gps-errors part-1

1. GNSS Surveying, GE 205
Kutubuddin ANSARI
kutubuddin.ansari@ikc.edu.tr
Lecture 3, March 8, 2015
GPS Error

2. X
GPS Errors
GPS measurements are
both affected by
several types of
random errors and
systematic errors which
affects the accuracy of
measurements

3. GPS Errors
Originating at the satellites
Originating signal propagation or
atmospheric refraction
Originating at the receiver

4. Satellites Errors
1. Ephemeris or Orbital
Error
•Satellite positions are a
function of time
•Forces acting on the GPS
satellites are not perfect
•Errors in the estimated
satellite positions known as
ephemeris errors.

5. Satellite Position Error Baseline Error
Range Satellite Baseline Length
=
Thumb Rule to estimate Orbital Error

6. 2. Selective Availability
Selective availability (SA) is a technique to deny accurate
real-time autonomous positioning to unauthorized users.
• δ -process is achieved by dithering the fundamental frequency of the
satellite clock.
• ε-error is the truncation of the orbital information in the transmitted
navigation message so that the coordinates of the satellites cannot
accurately be computed.
• SA turned on, nominal horizontal and vertical errors could be up to 100m
and 156m, respectively.

7. SA is on SA is off

8. Each GPS Block II and Block IIA satellite contains two cesium and
two rubidium atomic clocks.
The satellite clock error is about 8.64 to 17.28 nanoseconds per day.
The corresponding range error is 2.59 m to 5.18 m
GPS receivers, in contrast, use inexpensive crystal clocks.
The receiver clock error is much larger than that of the GPS satellite
clock.
3. Satellite and Receiver Clock Errors

9. B M O C
•A signal that bounces of a smooth object
and hits the receiver antenna.
•Increases the length of time for a signal to
reach the receiver.
•A big position error results.
•Gravel roads
•Open water
•Snow fields
•Rock walls
•Buildings
Receiver Error
Multipath Error

10. PHASE
Phase is the fraction of a wave cycle which has elapsed
relative to an arbitrary point

11. If a and φ denote the amplitude and the phase of the signal
direct signal = a cosφ
indirect signal =β a cos(φ+Δ φ)
Where β is a damping factor
a cos + a cos( + )
(1+ a cos ) a cos - ( sin ) a sin
ϕ β ϕ ϕ
β ϕ ϕ β ϕ ϕ
∆
∆ ∆
Multipath Error
The superposition of signals is represented by:
1+ cos cos
sin sin
M M
M M
β ϕ β ϕ
β ϕ β ϕ
∆ = ∆
∆ = ∆
Let us consider
where the subscript M indicates multipath

12. 2 sin
= 1+ +2 cos , tan
1 cos
M M
β ϕ
β β β ϕ ϕ
β ϕ
∆
∆ ∆ =
+ ∆
 β may vary between 0 and 1
 β = 0 (no reflected signal and no multipath)
βM = 1 and ΔφM =0
the resultant signal is identical to the direct signal
 β =1 (The strongest possible reflection )
= 2(1+cos ) =2 cos
2
sin 1
tan tan
1 cos 2 2
M
M M
ϕ
β ϕ
ϕ ϕ
ϕ ϕ ϕ
ϕ
∆
∆
∆ ∆
∆ = = → ∆ = ∆
+ ∆
 The best way to eliminate multipath error is to construct the observation
site with no reflecting surface and objects in its locality.

13. Earth’s
Atmosphere
Solid
Structures
Meta
l
Signal Propagation or Atmospheric
Refraction Errors

14. 1. Ionosphere
•The uppermost part of the
earth’s atmosphere (50 km
and 1000 km), ultraviolet and
X-ray radiations coming from
the sun interact with the gas
molecules and atoms.
•These interactions result in
gas ionization, a large number
of free, negatively charged,
electrons and positively
charged, atoms and
molecules, such a region of the
atmosphere where gas
ionization takes place is called
the ionosphere.

15. • The electron density within the ionospheric region is not
constant, it changes with altitude. As such, the
ionospheric region is divided into sub regions, or
layers, according to the electron density
• The altitude and thickness of those layers vary with time, as
a result of the changes in the sun’s radiation and the
Earth’s magnetic field.
• The ionosphere is a dispersive medium, which means
that it bends the GPS radio signal and changes its
speed as it passes through the various ionospheric layers to
reach a GPS receiver

16. The dispersion is the phenomenon in which the phase
velocity of a wave depends on its frequency and such
type of medium is called dispersive medium
Dispersive Medium
In a dispersive prism, material dispersion causes different
colors to refract at different angles, splitting white light into
a rainbow.

17. Phase and Group velocity
The phase velocity of a wave is the rate at which
the phase of the wave propagates in space (red dot).
The group velocity of a wave is the velocity with
which the overall shape of the wave’s amplitudes
propagates through space (green dot).

