GPS Positioning by an Estimation of the Measurement Variance

GPS Positioning by an Estimation of the Measurement Variance
GPS Positioning by an Estimation
of the Measurement Variance
Satellite Surveying and Navigation Laboratory
Center for Space and Remote Sensing Research
National Central University
320 Jhongli, Taiwan, R.O.C.
1
Outline







Preface
Mathematical Model
Least-Squares Adjustment
Variance-Component Estimator
Whitening Filter
Experiments
Summary and Outlook
2
Preface (1/2)

Ranges between several global positioning
system (GPS) satellites and a user can be
measured at a signal receiving epoch.

The ranges can be processed to determine
the user’s 3D position, in either a static or
on-the-fly mode.

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.

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)

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)

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)


 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
Least-Squares Adjustment

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)

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.

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)


 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)

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)

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)

The covariance of an a-vector floatambiguity solution is the Qa matrix.

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

An intU operation changes every entry of
U to integers.

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)

 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

The determinant of the T matrix is equal
to one, leading to a volume-preserving
transformation.

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)

 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening

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
Single-Epoch Weighting Adjustment
(1/3)

 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
Single-Epoch Weighting Adjustment
(2/3)
 Preface
 Model
 LeastSquares

A double-difference ambiguity covariance
matrix before a decorrelating transformation
will be shown.

The decorrelated ambiguity covariance
matrix is plotted for comparison.
 VarianceComponent
 Whitening
 Tests
Summary
21
Single-Epoch Weighting Adjustment
(3/3)

The dual-frequency observations were each
solved for the variances by using the
(unbiased) estimator, on a component-bycomponent basis.

The scaling (dimensionless) variancecomponent values indicated that they are
time-varying.
 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary
22
Zero-Baseline Test

 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

 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

For ManGo, a radius of about 10 km
is considered practical in a local area
surrounding a fixed reference receiver.

Kinematic experiments were conducted. A
dual-frequency Leica SR530 GPS receiver
was carried at a walking speed around a
track.
Summary
25
Local-Area GPS Positioning (2/9)
 Preface
 Model
 LeastSquares
 VarianceComponent

A reference receiver was placed at a
geodetic control point, separated from the
track by 2.5 km.

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)

The trajectory determined by Leica’s LGO
software served as a bench-mark.

Prior variances were made dependent on
the satellite elevation angle.

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

Both kinematic GPS positioning results
and the histograms of the ensemble
measurement residuals are illustrated.

Next, the estimated pseudorange and
carrier-phase variance components are
displayed for 250 epochs.
 Whitening
 Tests
Summary
28
Local-Area GPS Positioning (5/9)

 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)

It demonstrated the number of epochs in
each dataset required by ManGo for a
correct ambiguity-fixed positioning.
 Preface
 Model
 LeastSquares
 VarianceComponent

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

A simultaneous estimation of the coordinate,
the ambiguity and the ionospheric delay is
often prohibited by high correlation.

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)

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)

The parameter estimation and covariancematrix diagonalization remain the same.

The time series of the estimated doubledifference ionospheric delays are shown.

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

A variance-component estimator has been
formed, under the requirement that the
estimator’s expected value should reflect
a true scale.

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
adequate for a short-baseline solution.

For medium-length baselines, a doubledifference combination can contain
unmodeled errors, which still acts as a
challenge.
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
 Summary
35
Thank You for Your Attention
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
Weighting Adjustment of GPS Phase
Observables, Navigation, Vol. 52, No. 1, pp.
39-47.
Time and Setting




 Preface
 Model
 LeastSquares
 VarianceComponent
 Whitening
 Tests
Summary




Date: June 22, 2002
Time interval: 16:12 - 16:25
Baseline N091-SPP2: 7.5 km
Receiver(s): Leica SR530
Sampling rate: 1 Hz
Observed GPS satellites: 7
Dilution of precision (DOP): 3.9 - 5.0
Cut-off (elevation mask) angle: 15°
 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




 Preface

 Model
 LeastSquares

 VarianceComponent

 Whitening
 Tests
Summary

Date: March 24, 2001
Time interval: 00:36 - 01:04
Baseline N915-SPP1: 1.6 km
Receiver(s): Leica SR530
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




 Preface

 Model
 LeastSquares

 VarianceComponent

 Whitening
 Tests
Summary

Date: February 10, 2003
Time interval: 15:47:20 - 16:13:50
Zero-baseline at SPP1
Receiver(s): Leica SR530
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
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertising