Acta IMEKO, Title


ACTA IMEKO 
ISSN: 2221-870X 
June 2018, Volume 7, Number 2, 16-23 

 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 16 

Using local redundancy to improve GNSS absolute 
positioning in harsh scenario 
Antonio Angrisano1, Silvio Del Pizzo2, Salvatore Gaglione3, Salvatore Troisi3, Mario Vultaggio1  

1G.Fortunato University, Benevento, Italy 
2iNav Geomatics Solutions, Massa Lubrense (NA), Italy 
3Parthenope University, Naples, Italy 

 

 

Section: RESEARCH PAPER  

Keywords: GNSS; weighting schemes; redundancy matrix; local redundancy 

Citation: Antonio Angrisano, Silvio Del Pizzo, Salvatore Gaglione, Salvatore Troisi, Mario Vultaggio, Using local redundancy to improve GNSS absolute 
positioning in harsh scenario, Acta IMEKO, vol. 7, no. 2, article 4, June 2018, identifier: IMEKO-ACTA-07 (2018)-02-04 

Section Editor: Fabio Leccese, Università degli Studi di Roma Tre, Italy 

Received January 15, 2018; In final form March 26, 2018; Published June 2018 

Copyright: © 2018 IMEKO. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 License, which permits 
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited 

Corresponding author: Antonio Angrisano, e-mail: : a.angrisano@unifortunato.eu 

 

1. INTRODUCTION 
In GNSS context, scenarios characterized by bad signal 

quality and/or bad satellite geometry, usually due to natural or 
artificial obstacles, are considered harsh environments. 
Mountainous or highly urbanized areas are examples of harsh 
environments. In urban scenarios, multipath interference and 
non-line-of-sight (NLOS) phenomena typically affect the 
GNSS signals. The multipath interference happens when a 
signal is received through multiple paths; the combination of 
multiple signals causes a distortion of the correlation function 
(between received and locally generated signals) and yields a 
range error up to several tens of meters [1]. The NLOS 
reception happens when the direct signal is blocked and only 
reflected signals are received; this phenomenon is very common  

 
 

in urban scenario, yielding range errors of even km order [2]. 
The NLOS and multipath phenomena are even more 
widespread in high sensitivity receivers, which are able to 
receive very weak signals. 

Absolute positioning, also called Single Point Positioning 
(SPP), is the most common GNSS operational mode. In fact, 
SPP mode is usually adopted by mass-market receivers, which 
are mounted on tablet, smartphones, car navigation devices, 
UAVs [3]-[5]; on these devices, SPP is often combined with 
other information sources, such as maps, inertial sensors or 
vision systems [6], [7], [8]. 

The aforementioned phenomena are responsible for the 
presence of blunders among the measurements, which can 
cause large position error. Two approaches are usually adopted 
to cope with this problem: 

ABSTRACT 
In GNSS context, absolute positioning is a widespread operational mode, largely used in many field activities as automotive, aerospace 
and ships navigation. The functional model of absolute positioning, relating pseudorange measurements to unknowns, is well defined, 
while its stochastic model, defining the behavior of measurement error, is currently under investigation and is the core of this work.  
A realistic model of measurement error can be used to define a weighting scheme, able to reduce the positioning errors; a weighted 
approach is mainly necessary in signal-degraded scenario, where pseudorange accuracies are significantly different and equally 
weighting all the measurements would yield very large errors. 
The quality indicators, commonly adopted to define a pseudorange error model, are the signal-to-noise ratio and the satellite 
elevation angle. In this work, an additional indicator, the redundancy number, is introduced. 
The redundancy numbers are the diagonal elements of the redundancy matrix and they represent the degree of controllability of the 
measurements; a large value of the redundancy number corresponds to a well-controlled measurement, while a small one 
corresponds to a leverage observation, with a high potential to influence the solution. 
The benefits of using the proposed weighting schemes is demonstrated in urban canyon and with single (GPS only) or multiple 
(GPS/Glonass) constellations. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 17 

• Receiver Autonomous Integrity Monitoring (RAIM) 
techniques, able to detect and reject blunders [9]-[11]; 

• Robust estimation techniques, able to absorb the 
effects of the blunders [12]. 

In this work, it has been followed a different approach, 
consisting in defining a suitable weighting scheme, able to de-
weighting potentially dangerous measurements. A capillary 
survey has been carried out on the existing measurement 
weighting schemes, which are tested in urban environment; 
afterwards, an additional quality criterion is proposed in order 
to limit the effect of potential blunders. 

