gAGE

research group of Astronomy and Geomatics

Lecture 1
GNSS measurements
and their combinations

Contact: jaume.sanz@upc.edu
gAGE/UPC

Web site: http://www.gage.upc.edu

Master of Science in GNSS @ J. Sanz & J.M. Juan

1
gAGE
research group of Astronomy and Geomatics

Authorship statement

The authorship of this material and the Intellectual Property Rights are owned by
J. Sanz Subirana and J.M. Juan Zornoza.

These slides can be obtained either from the server http://www.gage.upc.edu,

or jaume.sanz@upc.edu. Any partial reproduction should be previously
authorized by the authors, clearly referring to the slides used.

This authorship statement must be kept intact and unchanged at all times.
gAGE/UPC

22 Jan 2015

Master of Science in GNSS @ J. Sanz & J.M. Juan

2
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

3
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

4
gAGE
research group of Astronomy and Geomatics GPS SIGNAL STRUCTURE
Two carriers in L-band:
• L1=154 fo=1575.42 MHz • C/A-code for civilian users [XC(t)]
• L2=120 fo=1227.60 MHz • P-code only for military and authorized
where fo=10.23 MHz users [XP(t)]
• Navigation message with satellite
ephemeris and clock corrections [D(t)]
C/A
P(Y) P(Y)

1227.6 MHz 1575.42 MHz

L2 L1
gAGE/UPC

=S L(1k ) (t ) aP X P( k ) (t ) D ( k ) (t ) sin(ω1t + φL1 ) + aC X C( k ) (t ) D ( k ) (t ) cos(ω1t + φL1 )

S L( 2k ) (t ) bP X P( k ) (t ) D ( k ) (t ) sin(ω2t + φL2 )

Master of Science in GNSS @ J. Sanz & J.M. Juan

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gAGE
GPS Code Pseudorange Measurements
=S L(1k ) (t ) aP X P( k ) (t ) D ( k ) (t ) sin(ω1t + ϕ L1 ) + aC X C( k ) (t ) D ( k ) (t ) cos(ω1t + ϕ L1 )
research group of Astronomy and Geomatics

S L( 2k ) (t ) bP X P( k ) (t ) D ( k ) (t ) sin(ω2t + ϕ L2 )
binary code XP(t)

C1,P1, P2
∆T

P(T ) = c ∆T = c trec (T ) − t sat (T − ∆T ) 

 

From hereafter we will call:

gAGE/UPC

• C1 pseudorange computed from XC(t) binary code (on frequency 1)

• P1 pseudorange computed from XP(t) binary code (on frequency 1)
• P2 pseudorange computed from XP(t) binary code (on frequency 2)
Master of Science in GNSS @ J. Sanz & J.M. Juan
6
 gAGE GPS Carrier Phase Measurements
=S L(1k ) (t ) aP X P( k ) (t ) D ( k ) (t ) sin(ω1t + ϕ L1 ) + aC X C( k ) (t ) D ( k ) (t ) cos(ω1t + ϕ L1 )
research group of Astronomy and Geomatics

S L( 2k ) (t ) bP X P( k ) (t ) D ( k ) (t ) sin(ω2t + ϕ L2 )
Carrier phase
Carrier beat phase:

φL (T ) φL rec (T ) − φLsat (T − ∆T )

=
C1,P1, P2
c
= ∆T + N Unknown ambiguity L1, L2
λ

From hereafter we will call:

• L1 =λ1φ L1 measur. computed from the carrier phase on frequency 1
• L2 =λ2φ L2 measur. computed from the carrier phase on frequency 2
gAGE/UPC

• C1 pseudorange computed from XC(t) binary code (on frequency 1)

• P1 pseudorange computed from XP(t) binary code (on frequency 1)
• P2 pseudorange computed from XP(t) binary code (on frequency 2)
Master of Science in GNSS @ J. Sanz & J.M. Juan
7
gAGE
research group of Astronomy and Geomatics Carrier and Code pseudorange measurements

P1 P1
P1= c ∆T= c [trec(T)-tsat(T-∆T)]
P1 ≈ ρ + clock offset
≈ 20.000 Km
P1 is basically the geometric range (ρ)
between satellite and receiver, plus the
relative clock offset.
The range varies in time due to the
satellite motion relative to the receiver.
gAGE/UPC

P1 is an absolute measurement (unambiguous)

Master of Science in GNSS @ J. Sanz & J.M. Juan
8
gAGE
Phase and Code pseudorange measurements

L1 (T ) = c ∆T + λ1 N1
research group of Astronomy and Geomatics

Relative measurement
(shifted by the unknown ambiguity “λN” )

Each time that the receiver loose the

phase lock, the unknown ambiguity
changes by an integer number of ”λ”
gAGE/UPC

L1 ≈ ρ + clock offset + λ1 N1
Master of Science in GNSS @ J. Sanz & J.M. Juan
9
 Code and Carrier Phase measurements
gAGE
research group of Astronomy and Geomatics