18. Phase and Group velocity
For a single electromagnetic wave propagating in
space with wavelength λ and frequency f
The phase velocity: phv fλ=
The group velocity:
2
gr
df
v
d
λ
λ
= −

19. Rayleigh Equation
2
1
,
,
ph
ph
ph ph
ph
gr ph
dvdf
dv df fd f
d d
dv dvdf df
f f
d d d d
dv
v v
d
λ λ
λ λ λ
λ λ λ λ
λ λ λ λ
λ
λ
 
= + = − 
 
= − = −
= − +

20. Refractive Index
c
n
v
=
The refractive index n of a
material is a dimensionless
number that describes how light
propagates through that
medium. It is defined as:
where c is the speed of
light in vacuum and v is
the phase velocity of
light in the medium

21. Depends on Refractive Index (n)
Modified Rayleigh Equation
ph gr
ph gr
c c
v and v
n n
= =
2 2
1
1
1 1 1
,
1 1 1 1
1 , 1
(1 ) 1
ph ph
gr ph ph gr ph ph
ph ph
gr ph
gr ph ph ph
ph
gr ph
dn dnc c c
n n n d n n n d
dn dn
n n
n n n d n d
dn
a a n n
d
λ λ
λ λ
λ λ
λ λ
λ
λ
−
−
= + = +
   
= + = + ÷  ÷ ÷  ÷
   
− = − ⇒ = −

22. Modified Rayleigh Equation
ph
gr ph
we have c f
d df
f
dn
n n f
df
λ
λ
λ
=
= −
⇒ = +

23. Ionospheric Refraction
The ionosphere extending in various layers from about 50
km to 1,000 km above earth is a dispersive medium with
respect to the GPS radio signal. Following Seeber series ..
Neglect higher order terms
32
2 3
1 ......ph
cc
n
f f
= + + +
2
2
2 2
3 2
1 ,
2 1
ph
ph
gr
c
n
f
dn c c
n
df f f
= +
= − → = −

24. Ionosphere dispersive relative to GPS Radio Signal
Ionospheric Refraction
2 2
2 2
2
2 2
gr ph gr ph
1 , 1
c = - 40.3 Ne (electron density)
40.3 40.3
1 , 1
n > n , v < v
ph gr
ph gr
c c
n n
f f
n Ne n Ne
f f
= + = −
= − = +

25. Pseudorange = Geometric range + Range correction
(Measured Range) = (Actual Range) + Error
Pseudorange

28. Total Electron Content (TEC)
TEC is an important
descriptive quantity for the
ionosphere of the Earth.
TEC is the total number
of electrons integrated
between two points, along a
tube of one meter squared
cross section. The TEC is
measured in a unit called
TECU, where
1TECU=1×1016
electrons/m2

29. GPS Frequencies
• Each satellite sends down exactly the same two radio
frequencies
L1 at 1575.43 MHz
L2 at 1227.60 MHz
•At these microwave frequencies the signal are highly
directional and hence are easily blocked, as well as highly
reflected by solid objects and water surfaces

30. For the dual frequency (L1, L2) observation, TEC in the
slant direction can be calculated from the pseudo range (P)
and phase observations ( ) asɸ
Total Electron Content (TEC)
2 2
1 2
2 12 2
1 2
2 2
1 2
1 22 2
1 2
1
( )
40.3
1
( )
40.3
f f
TEC P P
f f
f f
TEC
f f
φ φ
 
= − ÷
− 
 
= − ÷
− 
Here, P1 and P2 are pseudoranges and ɸ1 and ɸ2 are phases of
carriers L1 and L2 respectively. For simplification:
2 19.52 ( )TEC P P= −

31. STEC is the total electron content calculated on the path
different than local zenith. The value of TEC consists of
both the STEC along a satellite receiver ray path and
instrumental bias B (constant)
STEC= TEC + B
Slant Total Electron Content (STEC)

32. Vertical Total Electron Content (VTEC)
The electron content calculated on the local zenith path is
called Vertical Total Electron Content (VTEC)
2
E
max
R cosα
VTEC=STEC× 1-
Re+h
 
 ÷
 
where α is the elevation
angle, RE the radius of
the earth (RE=6378 km)
and hmax (=350 km) is
the height of the
ionospheric shell above
the surface of the earth

33. • TEC is a very complicated quantity depends on
sunspot activities line of sight (includes elevation
and azimuth of satellites) position of observation
etc.
• Determination of TEC is essential
• TEC effects needs to be measured, estimated,
modelled or eliminated
Total Electron Content

34.  Various time its very difficult to measure , estimate or
model the value of TEC
 Most efficient method is to eliminate the value of TEC
 Its very easy to eliminate it by using two signals with
different frequencies and this the main reason why the
GPS signal has two carrier waves L1 and L2
 It can be done by linear combination of pseudorange
models so that ionospheric refraction cancels out
Elimination

35. Start with Code Pseudorange model
with ionosphere affects
Elimination
L1 1
L2 2
R ( )
R ( )
Iono
L
Iono
L
c f
c f
ρ δ
ρ δ
= + ∆ + ∆
= + ∆ + ∆
fL1,fL2 frequency of the two carriers
Linear Combination
RL1, L2=n1RL1 + n2RL2

36. L1,L2 1 1 2 2
L1,L2 1 2 1 1 2 2
1 1 2 2
1
1 2
2
R [ ( )] [ ( )]
R ( )( ) ( ) ( )
For Elimanation
( ) ( ) 0
( )
Assuming n =1, n
( )
Iono Iono
L L
Iono Iono
L L
Iono Iono
L L
Iono
L
Iono
L
n c f n c f
n n c n f n f
n f n f
f
f
ρ δ ρ δ
ρ δ
= + ∆ + ∆ + + ∆ + ∆
= + + ∆ + ∆ + ∆
∆ + ∆ =
∆
→ =
∆
Elimination

37. 2
2
2
2 2
1
2
2
L1, L2 L1 L22
1
But
40.3
TEC
f
n
R =1.R .R
Iono
ph
L
L
L
L
f
f
f
f
∆ =−
=−
⇒ −
Elimination

38. Ionosphere free linear combinations
Code ranges
This is the ionospheric free linear combination for
code ranges. A similar ionospheric free linear
combination for carrier phase may be derived as
Carrier phases 2
L1, L2 L1 L2
1
=1. .L
L
f
f
Φ Φ − Φ
2
2
L1, L2 L1 L22
1
R =1.R .RL
L
f
f
−