In literature, two quality indicators are typically associated to 
the GNSS measurements, in order to differently weighting 
them: satellite elevation angle and carrier-to-noise ratio.  

Satellite elevation (El) is considered a quality indicator of a 
measurement, because satellites at low elevation are usually 
noisier, due to typical behaviour of multipath, tropospheric and 
ionospheric errors [1]. 

The Signal-to-Noise ratio (S/N), or better the Carrier-to-
Noise ratio (C/N0), is the ratio between the carrier power and 
the noise power per unit bandwidth and is usually expressed in 
Decibel-Hertz; C/N0 is a measure of signal strength, so it is a 
suitable indicator of measurement quality [10]. 

The El and C/N0 indicators are adoptable individually or 
synergistically to define a model of measurement error variance; 
a weighting scheme can be obtained, inverting the variances of 
the measurement errors. 

The positioning solution is differently influenced by each 
measurement of a dataset; indeed, some measurements, called 
leverage observations, have high potential to influence the 
solution, while others are less influent. If a leverage observation 
is affected by a gross error, a very large solution error could 
occur; for this reason, the leverage observations are critical. It 
can be demonstrated that the concept of leverage observation is 
related to the local redundancy [13], which is well represented 
by the diagonal elements of redundancy matrix, called 
redundancy numbers. 

In urban scenario, the satellite geometry is often weak and 
the presence of blunders is very common; in this conditions, a 
blunder on a leverage observation could strongly degrade the 
solution. 

In this work, an innovative weighting scheme, which takes 
into account the local redundancy information, is proposed and 
its effectiveness is shown in a very harsh environment like an 
urban canyon. The proposed scheme can be superimposed to 
the existing weighting strategies, which involve the indicators El 
and C/N0, and the benefits of this combined approach are 
demonstrated. 

The proposed weighting scheme is applied to GPS only and 
to GPS/Glonass cases, demonstrating its usefulness in both 
configurations. 

Below, the SPP operational mode is briefly described; 
afterwards, the existing weighting schemes, based on El and/or 
C/N0, are described. Then the redundancy matrix, whose 
diagonal elements are the redundancy numbers, is defined and 
is linked up to the concept of local redundancy. The possible 
contribution of the redundancy matrix to a weighting strategy is 
shown. At the end, the proposed weightings are applied to real 
data and the results are discussed. 

2. SINGLE POINT POSITIONING 
In SPP, the functional model, defining the relationship 

between measurements and unknowns, is 
𝒛 = 𝐻∆𝒙 + 𝜺 (1) 

where 𝒛 is the vector of measurements, defined as the 
difference between measured and computed (with a priori 
information) pseudorange, 𝐻 is the design matrix, ∆𝒙 is the 
state vector, containing the corrections to update the receiver 
coordinates and clock offset, ε is the measurement error vector. 

Equation (1), also referred to as measurement model, is a 
linearized equation; the receiver coordinates and clock offset 
are obtained correcting the a priori information with the 
estimated state vector. 

The stochastic model, expressing the uncertainty of the 
measurement model, is: 

𝐶 = �
𝜎12 ⋯ 0
⋮ ⋱ ⋮
0 ⋯ 𝜎𝑚2

� (2) 

where 𝐶 is the variance-covariance matrix of the measurement 
errors; 𝜎𝑖=1,…,𝑚

2  are the measurement error variances; 𝑚 is the 
number of measurements. 

The matrix 𝐶 is diagonal, under the usual assumption of 
independent measurement errors. 

The number of measurements is usually larger than the 
number of unknowns and the equation (1) is solved using 
weighted least squares (WLS) method. 

WLS solution is: 

∆𝒙� = (𝐻𝑇𝑊𝐻)−1𝐻𝑇𝑊𝒛 (3) 

where 𝑊 is the weighting matrix, usually set equal to the 
inverse of 𝐶. 

3. WEIGHTING SCHEMES REVIEW 
Most of weighting schemes in literature are based on C/N0, 

satellite elevation El or both; hence, the weighting strategies 
review is organized according to the considered quality 
indicators. 

3.1. Elevation-based weighting schemes 
The most widespread variance models for GPS carrier 

phase, based on satellite elevation, depend on sin (𝐸𝐸) or 
sin2(𝐸𝐸) functions [14], [15]; despite being designed for carrier 
phase, they were simply adapted for pseudorange observation 
[16], [17] as shown below: 

𝜎𝑃𝑃2 =
𝜎02

sin(𝐸𝐸)
 

 (4) 

𝜎𝑃𝑃2 =
𝜎02

sin2(𝐸𝐸)
 

  (5) 

where σ02 is the pseudorange error variance when 𝐸𝐸 = 90°. 
Slight modifications to the previously mentioned models are 

proposed in literature [18], maintaining the concept that low 
elevation satellites correspond to noisy measurements:  

𝜎𝑃𝑃2 = 𝐴2 +
𝐵2

sin2(𝐸𝐸)
 

(6) 

𝜎𝑃𝑃2 =
𝜎02

tan2(𝐸𝐸 − 𝜃)
 

(7) 

where 𝐴, 𝐵, 𝜃 are parameters to be defined according to the 
receiver and the operational scenario. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 18 

A more complex variance model, which takes into account 
the contributions from several measurement error sources, is 
defined in [19], [20]; it is designed for Wide Area Augmentation 
System (WAAS) and it is shown below: 

𝜎𝑃𝑃2 = 𝜎𝑈𝑈𝑃𝑈2 + 𝐹2(𝐸𝐸) ∙ 𝜎𝑈𝑈𝑈𝑈2 + 𝜎𝑆𝑆𝑃2 +
𝜎𝑚452

tan2(𝐸𝐸)
+

𝜎𝑡𝑡𝑡2

sin2(𝐸𝐸)
 

(8) 

where 𝜎𝑈𝑈𝑃𝑈 = 0.5 𝑚 is the variance of the supplied tropo-free 
and iono-free pseudorange correction, 𝜎𝑈𝑈𝑈𝑈 = 0.5 𝑚 is the 
variance of the vertical ionosphere correction, 𝐹 is the obliquity 
factor, projecting vertical ionospheric variance into receiver – 
satellite direction, 𝜎𝑆𝑆𝑃 = 0.22 𝑚 is the receiver noise variance, 
𝜎𝑚45 = 0.22 𝑚 is the variance of the multipath contribution at 
45 degrees, 𝜎𝑡𝑡𝑡 = 0.15 𝑚 is the variance of the vertical 
tropospheric delay estimate. 

The model (8) has been adapted to the case of GPS SPP in 
[21] as shown below: 

𝜎𝑃𝑃2 = 𝜎𝑈𝑃𝐴2 + 𝜎𝑈𝐼𝐼𝐼2 + 𝜎𝑇𝑡𝐼𝑇𝐼2 + 𝜎𝑚𝑇2  
(9) 

where 𝜎𝑈𝑃𝐴2  is the User Range Accuracy (URA) related to the 
satellite ephemeris and clock, broadcast in the GPS navigation 
message, 𝜎𝑈𝐼𝐼𝐼2  is the accuracy related to ionosphere delay after 
Klobouchar model application, 𝜎𝑇𝑡𝐼𝑇𝐼2  is the accuracy related to 
troposphere error after correction model application, 𝜎𝑚𝑇2  is the 
accuracy related to multipath error. 

3.2. C/N0-based weighting schemes 
In [14], a variance model for carrier phase observation is 

proposed; it is re-used and adapted in [2] for pseudorange 
measurement. The model is exclusively based on C/N0 and has 
the following form: 

𝜎𝑃𝑃2 = 𝑐 ∙ 10
−
𝐶/𝑆0
10  

(10) 

where c is a constant, empirically defined. 
The model of Hartinger and Brunner [15] has been modified 

by Wieser and Brunner [22], introducing an additive term, and 
making the model parameters estimation in view of their 
application to carrier phase observable. Kuusniemi et al. [23], 
[24] adapt the Wieser and Brunner [22] model to pseudorange, 
tuning the parameters according to actual measurement errors 
and used equipment; the obtained variance model is: 

𝜎𝑃𝑃2 = 𝑎 + 𝑏 ∙ 10
−
𝐶/𝑆0
10  (11) 

where a and b are the model parameters. 
Kuusniemi [10] has estimated the parameters of model (11) 

for different scenarios (lightly and heavily degraded signal 
environment). 

The Hartinger and Brunner [14] model has been the basis 
for Brunner et al. (1999) to build a carrier phase variance 
model, called sigma-Δ, whose expression is very similar to (10), 
but with at the exponent a correction term proportional to the 
difference between the measured and the predicted (according 
to satellite elevation) C/N0.  

Li and Wu [26] have modified the model (11), adopting 
Euler’s number as base of the exponentiation and the 
difference between the measured C/N0 and its minimum 
threshold as exponent: 

𝜎𝑃𝑃2 = 𝑎 + 𝑏 ∙ 𝑒𝑘[(𝐶/𝑆0)𝑚𝑚𝑚−𝐶/𝑆0] 
(12) 

The parameters of the model are a, b and k and are 
empirically determined. 

