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European Navigation Conference (ENC) 2019, Warsaw, Poland, 9-12 April 2019
Detection and De-weighting of Multipath-affected
Measurements in a GPS/Galileo Combined Solution
Ali Pirsiavash, Ali Broumandan, Gérard Lachapelle and Kyle O’Keefe
PLAN Group, Department of Geomatics Engineering
Schulich School of Engineering, University of Calgary
Calgary, AB, Canada
Abstract—Multipath is a major error source for Global
Navigation Satellite Systems (GNSS). Different multipath countermeasure techniques have been investigated in the literature, of which detection and exclusion (or de-weighting) of affected measurements is receiving significant attention. Thanks to the development of multi-GNSS constellation receivers, this approach is increasingly effective as measurement redundancy is now sufficiently large to detect and exclude (or de-weight) faulty measurements prior to being used in the navigation solution. Given the benefits of GNSS measurement monitoring in detecting multiple signal failures, this paper focuses on measurement level monitoring techniques to develop effective mechanisms for detection and de-weighting of the signals distorted by multipath. The combination of GPS and Galileo signals is investigated under height-constrained environments to increase measurement redundancy and observability, and consequently improve the performance of detection and de-weighting solutions. Results obtained for a real static multipath scenario show up to 60 percent horizontal improvement for different positioning approaches.
Keywords: Global Navigation Satellite Systems (GNSS),
Global Positioning System (GPS), Galileo positioning system,
multipath error, measurement quality monitoring, detection and
de-weighting of multipath-affected measurements, measurement
geometry, height-constrained GPS/Galileo combined solution
I. INTRODUCTION
Multipath is the combination of Line-of-Sight (LOS) and
a number of Non-Line-of-Sight (NLOS) signal components
reflected off surroundings before reaching the receiver
antenna. Fig. 1 shows a schematic view of one-ray multipath
resulting from a reflector nearby a Global Navigation Satellite
Systems (GNSS) receiver.
Fig. 1. A schematic view of one-ray multipath
Due to reflections, received signals differ in power, delay,
carrier phase and frequency, angle of arrival and
polarization, but have the same Pseudo-Random Noise
(PRN) codes. The combination of these signals in the
receiver antenna introduces multipath as discussed in the
literature. In multipath conditions, the receiver code tracking
procedure does not properly distinguish the actual peak of
the correlation curve between received and receiver
generated replica signals as required for an ideal code
alignment. This results in pseudorange (code phase) errors in
excess of tens of metres.
The first stage in protecting a receiver against multipath
is to isolate the receiver from multipath interference or
reduce the effect of the latter through antenna design (e.g.
“choke ring” antennas) or use physical or synthetic antenna
arrays. The latter case can be used either for estimating
signal angles of arrival and beamforming or antenna
diversity to mitigate the effect of multipath (e.g. [1]).
Polarization diversity is another multipath mitigation
approach which uses two receiver ports with orthogonal
polarizations to obtain diversity gain [2]. Generally, the
polarization of reflected signals may change depending on
reflector type and multipath channel. Exploiting two
complementary polarizations can provide useful information
about reflected signals to detect and/or mitigate the fading
effect of mismatched signals in the receiver (e.g. [3,4]). The
use of Narrow Correlator (NC) is another important solution
to reduce the effect of multipath. Reference [5] took
advantage of narrow spacing between early and late
correlators to alleviate multipath effects on tracking loops.
Following the concept of narrow correlators, other tracking
strategies such as High Resolution Correlator (HRC)
techniques [6] were developed where more than three
correlators are used in each tracking loop. Reference [7]
investigated a so-called Early-Late-Slope (ELS) technique
which uses correlator pairs on both early and late sides of the
correlation peak to estimate both side slopes and compute a
pseudorange correction metric. Multipath Estimating Delay
Lock Loop (MEDLL) [8,9] is another technique to mitigate
multipath which estimates multipath parameters using a bank
of correlators. Relatively high performance in estimating
LOS and NLOS components can be achieved using these
algorithms, but at the expense of a multi-correlator-based
tracking structure. The “vision correlator” [10] and “vector
tracking” [11] are other techniques proposed for multipath
detection and mitigation.
Multipath monitoring is another approach to detect and
isolate distorted measurements, which can be performed at
the tracking and navigation stages. At the navigation stage,
Receiver Autonomous Integrity Monitoring (RAIM) is the
most common technique [12] developed to detect faulty
measurement in all but the most harsh multipath
environments. Signal Quality Monitoring (SQM) algorithms
have been developed to detect GNSS multipath distortions
by incorporating monitoring correlators at the tracking level.
