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Page 1 of 29
August 2019
REPORT ON
GNSS Signal and Measurement Quality Monitoring
for Multipath Detection and Mitigation
Dr. Ali Pirsiavash*, Postdoctoral Associate
PLAN Group, Geomatics Engineering, University of Calgary
* Correspondence: [email protected]
Summary: Receiver level Global Navigation Satellite Systems (GNSS) Signal and Measurement
Quality Monitoring (SQM and MQM) to detect and de-weight measurements distorted by multipath
are investigated. SQM and MQM monitoring metrics are defined at the tracking and measurement
levels of the receiver; a new geometry-based de-weighting technique is developed. Following an
analytical discussion of the sensitivity and effectiveness of the metrics, field data analysis is provided
for static and kinematic modes to verify practical performance. Results obtained for the designed
SQM and MQM-based detection metrics show reliable performance of 3 to 5 m Minimum Detectable
Multipath Error (MDME). Although limited by multipath characteristics and measurement
geometry, when detected faulty measurements are de-weighted, positioning performance improves
by up to 53% for different multipath scenarios.
NOTE: This work consists of a part and an extension of Dr. Pirsiavash’s doctoral thesis completed in January 2019 on Receiver-level Signal and Measurement Quality Monitoring for Reliable GNSS-based Navigation
1. Introduction
Multipath (MP) is a significant source of error in Global Navigation Satellite System (GNSS)-based
navigation. Mitigation techniques can be generally divided into three major groups. Firstly, isolation
of the receiver from multipath interference or minimization of the effect by modifying the antenna (e.g.
“choke ring” antennas or using physical or synthetic antenna arrays [1]) or tracking design (e.g. the use
of Narrow Correlator (NC) [2] or High Resolution Correlator (HRC) [3] techniques). The second group
attempts to jointly estimate the multipath parameters and subsequently correct multipath errors or
mitigate their effects (e.g. Multipath Estimating Delay Lock Loop (MEDLL) techniques [4]). The third
group is based on multipath monitoring 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, this approach is increasingly effective as the
number of available measurements is sufficiently large to exclude or de-weight distorted ones; this is
the focus of this paper.
Figure 1 shows a high-level architecture of a generic receiver broadly divided into three stages: 1.
pre-despreading Radio Frequency (RF) front-end where received signals are first down-converted and
digitized, 2. Intermediate Frequency (IF) signal processing where digital signals are acquired and
tracked to extract range and range-rate information and 3. the navigation solution where position and
timing data is estimated. In the first stage, the output of the RF front-end can be monitored, although
challenging when signals are still buried under the noise floor. At the navigation stage, Receiver
Autonomous Integrity Monitoring (RAIM) has been the conventional technique since the mid-1980s
[5]. Although effective for maritime and aviation applications, poor geometry and multiplicity of error
sources make RAIM sub-optimal for land users, particularly in urban areas with high multipath.
Page 2 of 29
Multiple faulty measurements and poor geometry may result in large position (also velocity and
timing) errors, “masking” of existing outliers [6] and/or even “swamping” measurements less affected
by multipath [7] when statistical tests are performed at the navigation stage [8]. A similar argument
may also be applied to residual-based robust estimation techniques, which attempt to make the
navigation solution less sensitive to the effect of outliers by minimizing the L1-norm of the residuals
rather than the L2-norm, for example [9]. Antenna polarization and spatial diversity (e.g. [10,11]) are
also used to detect and mitigate the effect of multipath. These methods rely on different polarization
and arrival angles of the reflected signals and thus require special hardware (antenna) considerations
and modifications. Other solutions include three-Dimensional Building Models (3DBMs) in urban
environments (e.g. [12,13]) which require additional geospatial information about nearby reflectors.
Monitoring the quality of signals at the Intermediate Frequency (IF) signal processing stage allows
each PRN to be independently monitored, providing the capability of detecting multiple signal
failures. By incorporating monitoring correlators at the tracking level, different metrics can be defined
to monitor the correlation between the received signal and receiver-generated replica and issue alarms
when the correlation is distorted [e.g. 14-17]. Besides the correlation peak monitoring, signal strength
and corresponding measures such as Carrier-to-Noise density ratio (C/N0), code, carrier phase and
Doppler measurements can be used to detect the effect of multipath [e.g. 18-20]. Despite all these
investigations on how to detect multipath, one question still needs to be answered, namely how to use
detection results to improve ultimate positioning performance? An effective and throughout
multipath mitigation approach requires appropriate mechanisms to exclude or de-weight distorted
signals without significant degradation in measurement geometry. This is even more challenging in
harsh scenarios where multipath is detected simultaneously on multiple measurements. Exclusion or
de-weighting measurements under poor geometry and multiplicity of multipath errors may magnify
the effect of remaining errors and ultimately degrades navigation solution performance. Such
monitoring techniques to counter the above are developed herein and consist of two general steps,
namely Signal and Measurement Quality Monitoring (SQM and MQM) for multipath detection, and
geometry-based de-weighting of the detected faulty measurements. These steps are shown in Figure 1
in blue.
Figure 1. High-level architecture of a generic GNSS receiver upgraded by SQM and MQM units
Page 3 of 29
The term SQM is specifically predicated upon monitoring techniques developed at the tracking
level to check the distortion of the correlation peaks. Tracking outputs are used to extract range and
range rate measurements where MQM techniques are applied to monitor multipath-induced
measurement distortions. In addition to the analytical evaluation and practical implementation of the
different monitoring techniques, the main contributions of this paper are (a) development of effective
techniques to monitor the quality of GNSS signals at the tracking and measurement levels for
multipath detection and (b) development of new geometry-based de-weighting approaches to address
multiple simultaneous multipath scenarios. In addition to the analytical discussions and practical test
results, the underlying goal is to introduce the methodology of multipath monitoring at the receiver
signal processing stage. In Section 2, SQM techniques are investigated by incorporating early-late
monitoring correlators to define a symmetric zero-mean SQM metric at the tracking level. The
analytical discussion includes Binary Phase Shift Keying (BPSK) and Binary Offset Carrier (BOC)
modulations as the base signaling schemes used for satellite-based navigation approaches. The NC
and HRC tracking strategies are investigated as they are commonly used in many receivers to mitigate
multipath. MQM approaches are investigated in Section 3 where a combination of
Code-Minus-Carrier phase (CMC)-based error correction and a Geometry-Free (GF)-based detection
metric is investigated for a multi-frequency receiver. Detection and de-weighting strategies are
presented in Section 4. Along with an implementation of a fixed interval detection strategy named as
M of N technique, a new geometry-based SQM-based iterative change of measurement weights is
developed to de-weight distorted measurements and improve ultimate positioning performance.
Based on the combination of different techniques, the proposed SQM and MQM-based solutions are
presented. Data analysis is performed in Section 5 to evaluate the performance of the solutions under
real static and kinematic multipath scenarios. Section 6 includes a summary of the results and
conclusions.
2. Signal Quality Monitoring
2.1. Signal Model
In a GNSS receiver, the received IF signal can be modeled as a combination of digitized signals
corresponding to different Pseudo-Random Noise (PRN) codes. Assuming that received signal
parameters of each satellite including signal power, code delay, carrier phase and Doppler shift remain
unchanged during a coherent time period, the digitized IF signal received at epoch s
nT can be
modeled as
, ,(2 ( ) )
, , ,1
( ) ( ) ( ) ( )IF l k s l k
Lj f f nT
s l k l s l k l s l k fe sl
r nT C b nT c nT e nT
(1)
where
n is the sampling number,
sT is the sampling time interval,
L is the number of satellites in view,
k is the coherent interval index such that for 1,2,...k the corresponding interval is defined as
1coh coh
k N n kN , coh
N being the number of samples in each coherent interval,
,l k
C is the power of the signal received from the lth satellite during the kth coherent interval [the
received signal power is affected by transmission power, transmitter and receiver antenna gains,
Free-Space Path Loss (FSPL), atmospheric attenuation and depolarization loss],
, ,
andl k l k
f are the lth signal code delay and Doppler shift during the kth coherent interval
introduced by the communication channel and ,l k
is the corresponding carrier phase,
Page 4 of 29
IFf is the IF frequency and
( )fe s
nT is the front-end complex noise at time epoch snT .
At each receiver channel, a reference correlator multiplies the received signal by the corresponding
PRN code and carrier replica, and the samples pass through an Integration and Dump (I & D) filter
over each coherent interval. By doing so, the output of the lth receiver channel at the kth coherent
integration epoch (time instant coh s
kN T ) is given by
, ,
1 ˆ ˆ2
,( 1)
1ˆ( ) ( ) ( )
cohIF l k s l k
coh
kNj f f nT
l coh s s l s l kn k Ncoh
y kN T r nT c nT eN
(2)
where , , ,
ˆ ˆˆ , andl k l k l k
f are the code delay, carrier Doppler frequency and phase of the replica
generated by the lth reference correlator during the kth coherent integration interval. Assuming that the
binary data is also constant during each integration period, Equation 2 can be rewritten as [21]
, ,2 1 1,
, , ,
,
, , , ,
sin( )( ) ( ) ( )
sin( )
( , , )
l k coh s l kj f k N Tl k coh s
l coh s l k l k l k l coh s
coh l k s
l k l k l k l k
f N Ty kN T C b R e kN T
N f T
y f
(3)
where
,l k
b is the binary navigation data corresponding to the lth signal over the kth coherent integration
period,
, , ,
ˆ ,l k l k l k , , ,
ˆl k l k l k
f f f and , , ,
ˆl k l k l k
are code, frequency and phase offsets
between the lth received and generated replica signals at the kth integration epoch,
coh s
N T is equal to the coherent integration time also noted by I
T ,
( )l coh s
kN T consists of noise and residual cross correlation terms of the lth receiver channel at the
kth integration epoch, with approximately zero-mean Gaussian In-phase and Quadrature-phase (I/Q)
components,
( )R
is the code correlation function which is related to the choice of the signaling scheme
[15,16].
2.2. SQM Metrics
Referring to Equation 3, when tracking loops are locked and it is assumed that there are no
tracking code and phase offsets, the in-phase output of the ith early or late correlator of the lth receiver
channel at the kth coherent integration epoch can be defined in the code delay domain as follows:
, , , , , , ,,0,0
i i
I
l k u l k i c l k l k i c l k uI Re y uT C b R uT (4)
where
c
T is the chip duration
the absolute value of i c
u T denotes the spacing of the ith early (for 0i
u ) or late (for 0i
u )
correlator from the reference prompt correlator,
, , i
I
l k u is the corresponding in-phase noise with a zero-mean Gaussian distribution and variance
2
0/ 2
n IN T [21].
Page 5 of 29
Since the tracking loops are locked and the received signal is tracked in Phase Lock Loop (PLL)
mode, the in-phase branch is considered to monitor the correlation peak and thus for SQM metric
definition. Two types of correlators are considered, namely the tracking and monitoring correlators,
where the tracking correlators are used for tracking the signals and the monitoring correlators are used
for signal quality monitoring. SQM metrics are defined as the combination of different tracking and
monitoring correlators. The SQM metric, labeled as “Double-Delta” metric ( DDm ), is defined as the
difference between two pairs of early-late correlators normalized by the prompt correlator:
0
i i j ju u u u
DD
I I I Im
I
(5)
where
0
I is the in-phase output of the prompt correlator,
indices l, which refers to the lth PRN channel, and k, which refers to the kth integration epoch, have
been omitted for simplicity, and
it is assumed that iu and j
u are positive values; 2j c
u T and 2i c
uT are the tracking and
monitoring early-late correlator spacings.
Note that during each coherent integration period, the binary data is assumed constant. With that
in mind, at each correlation epoch, the constituent components of the SQM metric have the same
binary data with either a positive or negative sign (i.e. +1 or -1) in the corresponding numerator and
denominator. Therefore, the navigation data has no effect on SQM metric outputs. Under nominal
conditions such as low multipath open sky environments, the output of each SQM metric is a random
process whose statistics (e.g., mean and variance) are determined based on the location of the
constituent correlators and receiver noise. In [21], it is shown that by defining SQM metrics as the
linear combination of different early and late correlators normalized by the prompt value, the nominal
variance of each metric at each epoch is determined as
2
02
mm
I
h
C N T (6)
where m
h is a variance factor (determined based on the definition of the SQM metric, the correlator
spacing of the constituent correlators and the shape of the correlation function) and 0
C N is the
corresponding carrier-to-noise density ratio in Hz. For BPSK(1), it has been shown that with 0.2 and 1
chips tracking and monitoring early-late correlator spacings (as considered here), the value of m
h is
calculated as 1.6 for the Double-Delta metric [15]. For BOC(1,1), the corresponding m
h value for 0.2
and 1 chips tracking and monitoring early-late correlator spacings are calculated as 2.4 [16]. The
nominal mean values of the SQM metrics can be approximated by simply replacing the prompt
correlator with its mean value under nominal conditions. With this assumption, for symmetric
Double-Delta metric, the mean will be zero for the both signaling schemes as is discussed in [14].
2.3. Characterization and Performance Analysis
Characterization methodology is based on extracting “SQM variation profiles” and comparing
them with the “multipath tracking range error envelopes” for different discriminators and code
modulations. To extract multipath range error envelopes, a single reflection is considered and then the
relative delay of the reflected signal (with respect to direct signal) is swept through a range of values to
evaluate code tracking misalignment and the consequent range errors for in-phase and out-of-phase
Page 6 of 29
multipath components. These envelopes are shown in Figure 2 (right vertical axis - blue) for 3 and 6 dB
Signal-to-Multipath Ratio (SMR) values, NC and HRC discriminators with 0.2 chips as the primary
tracking correlator spacing (as discussed by [21]), and BPSK(1) and BOC(1,1) code modulations. Figure
2a defines the approximate range of short, medium and long delay multipath considered here.
Short-delay multipath is assumed for reflected signal delays less than 0.1 chips [or about 30 m for the
GPS L1 C/A case]; medium-delay multipath is considered for the range of 0.1 to 0.75 chips and
long-delay multipath covers the reflected signal delays longer than 0.75 chips. Figures 2 (left vertical
axis - red) also shows the SQM variation profiles for the defined Double-Delta SQM metric where the
relative delay of the reflected signal is swept from 0 to 1.5 chips to evaluate SQM metric outputs for
in-phase and out-of-phase multipath components.
To assess the performance of SQM approaches for multipath detection, the monitoring variation
profiles should be compared to the multipath error envelopes. According to Figure 2, the SQM metric
is sensitive where the SQM variation exceeds the detection threshold and multipath is detected, which
is the case for most medium and long-delay multipath scenarios. These detection results can be
potentially used to issue an alarm about multipath or reduce its effect by excluding (or de-weighting)
distorted measurements. The detection results and thus the SQM metric will not be effective when
multipath errors are negligible. For instance, as shown in Figure 2, when the HRC discriminator is
used under BPSK(1), the detection results obtained for medium-delay multipath is not effective as the
ranging error is negligible. In this scenario, relying on SQM detection to de-weight or exclude
measurements may even increase position errors due to geometry degradation. For short-delay
multipath, the effectiveness of SQM metrics is almost the same for all tracking strategies and signaling
schemes. In this area (especially for multipath delays less than 10 m) however, due to the low
sensitivity of all the SQM metrics, it is possible that the resulting SQM values do not exceed the
detection thresholds and thus multipath remains undetected for a realistic range of multipath power
values. For a given SMR value, a higher C/N0 or a higher coherent integration time increases SQM
sensitivity by reducing the nominal variance of SQM metrics and lowering the detection threshold. In
all cases, lower SMR values result in higher SQM sensitivity as expected.
(a) (b)
(c) (d)
Figure 2. SQM profiles and corresponding multipath error envelopes of Double-Delta metric for (a)
NC - BPSK(1), (b) NC - BOC(1,1), (c) HRC - BPSK(1) and (d) HRC - BOC(1,1) [22,23]
Page 7 of 29
The above discussion reveals that an effective SQM-based multipath monitoring requires a joint
analysis of both SQM sensitivity and effectiveness. For each tracking strategy and signaling scheme,
proper SQM metrics should be defined such that they provide acceptable detection sensitivity and
effectiveness. Since GPS L1 C/A (with BPSK(1) modulation) and NC tracking is the focus here, the
defined Double-Delta SQM is considered as it seems sensitive and effective for different ranges of
multipath delays (see Figure 2a). In Section 6, field data analysis is provided to examine practical
performance under static and kinematic multipath scenarios.
3. Measurement Quality Monitoring
Since SQM techniques have their own limitations, especially in detecting short-range multipath,
the other option is to monitor GNSS measurements (i.e., code phase, carrier phase, Doppler, and C/N0).
This is referred to as MQM and is the main subject of this section. Two steps are considered. First, the
multipath error is alleviated by a CMC-based error estimation and correction approach and then a
GF-based detection metric is applied to detect the remaining multipath error on pseudorange
measurements.
3.1. CMC-based Multipath Error Estimation and Measurement Correction
The monitoring metrics defined in these techniques are generally based on combinations of code
and carrier phase measurements to provide a direct measure of the code range multipath error,
resulting in the possibility of a pseudorange correction. The CMC metric is one of the well-known
monitoring metrics used to characterize and measure code multipath. CMC is computed by
subtracting the carrier phase measurements from the corresponding pseudoranges to remove the
effect of non-dispersive systematic errors such as receiver and satellite clock errors, orbital errors and
tropospheric delays. By doing so, for the lth satellite at the kth measurement epoch, the CMC metric is
calculated as
, , , , ,
( ) ( ) ( )
( ) 2 ( ) ( ) ( ) ( ) ( )
l l l
p l ion l l p l l l
CMC k p k k
MP k d k N k k MP k k
(7)
where
( )l
p k and ( )l
k are the corresponding code and carrier phase measurements (both converted to
units of length),
,( )
ion ld k is the ionospheric delay,
and ( )l
N k are wavelength and ambiguity, and
,( )
p lMP k , ,
( )p l
k , ,( )
lMP k
and ,
( )l
k
are code and carrier multipath and noise errors.
Given that the pseudorange multipath error is considerably larger than that of the carrier phase, if
the effect of ambiguity and ionosphere is somehow estimated and removed, the code minus carrier
measurement will be mostly an indication of pseudorange multipath. Although CMC can also be used
in other multipath mitigation approaches such as multipath detection and exclusion (e.g. [19]),
correction of affected measurements (when applicable) is preferred to mitigate the multipath error
without geometry degradation. When CMC is not used in real-time, the effect of ambiguity and
ionosphere can be estimated and removed. For real-time applications, considering that the integer
ambiguity is constant during a cycle slip-free period and ionosphere changes are low frequency, their
effects can be removed based on a simple moving average. Therefore, the pseudorange multipath error
can be extracted by the following CMC-based monitoring metric:
,( ) ( )
cmc l l l km k CMC k CMC (8)
Page 8 of 29
where l k
CMC denotes the mean value of the CMC metric for the lth measurement and at the kth
epoch computed by a moving average. The performance of such averaging will also be limited by low
elevated satellites in which case the ionospheric changes happen with higher amplitude and
frequency. In this case, the length of the moving average can be reduced accordingly. The output of the
CMC metric is then directly used for pseudorange correction as:
,ˆ ( ) ( ) ( )
l l cmc lp k p k m k (9)
where ˆl
p k 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 a re-estimation of the ambiguity.
3.2. GF-based Multipath Detection
Multipath can also be detected through the difference between pseudoranges on two frequencies.
This combination removes the geometric components of the measurements, but includes the
frequency-dependent parts, multipath and measurement noise. By doing so, for the lth satellite and kth
monitoring epoch, the corresponding Geometry-Free (GF) monitoring metric is defined as
1 2 1 2 1 2
,( ) ( ) ( ) ( )f f f f f f
GF l l l lm k p k p k d k (10)
where
1( )f
lp k and 2( )f
lp k are the corresponding pseudorange measurements for f1 and f2 frequencies and
1 2( )f f
ld k is the differential effect of frequency-dependent components between 1( )f
lp k and
2( )f
lp k (including the ionospheric effects and receiver/satellite inter-frequency biases) to be estimated
and removed either through modeling, the use of a nearby reference receiver or time-averaging; the
remainder can be used to monitor the code multipath.
The GF metric is a combination of errors on two frequencies and can thus be used only for
satellite-by-satellite detection. The GF monitoring metric is directly proportional to code multipath
errors and from this point of view outperforms SQM-based detection metrics which are not sensitive
to short-delay multipath even with significant range errors [see Section 2]. The main advantage of the
GF metric is its capability to be used after a CMC-based error correction. Multipath errors can be first
reduced by applying CMC-based error corrections and then the GF detection metric can be formed by
differencing (partially) corrected pseudoranges on two frequencies to detect the remaining multipath
errors. This will be discussed in Section 4.3 to perform a complementary combination of the
monitoring techniques.
4. Detection and Iterative De-weighting (D & I-D) Strategies
4.1. M of N Detection Filter
Herein, the detection procedure is defined based on a fixed interval detector called the M of N
detection strategy. The M of N detector is a fixed-lag sliding window which takes a window of N
samples (based on current and N−1 preceding samples) and compares 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 [24]
Page 9 of 29
1
0
1 1 1N MN n N n
n n
FA fa fa fa fan M n
N NP P P P P
n n
(11)
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 0.0027 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 variance of probability distribution for the null (when there is no or low multipath) and the
alternate (when multipath exists) hypothesis 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 and increasing the probability of detection in the case of the alternate hypothesis, 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 latency effect will be 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 position results. In Section 5, this strategy will be applied to samples of the monitoring metrics to
detect multipath. For each scenario, the appropriate parameters will be set based on the desired false
alarm probability and multipath conditions.
4.2. Iterative De-Weighting of Faulty Measurements
Once multipath is detected, the navigation solution can potentially be isolated from the effect of
multipath by excluding affected measurement(s). The critical point is the effect of exclusion on
measurement geometry. Poor geometry may magnify the effect of remaining errors and ultimately
degrades performance of navigation solution. Therefore, measurement geometry should be monitored
to control degradation when dealing with the distorted measurements. Instead of exclusion, all
measurements are retained but the contribution of faulty measurements is iteratively reduced when
multipath is detected. This differs from measurement stochastic weighting [Appendix A] which deals
with stochastic errors rather than gross errors caused by multipath. Based on the SQM or MQM
results, the procedure iteratively decreases the contribution of the detected measurements by
increasing the corresponding variance factors. Weighted DOP is considered as a geometry monitoring
metric [21]. Since position is the major concern, PDOP is adopted. Based on a pre-defined increasing
function, the detected measurements are de-weighted (by increasing the corresponding variance
factors) under the tolerable PDOP threshold. 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 de-weighting procedure continues until either the PDOP threshold or the maximum
number of iterations is exceeded.
4.2.1. Selection of PDOP Threshold
Limiting the PDOP value to a pre-defined threshold maintains geometry below a certain level, but
at the same time it prevents the system from being thoroughly isolated from the effect of faulty
measurements. Therefore, a trade-off is involved and the PDOP threshold should be selected carefully.
Assuming that detection of faulty measurements is sufficiently effective, the stochastic model of
undetected measurements can be modeled under low multipath conditions. For example, assuming a
priori variance factor of 1 m2 under low multipath conditions, the PDOP threshold can be simply set
below 5 to expect positioning accuracy better than 5 m under an effective detection and de-weighting
of faulty measurements. At the same time the PDOP threshold and the maximum number of iterations
should be large enough to ensure effective de-weighting procedure.
Page 10 of 29
4.2.2. 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 PDOP 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 two important factors into account. First is the trade-off between the resolution
and complexity of the de-weighting process. 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 whole process and thus a heavier computational burden. Second, the detected faulty
measurements have different error levels that should be considered in the de-weighting function.
Therefore, the variance of the lth measurement at the kth epoch and in the ni th iteration can be
calculated as
2 2
,
if MP is detected
( ) ( )
1 otherwisen
l iter n
l i l
b a i
k k (12)
where
2( )l
k is the initial variance of the corresponding measurement determined based on the
stochastic weighting model under low multipath condition,
iter
a is the de-weighting factor to be selected based on the desired de-weighting resolution and
maximum number of iterations, and
lb is a scale factor based on the level of multipath error (if known) for each measurement.
For undetected measurements, any selection of the conventional stochastic weighting models can
be adapted [see Appendix A]. Although not based on rigorous statistical theory, Equation 12 can be
effectively used to reduce the effect of multipath as it will be shown in Section 5 for static and
kinematic scenarios. For each data set, the proper selection of PDOP threshold, maximum number of
iterations, de-weighting and scale factors will be discussed based on multipath characteristics.
4.3. SQM and MQM-based Solutions
Figure 3 shows the SQM-based D & I-D approach. The symmetric zero-mean Double-Delta SQM
metric is considered for multipath detection (according to Equation 5), normalized by its nominal SD
calculated by Equation 6. The detected measurements are iteratively de-weighted and undetected
measurements are stochastically weighted based on the conventional constant, elevation or
C/N0-based models described in Appendix A.
Figure 3. SQM-based D & I-D scheme
Page 11 of 29
For MQM solutions, first, CMC-based error corrections are applied to pseudoranges to alleviate
multipath errors and then the GF-based detection metrics are used to iteratively de-weight remaining
errors below a designed PDOP threshold. In the case of included measurements or for those not
de-weighted iteratively, an elevation or C/N0-based model is applied to mitigate the effect of stochastic
errors [see Appendix A]. Since neither calibration, pre-defined models or reference receivers can be
used to remove the mean value of the CMC metric (affected by an unknown integer ambiguity), a
moving average is applied. The performance will be limited by two factors. First, as the code multipath
error is not zero-mean, low-frequency biases caused by multipath will be filtered out, which is not
desirable for multipath detection. Second, the performance of time averaging will be limited by the
sliding window length. The larger length provides larger sample-space and thus results in more
precise estimation of the mean value at the expense of a delayed response to multipath transition
states, geometry changes and other signal degradations such as signal attenuation and blockage. To
balance these two limitations, the moving average window is set to at least one multipath period,
which in static cases may reach several minutes. The nominal SD of the monitoring metrics in the
absence of multipath is determined using reference data collected in a low multipath environment.
Figure 4 shows the flowchart of this procedure. In this approach, the preceding CMC-based error
correction potentially reduces the number of measurements de-weighted in position solutions and
thus alleviates degradation in measurement geometry. However, since all GPS satellites do not
broadcast signals on all three frequencies at this time, the GF-based detection/de-weighting
performance is limited to the PRNs with signals on at least two different frequencies. Higher
performance is expected as the number of multi-frequency satellites increases.
Figure 4. MQM-based solution; a preceding CMC-based error correction followed by a GF-based
multipath detection and de-weighting
5. Demonstration Using Real Data
Field test results are used to evaluate the performance of developed monitoring techniques in real
static and kinematic multipath scenarios. In Section 5.1, SQM approaches are examined and in Section
5.2, data analysis is provided for MQM solutions. An overall comparison will be presented in Section
6.
5.1. SQM Test Scenarios
Static mode GPS L1 C/A data was collected using a NovAtel GPS-703-GGG antenna, surrounded
by buildings with smooth surfaces acting as short-range reflectors as shown in Figure 5a. IF samples
were collected using a National Instrument (NI) PXIe-1075 front-end with a 10 MHz sampling
frequency. For the kinematic mode, the antenna was mounted on a cart (Figure 5b) moving in a
sub-urban multipath environment. The data was down-converted and sampled with a Fraunhofer
GTEC RFFE Triple Band front-end with 20 MHz sampling frequency. In both cases, the IF samples
were processed by a software receiver to extract SQM outputs for different PRNs. A narrow correlator
discriminator with 0.2 chips early-late spacing was implemented. Monitoring correlators were placed
1 chip apart and the coherent integration time was set to 20 ms.