Aminian [27] proposed an alternative expression for the 
pseudorange error variance without exponential form, as shown 
below: 

𝜎𝑃𝑃2 = 𝜎02
𝐶/𝑁0(𝑍𝑍𝐼𝑖𝑡ℎ)

𝐶/𝑁0
 (13) 

where 𝜎02 and 𝐶/𝑁0(𝑍𝑍𝐼𝑖𝑡ℎ) are respectively the theoretic 
variance and carrier-to-noise ratio in case of satellite at zenith 
and open-sky scenario. 

3.3. Weighting schemes based on both satellite elevation and 
C/N0 

In literature, few weighting schemes, based on both satellite 
elevation and C/N0, are available. 

In [28] the following model is developed: 

𝜎𝑃𝑃2 = �
1

sin2(𝐸𝐸)
ᴦ,       𝑖𝑖 𝐶/𝑁0 < 𝑠1 

1,                    𝑖𝑖 𝐶/𝑁0 ≥ 𝑠1
 (14) 

where 

ᴦ = 10−
𝐶/𝑆0−𝑠1

𝐵 � �
𝐴

10−
𝑠0−𝑠1
𝐵

− 1�
𝐶/𝑁0 − 𝑠1
𝑠0 − 𝑠1

+ 1� (15) 

The term s1 is a threshold, used as comparison for the 
measured C/N0; if 𝐶/𝑁0 ≥ 𝑠1, the measurement is considered 
accurate and the corresponding weight is set to 1 (the 
maximum possible value for a weight), otherwise a complex 
formula is adopted for the weight. The parameters in (14) and 
(15) are determined empirically as 𝑠0 = 10, 𝐴 = 30, 𝐵 = 30. 

In [29] a model, based on both satellite elevation and C/N0, 
designed for land navigation, is defined: 

𝜎𝑃𝑃2 = 𝑘 ∙
10−

𝐶/𝑆0
10

sin2(𝐸𝐸)
 (16) 

The model (16) is clearly a fusion between the elevation-
based model (5) and the C/N0-based model (10). The constant 
k is 1 if the signal is received in line-of-sight (LOS) and 2 or 
+∞ if the signal is received after reflection by obstacles 
surrounding the antenna (NLOS); in [29], a fish-eye camera is 
used to distinguish LOS and NLOS signals. In this work, a 
modified model (16) is used, where k is always equal to 1. 

4. REDUNDANCY MATRIX 

The residuals 𝒗, defined as the difference between the actual 
measurements and their estimated values (𝒛�), are an indicator of 
the measurement mutual agreement [13]: 

𝒗 = 𝒛 − 𝒛� = 𝒛 − 𝐻∆𝒙� (17) 
It can be simply demonstrated that the relationship between 

the measurement errors 𝜺 and the LS residuals, considering 
equally weighted measurements, is: 
𝒗 = 𝑅𝜺 (18) 

The matrix 𝑅 is called “Redundancy Matrix” and has the 
following expression: 
𝑅 = 𝐼 − 𝐻(𝐻𝑇𝐻)−1𝐻𝑇 (19) 

The matrix 𝑅 has 𝑚 rows and 𝑚 columns, its trace is equal 
to the total redundancy (or the degree of freedom) of the 
equation, that is (𝑚 − 𝑛). The i-th diagonal element of 𝑅, 𝑟𝑖 
with 𝑖 = 1, … 𝑚, is called “redundancy number” and is the 
contribution of the i-th measurement to the total redundancy 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 19 

[31]. The redundancy number assumes values between 0 and 1. 
Small values of 𝑟𝑖 (near 0) correspond to measurements 
providing little contribution to total redundancy and so hardly 
controlled; on the other hand, approximately equal values of 𝑟𝑖 
are desirable, meaning that every measurement is controllable. 

Measurements with small 𝑟𝑖 values are leverage observations 
and have high potential to influence the solution; if a blunder or 
a large bias is present on a leverage observation, harmful effects 
can be evident on the positioning. In this context, de-weighting 
leverage observations could be a successful strategy, above all in 
environments, as urban canyon, characterized by frequent gross 
errors. 

In Table 1, an example of GPS geometry, in terms of 
satellite elevations (𝐸𝐸) and azimuths (𝐴𝐴), is shown; from 𝐸𝐸 
and 𝐴𝐴, the design matrix can be computed and, consequently, 
the redundancy matrix 𝑅 is obtained. The diagonal elements of 
𝑅 are the redundancy numbers. 