A review of the SQM techniques for detecting and mitigating
the effect of GNSS multipath has been presented in [13].
Received signal strength and corresponding measures such
as Signal-to-Noise Ratio (SNR) and Carrier-to-Noise
density ratio (C/N0) are also conventional quality metrics to
detect multipath and even mitigate its effect through
stochastic weighting models [14,15]. Combination of code
and carrier phase measurements is another approach for
pseudorange multipath correction [12]. Since the
pseudorange multipath error is considerably larger than that
of the carrier phase, the code-minus-carrier measurement is
mostly an (ambiguous) indication of pseudorange multipath,
which can be used for multipath error correction [16-18].
In addition to the above approaches, three-Dimensional
Building Models (3DBMs) have been investigated to
improve GNSS-based positioning performance in urban
multipath environments. These augmentation techniques
incorporate the geospatial information of nearby reflectors
using a 3D model of buildings and surroundings [19]. This
can be generally performed either through “shadow
matching” algorithms [20,21], modeling and exploiting path
delay of NLOS signals [19] or multipath fault detection and
exclusion [21].
Given the above, GNSS multipath countermeasures can
be generally divided in three major groups. Firstly, some
attempt to isolate the receiver from multipath interference or
minimize its effect by modifying the receiver antenna or
tracking design. The second group tries to jointly estimate
the multipath parameters and subsequently correct multipath
errors or mitigate their effects. The third group focusses on
detection techniques where gross errors caused by multipath
can be specifically reduced or eliminated by detecting and
excluding (or de-weighting) affected measurements. Thanks
to the development of multi-GNSS constellation receivers,
the third approach is increasingly effective as redundancy is
sufficiently large to detect and exclude (or de-weight) faulty
measurements. An effective and practical implementation of
such a solution requires the design and implementation of
appropriate algorithms to detect and de-weight distorted
signals without significant degradation in measurement
geometry, which is also a focus of this paper. In Section 2,
the methodology is described based on Measurement
Quality Monitoring (MQM) techniques in three general
steps. First, a multipath error correction is applied based on
combination of erroneous (but un-ambiguous) code phase
and precise (but ambiguous) carrier phase measurements. If
followed by a filtering approach, it is shown that the output
of such a combination provides a direct measure of the code-
phase multipath error under cycle slip free conditions, which
results in the possibility of partial pseudorange correction.
Second, detection metrics are defined by using partially
corrected measurements to detect remaining multipath errors.
A new iterative algorithm is then developed to de-weight
faulty measurements prior to being used in the navigation
solution. In Section 3, development of a height-aided
GPS/Galileo combined solution is investigated to improve
measurement redundancy and thus detection/de-weighting
performance. Since the performance of measurement-based
detection and de-weighting techniques is limited by
measurement geometry, higher performance is expected
when satellite geometry and redundancy of the
measurements is improved in a multi-constellation system.
This is investigated based on a GPS/Galileo combined
solution. A field test analysis is provided in Section 4 to
examine the performance of the above techniques under a
real multipath scenario. Section 5 deals with conclusions
reached in this research.
II. METHODOLOGY
Fig. 2 shows the procedure of the developed MQM-based
Detection and Iterative De-weighting (D & I-D) of
multipath-affected measurements in a height-constrained
GPS/Galileo combined solution. In order to reduce the
number of de-weighted measurements and thus preserve
measurement geometry, the D & I-D process is preceded by
a Code-Minus-Carrier (CMC)-based multipath correction as
discussed below.
Fig. 2. MQM-based D & I-D in a height-constrained GPS/Galileo combined solution
A. MQM for Multipath Correction
CMC is computed by subtracting the carrier phase
measurements from the corresponding pseudoranges. By
doing so, at each navigation epoch, the CMC metric is
calculated for the lth satellite as
, , , , ,2
l l l
p l ion l l p l l l
CMC p
MP d N MP
(1)
where
lp and
l are the corresponding code and carrier phase
measurements (both converted to units of length),
,ion ld is ionospheric delay,
and lN are wavelength and ambiguity, and
, , ,, ,p l p l lMP MP and
,l are code and carrier multipath
and noise errors [22]. Since integer ambiguities are constant during a cycle slip-
free period and ionosphere changes are minimal during short
periods, their effect are approximately estimated and
removed based on a moving time average sufficiently short
to avoid biases due to changing geometry and prevent the
accumulation of systematic errors. Since cycle slips may
result in a new unknown carrier phase ambiguity, the moving
average buffer is reset if a cycle slip is detected. Therefore,
the pseudorange multipath error can be extracted by the
following CMC-based monitoring metric:
,cmc l l lm CMC CMC (2)
where lCMC denotes the mean value of the CMC metric
computed by the moving average. The output of the CMC
metric is then directly used for pseudorange correction as
,ˆ
l l cmc lp p m (3)
where ˆlp is the corrected pseudorange. Due to the
dependency of the CMC on carrier phase measurements, the
major limitation is the need to restart the time averaging
process in the event of a cycle slip as it requires re-
estimation of the ambiguity [22].