Code (unambiguous but noisier)

Ambiguity

Carrier Phase (ambiguous but precise)

gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

10
gAGE GPS measurements: Code and Carrier Phase
∆
research group of Astronomy and Geomatics

Antispoofing (A/S):
The code P is encrypted to Y. Wavelength σ noise Main
 Only theGPS
code Csignal
at (chip-length) (1% of λ) [*] characteristics
frequency L1 is available.
Code measurements
C1 300 m 3m
Unambiguous
P1 (Y1): encrypted 30 m 30 cm
but noisier
P2 (Y2): encrypted 30 m 30 cm
Phase measurements
L1 19.05 cm 2 mm Precise
L2 24.45 cm 2 mm but ambiguous
gAGE/UPC

[*] the codes can be smoothed with the phases in order to reduce noise
(i.e, C 1 smoothed with L 1  50 cm noise)

Master of Science in GNSS @ J. Sanz & J.M. Juan

11
gAGE

RINEX FILES
research group of Astronomy and Geomatics
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

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GNSS Format Descriptions
 GNSS data files follow a well defined set of
standards formats: RINEX, ANTEX, SINEX…
 Understanding a format description is a tough
task.
 These standards are explained in a very easy and
friendly way through a set of html files.
 Described formats:
• Observation RINEX
• Navigation RINEX
• RINEX CLOCKS
• SP3 Version C
• ANTEX
Open GNSS Formats
with Firefox internet browser

More details at: http://www.gage.es/gLAB

gAGE/UPC Tutorial associated to the GNSS Data Processing book •13

Research group of Astronomy & Geomatics J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares
Technical University of Catalonia
gAGE
research group of Astronomy and Geomatics RINEX measurement file

HEADER

MEASUREMENTS
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

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gAGE
research group of Astronomy and Geomatics
gAGE/UPC RINEX measurement file

Master of Science in GNSS @ J. Sanz & J.M. Juan

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gAGE
research group of Astronomy and Geomatics RINEX measurement file

Measurement time
(receive time tags)

Number of tracked satellites

gAGE/UPC

One satellite per row

Epoch flag 0: OK
Master of Science in GNSS @ J. Sanz & J.M. Juan
16
gAGE
research group of Astronomy and Geomatics RINEX measurement file

Synthetic P2
(A/S=on)
gAGE/UPC

Master of Science in GNSS S/N indicatorsLoss of lock indicator

@ J. Sanz & J.M. Juan
17
gAGE
research group of Astronomy and Geomatics
Pseudorange modeling

P= c ∆T= c [trec(T)-tsat(T-∆T)]

P = ρ
sat
sat
rec
rec
sat
rec + c ⋅ (dtrec − dt sat
) + ∑δδ
Geometric range Clock offsets
gAGE/UPC

∑ δ= Trop sat
rec + Ion sat
rec + K rec + K sat
+ε
Ionospheric delay noise
Master of Science in GNSS
Tropospheric delay Instrumental delays
@ J. Sanz & J.M. Juan
18
gAGE
research group of Astronomy and Geomatics
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

19
gAGE
research group of Astronomy and Geomatics
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

20
gAGE

Exercise:
research group of Astronomy and Geomatics

a) Using Exercise 1:
the file 95oct18casa___r0.rnx, generate the “txt” file
95oct18casa.a (with data ordered in columns).
b) Plot code and phase measurements for satellite PRN28 and
discuss the results.

Resolution:
a) gLAB_linux -input:cfg meas.cfg -input:obs coco0090.97o
gAGE/UPC

b) See next plots:

Master of Science in GNSS @ J. Sanz & J.M. Juan

21
gAGE
research group of Astronomy and Geomatics An example of program to read the RINEX: gLAB

RINEX file  gLAB  txt file

sta Doy sec PRN L1 L2 C1/ P2

cambiar
gAGE/UPC

The RINEX file is convert to a “columnar format” to easily plot its

contents and to analyze the measurements (the public domain free
Master of Science in GNSS
tool “gnuplot” is used in the book to make the plots).
@ J. Sanz & J.M. Juan
22
 Code measurements The geometry “ρ” is the
gAGE
dominant term in the
plot. The pattern in the
research group of Astronomy and Geomatics

figures is due to the

P1 variation of “ρ”

P1
gAGE/UPC

= ρ sta
sat
P1sta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
Master of Science in GNSS
sat
+ Ion1sta
sat
+ K1sta + K1sat + ε231
@ J. Sanz & J.M. Juan
 Code measurements Similar plot for code
gAGE
measurements at f2.
research group of Astronomy and Geomatics

Notice that
P2 • Ionosphere (Ion) and
• Instrumental delays (K)
depend on frequency.
gAGE/UPC