In Figure 1, the sky-plot of the satellite geometry, detailed in 
Table 1, is shown; all the listed satellites are considered, so the 
number of visible GPS satellites is 10 and the PDOP (Position 
Dilution of Position) value is 1.59. The total redundancy of the 
measurement model is 6 (= 𝑚 − 𝑛), therefore the average 
redundancy number is 0.6. The sky-plot is accompanied by the 
values of the redundancy numbers and it is evident that all the 
values are near the average.  

In Figure 2, the sky-plot of the satellite geometry, detailed in 
Table 1, is shown, but satellites with PRNs 17, 19 and 25 are 
excluded; consequently, the number of visible GPS satellites is 
7 and the PDOP value raises to 2.15. The total redundancy is 3, 
so the average redundancy number is about 0.43. The 
redundancy numbers are all around or above the average value, 
except the one corresponding to satellite 22, which is very low 
and so can be considered a leverage observation. From Figure 
2, it is evident a geometric interpretation of leverage 

observation, for GPS SPP, as a measurement corresponding to 
a satellite “isolated” with respect to the others.  

In case of multi-constellation approach, a redundancy 
number is null when a single satellite from a system is present. 
For instance, in a GPS/Glonass combination, if there is only 
one GPS measurement and several Glonass, the redundancy 
number corresponding to the unique GPS observation is 0 and 
the observation is uncontrollable (because it is the only one able 
to estimate the clock offset to GPS time). The same concept is 
valid for any GNSS combination, including the European 
GNSS, Galileo [30], or the Chinese one BeiDou. 

5. CONTRIBUTION OF REDUNDANCY MATRIX TO 
WEIGHTING SCHEMES 

The diagonal elements of the redundancy matrix, the 
redundancy numbers 𝑟𝑖, represent the local reliability of the 
measurement model and so the controllability of the 
measurements [31]. 

Measurements with low 𝑟𝑖 are critical for the solution, 
because they can strongly influence it. For this reason, an 
effective weighting strategy should take into account the 
redundancy number, above all in scenarios with high probability 
of blunders. 

In [32] the redundancy number is used, together with several 
other parameters, as an indicator of the measurement quality 
for robust estimation. In [10] the extra-diagonal elements of the 
redundancy matrix are used to evaluate the correlation among 
the measurements, in order to avoid erroneous blunder 
rejections. Recently, Falco et al. [33] adopted a purely geometric 
strategy to identify measurements which could strongly 
influence the solution, in order to de-weight them. 

In this paper, the information contained in the redundancy 
matrix is used to correct the original weighting matrix, which 
could be defined according to one of the previously described 
methods. In general, the weight 𝑤𝑖, associated to the i-th 
measurement, is obtained inverting the pseudorange error 
variance 𝜎𝑃𝑃𝑖

2 . 
The contribution of a redundancy number to the weighting 

strategy could be introduced making 𝑤𝑖 proportional to 𝑟𝑖 as 
shown below: 

𝑤𝑖 =

⎩
⎨

⎧
𝑟𝑖
𝜎𝑃𝑃𝑖
2 , in case of redundant measurements

1
𝜎𝑃𝑃𝑖
2 , in case of lack of redundancy            

 (20) 

The redundancy numbers are 0 in case of lack of redundancy 
and the scheme (20) takes into account this possibility. 

 
Figure 2. Sky-plot of GPS satellite configuration; the total redundancy is 3. 

Table 1. Example of GPS geometry, in terms of satellite elevation and 
azimuth. 

PRN El deg Az deg 
2 12 30 
3 24 50 
5 45 80 
6 21 119 
9 65 130 

11 30 170 
17 81 190 
19 20 250 
22 19 282 
25 30 348 

 
 

 
Figure 1. Sky-plot of GPS satellite configuration; the total redundancy is 6. 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 20 

The strategy is tested on a multi-constellation static data 
collection, carried out in urban scenario, and demonstrated its 
efficiency in both GPS only and GPS/Glonass cases, exhibiting 
significant improvements on positioning accuracy. 

The same strategy could be simply extended to GNSS 
velocity estimation [34]. 

6. TEST 
To demonstrate the benefits of including the redundancy 

number into a weighting scheme in difficult environment, a 
static data collection is carried out in urban scenario; 
specifically, the test location is the “Centro Direzionale di 
Napoli” (CDN), a district of Naples (Italy) characterized by 
several skyscrapers. A specific point in CDN is surveyed with 
topographical methods, in order to determine its coordinates 
with mm order. The used equipment consists in a NVS 
receiver, a single frequency, double constellation 
(GPS/GLONASS) and high-sensitivity device, connected to a 
patch antenna; the device is placed on the defined point to 
collect pseudorange measurements for some hours; the 
collected data are processed in single point to test several 
weighting schemes. 

