes

)


Raw Code, Raw WL PhaseRaw Code, NetAdjust WL PhaseNetAdjust Code, NetAdjust WL Phase


Motivation• It’s difficult and costly to deploy a reference receiver network

• Differential network performance varies with– Number/location of reference receivers– Number/geometry of visible satellites– Type of measurement used (e.g., L1 or WL)– Characteristics (especially correlations) of DGPS errors

• May be possible to test small subset of network configurations

• Desirable to predict performance for other (untested) network configurations– “What if” scenarios– Based upon test results– Critical for final network design

Covariance Analysis Procedure

• Straightforward propagation of DGPS measurement error covariance into double-difference space:

function) covariance (from covariance error meas DGPS

receivers mobilereference/ between (matrix) operator DD

receivers reference between (matrix) operator DD

receivers mobilereference/ between covariance error DD

where

l

lerr

Tl

Tl

Tl

Tllerr

C

B

B

C

BCBBCBBCBBCBC

cp

cp

2

1

)(

21

1

111222)(

Validation of Covariance Function and Analysis Procedure

• Seven “test networks” selected– One receiver selected as “mobile” receiver– Remaining (or subset) form network– Closest reference receiver identified (for single

reference case)

• Double difference errors predicted by covariance analysis– Single reference (raw) case– Multiple reference (NetAdjust) case

• Prediction compared with actual results

Validation: Predicted and ActualL1 Phase

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

GE

IR-2

9

AR

ER

-67

GE

IR-2

23-s

pars

e

ALE

S-2

42

AR

ER

-0

GE

IR-1

64

ST

AV

-14

3

Dou

ble

Diff

ere

nce

Err

or R

MS

(L1

cyc

les)


Raw from DataRaw from Cov. AnalysisNetAdjust from DataNetAdjust from Cov. Analysis

Validation: Predicted and ActualWL Phase

0 50 100 150 200 2500

0.05

0.1

0.15

0.2

0.25

GE

IR-2

9

AR

ER

-67

GE

IR-2

23-s

pars

e

ALE

S-2

42

AR

ER

-0

GE

IR-1

64

ST

AV

-14

3

Dou

ble

Diff

ere

nce

Err

or R

MS

(W

L c

ycle

s)


Raw from DataRaw from Cov. AnalysisNetAdjust from DataNetAdjust from Cov. Analysis

Development of Network Performance Specification

• Primary emphasis is carrier-phase ambiguity resolution

• Develop relationship between double difference measurement error and distance between mobile and reference receivers

• Specification made in terms of distance from reference receiver (under “normal” conditions)– More intuitive than pure error statistics– Typically, already is distance specification established

• Convert distance specification into measurement error specification

Zenith DD Measurement Error vs. Baseline Distance

0 100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Baseline Distance (km)

Zen

ith D

oubl

e D

iffer

ence

Mea

s E

rror

Sta

ndar

d D

evia

tion

(L1

cycl

es)

L1 Phase

0 100 200 300 400 500 600 7000

0.05

0.1

0.15

0.2

Baseline Distance (km)

Zen

ith D

oubl

e D

iffer

ence

Mea

s E

rror

Sta

ndar

d D

evia

tion

(WL

cycl

es)

WL Phase

Specifications Chosen for Demonstration Purposes

• L1 Phase– Distance: 25 km– Zenith DD Meas Error Std Dev: 0.079 L1 cycles– DD Meas Error Std Dev: 0.182 L1 cycles

• WL Phase– Distance: 60 km– Zenith DD Meas Error Std Dev: 0.038 WL cycles– DD Meas Error Std Dev: 0.092

• Note: Assuming 7 SVs for plots that follow

WL Covariance Analysis Results

-200 0 200

-200

-100

0

100

200

300

0.1

0.08

0.12

0.04

0.06

0.06

0.06

0.06

0.06

0.08

0.08

0.08

0.08

0.08

0.080.08

0.08

0.08

WL - 98.1% Coverage

Easting (km)

Nor

thin

g (k

m)

L1 Covariance Analysis Results

-200 0 200

-200

-100

0

100

200

300

0.35

0.3

0.4

0.15

0.15

0.15 0

.2

0.2

0.2

0.2

0.2

0.2

0.2

0.25

0.25

0.30.25

0.25

0.25

0.25

0.25

0.25

L1 - 17.1% Coverage

Easting (km)

Nor

thin

g (k

m)

Effect of Repositioning Reference Receivers(WL)

-200 0 200

-200

-100

0

100

200

300

0.1

0.08

0.12

0.04

0.06

0.060.06

0.06

0.06

0.08

0.08

0.08

0.08

0.08

0.080.08

0.08

0.08

Original - 98.1% Coverage

Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.1

0.08

0.120.06

0.06

0.06

0.06

0.06

0.06

0.08

0.06 0.06

0.08

0.08

Repositioned - 100.0% Coverage

Easting (km)

Nor

thin

g (k

m)