B. MQM for Multipath Detection
For the case of a multi-frequency receiver, a Geometry-
Free (GF) detection metric is formed by differencing
pseudoranges on two frequencies as follows:
1 2 1 2 1 2
,
f f f f f f
GF l l l lm p p d (4)
where 1 2f f
ld includes the difference in the ionospheric
effects and inter-frequency receiver biases. This
combination removes the geometric components of the
measurements, but includes the frequency-dependent effects
(e.g. ionospheric delay), multipath and measurement noise.
1 2f f
ld is estimated and removed through time-averaging;
the remainder is used to monitor code-phase multipath.
Since the GF metric is a combination of errors on two
frequencies, it does not provide separate information about
multipath on each frequency and thus is used only for
satellite-by-satellite detection and de-weighting. The GF
monitoring metric is directly proportional to code phase
multipath errors and from this point of view outperforms the
conventional C/N0-based detection metrics which are more
sensitive to short-delay multipath (fading) with a small
impact on code tracking errors [23,24].
The main advantage of the GF metric is its capability to
be used after the CMC-based error correction as shown in
Fig. 2. Multipath errors are first alleviated by applying CMC-
based error estimation and correction. Although effective in
reducing the effect of multipath, this approach does not fully
remove multipath errors especially where the time averaging
process is restarted in the event of a cycle slip. In the next
step, the GF detection metrics are formed by differencing
(partially) corrected pseudoranges on two or more
frequencies to detect the remaining multipath errors.
To improve detection performance, a fixed-lag sliding
window is used to take a window of N samples (based on
current and N−1 preceding samples) and compare them to a
predefined threshold. If M ( M N ) or more samples exceed
the threshold, then the detection output is 1 and otherwise 0.
This procedure is then repeated for the next window. With
this detection strategy, the overall probability of false alarm
in N trials is given by [12]
1
0
1 1M
N nn
FA fa fa
n
NP P P
n
(5)
where N
n
is the number of combinations of N items taken
n at a time and faP is the false alarm probability in each trial
and equals 0.27% under a normal distribution and three
times Standard Deviation (SD) as the detection threshold. In
the case of ( , ) (1, 1),N M the detection strategy is
considered a general likelihood ratio test by comparing each
sample with the detection threshold at each epoch. By a
proper selection of N and M, the strategy behaves like a
moving average shrinking the probability distribution
variance for the null (when there is no or low multipath) and
the alternate (when multipath exists) hypotheses on the two
sides of the detection threshold. This can improve detection
performance by decreasing the false alarm probability in the
case of the null hypothesis while increasing the probability
of detection in the case of the alternate hypothesis, but at the
expense of latency in the transition from the null to the
alternate hypothesis and vice versa. Given the periodic
nature of GNSS multipath (or even when multipath behaves
more like noise in high dynamic scenarios), a side effect of
this latency is a reduction in the rate of change between the
two detection states. In the case of exclusion or de-
weighting, the lower rate of change has the benefit of
smoothing the resulting positions.
C. Measurement Weighting
Once multipath is detected, the position solution should
be protected from multipath through excluding or de-
weighting the affected measurement(s). The critical point is
the effect of exclusion or de-weighting on measurement
geometry. Poor geometry may magnify the effects of
remaining errors and worsen the ultimate position solution.
To counter the above, a geometry-based iterative de-
weighting approach is developed. In the navigation solution
all measurements are used, but the contribution of distorted
measurements is iteratively reduced. Weighted Dilution of
Precision (DOP) is considered as a geometry monitoring
metric. Since horizontal accuracy is the major concern, the
Horizontal DOP (HDOP) is adopted. At each epoch, the
HDOP is calculated based on the root sum square value of
the east and north variance factors of the estimated position.
When it is assumed that all the measurements have equal
variances and are uncorrelated, the variance factors of the
estimated parameters are shown to be only as a function of
satellite-receiver geometry and thus the calculated DOP is
referred to as “pure DOP”. Under different variances for
different measurements, the position variance factors are also
affected by the relative weights of the measurements and
used to determine a so-called “weighted DOP” [22] which is
considered here. Based on a pre-defined increasing function,
the detected measurements are de-weighted until a tolerable
HDOP threshold is exceeded. Due to practical
considerations, the maximum number of iterations is limited
to a number large enough to verify the appropriateness of the
de-weighting procedure. The remaining measurements (those
not detected as faulty) are stochastically weighted using
constant, elevation or C/N0-based weighting models [22],
developed under low-multipath conditions. The iterative de-
weighting procedure is shown in Fig. 3 where the variance of
measurements detected as faulty is increased to de-weight
these. For each measurement epoch,
- ni is the iteration number and
Maxi denotes the maximum
number of iterations,
- 0HDOP is the HDOP value before iterative de-weighting
of detected measurements,
- HDOPTh is the HDOP threshold,
- dN is the number of detected measurements,
- 1, 2, ..., dl N is the detected measurement index,
- , nl i
w is the weight of the thl detected measurement at the
th
ni iteration,
- 2
, nl i is the variance of the thl detected measurement at
the th
ni iteration,
- 2 2
,0 ,0l l lw is the initial weight of the thl detected
measurement, determined based on the stochastic
weighting model,
- ( )nF i is a pre-defined increasing function of iteration
number, and
- , nid iHDOP is the HDOP value after the th
ni iteration of the
de-weighting procedure applied to the detected
measurements.
Fig. 3. Geometry-based iterative change of measurement weights
Selection of HDOP Threshold and Stochastic Model
Parameters
Limiting the HDOP value to a pre-defined threshold
maintains horizontal geometry above a certain level, but at
the same time, it prevents the system from being thoroughly
isolated from the effect of faulty measurements. The
threshold boundary should be wide enough to ensure the
effectiveness of the de-weighting procedure. Therefore, a
trade-off is involved and the HDOP threshold should be
selected carefully. Under an effective detection process, the
stochastic model for those measurements not detected as
multipath can be modeled under assumed low multipath
conditions.
Selection of De-weighting Function
The main goal of iterative de-weighting is to minimize
the incorporation of detected measurements in the position
solution for a given HDOP threshold. Therefore, while any
linear or non-linear function of iteration numbers can be
used to de-weight detected measurements, a proper function
should take the trade-off between the resolution and
complexity of the de-weighting process into account. The
smaller size of de-weighting factor (at each iteration) will
control geometry degradation with a higher resolution at the
expense of a larger number of iterations for the entire
process and thus a heavier computational burden. Herein,
since the GF metric is a combination of errors on two
frequencies and the relation between detection statistics and
measurement errors is not known, when multipath is
detected, a scale factor is simply multiplied by the iteration
number to define the corresponding measurement variance
factor.
At each measurement epoch, the variance of the lth
measurement in the th
ni iteration is calculated as follows:
2 2
,
if multipath is detected
1 otherwisen
MQM n
l i l
a i
(6)
where 2
l is the initial variance of the corresponding
measurement determined based on the stochastic weighting
model under low multipath conditions, and MQMa is a scale
factor to be set based on the desired de-weighting resolution
and maximum number of iterations. For undetected
measurements, any selection of the conventional stochastic
weighting models such as constant, elevation or C/N0-based
algorithms can be adopted.
III. HEIGHT-CONSTRAINED GPS/GALILEO COMBINED
SOLUTION
A formulation is now provided to combine GPS and
Galileo signals in a pseudorange-based Least-squares (LS)
solution and constrain the height and hence improve the
measurement redundancy and the performance of the
detection and de-weighting algorithms. The height can be
constrained for a short or long duration depending on the
stability of the altimeter used; a correction for the geoid is
also applied to transform the altimetric height [above sea
level] to that above the reference ellipsoid selected. The
above procedure is done in three steps as follows.