In Figure 3, the considered point (indicated as P) is shown 
and the urban canyon scenario is evident. 

The data collection was carried out on 1st July 2016; the 
NVS receiver was placed on P point for about 2 hours. In 
Figure 4, the number of GPS/Glonass visible satellites and the 
corresponding PDOP values during the session are shown. 

The number of GPS visible satellites ranges between 0 and 
8, with an average of 6.4. The maximum number of Glonass 
visible satellites is 5, while the maximum number of combined 
GPS/Glonass satellites is 12. 

The GPS solution availability, defined as the time percentage 
where the solution can be computed (at least 4 satellites should 
be visible), is about 99.3%, while the GPS/Glonass solution 
availability is about 99.5%. 

Despite the urban canyon scenario, the solution availability 
is high, owing to the use of a high-sensitivity receiver, able to 
acquire very weak signals. The rapid oscillations of the number 
of visible satellites is due to frequent signal loss, typical of 
hostile environment. 

In GPS only case, the PDOP average value is about 2.7, with 
maximum and minimum respectively about 15 and 1.6; in 
GPS/Glonass case, minimum, maximum and mean values are 
respectively 1.5, 17 and 2.4. The maximum PDOP value in 
GPS/Glonass case happens when GPS satellites are less than 
four, so in that time window GPS DOP cannot be computed.  

7. RESULTS 
Several weighting strategies are described in section 2; 

among them, some techniques are chosen to process data 
collected in urban scenario, in order to compare their 
performance. 

The considered configurations, implementing GNSS SPP 
with a distinct weighting scheme, are: 
• The one using weights defined by formula (5), shortly 

indicated as ELV1; 
• The one using weights defined by formula (9), shortly 

indicated as ELV2; 
• The one using weights defined by formula (11), shortly 

indicated as CN01; 
• The one using weights defined by formula (13), shortly 

indicated as CN02; 
• The one using weights defined by formula (14), shortly 

indicated as ELVCN01; 
• The one using weights defined by formula (16), shortly 

indicated as ELVCN02. 
The previously mentioned configurations are compared with 

the baseline one, where all the measurements are equally 
weighted, shortly indicated as EQW. 

The figure of merits adopted for the comparison are the 
mean, RMS and maximum errors of both horizontal and 
vertical components of position. 

In Table 2 the configuration performance, in case of GPS 
only, are summarized. 

 
Figure 3. Test location. 

 
Figure 4. Number of visible GPS/Glonass satellites and corresponding PDOP 
during session. 

Table 2. Figure of merits obtained with GPS only, using classic weighting 
methods. 

 Mean m RMS m Maximum m 

 Hor. Vert. Hor. Vert. Hor. Vert. 

EQW 16,0 18,2 44,9 45,9 351,6 316,6 
ELV1 10,4 12,9 28,2 33,3 187,7 224,0 
ELV2 13,0 12,2 38,6 29,0 283,3 218,0 

CN01 15,2 17,2 43,1 44,3 340,1 300,9 
CN02 15,4 17,6 43,4 45,1 341,0 305,8 
ELVCN01 8,4 9,9 22,4 27,9 158,5 194,9 

ELVCN02 5,9 5,8 14,9 19,5 130,1 173,4 

 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 21 

The results obtained with the EQW configuration is clearly 
inaccurate; in fact, the errors, on both components and for each 
figure of merit, are very large owing to the frequent blunders in 
the dataset. The considered weighting schemes reduce 
drastically the errors; the models based uniquely on satellite 
elevation (ELV1 and ELV2) show better performance with 
respect to models based uniquely on carrier-to-noise ratio 
(CN01 and CN02). The methods which consider both quality 
indicators (ELVCN01 and ELVCN02) demonstrate significant 
improvements relatively to the other configurations. In 
particular, the model introduced by Tay and Marais [29], i.e. 
ELVCN02 configuration, shows the best performance for the 
considered data session; indeed RMS and mean errors are 
reduced, with respect to EQW configuration, between 58% and 
68%, and the maximum horizontal and vertical errors are 
reduced respectively by 63% and 45%. 

The considered weighting schemes are augmented, using 
information embedded in the redundancy matrix as described in 
previous sections; the configurations using this information are 
shortly indicated as RDM. For instance, a configuration using 
the elevation-based model (5), augmented with redundancy 
matrix-based approach, is shortly indicated as ELV1+RDM. In 
Table 3, the performance of the configurations, obtained 
combining the considered classical weighting schemes with 
redundancy matrix information, are resumed. Moreover, in the 
table, the percentage improvements, with respect to 
corresponding configurations without redundancy matrix use, 
are reported alongside the errors. It is evident that the 
introduction into the weighting schemes of information from 
redundancy matrix brings to significant improvements. The 
best configuration is ELVCN02+RDM, characterized by 
horizontal mean, RMS and maximum errors respectively about 
4.5, 8.8 and 87.2 meters, a very good performance considering 
the strongly degraded scenario. 

The same analysis, carried out for GPS only, is provided for 
GPS/Glonass multi-constellation case. In Table 4 the 
GPS/Glonass configuration performance are summarized. The 
addition of Glonass measurements to GPS ones does not 
improve all the considered figure of merits, as Glonass 
measurements are often very noisy; in detail, GPS/Glonass 
configurations have better performance in horizontal plane, but 
worse performance in vertical direction. The errors of EQW 
configuration are very large, because there are several blunders 
among the measurements, from both used systems; the Glonass 
use allows a reduction in maximum horizontal of about 130 

meters. As for GPS only case, the weighting schemes based 
uniquely on satellite elevation (ELV1 and ELV2) demonstrate 
better performance than models based uniquely on carrier-to-
noise ratio (CN01 and CN02); models exploiting both quality 
indicators show the best results. 

In Table 5, the figure of merits of the GPS/Glonass 
configurations, using classical weighting schemes augmented 
with redundancy matrix information, are resumed; in the table, 
the percentage improvements, with respect to corresponding 
configurations without redundancy matrix use, are reported 
alongside the errors. The benefits of using redundancy matrix in 
weighting schemes are evident even in GPS/Glonass case; 
indeed, all the considered errors are strongly reduced. The best 
configuration is ELVCN02+RDM, characterized by horizontal 
mean, RMS and maximum errors respectively about 4.5, 8.1 and 
73.6 meters, a very good performance considering the strongly 
degraded scenario and slightly lower with respect to GPS best 
configuration. 

The use of information derived from redundancy matrix 
demonstrates very good performance for both GPS only and 
GPS/Glonass cases. The error reductions are impressive for 
each considered configuration and figure of merit. In particular, 
the best configuration is ELVCN02+RDM, using 𝐶/𝑁0, 
satellite elevation and local redundancy as parameters for 
weighting; the obtained performances are characterized by 
mean and RMS horizontal errors respectively below 5 and 10 
meters, and maximum horizontal error about 80 meters. 

The considered approach, taking into account the 
information contained into the redundancy matrix, 
demonstrates its effectiveness, owing to the ability to de-
weighting the geometrically dangerous measurements.  

Table 4. Figure of merits obtained with GPS/Glonass, using classic weighting 
methods. 

 Mean m RMS m Maximum m 
 Horiz. Vert. Horiz. Vert. Horiz. Vert. 

EQW 14,6 28,0 33,5 62,3 212,9 343,2 
ELV1 8,7 19,0 17,9 40,4 112,0 205,7 
ELV2 12,7 12,7 37,4 29,7 268,9 217,7 
CN01 13,7 26,2 31,8 59,2 196,5 335,1 
CN02 14,0 26,9 32,3 60,5 202,0 339,0 
ELVCN01 7,1 14,4 14,4 32,0 97,5 178,9 

ELVCN02 5,3 8,4 10,4 21,2 92,4 162,0 

Table 5. Figure of merits obtained with GPS/Glonass, using classic weighting 
methods, augmented with information from redundancy matrix. 

 Mean m RMS m Maximum m 

 Horiz. Vert. Horiz. Vert. Horiz. Vert. 

EQW+RDM 
13,5 

(7,5%) 
19,8 

(29,4%) 
32,9 

(1,9%) 
44,1 

(29,1%) 
236,6 

(11,1%) 
266,0 

(22,5%) 

ELV1+RDM 
7,7 

(12,3%) 
12,1 

(36,0%) 
15,5 

(13,2%) 
25,9 

(36,0%) 
94,8 

(15,3%) 
153,1 

(25,5%) 

ELV2+RDM 
11,0 

(13,5%) 
10,0 

(21,1%) 
29,5 

(21,0%) 
23,0 

(22,5%) 
212,0 

(21,2%) 
164,8 

(24,3%) 