=
P2 sat ρ sat
+ c ⋅ ( dt
Master of Science in GNSS
sta sta sta − dt sat
) + Trop sat
sta + Ion sat
2 sta + K + K sat
+24ε 2
@ J. Sanz & J.M. Juan
2 sta 2
 Code
Ionosphere delays and
code and Phase measurements
gAGE advances phase measurements

Code measurements: C1,P1,P2

research group of Astronomy and Geomatics

sat
C1sta ; = ρ sta
sat
P1sta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
sat
+ Ion1sta
sat
+ K1sta + K1sat + ε1

=
P2 sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
2 sta + K 2 sta + K 2
sat
+ ε2

Frequency dependent
Phase measurements: L1,L2
=
L1sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w + ν 1
=
L2 sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta − Ion sat
2 sta + b2 sta + b2
sat
+ λ2 N 2 + λ2 w +ν 2

phase Ambiguities
gAGE/UPC

N1, N2 are integers

Wind Up

Master of Science in GNSS @ J. Sanz & J.M. Juan

25
 Carrier Phase measurements The geometry “ρ” is the
gAGE
dominant term in the
plot. The pattern in the
research group of Astronomy and Geomatics

figures is due to the

variation of “ρ”.
The curves are broken
when the receiver loss
the lock (cycle-slip).

When a cycle-slip happens, the

phase measurement “L” changes
by un unknown integer number
gAGE/UPC

of cycles (N)

=
sta ρ sta +
Master of Science
L1sat c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w +ν 1
@ J. Sanz & J.M. Juan
 Carrier Phase measurements The geometry “ρ” is the
gAGE
dominant term in the
plot. The pattern in the
research group of Astronomy and Geomatics

figures is due to the

variation of “ρ”.
The curves are broken
when the receiver loss
the lock (cycle-slip).

When a cycle-slip happens, the

phase measurement “L” changes
by un unknown integer number
of cycles (N)
gAGE/UPC

= ρ sta
L2 sta + c ⋅ (dtsta − dt sat ) + Tropsta
Master of Science
sat sat in GNSS sat
− Ion2 sat
sta + b2 sta + b2
sat
+ λ2 N 2 + λ2 w +ν 2
@ J. Sanz & J.M. Juan
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

28
gAGE
research group of Astronomy and Geomatics

Linear Combinations of measurements:

• Geometry-free (or Ionospheric) combination.

• Ionosphere-Free combination.
• Wide-lane and Narrow-lane combinations.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

29
 1. Geometry-free (or ionospheric) combination
gAGE 1 of Astronomy and Geomatics

PI= P2 – P1=Iono+ctt
LI= L1 –L2= Iono+ctt+Ambig
Ambiguity

Code measurements: C1,P1,P2

= ρ sta + c ⋅ (dtsta − dt sat ) + Tropsta + Ion1sta + K1sta + K1sat + ε1

2 group

sat sat sat sat sat

C1sta ; P1sta
P -P
research

sta= ρ sta + c ⋅ ( dt sta − dt

P2 sat ) + Tropsta + Ion2 sat
sta + K 2 sta + K 2 + ε2
sat sat sat sat

Carrier measurements: L1,L2

gAGE/UPC
L1-L2

=
L1sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w +ν 1

sta= ρ sta + c ⋅ ( dt sta − dt ) + Tropsta − Ion2 sta + b2 sta + b2

L2 sat + λ2 N 2 + λ2 w +ν 2
sat sat sat sat sat

Master of Science in GNSS @ J. Sanz & J.M. Juan

Carrier Ambiguities
30
 1. Geometry-free (or ionospheric) combination
gAGE
PGPS observables
I= P2 – P1=Iono+ctt • The pattern corresponds to the
ionospheric refraction (Ion),
research group of Astronomy and Geomatics

LI= L1 –L2= Iono+ctt+Ambig

because the other terms (K) are
constant.
• Notice that code measurements
are noisier.
Pij= c ∆t= c [trec(TR)-tems(TS)]

Ambiguity
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

31
gAGE
research group of Astronomy and Geomatics Ionospheric effects

The ionospheric refraction depends on:

• Geographic location
• Time of day
• Time with respect to solar cycle (11y)
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

32
gAGE
research group of Astronomy and Geomatics Ionospheric effects
The ionospheric delay (Ion) is
proportional to the electron density
integrated along the ray path (STEC)


40.3 r [ GPSreceiver ]

Ion = 2 STEC STEC = ∫ e , t )dr
N ( r
f 
r [ GPStransmitter ]

Ionosphere

Ambiguity
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

33
 2. Ionosphere-free
Ionospheric-Free
Combination
Combination(Pc,Lc)
gAGE

The ionospheric refraction depends on 40.3

research group of Astronomy and Geomatics

the inverse of the squared frequency Ion = 2 STEC

and can be removed up to 99.9% f
combining f1 and f2 signals:

f P1 − f P2 2 2
f L1 − f L2
2 2
Pc = 1 2
Lc = 1 2
f12 − f 2
2
f12 − f 2
2

Note: Ksat cancels in Pc

Pc = ρ
sat
sta
sat
sta + c ⋅ (dtsta − dt ) + Trop + ε c
sat sat
sta and Ksta included in dtsta