A. LS Adjustment for a Pseudorange-based Positioning in
Local Coordinate System
In a single constellation solution, the state vector includes
receiver’s three-dimensional (3D) position parameters and
receiver clock offset typically defined in the Cartesian Earth-
Centered Earth-Fixed (ECEF) coordinate system as
, , ,T
r r rx y z cdTx or equivalently in curvelinear
coordinate system as latitude, longitude and height (plus
receiver clock offset) noted by , , ,T
r r rh cdT x . By
this definition, the distance between receiver and lth
satellite
( 1, 2, ...,l L with L as the number of satellites tracked)
can be defined as
,
sl
l r rx x x (7a)
,
sl
l r ry y y (7b)
,
sl
l r rz z z (7c)
where , ,sl sl slx y z is the Cartesian position of the lth
satellite. To ultimately constrain the height, the XYZ
distance parameters are rotated into local East, North and
Up (i.e. vertical or height) or ENU system as
, ,
, ,
, ,
l r l r
l r rot l r
l r l r
e x
n y
u z
R (8)
where rotR is the rotation matrix and is determined as [25]
sin( ) cos( ) 0
sin( )cos( ) sin( )sin( ) cos( )
cos( )cos( ) cos( )sin( ) sin( )
r r
rot r r r r r
r r r r r
R (9)
By these definitions, at each measurement epoch, with L
satellites tracked, the relation between pseudorange
measurements and receiver-satellite geometry is defined as
2 2 2
1, 1, 1,
,11
2 2 2
,22 2, 2, 2,
,2 2 2
, , ,
( )
( )
r r r
p
pr r r
p LL
L r L r L r
h
e n up
p e n ucdT k
pe n u
p e
x
(10)
where lp and
,p l denote the corresponding pseudorange
measurement and pseudorange error with respect to the lth
satellite. By linearizing (10) with respect to an initial
estimate of the state vector [i.e.
0 0 0 0 0 0 0 0 0, , , , , , ,T T
r r r r r rx y z cdT h cdT x one
obtains
P H x e (11)
where
P is a 1L vector whose lth
row element is defined as
, 0
2 2 2
, 0 , 0 , 0
l r
l l l r l r l rp p e n u
(12)
0( )h
H x xx
is a 4L matrix whose lth
row element
( 1, 2, ...,l L ) in each column is defined as
, 0
,1 , 0 , 0
, 0
cos( )sin( )l r
l l r l r
l r
eh el az
(13a)
, 0
,2 , 0 , 0
, 0
cos( )cos( )l r
l l r l r
l r
nh el az
(13b)
, 0
,3 , 0
, 0
sin( )l r
l l r
l r
hh el
(13c)
,4 1lh (13d)
and ( , , , )Te n h cdT x is the error between actual
and initial estimate of the state vector to be estimated in
the ENU coordinate system.
By doing so, the LS-based estimation of the state error is
obtained by [22]
1
ˆ T T
x H WH H W p (14)
where W is a weight matrix determined based on the
stochastic model of measurement variances. Through ˆx ,
the initial estimates can be refined and the entire procedure
can be iterated until the state error converges to zero (e.g.
sub-metre level).
B. GPS-Galileo Combined Solution
A multi-constellation solution require consideration of
the differences in ephemeris data, coordinate frames and
reference time frames. Herein, this is done under a
GPS/Galileo combined solution where similarity of
ephemeris data and coordinate frames facilitates the
implementation of the combined solution.
Satellite Position and Clock Corrections
The first step in a position solution is to extract satellite
orbital information and satellite clock corrections provided
by ephemeris data. This requires design and implementation
of proper algorithms to decode ephemeris data for GPS and
Galileo satellites.
Coordinate Frames
GPS uses the World Geodetic System 1984 (WGS-84)
and Galileo uses the Galileo Terrestrial Reference System
(GTRF), both of which are different realizations of the
International Terrestrial Reference System (ITRS).
However, the offset between WGS-84 and GTRF is at the
sub-metre level, which is negligible for the present analysis
and thus the coordinate transformation is neglected [26].