CN01+RDM 
12,5 

(8,8%) 
18,0 

(31,3%) 
30,6 

(3,9%) 
41,1 

(30,7%) 
224,7 

(14,4%) 
242,2 

(27,7%) 

CN02+RDM 
12,8 

(8,5%) 
18,6 

(30,7%) 
31,2 

(3,4%) 
42,3 

(30,1%) 
224,3 

(11,1%) 
249,4 

(26,4%) 

ELVCN01+RDM 
6,1 

(14,1%) 
8,6 

(40,0%) 
11,9 

(17,6%) 
19,6 

(38,9%) 
86,1 

(11,7%) 
122,6 

(31,5%) 

ELVCN02+RDM 
4,5 

(14,8%) 
4,6 

(45,4%) 
8,1 

(22,8%) 
12,3 

(41,7%) 
73,6 

(20,4%) 
109,1 

(32,7%) 

Table 3. Figure of merits obtained with GPS only, using classic weighting 
methods, augmented with information from redundancy matrix. 

 Mean m RMS m Maximum m 

 Horiz. Vert. Horiz. Vert. Horiz. Vert. 

EQW+RDM 
14,0 

(12,2%) 
15,5 

(14,9%) 
36,4 

(19,0%) 
37,2 

(18,9%) 
276,5 

(21,4%) 
241,5 

(23,7%) 

ELV1+RDM 
8,0 

(22,9%) 
9,0 

(29,8%) 
18,1 

(35,7%) 
22,8 

(31,6%) 
121,1 

(35,5%) 
154,7 

(30,9%) 

ELV2+RDM 
10,6 

(18,4%) 
9,1 

(25,3%) 
27,8 

(27,8%) 
20,9 

(27,9%) 
186,2 

(34,3%) 
136,5 

(37,4%) 

CN01+RDM 
13,0 

(14,4%) 
14,2 

(17,6%) 
33,7 

(21,7%) 
34,7 

(21,6%) 
261,5 

(23,1%) 
228,1 

(24,2%) 

CN02+RDM 
13,3 

(13,7%) 
14,7 

(16,6%) 
34,4 

(20,9%) 
35,8 

(20,6%) 
261,7 

(23,2%) 
230,3 

(24,7%) 

ELVCN01+RDM 
6,3 

(24,7%) 
6,3 

(35,7%) 
13,6 

(39,4%) 
17,6 

(36,9%) 
108,1 

(31,8%) 
125,0 

(35,9%) 

ELVCN02+RDM 
4,5 

(23,6%) 
3,3 

(43,8%) 
8,8 

(41,1%) 
11,4 

(41,7%) 
87,2 ( 

33,0%) 
115,9 

(33,2%) 



 

ACTA IMEKO | www.imeko.org June 2018 | Volume 7 | Number 2 | 22 

8. CONCLUSIONS 
In GNSS context, some environments, such as urban 

canyons, are critical, owing to the likely presence of blunders 
among the measurements, which yields very large position 
errors. In these conditions, it is essential to adopt a suitable 
method to differently weight the measurements, in order to 
limit the blunder effects. In literature, several schemes exist, 
based on two quality indicators: satellite elevation and carrier-
to-noise ratio. 

In this paper, the most representative weighting schemes for 
GNSS absolute positioning are tested in urban scenario and 
their performance are compared. The best results are obtained 
with the weighting strategies based on both the 
abovementioned quality indicators. 

The concept of local redundancy refers to the degree of 
controllability of the observations; for instance, a measurement 
with low local redundancy is difficultly controlled and, if 
affected by gross errors, it strongly influences the solution. The 
redundancy matrix relates the measurement errors to the 
residuals and its diagonal elements are the redundancy numbers, 
representing the local redundancy. In this work, a strategy to 
include the local redundancy into the existing weighting 
schemes is suggested. In the proposed approach, each weight 
value is set proportionally to the corresponding local 
redundancy number. In this way, the most influencing 
measurements (i.e. the leverage measurements) are de-weighted 
and their effects on the solution are limited. 

The considered weighting schemes are applied to process in 
SPP a 2-hour dataset, collected in urban scenario; both GPS 
only and multi-constellation GPS/Glonass configurations are 
analysed. The obtained results demonstrate the effectiveness of 
the proposed weighting strategy; indeed, the position errors, 
using the weighting scheme based on redundancy numbers, are 
significantly reduced for all the considered configurations. 

The best performances are attained with a weighting 
scheme, comprising the two classical quality indicators, satellite 
elevation and carrier-to-noise ratio, and the redundancy 
number; in single point positioning, a horizontal accuracy below 
10 meters is obtained in a severe urban scenario. 

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