# GPS Positioning by an Estimation of the Measurement Variance ```GPS Positioning by an Estimation
of the Measurement Variance
Center for Space and Remote Sensing Research
National Central University
320 Jhongli, Taiwan, R.O.C.
1
Outline
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Preface
Mathematical Model
Variance-Component Estimator
Whitening Filter
Experiments
Summary and Outlook
2
Preface (1/2)
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Ranges between several global positioning
system (GPS) satellites and a user can be
measured at a signal receiving epoch.
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The ranges can be processed to determine
the user’s 3D position, in either a static or
on-the-fly mode.
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Basically, the quality of GPS positioning
depends on the range precision, satellite
geometry, etc.
 Preface
Model
LeastSquares
VarianceComponent
Whitening
Tests
Summary
3
Preface (2/2)
 Preface
Model
LeastSquares
VarianceComponent
Whitening

A double-difference combination of the
observations is very often carried out. In a
local area, such a differencing operation
can reduce many errors effectively.
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As far as a resolution of the whole-cycle
ambiguity parameter is concerned, the
estimated covariance matrix plays an
important role.
Tests
Summary
4
Mathematical Model (1/4)
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Provided that unmodeled errors are negligibly
small, the pseudorange ρ (m) and the carrier
phase  (cycles) can be written as:
( ρ  ν ρ )ijgh  Rijgh  0,
 Preface
 Model
LeastSquares
VarianceComponent
Whitening
Tests
Summary
(φ  νφ )ijgh
1 gh
 Rij  N ijgh  0,
λ
where Rijgh  Rig  R gj  Rih  R hj ,
gh
ρij

g
ρi

g
ρj

h
ρi

h
ρj ,
5
Mathematical Model (2/4)
N ijgh  N ig  N gj  N ih  N hj ;
and i, j : reference and user receivers,
 Preface
 Model
LeastSquares
VarianceComponent
Whitening
Tests
Summary
g, h :
λ:
R:
N:
ν ρ , νφ :
reference and other satellites,
the wavelength (m),
Euclidean topocentric distance (m),
the integer ambiguity (cycles),
the stochastic zero-mean pseudorange
and phase errors, respectively.
6
Mathematical Model (3/4)
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By defining a measurement residual vector,
T
v  (ν ρ g , ν ρ g , νφ g , νφ g , ν ρ h , ν ρ h , νφh , νφh ) ,
i
j
i
j
i
j
i
j
and a parameter correction vector,
 Preface
 Model
LeastSquares
VarianceComponent
Whitening
Tests
Summary
x  (dX j , dY j , dZ j , N ijgh )T ,
we get Bv  Ax  l , where B is a linear
coefficient matrix, A the design matrix,
and l a misclosure vector.
7
Mathematical Model (4/4)
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 Preface
 Model
LeastSquares
VarianceComponent
Whitening
Tests
In addition, the covariance matrix of the vvector is a positive definite σ 02 Q matrix,
presumably with σ 02  1.
2
σ0 Q
If the ranges are independent,
is
diagonal, but has different V () variances,
resulting in
σ 02 Q  diag(V ( ρig ), V ( ρ gj ), V (φig ), V (φ gj ),
V ( ρih ), V ( ρ hj ), V (φih ), V (φ hj )).
Summary
8
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Now, let Qx stand for the scaled covariance
matrix of the x-vector, Ql and Qv for the
respective covariance matrices of l and v,
one gets
 Preface
 Model
 LeastSquares
VarianceComponent
Whitening
x  Qx A
1
Ql l ,
T
1
Ql l .
T
v  Qv B
Tests
Summary
9
Variance-Component Estimator (1/5)
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In order to check whether the prior Q is
built correctly, an estimator is derived from
the expected value of the v T Q 1v quadraticform.
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Since zero-mean error is assumed, the
expectation of v is zero, E (v )  0.
 Preface
 Model
 LeastSquares
 VarianceComponent
Whitening
Tests
Summary
10
Variance-Component Estimator (2/5)
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 Preface
 Model
 LeastSquares
T
The expectation of v Q v becomes
1
tr(Qv Q ).
A decomposition of the covariance matrix
into m components is considered,
m
σ 02Q   σ i2C i ,
i 1
 VarianceComponent
Whitening
Tests
Summary
1
where
factor.
2
σ i stands
for an unknown scale
11
Variance-Component Estimator (3/5)
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The four accompanying matrices are:
g
g
C1  diag( V ( ρ1,i ), V ( ρ1, j ),
V ( ρ1h,i ), V ( ρ1h, j ), 0, 0,
0, 0 , 0, 0, 0, 0 ,
0, 0, 0, 0, ...),
 Preface
 Model
g
g
C 2  diag( 0, 0, V ( ρ2 ,i ), V ( ρ2 , j ), 0, 0, 0, 0,
 Variance0, 0, V ( ρ2h,i ), V ( ρ2h, j ), 0, 0, 0, 0, ...),
Component
 LeastSquares
Whitening
Tests
Summary
12
Variance-Component Estimator (4/5)
C 3  diag(0, 0, 0, 0, V (φ1g,i ), V (φ1g, j ), 0, 0,
0, 0, 0, 0,
h
V (φ1,i ),
h
V (φ1, j ),
0, 0, ...),
 Preface
 Model
 LeastSquares
C 4  diag(0, 0, 0, 0,
 VarianceComponent
0, 0, 0, 0, 0, 0,
g
g
0, 0, V (φ2 ,i ), V (φ2 , j ),
h
h
V (φ2 ,i ), V (φ2 , j ), ...).
Whitening
Tests
Summary
13
Variance-Component Estimator (5/5)
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If one-to-one correspondence holds, an
unbiased variance-component estimator can
be defined as follows:
 Preface
 Model
 LeastSquares
 VarianceComponent
T
σˆi2
1
1
v Q Ci Q v