Effect of Repositioning Reference Receivers(L1)

-200 0 200

-200

-100

0

100

200

300

0.35

0.3

0.4

0.15

0.15

0.15 0

.2

0.2

0.2

0.2

0.2

0.2

0.2

0.25

0.25

0.30.25

0.25

0.25

0.25

0.25

0.25


Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.2

0.2

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2


Easting (km)

Nor

thin

g (k

m)

Effect of Repositioning/Adding Reference Receivers (L1)

-200 0 200

-200

-100

0

100

200

300

0.35

0.3

0.4

0.15

0.15

0.15 0

.2

0.2

0.2

0.2

0.2

0.2

0.2

0.25

0.25

0.30.25

0.25

0.25

0.25

0.25

0.25


Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.35

0.25

0.3

0.15 0.

15

0.15

0.15

0.15 0.15

0.15

0.15

0.15

0.2

0.2

0.2

0.2

0.2 0.2

0.2

0.25

0.25

0.3

22 Ref Rcvrs - 91.4% Coverage

Nor

thin

g (k

m)

Easting (km)

Effect of Varying Satellite Constellation (L1)

-200 0 200

-200

-100

0

100

200

3000.

3

0.2

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

6 SVs - 18.6% Coverage

Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.2

0.2

0.3

0.2

0.2

0.2

0.2

0.2

0.2 0.2

0.2

0.2

0.2


Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.3

0.25

0.250.15

0.15

0.15

0.2

0.2

0.2

0.2

0.2

0.2 0.2

0.25


Nor

thin

g (k

m)

Easting (km)

Analysis of Day/Night VariationWL Covariance Function

0 100 200 300 400 500 6000

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

Distance between receivers (km)

DD

Cor

rela

ted

Var

ianc

e (W

L cy

cles

2 )

Zenith DD Error Correlated Variance of WL Phase (cycles2)

Day

NightAve

rage

Analysis of Day Night VariationL1

-200 0 200

-200

-100

0

100

200

300

Easting (km)

Nor

thin

g (k

m)

Average - 17.1% Coverage

0.15 0.2

0.2

0.2

0.2

0.2

0.2

0.2 0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.25

0.3

0.3 0.3

0.35

-200 0 200

-200

-100

0

100

200

300

Easting (km)

Nor

thin

g (k

m)

Night - 81.5% Coverage

0.1

0.15

0.15

0.15

0.150.15

0.15

0.15

0.15 0.2

0.2

0.2

0.2

0.25

-200 0 200

-200

-100

0

100

200

300

Easting (km)

Nor

thin

g (k

m)

Day - 12.7% Coverage

0.2

0.2

0.2

0.2

0.2

0.3

0.3

0.3

0.4

Prediction of Effect of Increased Ionospheric Activity

• Covariance analysis technique can be used to predict (simulate) high ionospheric activity– Covariance function represents combination of all errors

(including ionosphere)– If ionosphere increases by some percentage, then total

errors increase by a lesser percentage• Depends on the ratio of the ionospheric errors to all other

error sources• Relatively easy to determine this ratio by using various

L1/L2 combinations

• Measurement errors amplified by 1.5 (variance by 2.25) to simulate increased ionospheric activity

Effect of Increased Ionosphere(L1)

-200 0 200

-200

-100

0

100

200

300

0.2

0.2

0.3

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2


Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.6

0.4

0.4

0.2

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.3

0.5

Increased Ionosphere - L1 - 4.1% Coverage

Nor

thin

g (k

m)

Easting (km)

Effect of Increased Ionosphere(WL)

-200 0 200

-200

-100

0

100

200

300

0.1

0.08

0.120.06

0.06

0.06

0.06

0.06

0.06

0.08

0.06 0.06

0.08

0.08


Easting (km)

Nor

thin

g (k

m)

-200 0 200

-200

-100

0

100

200

300

0.1

0.1

0.15

0.1

0.1

0.1

0.1 0.1

0.1 0.1

Increased Ionosphere - WL - 49.3% Coverage

Easting (km)

Nor

thin

g (k

m)


Tests Accomplished

• Many post-processing tests have been performed– Holloman AFB, Aug 96– Norway, Sep 97– Norway, Sep 98– St. Lawrence Seaway, Nov 98– St. Lawrence Seaway, Aug 99

• Real-time implementation and testing underway– Norway– Japan

Conclusions• Network approach shows promising results

– Significantly educes both L1 and widelane errors– More effective with widelane ambiguity resolution, in

tested networks– Not a cure-all

• Depends on network spacing

• Depends on error characteristics (especially ionosphere)

• Are many areas of ongoing work– Real-time network ambiguity resolution– Correction transmission schemes– Use of fixed and floating ambiguities

Additional Information

Dissertation and Related Papers:

http:/www.ensu.ucalgary.ca/GPSRes/multiref.html

My e-mail address:

[email protected]

Documents

Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network Maj John Raquet [email protected] Air Force Institute of Technology (and