= ρ sta
sat
Lcsta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
sat
+ bc , sta + bc sat + λN Rc + λN w +ν c

• The ionospheric refraction has been removed in Lc and Pc

gAGE/UPC

λN = 10.7 cm, λW =86.2cm

λW
The Rc ambiguities are NOT integers!! Rc =N1 − ( N − N2 )
λ2 1
Master of Science in GNSS @ J. Sanz & J.M. Juan
34
gAGE
research group of Astronomy and Geomatics

Comments:

Two-frequency receivers are needed to apply the

ionosphere-free combination.

If a one-frequency receiver is used, a ionospheric

model must be applied to remove the ionospheric
refraction. The GPS navigation message provides the
parameters of the Klobuchar model which accounts
for more than 50% (RMS) of the ionospheric delay.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

35
 3.- Narrow-lane (PN) and Wide-lane
Ionospheric-Free Combination (LW)
Combination
gAGE

The wide-lane combination LW provides a signal with a large wave-

research group of Astronomy and Geomatics

length (λW=86.2cm ~ 4*λ1). This makes it very useful for detecting

cycle-slips through the Melbourne-Wübbena combination: LW – PN

f1 P1 + f 2 P2 f1 L1 − f 2 L2
PN = LW =
f1 + f 2 f1 − f 2

The same sign

=
PN sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
w sta + K w sta + K w
sat
+εN
=
LW sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
w sta + bw sta + bw
sat
+ λw N w +ν w
gAGE/UPC

The ambiguities NW are INTEGERS!

Master of Science in GNSS No@ wind-up
J. Sanz & J.M. Juan
36
gAGE
Exercises:
research group of Astronomy and Geomatics

1) Consider the wide-lane combination of carrier phase measurements

f1 L1 − f 2 L2
LW = , where LW is given in length units (i.e. Li= λi φi ).
f1 − f 2
c
Show that the corresponding wavelength is: λW =
f1 − f 2

Hint:
LW= λW φW ; φW = φ1 – φ2

2) Assuming L1, L2 uncorrelated measurements with equal noise σL,

show that:
γ 12 + 1
2
 f 
=σ LW = σ L ; γ 12  1 
γ 12 − 1  f2 
gAGE/UPC

Master of Science in GNSS Slides associated to the GNSS Data Processing book
J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares •37
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

38
gAGE
Detecting cycle-slips
This cycle-slip
research group of Astronomy and Geomatics

involves millions
of cycles  it is
easy to detect!!

There is a cycle-
slip of only one
gAGE/UPC

cycle (~20cm) 
How to detect it?
Master of Science in GNSS @ J. Sanz & J.M. Juan
39
gAGE
Exercise:
research group of Astronomy and Geomatics

a) Using the file 95oct18casa___r0.rnx, generate the “txt” file

95oct18casa.a (with data ordered in columns).
b) Insert a cycle-slip of “one wavelength” (19cm) in L1
measurement at t=5000 s (and no cycle-slip in L2).
c) Plot the measurements “L1, L1-P1, LC-PC, Lw-PN and L1-L2”
and discuss which combination/s should be used to detect the
cycle-slip.
Resolution:
a) gLAB_linux -input:cfg meas.cfg -input:obs 95oct18casa_r0.rnx
b) cat 95oct18casa.a | gawk ‘{if ($4==18)
print $3,$5,$6,$7,$8}’ > s18.org
gAGE/UPC

cat s18.org | gawk ‘{if ($1>=5000) $2=$2+0.19;

printf “%s %f %f %f %f \n”, $1,$2,$3,$4,$5}’ > s18.cl
c) See next plots:
Master of Science in GNSS @ J. Sanz & J.M. Juan
40
 The geometry “ρ” is the dominant term in the plot. The variation
gAGE
of “ρ” in 1 sec may be hundreds of meters, many times greater than
the cycle-slip (19 cm)  the variation of ρ shadows the cycle-slip!
research group of Astronomy and Geomatics

L1 (without the cycle-slip)

L1 (with the cycle-slip)

A jump of λ=19 cm (one cycle in L1)

has been introduced in L1 at t=5000s
1 unit = 19 cm
(L1 cycles)

ρ
gAGE/UPC

=
sta ρ
sat of Science
L1Master sta + c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w +41ν 1
@ J. Sanz & J.M. Juan
 The geometry and clock offsets have been removed.
gAGE
The trend is due to the Ionosphere. The P1 code noise shadows
the cycle-slip, and without the reference (in blue), the time where
research group of Astronomy and Geomatics

the cycle-slip happens could not be identified.

L1-P1 (with the cycle-slip)

A jump of λ=19 cm (one cycle in L1)

cm
19cm

has been introduced in L1 at t=5000s

(L1 cycles)
= 19
unit =
11 unit

L1sat − =
− 1sta + ctt + ambig + ε
sat sat
sta P1sta 2 Ion
gAGE/UPC

=
P1sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
1sta + K1sta + K1
sat
+ ε1
=
sta ρ
sat of Science
L1Master sta + c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w +42ν 1
@ J. Sanz & J.M. Juan
 The geometry and clock offsets have been removed.
gAGE
The trend is due to the Ionosphere. The P1 code noise shadows
the cycle-slip, and without the reference (in blue), the time where
research group of Astronomy and Geomatics

the cycle-slip happens could not be identified.

L1-P1 (without the cycle-slip)

L1-P1 (with the cycle-slip)

A jump of λ=19 cm (one cycle in L1)

cm
19cm

has been introduced in L1 at t=5000s

(L1 cycles)
= 19
unit =
11 unit

L1sat − =
− 1sta + ctt + ambig + ε
sat sat
sta P1sta 2 Ion
gAGE/UPC

=
P1sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
1sta + K1sta + K1
sat
+ ε1
=
sta ρ
sat of Science
L1Master sta + c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta − Ion sat
1sta + b1sta + b1
sat
+ λ1 N1 + λ1w +43ν 1
@ J. Sanz & J.M. Juan
gAGE
The geometry, clock offsets and iono have been removed.
There is a constant pattern plus noise. The P1 code noise also shadows
the cycle-slip, and without the reference (in blue), the time where the
research group of Astronomy and Geomatics

cycle-slip happens could not be identified.

LC-PC (without the cycle-slip)
LC-PC (with the cycle-slip)
A jump of λ=19 cm (one cycle in L1)
has been introduced in L1 at t=5000s
1 unit = 10.7 cm
(Lc cycles)

sat
Lcsta − Pcsta
sat
=ctt + ambig + ε
gAGE/UPC

= ρ sta
sat
Pcsta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
sat
+ εc
= ρ sta
sat
Lcsta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
Master of Science in GNSS
sat
+ bC , sta + bC sat + λN Rc + λN w +ν c
@ J. Sanz & J.M. Juan
44
gAGE
The geometry, clock offsets and iono have been removed.
There is a constant pattern plus noise. The P1 code noise also shadows
the cycle-slip, and without the reference (in blue), the time where the
research group of Astronomy and Geomatics

cycle-slip happens could not be identified.

LI-PI (without the cycle-slip)
LI-PI (with the cycle-slip)
A jump of λ=19 cm (one cycle in L1)
has been introduced in L1 at t=5000s
1 unit = 5.4 cm

sat
LI sta − PI sta
sat
=ctt + ambig + ε
gAGE/UPC

sat
PI sta = IonI + K I sta + K I sat + ε I
Master of Science in GNSS
sat
LI sta = IonI + bI sta + bIsat + λ1 N1 − λ2 N 2 + (λ1 − λ2 ) w +ν I45
@ J. Sanz & J.M. Juan
 The geometry , clock offsets and iono have been removed.
gAGE
There is a constant pattern plus noise. The PN code noise is under one
cycle of Lw. Thence, the cycle-slip is clearly detected
research group of Astronomy and Geomatics

Lw-PN (without the cycle-slip)

Lw-PN (with the cycle-slip)
A jump of λ=19 cm (one cycle in L1)
has been introduced in L1 at t=5000s
86.2 cm
= 86.2cm
(Lw cycles)
unit =
1 unit
1

LW sat
sta − PN sta =
sat
ctt + ambig + ε
gAGE/UPC

=
PN sat
sta ρ sat
sta + c ⋅ ( dt sta − dt sat
) + Trop sat
sta + Ion sat
w sta + K w sta + K w
sat
+εN
=
sta ρ
sat of Science
LWMaster sta + c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta + Ion sat
w sta + bw sta + b sat
+ λw N w +ν46w
@ J. Sanz & J.M. Juan
w
gAGE
The geometry and clock offsets have been removed.
The trend is due to the Iono. The L1 code noise is few mm, and
the variation of the ionosphere in 1 second is lower than λ1 =19 cm
research group of Astronomy and Geomatics

Thence, the cycle-slip is detected.

LI (without the cycle-slip)
LI (with the cycle-slip)
A jump of λ=19 cm (one cycle in L1)
has been introduced in L1 at t=5000s
1 unit = 5.4 cm

ε  mm
(λ1 − λ2 )w <<

L1,satsta − Lsat= − 2, sta + ctt + ambig + ε

sat sat
2, sta Ion1, sta Ion
gAGE/UPC

= ρ sta
sat
L1sta sat
+ c ⋅ (dtsta − dt sat ) + Tropsta
sat
− Ion1sat
sta + b1sta + b1
sat
+ λ1 N1 + λ1w +ν 1
=
sta ρ
sat of Science
L2Master sta + c ⋅
sat in GNSS
( dt sta − dt sat
) + Trop sat
sta − Ion sat
2 sta + b2 sta + b2
sat
+ λ2 N 2 + λ2 w +47ν 2
@ J. Sanz & J.M. Juan
 Summary
gAGE

L1
research group of Astronomy and Geomatics

L1-P1

LI-PI LC-PC
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

48
 The cycle-slips are detected by Combination
Ionospheric-Free the Ionospheric combination
gAGE
research group of Astronomy and Geomatics
(LI=L1-L2) and the Melbourne Wübbena (W=Lw-PN)

LI Lw-PN

Two independent combinations, LI and Lw, allow to detect

two independent cycle-slips (in L1 and L2 phase measur.).

L1 L2
gAGE/UPC

Notice that, from L1, L2 is not possible to detect short cycle-slips

Master of Science in GNSS @ J. Sanz & J.M. Juan

49
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
3.1 Cycle-slip Detection Algorithms
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

50
 Cycle-slip detector based on carrier phase data:
gAGE
The Geometry-free combination
research group of Astronomy and Geomatics

=LI L1 ( s; k ) − L2 ( s; k ) The detection is based on fitting a

second order polynomial over a
sliding window of NI samples.
The predicted value is compared with
the observed one to detect cycle-slip.

p ( s; k )
[ LI ( s; k − N I ), , LI ( s; k − 1)]
LI ( s; k )
p ( s; k )

LI ( s; k ) − p ( s; k ) > threshold
gAGE/UPC

LI ( s; k )

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics
LI ( s; k )
Carrier
measurements

p ( s; k )
Fitted Polynomial

Under not disturbed ionospheric conditions,

the geometry-free combination performs as a
very precise and smooth test signal,
driven by the ionospheric refraction.
Although, for instance, the jump produced by
a simultaneous one-cycle slip in both signals
gAGE/UPC

is smaller in this combination than in the

original signals (λ2 -λ1 =5.4cm), it can provide
reliable detection even for small jumps
Master of Science in GNSS @ J. Sanz & J.M. Juan
 Cycle-slip detector based on code and carrier phase
gAGE
data: The Melbourne-Wübbena combination
BW = LW − PN = λW NW + ε
research group of Astronomy and Geomatics

The detection is based on real-time

computation of mean (mBW) and sigma (SBW)
values of the measurement test data Bw.

A cycle-slip is
declared when
the measurement
differs form the
mean value by a
gAGE/UPC

predefined
number of
standard
deviations (SBW)

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE The Melbourne-Wübbena
combination has a double
benefit:
research group of Astronomy and Geomatics

Threshold
• The enlargement of the
Moving sigma
ambiguity spacing, thanks to
Moving average the larger wavelength
λW =80.4cm.
• The noise is reduced by
the narrow-lane combination
of code measurement

Cycle-slip detection

Nevertheless, in spite of these

benefits, the performance is
worse than in the previous
gAGE/UPC

carrier-phase-only based
detector and it is used as a
secondary test.

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE Exercises:

1) Show that ∆=N1 9 and ∆=

research group of Astronomy and Geomatics

N2 7
produces jumps of few millimetres in the geometry-free combination.

2) Show that no jump happens in the geometry-free combination when

∆N1 / ∆N 2 = 77 / 60 . In particular when ∆N1=77 and ∆N2=60 the
jump in the wide-lane combination is: 17 λw  15 m

Hint: Consider the following relationships (from [RD-1]):

gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

55
 Example of Single frequency Cycle-slip detector
gAGE
( P1 ) ( L1 )
research group of Astronomy and Geomatics

The detection is based on

real-time computation of
mean and sigma values of
the differences (d=L1-P1) of
the code pseudorange and
carrier over a sliding window
of N samples (e.g. N=100).
d=
( s; k ) L1 ( s; k ) − P1 ( s; k )
A cycle-slip is declared when
a measurement differs from
the mean bias value over a
predefined threshold.
gAGE/UPC

Missed detection
Master of Science in GNSS @ J. Sanz & J.M. Juan
 This detector is affected by the
gAGE code pseudorange noise and
multipath as well as the
divergence of the ionosphere.
research group of Astronomy and Geomatics

Higher sampling rate improves

detection performance, but
shortest jumps can still escape
from this detector.
On the other hand, a minimum
number of samples is needed
for filter initialization in order to
ensure a reliable value of sigma
for the detection threshold

More details, exercises and examples

of software code implementation of
gAGE/UPC

these detectors can be found in

[RD-1] and [RD-2].

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

58
gAGE Carrier smoothing of code pseudorange
The noisy (but unambiguous) code pseudorange can be smoothed with
research group of Astronomy and Geomatics

the precise (but ambiguous) carrier. A simple algorithm is given next:

Hatch filter:

n −1 ˆ
1
Pˆ=
n
P(k ) +
(k )
n
(
P (k − 1) + L(k ) − L(k − 1) )
where Pˆ (1) = P (1) and
=n k; k < N
=n N; k ≥ N

This algorithm can be interpreted as a real-time alignment of the

carrier phase with the code measurement:

n −1 ˆ
1
( )
gAGE/UPC

P(k=
ˆ ) P(k ) + P (k − 1) + L(k ) − L(k − 1)= L(k ) + P − L (k )
n n

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics

L P−L

This algorithm can be interpreted as a real-time alignment of the

carrier phase with the code measurement:
gAGE/UPC

n −1 ˆ
Pˆ (k=
)
1
n
P(k ) +
n
( )
P (k − 1) + L(k ) − L(k − 1)= L(k ) + P − L (k )

Master of Science in GNSS @ J. Sanz & J.M. Juan

60
gAGE
research group of Astronomy and Geomatics

P
P̂

P−L
L
gAGE/UPC

n −1 ˆ
Pˆ (k=
)
1
n
P(k ) +
n
( )
P (k − 1) + L(k ) − L(k − 1)= L(k ) + P − L (k )

Master of Science in GNSS @ J. Sanz & J.M. Juan

61
gAGE
research group of Astronomy and Geomatics

P
P̂

P−L
L
gAGE/UPC

n −1 ˆ
Pˆ (k=
)
1
n
P(k ) +
n
( )
P (k − 1) + L(k ) − L(k − 1)= L(k ) + P − L (k )

Master of Science in GNSS @ J. Sanz & J.M. Juan

62
gAGE Code-carrier divergence: SF smoother
Time varying ionosphere induces a bias in the single frequency (SF)
research group of Astronomy and Geomatics

smoothed code when it is averaged in the smoothing filter (Hatch filter).

Let: Where ρ includes all non dispersive terms (geometric range,
P1 = ρ + I1 + ε1 clock offsets, troposphere) and I1 represents the frequency
dependent terms (ionosphere and DCBs). B1 Is the carrier
L1 = ρ − I1 + B1 + ς 1 ambiguity, which is constant along continuous carrier phase
arcs and ε1 , ς 1 account for code and carrier multipath and
thermal noise.
thence,

P1 − L1 = 2 I1 − B + ε1 ⇒ 2 I1 : Code-carrier divergence
Substituting P1 − L1 in Hatch filter equation
Pˆ (k )= L(k ) + P − L ( k )= ρ (k ) − I1 (k ) + B1 + 2 I1 − B1 =
(k )

( )
= ρ (k ) + I1 (k ) + 2 I1 ( k ) − I1 (k )
⇒ Pˆ1 = ρ + I1 + biasI + υ1
gAGE/UPC


biasI
where υ1 is the noise term
where, being the ambiguity term B1 a constant bias, after smoothing.
thence B1 ( k )  B1 , and cancels in the previous expression.
Master of Science in GNSS @ J. Sanz & J.M. Juan
63
gAGE
research group of Astronomy and Geomatics

N=3600 s

Iono
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

65
gAGE
research group of Astronomy and Geomatics
Halloween storm
Data File: amc23030.03o_1Hz

STEC
gAGE/UPC

N=100 (i.e. filter smoothing time constant τ=100 sec ).

Master of Science in GNSS @ J. Sanz & J.M. Juan

66
gAGE
research group of Astronomy and Geomatics Carrier-smoothed pseudorange: DFree
Divergence-Free (Dfree) smoother:
With two frequency carrier measurements a combination of carriers
with the same ionospheric delay (the same sign) as the code can be
generated: 2
f2 1
α1
= = = 1.545
L1, DF = L1 + 2α1 ( L1 − L2 ) = ρ + I1 + B1, DF + ς 1, DF f − f2 γ −1
2 2
1
2
 77 
γ = 
 60 
With this new combination we have:
This smoothed code is immune to temporal
P1 = ρ + I1 + ε1 gradients (unlike the SF smoother), being
L1, DF = ρ + I1 + B1, DF + ς 1, DF the same ionospheric delay as in the
original raw code (i.e. I1). Nevertheless, as
it is still affected by the ionosphere, its
Thence, spatial decorrelation must be taken into
gAGE/UPC

account in differential positioning.

P1 − L1, DF = B1, DF + ε1
⇒ No Code-carrier divergence! ⇒ Pˆ1, DF = ρ + I1 + υ12
Master of Science in GNSS @ J. Sanz & J.M. Juan
gAGE
research group of Astronomy and Geomatics
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

68
gAGE
research group of Astronomy and Geomatics Carrier-smoothed pseudorange: IFree
Ionosphere-Free (Ifree) smoother:
Using both code and carrier dual-frequency measurements, it is possible
to remove the frequency dependent effects using the ionosphere-free
combination of code and carriers (PC and LC). Thence:

PC= ρ + ε PC γ P1 − P2 γ L1 − L2  77 
2

=
PIFree ≡ PC =
; LIFree ≡ LC γ = 
LC =ρ + BLC + υ LC γ −1 γ −1  60 

Thence, γ 2 +1
σ Pc = σ  3σ P
γ −1 P 1 1

PC − LC = BC + ε PC ⇒ PˆIFree ≡ PˆC =ρ + υ IFree

This smoothed is based on the ionosphere-free combination of

gAGE/UPC

measurements, and therefore it is unaffected by either the spatial and

temporal inospheric gradients, but has the disadvantage that the noise
in amplified by a factor 3 (using the legacy GPS signals).
Master of Science in GNSS @ J. Sanz & J.M. Juan
gAGE
research group of Astronomy and Geomatics

Vertical range: [-5 : 5] Vertical range: [-15:15]

gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

70
 C1, L1 PC, LC
gAGE

N=100 N=100
research group of Astronomy and Geomatics

Exercise:
STEC
Justify that the ionosphere-free
combination (PC) is (obviously) not
affected by the code-carrier
gAGE/UPC

divergence, but it is 3 times noisier.

Master of Science in GNSS @ J. Sanz & J.M. Juan

 C1, L1 PC, LC
gAGE

N=360 N=360
research group of Astronomy and Geomatics

N=100 N=100
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

 C1, L1 PC, LC
gAGE
research group of Astronomy and Geomatics

N=3600 N=3600

N=360 N=360
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics
Halloween storm
Data File: amc23030.03o_1Hz

STEC
gAGE/UPC

N=100 (i.e. filter smoothing time constant τ=100 sec ).

Master of Science in GNSS @ J. Sanz & J.M. Juan

74
gAGE
Contents
research group of Astronomy and Geomatics

1. Review of GNSS measurements.

2. Linear combinations of measurements.
3. Carrier cycle-slips detection.
4. Carrier smoothing of code pseudorange.
5. Code Multipath.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

75
gAGE
Multipath
One or more reflected signals reach the antenna in addition
to the direct signal. Reflective objects can be earth surface
research group of Astronomy and Geomatics

(ground and water), buildings, trees, hills, etc.

It affects both code and carrier phase measurements, and it
is more important at low elevation angles.

Butterfly shape

Code: up to 1.5 chip-length  up to 450m for C1 [theoretically]

Typically: less than 2-3 m.
gAGE/UPC

Phase: up to λ/4  up to 5 cm for L1 and L2 [theoretically]

Typically: less than 1 cm
Master of Science in GNSS @ J. Sanz & J.M. Juan
76
 Exercise
gAGE
Plot code and phase geometry-free combination for satellite PRN 15
of file 97jan09coco_r0.rnx and discuss the results.
research group of Astronomy and Geomatics
gAGE/UPC

Butterfly shape:
Master of Science in GNSS
High multipath for low elevation rays (when satellite rises and sets)
@ J. Sanz & J.M. Juan
77
gAGE
research group of Astronomy and Geomatics

= Pc − Lc
M Pc
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics

= Pc − Lc
M Pc
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

gAGE
research group of Astronomy and Geomatics

M MW= PN − LW
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

80
gAGE
research group of Astronomy and Geomatics

M MW= PN − LW
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

81
gAGE
research group of Astronomy and Geomatics

After one year, the directions of the Sun and Aries coincide again, but
the number of laps relative to the Sun (solar days) is one less than
those relative to Aries (sidereal days).
24h
gAGE/UPC

 3m56s
365.2422
Thus, a sidereal day is shorter than a solar day for about 3m 56s
Master of Science in GNSS @ J. Sanz & J.M. Juan
82
 Receiver and multipath noise
gAGE
research group of Astronomy and Geomatics
gAGE/UPC

GPS standalone (C1 code) 10,000 €

Master of Science in GNSS @ J. Sanz & J.M. Juan

83
 Receiver and multipath noise
gAGE
research group of Astronomy and Geomatics

Same environment!
gAGE/UPC

GPS standalone (C1 code) 100 €

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84
gAGE
research group of Astronomy and Geomatics References

[RD-1] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS

Data processing. Volume 1: Fundamentals and Algorithms. ESA TM-
23/1. ESA Communications, 2013.
[RD-2] J. Sanz Subirana, J.M. Juan Zornoza, M. Hernández-Pajares, GNSS
Data processing. Volume 2: Laboratory Exercises. ESA TM-23/2. ESA
Communications, 2013.
[RD-3] Pratap Misra, Per Enge. Global Positioning System. Signals,
Measurements, and Performance. Ganga-Jamuna Press, 2004.
[RD-4] B. Hofmann-Wellenhof et al. GPS, Theory and Practice. Springer-Verlag.
Wien, New York, 1994.
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan

85
gAGE
research group of Astronomy and Geomatics

Thank you!
gAGE/UPC

Master of Science in GNSS @ J. Sanz & J.M. Juan