gnss signal and measurement quality monitoring …...monitoring techniques to counter the above are...

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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.

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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

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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,

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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].

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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

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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]

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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)

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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]

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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.

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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

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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.

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(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

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(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.

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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

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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

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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 &

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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)

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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

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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.

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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

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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)

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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.

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(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.

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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

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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].

https://en.wikipedia.org/wiki/DB-Hz

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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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