Reference Time Frames
GPS and Galileo use different time frames namely
Galileo System Time (GST) and GPS use GPS Time
(GPST). The receiver clock offsets need to be estimated
along with position parameters. The receiver clock offsets
are common for all corresponding pseudoranges and hence
can be estimated as an unknown parameter through the
navigation solution. Therefore, in a GPS/Galileo combined
solution, there will be two different clock offsets with
respect to each timeframe which can be estimated along
with three position parameters. This will increase the
number of unknowns from 4 to 5 but with the advantage of
improved geometry and a higher number of available
measurements provided by different GPS and Galileo
satellites [26]. Under this scenario, with GPSL GPS and GALL
Galileo measurements and using corresponding indices, (11)
is rewritten as
1 1
1 1
GPS GPS
GAL GAL
GPS GPSL L
GAL GALL LGPS
GAL
e
n
h
cdT
cdT
P eH
P e (15)
and H will be a 5GPS GALL L matrix whose lth
row
element in each column is defined as follows
,1 , 0 , 0cos( )sin( ) for 1, ...,l l r l r GPS GALh el az l L L (16a)
,2 , 0 , 0cos( )cos( ) for 1, ...,l l r l r GPS GALh el az l L L (16b)
,3 , 0sin( ) for 1, ...,l l r GPS GALh el l L L (16c)
,4
1 for 1, ...,
0 for 1, ...,
GPS
l
GPS GPS GAL
l Lh
l L L L
(16d)
,5
0 for 1, ...,
1 for 1, ...,
GPS
l
GPS GPS GAL
l Lh
l L L L
(16e)
C. Height-constrained GPS/Galileo Solution
In order to add redundancy, one solution is to constrain
the receiver height when the latter is obtained independently.
When height information is available, the position solution is
reduced to a two-dimensional horizontal solution with
improved measurement redundancy and higher performance
for de-weighting of faulty measurements. In this scenario,
(15) is rewritten as
1 1
1 1
GPS GPS
GAL GAL
GPS GPSL L
GPSGAL GALL L
GAL
e
n
cdT
cdT
P eH
P e (17)
and H will be a 4GPS GALL L matrix whose lth
row
element in each column is defined as follows:
,1 , 0 , 0cos( )sin( ) for 1, ...,l l r l r GPS GALh el az l L L (18a)
,2 , 0 , 0cos( )cos( ) for 1, ...,l l r l r GPS GALh el az l L L (18b)
,3
1 for 1, ...,
0 for 1,...,
GPS
l
GPS GPS GAL
l Lh
l L L L
(18c)
,4
0 for 1, ...,
1 for 1, ...,
GPS
l
GPS GPS GAL
l Lh
l L L L
(18d)
IV. FIELD DATA ANALYSIS
A field test was performed using GPS and Galileo signals
under a static multipath scenario. A Trimble R10 GNSS
receiver was used in a high multipath environment to collect
GPS L1 (L1 C/A)/L2 (L2C M+L)/L5 and Galileo E1 (E1
B+C)/E5a/E5b signals with a 1 sample/second sampling rate.
Fig. 4 shows the test site location near reflective surfaces and
the satellite geometry (using a 5 degree elevation mask) at
the beginning of the test.
(a)
(b)
Fig. 4. Multipath data collection; (a) a Trimble R10 antenna-receiver
surrounded by reflectors; (b) Satellite geometry with a 5 degree of elevation
mask
A. Monitoring Results
Time-averaging was used to estimate the mean value of
the different monitoring metrics. A simple moving average
was used with a length of 10 minutes (min) selected based
on twice the average period observed for “quasi-periodic”
oscillations of PRNs exhibiting static multipath. Fig. 5
shows the CMC-based monitoring metrics and their
corresponding Root Mean Square (RMS) values [in metre
(m)] for a sample of GPS signals (PRN 09) with different
frequencies. For PRN 09, it is shown that multipath is
relatively low in the first half of the data while it is higher
during the rest of the test on all three frequencies. Besides
multipath characterization, the CMC-based monitoring
metric is used to correct the corresponding pseudoranges
according to (3).
In addition to affecting code and carrier phase
measurements (and thus CMC metric), multipath affects the
measured signal power and C/N0 values as shown in Fig. 6
[in decibel-Hertz (dB-Hz)]. The satellite elevation angle
change is also shown on the right vertical axis [in degree
(deg)]. The power of the combined signal (and consequently
the measured C/N0) fluctuates with time because it is
affected by the time-varying phase lag between the direct
and reflected signal(s), as these add constructively or
destructively with each other. Since the phase lag is
frequency dependent, the C/N0 is affected differently by
different frequencies.
Fig. 5. CMC-based monitoring metric for GPS L1 (blue), L2 (red) and L5
(green); PRN 09 affected by different levels of multipath error
Fig. 6. C/N0 measurements for PRN 09, GPS L1 (blue), L2 (red) and L5
(green)
As discussed in Section 2B, for detection purposes, GF
measurements were first obtained by differencing
pseudorange measurements on different pairs of the
available frequencies. At this stage, the pseudorange
measurements used to form GF detection metrics are still
not corrected by CMC to examine the detection performance
of the GF metric under multipath occurrence. The
corresponding mean values are then filtered (by the same
moving average discussed before) and resulting outputs are
normalized based on an estimation of the noise variance
calibrated under low-multipath conditions. For the resulting
unitless monitoring metric, the detection thresholds were set
to ±3 as three times the normalized nominal SD for GPS L1,
L2 and L5 signals. Referring to Section 2C, the N value was
set equal to half of the moving average length (i.e. 5 minutes
or 300 samples of measurements with the input rate of 1
sample/second) and M was set to 10 to satisfy the false
alarm probability of 81 .41 10 under multipath-free
conditions and normal distribution of the detection metric
[see (5)]. Fig. 7 shows the normalized GF-based monitoring
metrics, threshold excess and corresponding M of N
detection results. It is observed that when multipath error
exceeds 3 m, it is effectively detected most of the time. The
GF-based detection performance is limited to the PRNs with
signals on at least two different frequencies. For example,
when detection of multipath on GPS L1 is desired, the
multipath errors (not corrected by CMC) without
corresponding L2 and/or L5 measurements will remain
undetected.
(a)
(b)
Fig. 7. (a) Normalized GF-based detection metric and (b) detection outputs
for GPS L1/L2/L5 signals; PRN 09
Fig. 8 to Fig. 10 show corresponding results for Galileo
PRN 30 where the constructive and destructive effects of
multipath are observed on each frequency with different
phase lags. It is also observed that there are intervals in
which the detection metrics are within the detection
boundaries but multipath is detected. This is a result of the
M of N latency discussed in Section 2B. Given the periodic
nature of GNSS multipath, a side effect of the M of N
latency is to smooth detection and ultimately position results
as shown in the next sub-section.
Fig. 8. CMC-based monitoring metric for Galileo E1 (blue), E5a (red) and
E5b (green); PRN 30 affected by different levels of multipath error
Fig. 9. C/N0 measurements for Galileo PRN 30; Galileo E1 (blue), E5a (red) and E5b (green)
(a)
(b)
Fig. 10. (a) Normalized GF-based detection metric and (b) detection outputs
for Galileo PRN 30 E1/E5a/E5b signals
B. Positioning Performance
Resulting height-constrained GPS/Galileo pseudorange-
based LS solutions are now discussed. First, the GPS
L1/Galileo E1 [both with 1575.42 Megahertz (MHz) carrier
frequency] combined solution was obtained under a constant
weighting scheme. Since all satellites do not broadcast
signals on all frequencies at this time, single frequency
positioning was considered, whose accuracy is sufficient for
the current analysis. The effect of the ionosphere on
positions during the test was evaluated separately and was
below 1 m. For monitoring purposes, the GF-based detection
metrics were extracted on as many PRN (with signals on at
least two frequencies) as possible.
The performance of the position solution with no
correction and no measurement de-weighting (A0) was
compared to the case when the GF-based D & I-D was
applied (A1). For GPS L1/Galileo E1 positioning (when GPS
and Galileo signals are monitored on all three frequencies,
but only GPS L1 and Galileo E1 pseudoranges are used for
positioning), GF-based results used for multipath detection
include those provided by the GPS L1-L2 and L1-L5, and
Galileo E1-E5a and E1-E5b which involve either L1 or E1 in
their definitions. For GPS L1 measurements used in the
navigation solution, multipath is detected if at least one of
the corresponding L1-L2 or L1-L5 M of N detection outputs
is one; for Galileo E1 measurements, the occurrence of
multipath is considered when either E1-E5a or E1-E5b GF
metric (or both) detects multipath. A similar argument is
used for GPS L2/Galileo E5b and GPS L5/Galileo E5a
positioning solutions which will be investigated later in this
section. Referring to Section 2C, an empirically tested value
of 5 was chosen as an appropriate weighted HDOP threshold
to avoid geometrically poor solutions that would likely not
benefit from measurement de-weighting. With the maximum
number of iteration equal to 10, scale factor is empirically set
to 10 [see (6)] to perform an effective and relatively low
complexity de-weighting procedure. The performance of the
CMC-based error correction (A2) and its combination with
the GF-based D & I-D (A3) were investigated next.
Fig. 11 shows the number of satellites tracked along with
the weighted HDOP values for a constant weight LS solution
and the positioning approaches discussed. It is observed that
while iterative de-weighting of detected measurements
degrades the HDOP values (under the defined HDOP
threshold), the preceding CMC-based correction can reduce
the level of degradation. Horizontal position errors have been
presented in Fig. 12 where the complementary combination
of CMC-based error correction and GF-based D & I-D
shows improvement over each single monitoring approach.
Fig. 11. Number of satellites tracked and HDOP values for a constant
weight LS solution; A0 (red): no correction, no de-weighting; A1 (blue):
with D & I-D; A2 (same as A0 ploted in red): with CMC-based error correction with no de-weighting; A4 (light green): combination of a
preceding CMC-based error correction followed by the GF-based D & I-D
Fig. 12. Horizontal position errors for constant weighting and GPS L1 &
Galileo E1 combined solution; A0 (red): no correction, no de-weighting
(benchmark); A1 (blue): with D & I-D; A3 (dark green): with CMC-based error correction; A4 (light green): combination of a preceding CMC-based
error correction followed by the GF-based D & I-D
Table 1 shows the numerical results for GPS L1/Galileo
E1 (1575.42 MHz carrier frequency), GPS L2/Galileo E5b
(1227.60 and 1207.14 MHz carrier frequency) and GPS
L5/Galileo E5a (1176.45 MHz carrier frequency)
positioning solutions. Position errors are compared for
different scenarios including constant, elevation and C/N0-
based stochastic models and for different monitoring
approaches from A0 to A3. It is generally shown that while
improvement is limited (and not always the case) for each
single MQM approach (A1 and A2), the combination of the
CMC-based error correction followed by the GF-based D &
I-D (A3) generally shows higher performance. Comparing
horizontal RMS errors for the GPS L1/Galileo E1 solution,
A3 shows 59, 16 and 19 percent improvement (over A0 as
the benchmark) when conventional constant, elevation and
C/N0-based stochastic models are used in the position
solution. These values for the GPS L2/Galileo E5b solution
are 20, 12 and 10 percent and for GPS L5/Galileo E5a
solution, are 51, 27 and 28 percent under the designed
multipath test scenario.
TABLE I. HORIZONTAL PERFORMANCE FOR DIFFERENT POSITIONING
APPROACHES
Combined
Signals
Weighting
Model MQM
Horizontal
RMS Error
(m)
Improvement
A3 Over A0
GPS L1 &
GalileoE1
(1575.42
MHz)
Constant
A0 3.24
59% A1 1.86
A2 2.85
A3 1.33
Elevation-
based
A0 1.24
16% A1 1.78
A2 1.04
A3 1.04
C/N0-based
A0 1.75
19% A1 1.92
A2 1.68
A3 1.42
GPS L2 &
Galileo E5b
(1227.60
and 1207.14
MHz)
Constant
A0 2.66
20% A1 4.27
A2 2.17
A3 2.12
Elevation-
based
A0 2.98
12% A1 5.76
A2 2.52
A3 2.62
C/N0-based
A0 2.84
10% A1 4.47
A2 2.30
A3 2.57
GPS L5 &
Galileo E5a
(1176.45
MHz)
Constant
A0 5.10
51% A1 3.69
A2 7.02
A3 2.52
Elevation-based
A0 2.41
27% A1 2.89
A2 2.80
A3 1.75
C/N0-based
A0 2.76
28% A1 2.94
A2 4.01
A3 1.99
A0: No correction, No De-weighting (Benchmark)
A1: GF-based D & I-D
A2: CMC-based error correction
A3: CMC-based error correction and GF-based D & I-D (Combined
Method)
V. CONCLUSIONS
For the case of a dual-frequency receiver, a GF-based
detection metric was investigated herein, given its capability
to be combined with a preceding CMC-based error
correction. GPS and Galileo measurements were combined
with the addition of a height-constrained horizontal solution
to improve redundancy and consequently, the performance
of de-weighting faulty measurements. While de-weighting
of faulty measurements is challenging due to its effect on
satellite geometry, a geometry-based iterative de-weighting
algorithm was used to reduce the effect of multiple
multipath signals below a tolerable HDOP threshold.
Analysis of Galileo and GPS data showed that while each
monitoring technique has its own limitations, a
complementary combination of them can be used for
reliable multipath mitigation. Multipath errors were first
alleviated by applying CMC-based error corrections on
pseudorange measurements and then the GF-based detection
metrics were performed by differencing partially corrected
pseudoranges on each pair of available frequencies to detect
and de-weight remaining code multipath errors. Field data
results obtained for a static multipath scenario showed 10 to
almost 60 percent improvement in horizontal positioning
performance.
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