GNSS Surveying, GE 205

Kutubuddin ANSARIkutubuddin.ansari@ikc.edu.tr

Lecture 3, March 8, 2015

GPS Error

X

GPS Errors

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

GPS Errors

Originating at the satellites

Originating signal propagation or atmospheric refraction

Originating at the receiver

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.

Satellite Position Error Baseline ErrorRange Satellite Baseline Length

Thumb Rule to estimate Orbital Error

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.

SA is on SA is off

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

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 ErrorMultipath Error

PHASE

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

If a and φ denote the amplitude and the phase of the signal

direct signal = a cosφindirect 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

Multipath Error

The superposition of signals is represented by:

1+ cos cos sin sin

M M

M M

Let us consider

where the subscript M indicates multipath

2 sin= 1+ +2 cos , tan1 cosM M

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 cos2

sin 1tan tan1 cos 2 2

M

M M

The best way to eliminate multipath error is to construct the observation site with no reflecting surface and objects in its locality.

Earth’s Atmosphere

Solid Structures

Metal

Signal Propagation or Atmospheric Refraction Errors

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.

• 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

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.

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).

Phase and Group velocity

For a single electromagnetic wave propagating in space with wavelength λ and frequency f

The phase velocity: phv f

The group velocity:2

grdfvd

Rayleigh Equation

2

1,

,

phph

ph ph

phgr ph

dvdfdv df fd fd d

dv dvdf dff fd d d d

dvv v

d

Refractive Index

cnv

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

Depends on Refractive Index (n)

Modified Rayleigh Equation

ph grph gr

c cv and vn n

2 2

1

1

1 1 1,

1 1 1 11 , 1

(1 ) 1

ph ph

gr ph ph gr ph ph

ph phgr ph

gr ph ph ph

phgr ph

dn dnc c cn n n d n n n d

dn dnn n

n n n d n d

dna a n n

d

Modified Rayleigh Equation

phgr ph

we have c f

d dff

dnn n f

df

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

322 31 ......ph

ccnf f

22

2 23 2

1 ,

2 1

ph

phgr

cnf

dn c cndf f f

Ionosphere dispersive relative to GPS Radio Signal

Ionospheric Refraction

2 22 2

2

2 2

gr ph gr ph

1 , 1

c = - 40.3 Ne (electron density)40.3 40.31 , 1

n > n , v < v

ph gr

ph gr

c cn nf f

n Ne n Nef f

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

Pseudorange

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=1×1016electrons/m2

GPS Frequencies

• Each satellite sends down exactly the same two radio frequenciesL1 at 1575.43 MHzL2 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

For the dual frequency (L1, L2) observation, TEC in the slant direction can be calculated from the pseudo range (P) and phase observations (ɸ) as

Total Electron Content (TEC)

2 21 2

2 12 21 2

2 21 2

1 22 21 2

1 ( )40.3

1 ( )40.3

f fTEC P Pf f

f fTECf f

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

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)

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 cosαVTEC=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

• 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

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

Start with Code Pseudorange modelwith ionosphere affects

Elimination

L1 1

L2 2

R ( )

R ( )

IonoL

IonoL

c f

c f

fL1, fL2 frequency of the two carriers

Linear CombinationRL1, L2=n1RL1 + n2RL2

L1,L2 1 1 2 2

L1,L2 1 2 1 1 2 2

1 1 2 2

11 2

2

R [ ( )] [ ( )]

R ( )( ) ( ) ( )

For Elimanation

( ) ( ) 0

( )Assuming n =1, n ( )

Iono IonoL L

Iono IonoL L

Iono IonoL L

IonoL

IonoL

n c f n c f

n n c n f n f

n f n f

ff

Elimination

2

22

2 21

22

L1, L2 L1 L221

But40.3 TECf

n

R =1.R .R

Iono

ph

L

L

L

L

ff

ff

Elimination

Ionosphere free linear combinations Code ranges

This is the ionospheric free linear combination for code ranges. A similar ionospheric free linearcombination for carrier phase may be derived as

Carrier phases2

L1, L2 L1 L21

=1. .L

L

ff

22

L1, L2 L1 L221

R =1.R .RL

L

ff