Page 12 of 29
(a) (b)
Figure 5. Multipath data collection for (a) static and (b) kinematic test scenarios [22,23]
5.1.1. Detection Results – Static Test
Figure 6 shows CMC and C/N0 measurements for selected PRNs during the static test. PRN 23 is
under low multipath conditions while PRN 22 is heavily affected as shown by corresponding CMC
values. Figure 7 shows corresponding monitoring results for the Double-Delta SQM metric calibrated
and normalized using its nominal standard deviation. In this normalization, the C/N0 values were
smoothed by a moving average with a 10 second length. Detection thresholds were fixed to ±3 for the
normalized metric. The M of N detection strategy was used by taking windows of N samples and
comparing them to the predefined threshold. Figure 7 also shows detection results for PRNs 23 and 22
when N = 500 (samples, equal to 10 s under 20 ms coherent integration time). M = 10, 15 and 20 chosen
to satisfy the theoretical probability of false alarm below 1.53×10-6. It is noted that for NC and BPSK(1)
(which is the case here), the sign of Double-Delta SQM variations is opposite to the corresponding
multipath errors for in-phase and out-of-phase components. Therefore, the sign of Double-Delta
variations has been reversed for a better comparison of in-phase/out-of-phase effects of multipath on
CMC and SQM metrics. As shown in Figure 6a, for M ≥ 10, the output of the detection algorithm for
PRN 23 is mostly zero, identifying the low effect of multipath. In the case of PRN 22 (shown in Figure
7b), multipath with less than a 1 m ranging error remains buried under the SQM metric noise and thus
mostly undetected. Nevertheless, when multipath error is in the order of 5 m or more, the SQM metric
is sensitive and the detection output shows the occurrence of multipath most of the time.
Figure 6. CMC and C/N0 measurements for PRN 23 and 22 with different levels of multipath error,
static test
Page 13 of 29
(a)
(b)
Figure 7. SQM-based monitoring results and corresponding detection outputs for (a) PRN 23 and (b)
PRN 22, static test
Figure 8 shows probabilities of false alarm (PFA) for different values of M. The green line shows the
theoretical results (multipath-free conditions) obtained using Equation 11. To examine practical
performance, the null hypothesis was defined based on low multipath measurements (i.e. multipath
errors less than 1 m according to the CMC values) and then the number of false alarms was evaluated
over the entire period as a measure of practical false alarm probability. In Figure 8, the blue line
represents the corresponding values obtained from PRN 23. For instance, for M = 10, it is shown that
the detection system distinguishes the low multipath measurements of PRN 23 as clean measurements
with less than 2% false alarm probability. For PRN 22, the corresponding values have been presented
in red and show a significantly higher rate of false alarms (about 30% false alarm probability for M =
10) than that of PRN 23. This is due to the latency imposed by the M of N filter degrading the
“resolution” of the detection system during transition between null and alternate hypotheses as
discussed in Section 4.1.
Page 14 of 29
Figure 8. Probability of false alarm for N = 500 (samples) and different values of selected M
Figure 9 shows the probability of detecting (PD) multipath when its error exceeds 1, 3 and 5 m for
different measurements of PRN 22. For M = 10, it is observed that multipath errors higher than 5 m are
detected 93% of the time. Therefore, considering the practical false alarm probabilities of Figure 8, it
can be stated that for (N, M) = (500, 10), a 5 m Minimum Detectable Multipath Error (MDME) is
detected 93% of the time with a 98% confidence for PRN 23 and a 70% confidence for PRN 22. While
this detection performance is considered sufficient for the current research, for a desired detection
probability and for a specific value of N, a smaller MDME can be obtained by either decreasing the
value of M or lowering the primary detection threshold at the expense of higher false alarm
probabilities. Comparing detection probabilities of PRN 22 (heavily affected by multipath) and
practical false alarm probabilities obtained from PRN 23 (with generally low multipath), Figure 10
shows Receiver Operating Characteristic (ROC) of the detection system for different values of N. As
observed, by increasing the value of N and thus more smoothing detection results, the overall
performance improves gradually, again at the expense of a lower detection resolution in transition
between null and alternate hypotheses.
Figure 9. Probability of detection for MDME = 1, 3 and 5 (m), N = 500 (samples) and different values of
Page 15 of 29
selected M; PRN 22
(a)
(b) (c)
(d) (e)
Figure 10. ROC of the detection system for N = (a) 500, (b) 400, (c) 300, (d) 200 and (e) 100 (samples)
5.1.2. Detection Results – Kinematic Test
Figures 11 and 12 show sample monitoring results for the kinematic test shown in Figure 5b. The
CMC measurements indicate that multipath is generally low and errors do not exceed 5 m except for
some epochs between 20 s and 60 s. The corresponding SQM results were then extracted for each 20 ms
of coherent integration time. The metric was calibrated and normalized with its nominal standard
deviation. Compared to the static scenario, for similar detection performance, the length of the
smoothing window was reduced to 4 s due to the dynamic characteristic of the data and consequently
faster multipath variations as a function of time. The detection thresholds were fixed to ±3 times the
normalized SD. The length of the sliding search window (N) was chosen equal to the smoothing
window length (i.e. 4 seconds or 200 samples) and M = 7, 8 and 9 (samples) were chosen to detect
multipath with theoretical false alarm probability below 61.52 10 . As shown in Figures 11 and 12,
the sensitivity of the SQM metric is limited to epochs whose multipath errors are in the order of 5 m or
Page 16 of 29
more. For M ≥ 9, the SQM detection output is zero for all epochs and multipath errors, if present,
remain undetected.
As the relative velocity between receiver and reflector(s) increases, the direct and reflected
correlation peaks separates in the Doppler domain, resulting in generally less distortion and
consequently lower multipath error. In any case, the multipath errors higher than 5 m are effectively
detected by a proper selection of M and N values. Moreover, results shown in Figures 11 and 12 relates
to a very slow kinematic scenario where the receiver setup was moving at a velocity of 0.5 m/s. By
increasing the motion speed, the performance of the SQM, as a detection solution, degrades
dramatically. Working with IF samples at the tracking level makes SQM metrics vulnerable to tracking
phase loop instability under such dynamic stress conditions.
Figure 11. CMC and C/N0 measurements for PRN 7, kinematic test
Figure 12. SQM-based monitoring results and detection outputs for PRN 7, kinematic test
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5.1.3. Position Results
In the kinematic scenarios, detection results were limited to a few epochs and thus no significant
improvement was observed by de-weighting detected measurements. Therefore, position results only
for the static mode are presented here. A pseudorange-based LS solution was used. The PDOP
threshold was set to 5 and Equation 12 was used to iteratively de-weight detected faulty
measurements. The empirical value of 5 was tested and chosen as an appropriate PDOP threshold to
sort out solutions with poor geometry and benefit from excluding multipath-affected measurements.
Referring to Section 2, for most multipath delay cases, the direct relationship observed between
multipath error and SQM variation profiles can be used to define the de-weighting function. To this
end, when multipath (MP) is detected, the Mean Square Deviation (MSD) of the normalized SQM
metric (calculated over 10 s of the sliding window) is scaled and multiplied by the iteration number to
iteratively increase the corresponding measurement variance factor as shown in Equation 12. Note that
the MSD of the normalized SQM metric is a unit-free factor whose expected value under
multipath-free conditions is 1. The de-weighting resolution was set to 1 and the maximum number of
iterations was chosen as 10, observed as sufficient to de-weight distorted measurements under the
defined PDOP threshold. Figure 13 shows PDOP and positioning error values for the static data set.
For a constant weight LS solution, the blue and red plots represent corresponding results before and
after applying SQM-based D & I-D. Although the SQM-based D & I-D does not work satisfactorily at
some epochs, the overall positioning performance has improved at epochs heavily affected by
multipath.
(a)
(b)
Figure 13. (a) PDOP values and (b) position errors with (blue) and without (red) SQM-based D &
Page 18 of 29
I-D, static test
Position results were also investigated using other conventional weighting models, namely
elevation and C/N0-based weighting algorithms discussed in Appendix A. The numerical results of
position Root Mean Square (RMS) errors are presented in Table 1 for the defined weighting algorithms
with and without SQM-based D & I-D. Improvement percentages are added for 3D positioning. By
applying the SQM-based D & I-D, although in some cases (e.g. east direction) RMS errors slightly
increase, the overall positioning performance improves from 14% to 35% for the constant, elevation
and C/N0-based LS solutions.
Table 1. Position errors for different positioning approaches; SQM static test
SQM-based
D & I-D
Stochastic
Weighting
Approach
Position RMS Errors (m) 3D Positioning
Improvement
Percentage
(YES over NO) East North Height
Sta
tic
Tes
t NO
A1 1.18 2.37 6.97 -
A2 1.11 1.85 3.54 -
A3 1.17 1.98 3.75 -
YES
A1 1.15 1.80 4.35 35%
A2 1.09 1.72 2.68 19%
A3 1.15 1.85 3.08 14%
Al: Constant Weighting (No Weighting)
A2: Elevation-based Weighting
A3: C/N0-based Weighting
5.2. MQM Test Scenarios
5.2.1. Static Test
Using a Trimble R10 receiver, GPS L1 (C/A), L2C (M + L) and L5 (I + Q) code, carrier and C/N0
measurements were collected every second in the multipath environment shown in Figure 5a. While
similar results were observed for all PRNs, PRN 10, affected by different levels of multipath, was used
to examine the sensitivity of the MQM-based detection metrics. The time-averaging approach was
used to estimate the mean value of the monitoring metrics. A simple moving average was used with a
length of 5 minutes, based on the average period observed for “quasi-periodic” oscillations of PRNs
exhibiting static multipath. At each epoch, the CMC metric was obtained by subtracting carrier phase
measurements from corresponding pseudoranges according to Equation 7. The nominal mean value,
estimated by the moving average, was then filtered to extract the CMC-based monitoring metric
defined in Equation 8. Since cycle slips may result in new unknown carrier phase ambiguities, the
moving average buffer was reset if a cycle slip was detected. The cycle slip detection procedure was
performed based on the “phase velocity trend” method with a threshold of 1 cycle. For PRN 10, Figure
14 shows the L1, L2C and L5 CMC-based monitoring metrics as an indication of the corresponding
code multipath errors. The RMS values have been also presented for three segments of the data set
with low, medium and high level of multipath errors. The estimated CMC values are also affected by
the moving average and how the mean value is estimated and removed and thus may be inaccurate
especially when the occurrence of cycle slips resets the buffer. This limits the performance of the
CMC-based multipath error correction as will be discussed in the sequel.
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Figure 14. CMC-based monitoring metric for GPS L1 (blue), L2C (red) and L5 (green); PRN 10
affected by different multipath error levels
In addition to code and carrier phase measurements (and thus CMC metric), multipath affects the
measured signal power and C/N0 values as shown in Figure 15. 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 on each frequency.
For each signal, the measured C/N0 was used to perform the stochastic weight model discussed in
Appendix A.
Figure 15. C/N0 measurements for PRN 10, GPS L1 (blue), L2C (red) and L5 (green)
Page 20 of 29
For detection purposes, GF measurements were first filtered to remove their mean values and
were then normalized based on an estimation of the measurement noise standard deviation under the
null hypothesis. For the resulting unit-free monitoring metric, the detection thresholds were set to ±3
as three times the normalized nominal SD for L1, L2C and L5 signals. Figure 16 shows the normalized
GF-based monitoring metric and corresponding M of N detection results. The N was set equal to the
length of the moving average as 5 min or 300 samples (with the rate of 1 sample/second). With that
selection, M was chosen as 10 to satisfy the overall probability of false alarm of 1.41 x 10-8 under
multipath-free conditions and normal distribution of the detection metrics. Figures 16b shows the
GF-based detection metrics and the corresponding results based on the differences between L1, L2C
and L5 GPS signals. For GPS L1 positioning, detection results include those provided by the GF-based
detection metrics using the L1-L2C and L1-L5 combinations. For L2C positioning, detection results are
related to the L1-L2C and L5-L2C combinations. Similarly, for L5 positioning, detection results are
based on differential metrics on L1-L5 and L5-L2C combinations.
(a)
(b)
Figure 16. (a) GF monitoring and (b) corresponding threshold excesses and M of N outputs for PRN
32
Page 21 of 29
Comparing detection results with the corresponding CMC values shown in Figure 18, it is
observed that the detection output remains zero for the first 35 min with generally low multipath
measurements. When the multipath error exceeds 3 m, all detection metrics exceed their respective
thresholds most of the time. Regarding PRN 10, Figure 21 shows the overall detection probability for
multipath errors higher than 1, 3 and 5 m for N = 300 (samples) and different values of M. For M = 10
(samples) and when multipath error exceeds 1 m, errors detected by the GF-based detection metric
more than 88% of the time. For a specific value of N, a higher detection probability could be attained by
decreasing the value of M (or even lowering the primary detection thresholds) at the expense of higher
false alarm probability when multipath is low.
Figure 17. Probability of multipath detection for MDME = 1, 3 and 5 (m), N = 300 (samples) and
different values of M for PRN 32
The performance of the MQM techniques was evaluated in the position domain where a LS
solution was used to provide epoch-by-epoch positions. GPS L1 positioning with constant weighting
model was considered first. Figure 18 shows the PDOP and position error values for four different
MQM approaches; A0 was the benchmark with no correction or de-weighting solution. The
corresponding results have been shown in red. A1 was considered when MQM D & I-D approach was
applied. For multipath detection, the combination of all the GF-based detection results related to the
L1-L2C and L1-L5 combinations was considered to determine whether a GPS L1 measurement is
affected by multipath or not. Equation 12 was used to iteratively de-weight distorted measurements
under the PDOP threshold with the value of 5. Similar to the SQM case, the maximum number of
iterations and de-weighting factor were set to 10 and 1, sufficient to de-weight distorted measurements
under the defined PDOP threshold. Since there is no direct relationship between detection statistics
and multipath error associated with each single frequency, the scale factor is set to unity. The
corresponding results are shown in blue in Figure 18. By de-weighting distorted measurements, PDOP
values increase, but position results improves in overall especially for the epochs with strong
multipath. CMC-based multipath corrections were examined where the estimated zero mean CMC
values were applied to pseudoranges to alleviate code multipath errors. For this scenario, labeled as
A2, position errors are shown in yellow. It is observed that CMC-based error corrections effectively
smooth position errors except during intervals when cycle slips degrade the performance of the
monitoring approach [e.g. around t = 60 (s)]. The combination of CMC error corrections and GF-based
D & I-D was investigated next (A3). Multipath errors were first alleviated by applying CMC-based
error corrections on pseudoranges and then the GF detection metrics were performed on L1-L2C and
L1-L5 combinations using the (partially) corrected measurements.
Page 22 of 29
Results are plotted in green in Figure 18. The primary CMC-based error correction reduces the
number of measurements de-weighted and consequently preserves measurement geometry which can
be concluded from lower level of PDOP values. While iterative de-weighting of detected
measurements degrades the PDOP values (under the defined PDOP threshold), the preceding
CMC-based correction can reduce the level of degradation. In the position domain, as shown in Figure
18b, the complementary combination of CMC-based error correction and GF-based D & I-D shows
improvement over each single monitoring approach.
(a)
(b)
Figure 18. (a) PDOP values and (b) position errors for GPS L1 positioning and different MQM
approaches. A0 (red): No correction, No de-weighting (Benchmark), A1 (blue): GF-based D & I-D,
(yellow) A2: CMC-based error correction and A3 (green): CMC-based error correction and
GF-based D & I-D (Combined Method), MQM static test
Page 23 of 29
Table 2 shows numerical RMS error values for GPS L1 and L2C and L5 solutions for constant,
elevation and C/N0-based stochastic models. The RMS error values have been extracted for the
intervals where the number of satellites is four or more and the position solution has converged.
Table 2. Position errors for different positioning approaches, MQM static test
Combined
Signals
Weighting
Model MQM
East RMS
Error (m)
North RMS
Error (m)
Height RMS
Error (m)
3D
Improvement
(A3 Over A0)
GPS L1
(C/A)
Constant
A0 0.75 1.71 2.85
53% A1 0.66 1.17 1.99
A2 0.52 1.09 2.17
A3 0.43 0.75 1.36
Elevation-
based
A0 0.64 1.23 2.29
35% A1 0.65 1.45 2.25
A2 0.43 0.78 1.57
A3 0.38 0.89 1.45
C/N0-based
A0 0.65 1.62 2.92
35% A1 0.77 1.74 2.15
A2 0.48 1.28 2.31
A3 0.56 1.39 1.64
GPS L2C
(M+L)
Constant
A0 1.21 2.42 6.33
17% A1 1.21 2.90 6.04
A2 0.88 2.11 4.83
A3 0.99 2.46 5.03
Elevation-
based
A0 1.14 2.35 6.14
4% A1 1.28 2.75 6.31
A2 0.89 2.06 4.70
A3 1.13 2.28 5.89
C/N0-based
A0 1.38 3.81 7.66
23% A1 1.35 4.02 7.57
A2 0.89 2.49 5.62
A3 0.98 2.64 6.01
GPS L5
(I+Q)
Constant
A0 3.60 4.78 8.41
20% A1 3.59 4.78 8.41
A2 1.42 4.04 7.06
A3 1.41 4.06 7.04
Elevation-
based
A0 3.59 4.79 7.98
17% A1 3.59 4.81 7.96
A2 1.41 4.06 7.04
A3 1.41 4.08 7.04
C/N0-based
A0 3.63 4.82 8.09
18% A1 3.63 4.82 8.09
A2 1.42 4.06 7.10
A3 1.42 4.06 7.08
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)
Page 24 of 29
The number of satellites broadcasting signals on L2C and L5 frequencies is generally lower than
L1, hence geometry is poorer and position errors are higher; the detection/de-weighting techniques
therefore yield lower performance compared to the L1 solution. This is obvious in theL5 A1
positioning approach where the de-weighting process does not iterate more than once or twice at each
epoch and thus is not effective. In this case, even by increasing the PDOP threshold to a higher number
such as 8 or 10, no significant improvement was obtained. A2 is however effective as it does not affect
measurements geometry. In general, the lowest RMS error values relate to the combined method (A3)
where multipath errors are first alleviated by applying CMC-based error corrections and then the
corresponding GF detection metrics are used to detect and de-weight remaining multipath errors.
Comparing three-dimensional (3D) RMS values, the combined MQM approach (A3) shows 53% (L1),
17% (L2C) and 19% (L5) improvement over A0 (the benchmark), when the constant model is applied
as the initial weighting model. These values are reduced to 35%, 4% and 18% for the elevation-based
model and 35%, 23% and 19% for the C/N0-based weighting model. This is because the effect of
multipath has been already mitigated in A0 by applying lower weight to low-elevated satellites or
those with lower C/N0 values, which are typically due to multipath.
5.2.2. Kinematic Test
Figure 19 shows the data collection setup, trajectory and multipath environment for the MQM
kinematic test. The Trimble R10 receiver was mounted on a cart moving at a velocity of 0.5 to 2 m/s
through an area surrounded by buildings with reflecting surfaces. A reference trajectory was obtained
using a NovAtel SPAN system consisting of a tactical grade Inertial Measurement Unit (IMU) and a
GNSS receiver. The phase velocity trend method was used for cycle slip detection with two cycles as
the detection threshold. Compared to the static scenario, the length of the moving average was
empirically reduced to 60 s to account for the dynamic characteristics of the data, which result in faster
multipath variations. The N and M were chosen as 10 and 4 to satisfy the overall false alarm
probability of 1.1×10-8. Only GPS L1 was investigated as the mean number of measurements on L2C,
and L5 was lower than four, limiting positioning outputs to a few epochs. Compared to the static test
and due to poorer geometry, the PDOP threshold was increased to 8. With the maximum number of
iterations equal to 10, the scale factor was empirically set to 10 to perform an effective and relatively
low complexity de-weighting procedure under the designed PDOP threshold. Figure 20 shows
horizontal position results for A0 to A3 under a constant weight stochastic model. For A0, position
errors are high, especially when passing by the buildings with reflecting surfaces. In this area, by
detecting and de-weighting distorted measurements (A1), position results improve for epochs with
strong multipath. The CMC-based error correction method (A2) generally smooths out position results
but magnifies the errors when cycle slips occur or when position becomes biased as a result of
multipath. The lowest position errors were obtained by combining A1 and A2 where multipath errors
are first mitigated by applying the CMC-based error correction method and then GF-based detection
results are used to de-weight remaining distorted measurements.
Page 25 of 29
(a) (b)
Figure 19. MQM kinematic test; (a) multipath environment and reference trajectory, (Map data © 2019
Imagery © 2019, Google) and (b) data collection setup
(a) (b)
(c) (d)
Figure 20. Horizontal position results for (a) A0: No correction, No de-weighting (Benchmark), (b) A1:
GF-based D & I-D, (c) A2: CMC-based error correction and (d) A3: CMC-based error correction and
GF-based D & I-D (Combined Method), MQM kinematic test
The numerical results including the height (vertical) RMS errors are given in Table 3 for different
positioning approaches and for different stochastic weighting models. While MQM D & I-D (A1) and
CMC-based error corrections (A2) do not provide absolute solutions, the complementary combination
of monitoring techniques (A3) shows solid performance for all scenarios. Comparing 3D RMS error
values, A4 outperforms A0 by 29%, 22% and 23% improvement for constant, elevation and C/N0-based
weighting models, respectively.
Page 26 of 29
Table 3. Position errors for GPS L1 position solutions, MQM kinematic test
Combined
Signals
Weighting
Model MQM
East RMS
Error (m)
North RMS
Error (m)
Height RMS
Error (m)
3D
Improvement
(A3 Over A0)
GPS L1
(C/A)
Constant
A0 1.27 2.21 3.16
29% A1 1.12 1.08 2.92
A2 1.51 1.54 3.64
A3 1.00 1.01 2.50
Elevation-
based
A0 1.15 0.92 2.99
22% A1 1.17 1.14 2.99
A2 1.37 0.97 3.32
A3 1.03 0.98 2.16
C/N0-based
A0 1.17 1.01 2.90
23% A1 1.30 1.21 2.98
A2 1.27 1.00 3.04
A3 1.10 1.02 2.05
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)
6. Conclusions
The SQM and MQM techniques were investigated to detect and mitigate the effect of code phase
multipath. While the statistics of the SQM metric is mathematically defined based on receiver tracking
setup, the MQM approach uses moving average to provide normalized zero-mean monitoring metrics.
This is more crucial in the case of CMC-based error correction when the effect of carrier phase integer
ambiguity is unknown. Against, the SQM detection metric which can be defined on each single
frequency, the GF-MQM metric requires at least a two-frequency receiver, but generally shows higher
detection performance than the SQM approach, especially in short-range multipath scenarios. The
SQM metrics need the use of additional monitoring correlators at the tracking level and thus
modification is required in the receiver signal structure, while the MQM metrics use the typical code
and carrier-phase measurements and thus are more compatible with current receivers. The results
obtained for the defined Double-Delta SQM metric, the receiver setups and GPS L1 C/A data used,
revealed that multipath errors higher than 5 m are effectively detected for multipath delays
theoretically higher than 0.03 chips. By applying the SQM-based D & I-D in a static multipath scenario,
the overall 3D positioning performance improved by 35%, 19% and 14% for the conventional constant,
elevation and C/N0-based LS solutions. Complementary to the SQM techniques, MQM was
investigated based on the CMC-based multipath error correction followed by the GF-based detection
and iterative de-weighting of distorted measurements. The combination of monitoring approaches
shows over 22% improvement for GPS L1 C/A positioning when the conventional constant, elevation
and C/N0-weighted LS solution is used under static and kinematic test scenarios.
The performance of both SQM and MQM techniques is strictly limited by multipath dynamics and
measurement geometry. Investigation of other kinematic data sets, revealed that as the receiver
motion increases, the SQM metrics, defined at the tracking level, generally fail to due to tracking phase
loop instability under high dynamic stress conditions. The improvement obtained with the
CMC-based corrections also decreases from static to kinematic as the occurrence of cycle slips
increases due to obstructions and antenna motion. Moreover, the detection delay imposed by the
averaging filter limits the effectiveness of the monitoring approaches when the multipath dynamics
Page 27 of 29
increase, which is the case of vehicular applications. The performance of the SQM and MQM-based
de-weighting techniques is also limited by measurement geometry. Under poor geometry, an effective
de-weighting of faulty measurements is hardly achievable as the remaining errors may magnify
ultimate positioning errors. Results also show that the de-weighting of distorted measurements
breakdown when more than 50% of measurements are distorted. Therefore, future work may include
definition and evaluation of new metrics based on upcoming GNSS signals where higher performance
is expected when satellite geometry and redundancy of the measurements improve in a
multi-constellation solution.
Appendix A: Stochastic Model of Measurements and Weighting Approach
Besides the constant model which simply assigns an identical weight to all measurements,
elevation and C/N0-based models are used for the purpose of measurement weighting.
A.1. Elevation-based Weighting
This model uses satellite elevation angles to weight measurements. Signals received from lower
elevation angles suffer from additional signal attenuation, antenna gain pattern loss, atmospheric and
multipath effects. The relationship between satellite elevation angle and corresponding measurement
precision can be simply modelled as
2 2
0 2
,
( )sin ( )l
l k
ak
El
(A1)
where
2( )l
k is the variance of the lth measurement at the kth measurement epoch,
,l k
El is the elevation of the lth satellite at the kth measurement epoch, and
a is a unit-free model parameter determined such that 2
0a corresponds to the variance of the
measurement at the zenith.
One of the major limitations of elevation-based weighting is that multipath propagation can affect
higher elevation satellites, which is not considered here.
A.2. C/N0-based Weighting
SNR and equivalently C/N0 are other quality metrics used for the purpose of GNSS measurement
weighting. Although initially developed for carrier phase measurements, it has been shown that these
models are also beneficial for pseudoranges as discussed by [25,26]. The pseudorange measurements
can be then weighted as
0 ,
/
2 2 100
( ) 10l k
C N
lk a b
(A2)
where a (unit-free) and b [in Hertz (Hz)] are model parameters (set based on the multipath
environment and user equipment) and 0 ,/
l kC N represents the measured C/N0 (either smoothed or
not) for the lth measurement at the kth epoch, in dB-Hz. Although C/N0-based solutions generally show
improvement in positioning performance, they do not thoroughly deal with gross errors caused by
multipath [21].
Page 28 of 29
A.3. Calibration of Weighting Model Parameters
The calibration process was applied through collecting and processing pseudorange
measurements in a pre-surveyed position (as discussed by [25,26]). First, the receiver operates in a
pre-surveyed reference location within the target environment. The data is processed by the
navigation solution (pseudorange-based LS solution) and the residuals are extracted by fixing the
receiver position to known coordinates. Neglecting the effect of clock bias estimate errors, the
residuals can be approximated by the pseudorange measurement errors to be used in the calibration
process. Figure A1 shows a sample of calibration results obtained using GPS L1 C/A data collected for
about half an hour in fairly low multipath conditions. For a constant weight model, a priori variance
factor was estimated over all measurements collected over time equal to 1.64 m2; the standard
deviation of pseudoranges 1.28 m is for low multipath conditions. Figure A1a shows absolute
pseudorange errors versus their corresponding elevation angles. In this approach, estimated
pseudorange errors were first sorted based on corresponding elevation angles and then categorized in
different groups with a resolution of 1 deg. The standard deviation of each group was then calculated
and shown in light green. The standard deviation of the measurements above 80 degrees was
estimated as 0.3 m and used to tune the model parameter of Equation A1. The dashed black line shows
the elevation-based weighting model. As observed, it matches the estimated standard deviation values
properly. Figure A1b shows absolute pseudorange errors versus their corresponding C/N0 values used
to tune the C/N0-based weighting model parameters. With a similar methodology, estimated errors
was first sorted based on corresponding C/N0 values and then categorized in different groups with a
resolution of 1 dB-Hz. The standard deviation of each group was then calculated and used to tune the
model parameters of Equation A2. It was observed that when the a priori variance factor is set to 1 m,
with the selection of a = 0 and b = 2002 (Hz), an appropriate weighting model is obtained.
(a) (b)
Figure A1. (a) Elevation and (b) C/N0-based -based pseudorange weighting model under low
multipath conditions
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