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Estimation of DGPS Carrier-Phase Errors Using a Reference Receiver Network
Maj John Raquet
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
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
Reference Receiver Network Motivation (2/3)
• 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)
Desired Coverage Area
Reference Receiver Network Motivation (3/3)
• 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)
Desired Coverage Area
Phase Measurements
• Measurement with errors
• Double-differencing
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1
ambiguity integer
multipath phase
noise meas phase
errorc ionospheri
erroric tropospher
bias clock satellite
bias clock receiver
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abababababab
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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-
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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
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
Setting Up the Problem
x
ComputationPoint
Sample 5-receiver network:
Ref 1
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Ref 3
Ref 4
Ref 5
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Setting Up the ProblemMeasurement errors:
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edUncorrelat Position)SV Tropo, (Iono,
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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
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
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1YYx
,
,
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then,if or, EE
EE
nll
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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!
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nncp
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,ˆ
nnnTnln
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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
Implementation Approach
n
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2
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2
1
n
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l
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l
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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
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2
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2
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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
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
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
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
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
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
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)
Distance To Nearest Reference Receiver (km)
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
Distance To Nearest Reference Receiver (km)
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
Distance To Nearest Reference Receiver (km)
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
Distance to Nearest Reference Receiver (km)
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
)
Distance to Nearest Reference Receiver (km)
Raw Code, Raw WL PhaseRaw Code, NetAdjust WL PhaseNetAdjust Code, NetAdjust WL Phase
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
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)
Distance to Nearest Reference Receiver (km)
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)
Distance to Nearest Reference Receiver (km)
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
Original - 17.1% 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
Repositioned - 30.7% Coverage
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
Original - 17.1% Coverage
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
7 SVs - 30.7% Coverage
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
8 SVs - 46.2% Coverage
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
Original - 30.7% Coverage
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
Repositioned - 100.0% Coverage
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)
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
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: