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European Navigation Conference (ENC) 2019, Warsaw, Poland, 9-12 April 2019

Detection and De-weighting of Multipath-affected

Measurements in a GPS/Galileo Combined Solution

Ali Pirsiavash, Ali Broumandan, Gérard Lachapelle and Kyle O’Keefe

PLAN Group, Department of Geomatics Engineering

Schulich School of Engineering, University of Calgary

Calgary, AB, Canada

Abstract—Multipath is a major error source for Global

Navigation Satellite Systems (GNSS). Different multipath countermeasure techniques have been investigated in the literature, of which detection and exclusion (or de-weighting) of affected measurements is receiving significant attention. Thanks to the development of multi-GNSS constellation receivers, this approach is increasingly effective as measurement redundancy is now sufficiently large to detect and exclude (or de-weight) faulty measurements prior to being used in the navigation solution. Given the benefits of GNSS measurement monitoring in detecting multiple signal failures, this paper focuses on measurement level monitoring techniques to develop effective mechanisms for detection and de-weighting of the signals distorted by multipath. The combination of GPS and Galileo signals is investigated under height-constrained environments to increase measurement redundancy and observability, and consequently improve the performance of detection and de-weighting solutions. Results obtained for a real static multipath scenario show up to 60 percent horizontal improvement for different positioning approaches.

Keywords: Global Navigation Satellite Systems (GNSS),

Global Positioning System (GPS), Galileo positioning system,

multipath error, measurement quality monitoring, detection and

de-weighting of multipath-affected measurements, measurement

geometry, height-constrained GPS/Galileo combined solution

I. INTRODUCTION

Multipath is the combination of Line-of-Sight (LOS) and

a number of Non-Line-of-Sight (NLOS) signal components

reflected off surroundings before reaching the receiver

antenna. Fig. 1 shows a schematic view of one-ray multipath

resulting from a reflector nearby a Global Navigation Satellite

Systems (GNSS) receiver.

Fig. 1. A schematic view of one-ray multipath

Due to reflections, received signals differ in power, delay,

carrier phase and frequency, angle of arrival and

polarization, but have the same Pseudo-Random Noise

(PRN) codes. The combination of these signals in the

receiver antenna introduces multipath as discussed in the

literature. In multipath conditions, the receiver code tracking

procedure does not properly distinguish the actual peak of

the correlation curve between received and receiver

generated replica signals as required for an ideal code

alignment. This results in pseudorange (code phase) errors in

excess of tens of metres.

The first stage in protecting a receiver against multipath

is to isolate the receiver from multipath interference or

reduce the effect of the latter through antenna design (e.g.

“choke ring” antennas) or use physical or synthetic antenna

arrays. The latter case can be used either for estimating

signal angles of arrival and beamforming or antenna

diversity to mitigate the effect of multipath (e.g. [1]).

Polarization diversity is another multipath mitigation

approach which uses two receiver ports with orthogonal

polarizations to obtain diversity gain [2]. Generally, the

polarization of reflected signals may change depending on

reflector type and multipath channel. Exploiting two

complementary polarizations can provide useful information

about reflected signals to detect and/or mitigate the fading

effect of mismatched signals in the receiver (e.g. [3,4]). The

use of Narrow Correlator (NC) is another important solution

to reduce the effect of multipath. Reference [5] took

advantage of narrow spacing between early and late

correlators to alleviate multipath effects on tracking loops.

Following the concept of narrow correlators, other tracking

strategies such as High Resolution Correlator (HRC)

techniques [6] were developed where more than three

correlators are used in each tracking loop. Reference [7]

investigated a so-called Early-Late-Slope (ELS) technique

which uses correlator pairs on both early and late sides of the

correlation peak to estimate both side slopes and compute a

pseudorange correction metric. Multipath Estimating Delay

Lock Loop (MEDLL) [8,9] is another technique to mitigate

multipath which estimates multipath parameters using a bank

of correlators. Relatively high performance in estimating

LOS and NLOS components can be achieved using these

algorithms, but at the expense of a multi-correlator-based

tracking structure. The “vision correlator” [10] and “vector

tracking” [11] are other techniques proposed for multipath

detection and mitigation.

Multipath monitoring is another approach to detect and

isolate distorted measurements, which can be performed at

the tracking and navigation stages. At the navigation stage,

Receiver Autonomous Integrity Monitoring (RAIM) is the

most common technique [12] developed to detect faulty

measurement in all but the most harsh multipath

environments. Signal Quality Monitoring (SQM) algorithms

have been developed to detect GNSS multipath distortions

by incorporating monitoring correlators at the tracking level.

A review of the SQM techniques for detecting and mitigating

the effect of GNSS multipath has been presented in [13].

Received signal strength and corresponding measures such

as Signal-to-Noise Ratio (SNR) and Carrier-to-Noise

density ratio (C/N0) are also conventional quality metrics to

detect multipath and even mitigate its effect through

stochastic weighting models [14,15]. Combination of code

and carrier phase measurements is another approach for

pseudorange multipath correction [12]. Since the

pseudorange multipath error is considerably larger than that

of the carrier phase, the code-minus-carrier measurement is

mostly an (ambiguous) indication of pseudorange multipath,

which can be used for multipath error correction [16-18].

In addition to the above approaches, three-Dimensional

Building Models (3DBMs) have been investigated to

improve GNSS-based positioning performance in urban

multipath environments. These augmentation techniques

incorporate the geospatial information of nearby reflectors

using a 3D model of buildings and surroundings [19]. This

can be generally performed either through “shadow

matching” algorithms [20,21], modeling and exploiting path

delay of NLOS signals [19] or multipath fault detection and

exclusion [21].

Given the above, GNSS multipath countermeasures can

be generally divided in three major groups. Firstly, some

attempt to isolate the receiver from multipath interference or

minimize its effect by modifying the receiver antenna or

tracking design. The second group tries to jointly estimate

the multipath parameters and subsequently correct multipath

errors or mitigate their effects. The third group focusses on

detection techniques where gross errors caused by multipath

can be specifically reduced or eliminated by detecting and

excluding (or de-weighting) affected measurements. Thanks

to the development of multi-GNSS constellation receivers,

the third approach is increasingly effective as redundancy is

sufficiently large to detect and exclude (or de-weight) faulty

measurements. An effective and practical implementation of

such a solution requires the design and implementation of

appropriate algorithms to detect and de-weight distorted

signals without significant degradation in measurement

geometry, which is also a focus of this paper. In Section 2,

the methodology is described based on Measurement

Quality Monitoring (MQM) techniques in three general

steps. First, a multipath error correction is applied based on

combination of erroneous (but un-ambiguous) code phase

and precise (but ambiguous) carrier phase measurements. If

followed by a filtering approach, it is shown that the output

of such a combination provides a direct measure of the code-

phase multipath error under cycle slip free conditions, which

results in the possibility of partial pseudorange correction.

Second, detection metrics are defined by using partially

corrected measurements to detect remaining multipath errors.

A new iterative algorithm is then developed to de-weight

faulty measurements prior to being used in the navigation

solution. In Section 3, development of a height-aided

GPS/Galileo combined solution is investigated to improve

measurement redundancy and thus detection/de-weighting

performance. Since the performance of measurement-based

detection and de-weighting techniques is limited by

measurement geometry, higher performance is expected

when satellite geometry and redundancy of the

measurements is improved in a multi-constellation system.

This is investigated based on a GPS/Galileo combined

solution. A field test analysis is provided in Section 4 to

examine the performance of the above techniques under a

real multipath scenario. Section 5 deals with conclusions

reached in this research.

II. METHODOLOGY

Fig. 2 shows the procedure of the developed MQM-based

Detection and Iterative De-weighting (D & I-D) of

multipath-affected measurements in a height-constrained

GPS/Galileo combined solution. In order to reduce the

number of de-weighted measurements and thus preserve

measurement geometry, the D & I-D process is preceded by

a Code-Minus-Carrier (CMC)-based multipath correction as

discussed below.

Fig. 2. MQM-based D & I-D in a height-constrained GPS/Galileo combined solution

A. MQM for Multipath Correction

CMC is computed by subtracting the carrier phase

measurements from the corresponding pseudoranges. By

doing so, at each navigation epoch, the CMC metric is

calculated for the lth satellite as

, , , , ,2

l l l

p l ion l l p l l l

CMC p

MP d N MP

(1)

where

lp and

l are the corresponding code and carrier phase

measurements (both converted to units of length),

,ion ld is ionospheric delay,

and lN are wavelength and ambiguity, and

, , ,, ,p l p l lMP MP and

,l are code and carrier multipath

and noise errors [22]. Since integer ambiguities are constant during a cycle slip-

free period and ionosphere changes are minimal during short

periods, their effect are approximately estimated and

removed based on a moving time average sufficiently short

to avoid biases due to changing geometry and prevent the

accumulation of systematic errors. Since cycle slips may

result in a new unknown carrier phase ambiguity, the moving

average buffer is reset if a cycle slip is detected. Therefore,

the pseudorange multipath error can be extracted by the

following CMC-based monitoring metric:

,cmc l l lm CMC CMC (2)

where lCMC denotes the mean value of the CMC metric

computed by the moving average. The output of the CMC

metric is then directly used for pseudorange correction as

,ˆ

l l cmc lp p m (3)

where ˆlp is the corrected pseudorange. Due to the

dependency of the CMC on carrier phase measurements, the

major limitation is the need to restart the time averaging

process in the event of a cycle slip as it requires re-

estimation of the ambiguity [22].

B. MQM for Multipath Detection

For the case of a multi-frequency receiver, a Geometry-

Free (GF) detection metric is formed by differencing

pseudoranges on two frequencies as follows:

1 2 1 2 1 2

,

f f f f f f

GF l l l lm p p d (4)

where 1 2f f

ld includes the difference in the ionospheric

effects and inter-frequency receiver biases. This

combination removes the geometric components of the

measurements, but includes the frequency-dependent effects

(e.g. ionospheric delay), multipath and measurement noise.

1 2f f

ld is estimated and removed through time-averaging;

the remainder is used to monitor code-phase multipath.

Since the GF metric is a combination of errors on two

frequencies, it does not provide separate information about

multipath on each frequency and thus is used only for

satellite-by-satellite detection and de-weighting. The GF

monitoring metric is directly proportional to code phase

multipath errors and from this point of view outperforms the

conventional C/N0-based detection metrics which are more

sensitive to short-delay multipath (fading) with a small

impact on code tracking errors [23,24].

The main advantage of the GF metric is its capability to

be used after the CMC-based error correction as shown in

Fig. 2. Multipath errors are first alleviated by applying CMC-

based error estimation and correction. Although effective in

reducing the effect of multipath, this approach does not fully

remove multipath errors especially where the time averaging

process is restarted in the event of a cycle slip. In the next

step, the GF detection metrics are formed by differencing

(partially) corrected pseudoranges on two or more

frequencies to detect the remaining multipath errors.

To improve detection performance, a fixed-lag sliding

window is used to take a window of N samples (based on

current and N−1 preceding samples) and compare them to a

predefined threshold. If M ( M N ) or more samples exceed

the threshold, then the detection output is 1 and otherwise 0.

This procedure is then repeated for the next window. With

this detection strategy, the overall probability of false alarm

in N trials is given by [12]

1

0

1 1M

N nn

FA fa fa

n

NP P P

n

(5)

where N

n

is the number of combinations of N items taken

n at a time and faP is the false alarm probability in each trial

and equals 0.27% under a normal distribution and three

times Standard Deviation (SD) as the detection threshold. In

the case of ( , ) (1, 1),N M the detection strategy is

considered a general likelihood ratio test by comparing each

sample with the detection threshold at each epoch. By a

proper selection of N and M, the strategy behaves like a

moving average shrinking the probability distribution

variance for the null (when there is no or low multipath) and

the alternate (when multipath exists) hypotheses on the two

sides of the detection threshold. This can improve detection

performance by decreasing the false alarm probability in the

case of the null hypothesis while increasing the probability

of detection in the case of the alternate hypothesis, but at the

expense of latency in the transition from the null to the

alternate hypothesis and vice versa. Given the periodic

nature of GNSS multipath (or even when multipath behaves

more like noise in high dynamic scenarios), a side effect of

this latency is a reduction in the rate of change between the

two detection states. In the case of exclusion or de-

weighting, the lower rate of change has the benefit of

smoothing the resulting positions.

C. Measurement Weighting

Once multipath is detected, the position solution should

be protected from multipath through excluding or de-

weighting the affected measurement(s). The critical point is

the effect of exclusion or de-weighting on measurement

geometry. Poor geometry may magnify the effects of

remaining errors and worsen the ultimate position solution.

To counter the above, a geometry-based iterative de-

weighting approach is developed. In the navigation solution

all measurements are used, but the contribution of distorted

measurements is iteratively reduced. Weighted Dilution of

Precision (DOP) is considered as a geometry monitoring

metric. Since horizontal accuracy is the major concern, the

Horizontal DOP (HDOP) is adopted. At each epoch, the

HDOP is calculated based on the root sum square value of

the east and north variance factors of the estimated position.

When it is assumed that all the measurements have equal

variances and are uncorrelated, the variance factors of the

estimated parameters are shown to be only as a function of

satellite-receiver geometry and thus the calculated DOP is

referred to as “pure DOP”. Under different variances for

different measurements, the position variance factors are also

affected by the relative weights of the measurements and

used to determine a so-called “weighted DOP” [22] which is

considered here. Based on a pre-defined increasing function,

the detected measurements are de-weighted until a tolerable

HDOP threshold is exceeded. Due to practical

considerations, the maximum number of iterations is limited

to a number large enough to verify the appropriateness of the

de-weighting procedure. The remaining measurements (those

not detected as faulty) are stochastically weighted using

constant, elevation or C/N0-based weighting models [22],

developed under low-multipath conditions. The iterative de-

weighting procedure is shown in Fig. 3 where the variance of

measurements detected as faulty is increased to de-weight

these. For each measurement epoch,

- ni is the iteration number and

Maxi denotes the maximum

number of iterations,

- 0HDOP is the HDOP value before iterative de-weighting

of detected measurements,

- HDOPTh is the HDOP threshold,

- dN is the number of detected measurements,

- 1, 2, ..., dl N is the detected measurement index,

- , nl i

w is the weight of the thl detected measurement at the

th

ni iteration,

- 2

, nl i is the variance of the thl detected measurement at

the th

ni iteration,

- 2 2

,0 ,0l l lw is the initial weight of the thl detected

measurement, determined based on the stochastic

weighting model,

- ( )nF i is a pre-defined increasing function of iteration

number, and

- , nid iHDOP is the HDOP value after the th

ni iteration of the

de-weighting procedure applied to the detected

measurements.

Fig. 3. Geometry-based iterative change of measurement weights

Selection of HDOP Threshold and Stochastic Model

Parameters

Limiting the HDOP value to a pre-defined threshold

maintains horizontal geometry above a certain level, but at

the same time, it prevents the system from being thoroughly

isolated from the effect of faulty measurements. The

threshold boundary should be wide enough to ensure the

effectiveness of the de-weighting procedure. Therefore, a

trade-off is involved and the HDOP threshold should be

selected carefully. Under an effective detection process, the

stochastic model for those measurements not detected as

multipath can be modeled under assumed low multipath

conditions.

Selection of De-weighting Function

The main goal of iterative de-weighting is to minimize

the incorporation of detected measurements in the position

solution for a given HDOP threshold. Therefore, while any

linear or non-linear function of iteration numbers can be

used to de-weight detected measurements, a proper function

should take the trade-off between the resolution and

complexity of the de-weighting process into account. The

smaller size of de-weighting factor (at each iteration) will

control geometry degradation with a higher resolution at the

expense of a larger number of iterations for the entire

process and thus a heavier computational burden. Herein,

since the GF metric is a combination of errors on two

frequencies and the relation between detection statistics and

measurement errors is not known, when multipath is

detected, a scale factor is simply multiplied by the iteration

number to define the corresponding measurement variance

factor.

At each measurement epoch, the variance of the lth

measurement in the th

ni iteration is calculated as follows:

2 2

,

if multipath is detected

1 otherwisen

MQM n

l i l

a i

(6)

where 2

l is the initial variance of the corresponding

measurement determined based on the stochastic weighting

model under low multipath conditions, and MQMa is a scale

factor to be set based on the desired de-weighting resolution

and maximum number of iterations. For undetected

measurements, any selection of the conventional stochastic

weighting models such as constant, elevation or C/N0-based

algorithms can be adopted.

III. HEIGHT-CONSTRAINED GPS/GALILEO COMBINED

SOLUTION

A formulation is now provided to combine GPS and

Galileo signals in a pseudorange-based Least-squares (LS)

solution and constrain the height and hence improve the

measurement redundancy and the performance of the

detection and de-weighting algorithms. The height can be

constrained for a short or long duration depending on the

stability of the altimeter used; a correction for the geoid is

also applied to transform the altimetric height [above sea

level] to that above the reference ellipsoid selected. The

above procedure is done in three steps as follows.

A. LS Adjustment for a Pseudorange-based Positioning in

Local Coordinate System

In a single constellation solution, the state vector includes

receiver’s three-dimensional (3D) position parameters and

receiver clock offset typically defined in the Cartesian Earth-

Centered Earth-Fixed (ECEF) coordinate system as

, , ,T

r r rx y z cdTx or equivalently in curvelinear

coordinate system as latitude, longitude and height (plus

receiver clock offset) noted by , , ,T

r r rh cdT x . By

this definition, the distance between receiver and lth

satellite

( 1, 2, ...,l L with L as the number of satellites tracked)

can be defined as

,

sl

l r rx x x (7a)

,

sl

l r ry y y (7b)

,

sl

l r rz z z (7c)

where , ,sl sl slx y z is the Cartesian position of the lth

satellite. To ultimately constrain the height, the XYZ

distance parameters are rotated into local East, North and

Up (i.e. vertical or height) or ENU system as

, ,

, ,

, ,

l r l r

l r rot l r

l r l r

e x

n y

u z

R (8)

where rotR is the rotation matrix and is determined as [25]

sin( ) cos( ) 0

sin( )cos( ) sin( )sin( ) cos( )

cos( )cos( ) cos( )sin( ) sin( )

r r

rot r r r r r

r r r r r

R (9)

By these definitions, at each measurement epoch, with L

satellites tracked, the relation between pseudorange

measurements and receiver-satellite geometry is defined as

2 2 2

1, 1, 1,

,11

2 2 2

,22 2, 2, 2,

,2 2 2

, , ,

( )

( )

r r r

p

pr r r

p LL

L r L r L r

h

e n up

p e n ucdT k

pe n u

p e

x

(10)

where lp and

,p l denote the corresponding pseudorange

measurement and pseudorange error with respect to the lth

satellite. By linearizing (10) with respect to an initial

estimate of the state vector [i.e.

0 0 0 0 0 0 0 0 0, , , , , , ,T T

r r r r r rx y z cdT h cdT x one

obtains

P H x e (11)

where

P is a 1L vector whose lth

row element is defined as

, 0

2 2 2

, 0 , 0 , 0

l r

l l l r l r l rp p e n u

(12)

0( )h

H x xx

is a 4L matrix whose lth

row element

( 1, 2, ...,l L ) in each column is defined as

, 0

,1 , 0 , 0

, 0

cos( )sin( )l r

l l r l r

l r

eh el az

(13a)

, 0

,2 , 0 , 0

, 0

cos( )cos( )l r

l l r l r

l r

nh el az

(13b)

, 0

,3 , 0

, 0

sin( )l r

l l r

l r

hh el

(13c)

,4 1lh (13d)

and ( , , , )Te n h cdT x is the error between actual

and initial estimate of the state vector to be estimated in

the ENU coordinate system.

By doing so, the LS-based estimation of the state error is

obtained by [22]

1

ˆ T T

x H WH H W p (14)

where W is a weight matrix determined based on the

stochastic model of measurement variances. Through ˆx ,

the initial estimates can be refined and the entire procedure

can be iterated until the state error converges to zero (e.g.

sub-metre level).

B. GPS-Galileo Combined Solution

A multi-constellation solution require consideration of

the differences in ephemeris data, coordinate frames and

reference time frames. Herein, this is done under a

GPS/Galileo combined solution where similarity of

ephemeris data and coordinate frames facilitates the

implementation of the combined solution.

Satellite Position and Clock Corrections

The first step in a position solution is to extract satellite

orbital information and satellite clock corrections provided

by ephemeris data. This requires design and implementation

of proper algorithms to decode ephemeris data for GPS and

Galileo satellites.

Coordinate Frames

GPS uses the World Geodetic System 1984 (WGS-84)

and Galileo uses the Galileo Terrestrial Reference System

(GTRF), both of which are different realizations of the

International Terrestrial Reference System (ITRS).

However, the offset between WGS-84 and GTRF is at the

sub-metre level, which is negligible for the present analysis

and thus the coordinate transformation is neglected [26].

Reference Time Frames

GPS and Galileo use different time frames namely

Galileo System Time (GST) and GPS use GPS Time

(GPST). The receiver clock offsets need to be estimated

along with position parameters. The receiver clock offsets

are common for all corresponding pseudoranges and hence

can be estimated as an unknown parameter through the

navigation solution. Therefore, in a GPS/Galileo combined

solution, there will be two different clock offsets with

respect to each timeframe which can be estimated along

with three position parameters. This will increase the

number of unknowns from 4 to 5 but with the advantage of

improved geometry and a higher number of available

measurements provided by different GPS and Galileo

satellites [26]. Under this scenario, with GPSL GPS and GALL

Galileo measurements and using corresponding indices, (11)

is rewritten as

1 1

1 1

GPS GPS

GAL GAL

GPS GPSL L

GAL GALL LGPS

GAL

e

n

h

cdT

cdT

P eH

P e (15)

and H will be a 5GPS GALL L matrix whose lth

row

element in each column is defined as follows

,1 , 0 , 0cos( )sin( ) for 1, ...,l l r l r GPS GALh el az l L L (16a)

,2 , 0 , 0cos( )cos( ) for 1, ...,l l r l r GPS GALh el az l L L (16b)

,3 , 0sin( ) for 1, ...,l l r GPS GALh el l L L (16c)

,4

1 for 1, ...,

0 for 1, ...,

GPS

l

GPS GPS GAL

l Lh

l L L L

(16d)

,5

0 for 1, ...,

1 for 1, ...,

GPS

l

GPS GPS GAL

l Lh

l L L L

(16e)

C. Height-constrained GPS/Galileo Solution

In order to add redundancy, one solution is to constrain

the receiver height when the latter is obtained independently.

When height information is available, the position solution is

reduced to a two-dimensional horizontal solution with

improved measurement redundancy and higher performance

for de-weighting of faulty measurements. In this scenario,

(15) is rewritten as

1 1

1 1

GPS GPS

GAL GAL

GPS GPSL L

GPSGAL GALL L

GAL

e

n

cdT

cdT

P eH

P e (17)

and H will be a 4GPS GALL L matrix whose lth

row

element in each column is defined as follows:

,1 , 0 , 0cos( )sin( ) for 1, ...,l l r l r GPS GALh el az l L L (18a)

,2 , 0 , 0cos( )cos( ) for 1, ...,l l r l r GPS GALh el az l L L (18b)

,3

1 for 1, ...,

0 for 1,...,

GPS

l

GPS GPS GAL

l Lh

l L L L

(18c)

,4

0 for 1, ...,

1 for 1, ...,

GPS

l

GPS GPS GAL

l Lh

l L L L

(18d)

IV. FIELD DATA ANALYSIS

A field test was performed using GPS and Galileo signals

under a static multipath scenario. A Trimble R10 GNSS

receiver was used in a high multipath environment to collect

GPS L1 (L1 C/A)/L2 (L2C M+L)/L5 and Galileo E1 (E1

B+C)/E5a/E5b signals with a 1 sample/second sampling rate.

Fig. 4 shows the test site location near reflective surfaces and

the satellite geometry (using a 5 degree elevation mask) at

the beginning of the test.

(a)

(b)

Fig. 4. Multipath data collection; (a) a Trimble R10 antenna-receiver

surrounded by reflectors; (b) Satellite geometry with a 5 degree of elevation

mask

A. Monitoring Results

Time-averaging was used to estimate the mean value of

the different monitoring metrics. A simple moving average

was used with a length of 10 minutes (min) selected based

on twice the average period observed for “quasi-periodic”

oscillations of PRNs exhibiting static multipath. Fig. 5

shows the CMC-based monitoring metrics and their

corresponding Root Mean Square (RMS) values [in metre

(m)] for a sample of GPS signals (PRN 09) with different

frequencies. For PRN 09, it is shown that multipath is

relatively low in the first half of the data while it is higher

during the rest of the test on all three frequencies. Besides

multipath characterization, the CMC-based monitoring

metric is used to correct the corresponding pseudoranges

according to (3).

In addition to affecting code and carrier phase

measurements (and thus CMC metric), multipath affects the

measured signal power and C/N0 values as shown in Fig. 6

[in decibel-Hertz (dB-Hz)]. The satellite elevation angle

change is also shown on the right vertical axis [in degree

(deg)]. The power of the combined signal (and consequently

the measured C/N0) fluctuates with time because it is

affected by the time-varying phase lag between the direct

and reflected signal(s), as these add constructively or

destructively with each other. Since the phase lag is

frequency dependent, the C/N0 is affected differently by

different frequencies.

Fig. 5. CMC-based monitoring metric for GPS L1 (blue), L2 (red) and L5

(green); PRN 09 affected by different levels of multipath error

Fig. 6. C/N0 measurements for PRN 09, GPS L1 (blue), L2 (red) and L5

(green)

As discussed in Section 2B, for detection purposes, GF

measurements were first obtained by differencing

pseudorange measurements on different pairs of the

available frequencies. At this stage, the pseudorange

measurements used to form GF detection metrics are still

not corrected by CMC to examine the detection performance

of the GF metric under multipath occurrence. The

corresponding mean values are then filtered (by the same

moving average discussed before) and resulting outputs are

normalized based on an estimation of the noise variance

calibrated under low-multipath conditions. For the resulting

unitless monitoring metric, the detection thresholds were set

to ±3 as three times the normalized nominal SD for GPS L1,

L2 and L5 signals. Referring to Section 2C, the N value was

set equal to half of the moving average length (i.e. 5 minutes

or 300 samples of measurements with the input rate of 1

sample/second) and M was set to 10 to satisfy the false

alarm probability of 81     .41  10 under multipath-free

conditions and normal distribution of the detection metric

[see (5)]. Fig. 7 shows the normalized GF-based monitoring

metrics, threshold excess and corresponding M of N

detection results. It is observed that when multipath error

exceeds 3 m, it is effectively detected most of the time. The

GF-based detection performance is limited to the PRNs with

signals on at least two different frequencies. For example,

when detection of multipath on GPS L1 is desired, the

multipath errors (not corrected by CMC) without

corresponding L2 and/or L5 measurements will remain

undetected.

(a)

(b)

Fig. 7. (a) Normalized GF-based detection metric and (b) detection outputs

for GPS L1/L2/L5 signals; PRN 09

Fig. 8 to Fig. 10 show corresponding results for Galileo

PRN 30 where the constructive and destructive effects of

multipath are observed on each frequency with different

phase lags. It is also observed that there are intervals in

which the detection metrics are within the detection

boundaries but multipath is detected. This is a result of the

M of N latency discussed in Section 2B. Given the periodic

nature of GNSS multipath, a side effect of the M of N

latency is to smooth detection and ultimately position results

as shown in the next sub-section.

Fig. 8. CMC-based monitoring metric for Galileo E1 (blue), E5a (red) and

E5b (green); PRN 30 affected by different levels of multipath error

Fig. 9. C/N0 measurements for Galileo PRN 30; Galileo E1 (blue), E5a (red) and E5b (green)

(a)

(b)

Fig. 10. (a) Normalized GF-based detection metric and (b) detection outputs

for Galileo PRN 30 E1/E5a/E5b signals

B. Positioning Performance

Resulting height-constrained GPS/Galileo pseudorange-

based LS solutions are now discussed. First, the GPS

L1/Galileo E1 [both with 1575.42 Megahertz (MHz) carrier

frequency] combined solution was obtained under a constant

weighting scheme. Since all satellites do not broadcast

signals on all frequencies at this time, single frequency

positioning was considered, whose accuracy is sufficient for

the current analysis. The effect of the ionosphere on

positions during the test was evaluated separately and was

below 1 m. For monitoring purposes, the GF-based detection

metrics were extracted on as many PRN (with signals on at

least two frequencies) as possible.

The performance of the position solution with no

correction and no measurement de-weighting (A0) was

compared to the case when the GF-based D & I-D was

applied (A1). For GPS L1/Galileo E1 positioning (when GPS

and Galileo signals are monitored on all three frequencies,

but only GPS L1 and Galileo E1 pseudoranges are used for

positioning), GF-based results used for multipath detection

include those provided by the GPS L1-L2 and L1-L5, and

Galileo E1-E5a and E1-E5b which involve either L1 or E1 in

their definitions. For GPS L1 measurements used in the

navigation solution, multipath is detected if at least one of

the corresponding L1-L2 or L1-L5 M of N detection outputs

is one; for Galileo E1 measurements, the occurrence of

multipath is considered when either E1-E5a or E1-E5b GF

metric (or both) detects multipath. A similar argument is

used for GPS L2/Galileo E5b and GPS L5/Galileo E5a

positioning solutions which will be investigated later in this

section. Referring to Section 2C, an empirically tested value

of 5 was chosen as an appropriate weighted HDOP threshold

to avoid geometrically poor solutions that would likely not

benefit from measurement de-weighting. With the maximum

number of iteration equal to 10, scale factor is empirically set

to 10 [see (6)] to perform an effective and relatively low

complexity de-weighting procedure. The performance of the

CMC-based error correction (A2) and its combination with

the GF-based D & I-D (A3) were investigated next.

Fig. 11 shows the number of satellites tracked along with

the weighted HDOP values for a constant weight LS solution

and the positioning approaches discussed. It is observed that

while iterative de-weighting of detected measurements

degrades the HDOP values (under the defined HDOP

threshold), the preceding CMC-based correction can reduce

the level of degradation. Horizontal position errors have been

presented in Fig. 12 where the complementary combination

of CMC-based error correction and GF-based D & I-D

shows improvement over each single monitoring approach.

Fig. 11. Number of satellites tracked and HDOP values for a constant

weight LS solution; A0 (red): no correction, no de-weighting; A1 (blue):

with D & I-D; A2 (same as A0 ploted in red): with CMC-based error correction with no de-weighting; A4 (light green): combination of a

preceding CMC-based error correction followed by the GF-based D & I-D

Fig. 12. Horizontal position errors for constant weighting and GPS L1 &

Galileo E1 combined solution; A0 (red): no correction, no de-weighting

(benchmark); A1 (blue): with D & I-D; A3 (dark green): with CMC-based error correction; A4 (light green): combination of a preceding CMC-based

error correction followed by the GF-based D & I-D

Table 1 shows the numerical results for GPS L1/Galileo

E1 (1575.42 MHz carrier frequency), GPS L2/Galileo E5b

(1227.60 and 1207.14 MHz carrier frequency) and GPS

L5/Galileo E5a (1176.45 MHz carrier frequency)

positioning solutions. Position errors are compared for

different scenarios including constant, elevation and C/N0-

based stochastic models and for different monitoring

approaches from A0 to A3. It is generally shown that while

improvement is limited (and not always the case) for each

single MQM approach (A1 and A2), the combination of the

CMC-based error correction followed by the GF-based D &

I-D (A3) generally shows higher performance. Comparing

horizontal RMS errors for the GPS L1/Galileo E1 solution,

A3 shows 59, 16 and 19 percent improvement (over A0 as

the benchmark) when conventional constant, elevation and

C/N0-based stochastic models are used in the position

solution. These values for the GPS L2/Galileo E5b solution

are 20, 12 and 10 percent and for GPS L5/Galileo E5a

solution, are 51, 27 and 28 percent under the designed

multipath test scenario.

TABLE I. HORIZONTAL PERFORMANCE FOR DIFFERENT POSITIONING

APPROACHES

Combined

Signals

Weighting

Model MQM

Horizontal

RMS Error

(m)

Improvement

A3 Over A0

GPS L1 &

GalileoE1

(1575.42

MHz)

Constant

A0 3.24

59% A1 1.86

A2 2.85

A3 1.33

Elevation-

based

A0 1.24

16% A1 1.78

A2 1.04

A3 1.04

C/N0-based

A0 1.75

19% A1 1.92

A2 1.68

A3 1.42

GPS L2 &

Galileo E5b

(1227.60

and 1207.14

MHz)

Constant

A0 2.66

20% A1 4.27

A2 2.17

A3 2.12

Elevation-

based

A0 2.98

12% A1 5.76

A2 2.52

A3 2.62

C/N0-based

A0 2.84

10% A1 4.47

A2 2.30

A3 2.57

GPS L5 &

Galileo E5a

(1176.45

MHz)

Constant

A0 5.10

51% A1 3.69

A2 7.02

A3 2.52

Elevation-based

A0 2.41

27% A1 2.89

A2 2.80

A3 1.75

C/N0-based

A0 2.76

28% A1 2.94

A2 4.01

A3 1.99

A0: No correction, No De-weighting (Benchmark)

A1: GF-based D & I-D

A2: CMC-based error correction

A3: CMC-based error correction and GF-based D & I-D (Combined

Method)

V. CONCLUSIONS

For the case of a dual-frequency receiver, a GF-based

detection metric was investigated herein, given its capability

to be combined with a preceding CMC-based error

correction. GPS and Galileo measurements were combined

with the addition of a height-constrained horizontal solution

to improve redundancy and consequently, the performance

of de-weighting faulty measurements. While de-weighting

of faulty measurements is challenging due to its effect on

satellite geometry, a geometry-based iterative de-weighting

algorithm was used to reduce the effect of multiple

multipath signals below a tolerable HDOP threshold.

Analysis of Galileo and GPS data showed that while each

monitoring technique has its own limitations, a

complementary combination of them can be used for

reliable multipath mitigation. Multipath errors were first

alleviated by applying CMC-based error corrections on

pseudorange measurements and then the GF-based detection

metrics were performed by differencing partially corrected

pseudoranges on each pair of available frequencies to detect

and de-weight remaining code multipath errors. Field data

results obtained for a static multipath scenario showed 10 to

almost 60 percent improvement in horizontal positioning

performance.

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