, i   1, 2, ..., m .
T
tr(BCi B Q k )
Whitening
Tests
Summary
14
Whitening Filter (1/5)
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The covariance of an a-vector floatambiguity solution is the Qa matrix.
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A matrix factorization can lead to the
T
expression Qa  UDU , where U is an
upper unit triangular matrix and D is a
diagonal one.
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
Testts
Summary
15
Whitening Filter (2/5)
 Preface
 Model
 LeastSquares
 VarianceComponent
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An intU operation changes every entry of
U to integers.
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The matrix is transformed such that it,
1
T
Q a  (intU ) Qa (intU ) ,
becomes more diagonally dominant.
 Whitening
Tests
Summary
16
Whitening Filter (3/5)
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 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
Tests
Alternatively, a lower triangular L-matrix is
used. If at the k-th iteration, either int U k or
int Lk changes to an identity matrix, then
T 1  (int Lk ) 1 (int U k ) 1 (int Lk 1 ) 1
1
1
1
(int U k 1 ) (int L1 ) (int U1 ) ,
represents a decorrelating transformation
matrix.
Summary
17
Whitening Filter (4/5)
 Preface
 Model
 LeastSquares
 VarianceComponent
1
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The determinant of the T matrix is equal
to one, leading to a volume-preserving
transformation.
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The ambiguity a-vector and its covariance
σ 02 Qa - matrix are mapped into new ones,
1
such that z  T a and Q z  T QaT
1
T
.
 Whitening
Tests
Summary
18
Whitening Filter (5/5)
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 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
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2
σ
z
When centered at , the new 0 Q z hyperellipsoid is aligned with the coordinate axis,
due to diagonal dominance.
Some int ( z ) vectors are chosen and
transformed back, in terms of
1
T int ( z )  TT int (a )  int (a ).
Tests
Summary
19
(1/3)
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 Preface
 Model
A software, called ManGo (Managing
GNSS-data for orientation), has been
developed using the parameter estimators
mentioned above.
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
20
(2/3)
 Preface
 Model
 LeastSquares
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A double-difference ambiguity covariance
matrix before a decorrelating transformation
will be shown.
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The decorrelated ambiguity covariance
matrix is plotted for comparison.
 VarianceComponent
 Whitening
 Tests
Summary
21
(3/3)
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The dual-frequency observations were each
solved for the variances by using the
(unbiased) estimator, on a component-bycomponent basis.
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The scaling (dimensionless) variancecomponent values indicated that they are
time-varying.
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
22
Zero-Baseline Test
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 Preface
 Model
ManGo yields single-epoch position
solutions. Also plotted are the results from
SKI-Pro (the predecessor of LGO, Leica’s
proprietary software).
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
23
Effectiveness in Resolving the
Ambiguity
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 Preface
The effectiveness consists in reducing the
number of some candidate integerambiguity vectors.
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
24
Local-Area GPS Positioning (1/9)
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
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is considered practical in a local area
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Kinematic experiments were conducted. A
was carried at a walking speed around a
track.
Summary
25
Local-Area GPS Positioning (2/9)
 Preface
 Model
 LeastSquares
 VarianceComponent
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A reference receiver was placed at a
geodetic control point, separated from the
track by 2.5 km.
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The cut-off angle was 15º, the sampling
rate at 1-Hz, the average positional DOP
was 3.7.
 Whitening
 Tests
Summary
26
Local-Area GPS Positioning (3/9)
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The trajectory determined by Leica’s LGO
software served as a bench-mark.
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Prior variances were made dependent on
the satellite elevation angle.
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The variances were employed in
Bv  Ax  l to initialize the error
2
covariance matrix σ 0 Q .
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
27
Local-Area GPS Positioning (4/9)
 Preface
 Model
 LeastSquares
 VarianceComponent
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Both kinematic GPS positioning results
and the histograms of the ensemble
measurement residuals are illustrated.
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Next, the estimated pseudorange and
carrier-phase variance components are
displayed for 250 epochs.
 Whitening
 Tests
Summary
28
Local-Area GPS Positioning (5/9)
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 Preface
 Model
 LeastSquares
The figures showed that, with the decorrelating
transformation, the estimation of variance
factors has made an impact on the ambiguity
resolution.
 VarianceComponent
 Whitening
 Tests
Summary
29
Local-Area GPS Positioning (6/9)
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It demonstrated the number of epochs in
each dataset required by ManGo for a
correct ambiguity-fixed positioning.
 Preface
 Model
 LeastSquares
 VarianceComponent
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A distinction between the ambiguity-fixed
and -float processing schemes is made.
 Whitening
 Tests
Summary
30
Local-Area GPS Positioning (7/9)
 Preface
 Model
 LeastSquares
 VarianceComponent
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A simultaneous estimation of the coordinate,
the ambiguity and the ionospheric delay is
often prohibited by high correlation.
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Despite this, an analyst can gain an insight
into the path delay by treating the station
coordinates as constant.
 Whitening
 Tests
Summary
31
κ
κijgh f 2
Local-Area GPS Positioning (8/9)
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The first-order ionospheric path delay (m)
2
gh
can be expressed as κij f , and
κ : a function of the total electron content,
f : the carrier frequency (Hz).
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests

The x vector is arranged as:
gh
T
(κijgh f12 , ..., N1gh
,
N
,
...)
, h  {1, 2, ..., nh }.
,ij
2 ,ij
Summary
32
κ
κijgh f 2
Local-Area GPS Positioning (9/9)
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The parameter estimation and covariancematrix diagonalization remain the same.
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The time series of the estimated doubledifference ionospheric delays are shown.
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It becomes evident that some delays have
a decimeter-level order of magnitude and
reflect no stationary behavior.
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
33
Summary and Outlook (1/2)
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
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A variance-component estimator has been
formed, under the requirement that the
estimator’s expected value should reflect
a true scale.
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The volume-preserving, integer-valued
transformation technique can function as
a powerful tool for ambiguity resolution.
 Tests
 Summary
34
Summary and Outlook (2/2)
 Preface

Single-epoch GPS L1-, L2-carrier phases
and code pseudoranges are usually
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For medium-length baselines, a doubledifference combination can contain
unmodeled errors, which still acts as a
challenge.
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
 Summary
35
36
References

Wu, J. and Hsieh, C.H., 2008. GPS On-TheFly Medium-Length Positioning by an
Estimation of the Measurement Variance,
Journal of the Chinese Institute of Engineers,
Vol. 31, No. 3, pp. 459-468.

Wu, J. and Yeh, T.F., 2005. Single-Epoch
Observables, Navigation, Vol. 52, No. 1, pp.
39-47.
Time and Setting
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 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
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Date: June 22, 2002
Time interval: 16:12 - 16:25
Baseline N091-SPP2: 7.5 km
Sampling rate: 1 Hz
Observed GPS satellites: 7
Dilution of precision (DOP): 3.9 - 5.0
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Values of a single-epoch 1212 ambiguity covariance
matrix before the decorrelation
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Values of a single-epoch 1212 ambiguity covariance
matrix after the decorrelation
Time and Setting
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 Preface
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 Model
 LeastSquares
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 VarianceComponent
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 Whitening
 Tests
Summary
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Date: March 24, 2001
Time interval: 00:36 - 01:04
Baseline N915-SPP1: 1.6 km
Sampling rate: 1 Hz
Observed GPS satellites: 9
DOP: 2.3 - 2.8
Cut-off angle: 15°
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Best linear unbiased estimates of scaling factors for
the L1 and L2 phases
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Best linear unbiased estimates of scaling factors for
the C/A- and P-code pseudoranges
Time and Setting
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 Preface
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 Model
 LeastSquares
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 VarianceComponent
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 Whitening
 Tests
Summary

Date: February 10, 2003
Time interval: 15:47:20 - 16:13:50
Zero-baseline at SPP1
Sampling rate: 0.1 Hz
Observed GPS satellites: 9
DOP: 2.3 - 2.8
Cut-off angle: 15°
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Northings from the single-epoch GPS positioning
of a zero-baseline vector
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Eastings from the single-epoch GPS positioning
of a zero-baseline vector
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Heights from the single-epoch GPS positioning
of a zero-baseline vector
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Relative frequency of change (decreased: 98, no
change: 153, and increased: 44 cases; on a 1.6 km
baseline)
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
The kinematic GPS positioning results with (left) and
without (right) the variance-component estimator
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
The histograms of the residuals with the variancecomponent estimator (left) and without (right)
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
The histograms of the residuals with the variancecomponent estimator (left) and without (right)
ρ1
ρ2
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Variance components versus each 1-s epoch (ensemble
average in sigma: 9.3 and 15.5 cm for the pseudoranges)
φ1
φ2
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Variance components versus each 1-s epoch (ensemble
average in sigma: 0.01 and 0.03 cycles for the carrier
phases)
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Trial-and-error vector sets of the integer-valued phase
ambiguity
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Trial-and-error vector sets of the integer-valued phase
ambiguity
2-epoch
3-epoch
4-epoch
5-epoch
41%
69%
78%
90%
200
300
400
Sets of multi-epoch data
500
600
Percentage in
accumulated success-rate
Number of epochs
5
4
3
2
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
1
0
100
Minimum number of epochs that lead to a determination
of the whole-cycle ambiguity
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
GPS positioning precision and accuracy values in
north-component for the 29.6 km baseline
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
GPS positioning precision and accuracy values in eastcomponent for the 29.6 km baseline
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
GPS precision and accuracy in height for the 29.6 km
baseline
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Ionospheric path delays for the 29.6 km control baseline,
with pseudo-random noise (PRN) 18 acting as a reference
satellite
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
Ionospheric delays for the 29.6 km control baseline
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