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GPS: Errors in Pseudorange Measurements and Their Modelling

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Announcements:Quiz on 13th April (Friday)Final Exam : 50 marks / 10 questions1-DIP, 4 GIS, 5 GPS

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Uses measurements from 4+ satellitesdistance = travel time x speed of light

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Sources Of Errors in GPS Data

Ephemeris DataErrors in the Transmitted Location of the satellite

Satellite ClockErrors in transmitted Clock signal

IonosphereErrors in Pseudorange caused by Ionospheric effect

TroposphereErrors in Pseudorange caused by Tropospheric effect

MultipathErrors due to reflected signal sensed by the receiver Antena

ReceiverErrors due to Thermal Noise, Inter-channel biases and Software accuracy

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Troposphere Begins from Surface of Earth and Ends where the Temperature in the Atmosphere Stops decreasing (10 Km altitude)

IONOSPHERE

Ionosphere has a significant influence on advanced communication and navigation systems like GPS

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Atmospheric Layers

Passage of GPS Signal Through Atmosphere

Space to Ground and Space to Space GPS path delay measurement setup

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Generalised Atmospheric Circulation Model

AttenuationScintillation 0.5 dbDelay (2 - 25 M)

EFFECT OF TROPOSHERE

75% of Total, rest 25% comes from Tropopause region

Also varies with Elevation Angle and Gas density profile with altitude

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The variation of pressure with altitude

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=

v

cn=

The refractive index of a medium, n, is defined as the ratio of the speed of propagation of an electromagnetic wave in a vacuum, c, to the speed of propagation in this medium, v:

Usually instead of the refractive index n, the refractivity N is used as the electromagnetic waves in the atmosphere propagate just slightly slower than in a vacuum and it is given by

)1(106 = nN Trop

Hopfield (1969) shows the possibility of separating N into a dry and a wet component

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The dry refractivity is directly proportional to atmospheric pressure and can be accurately modeled. The hydrostatic refractivity can be expressed as

where ,

P is the hydrostatic air pressure in mb and T is temperature in Kelvins

In the presence of water vapor and hydrometeors, refractivity can be modeled with the aid of additional measurements. The wet refractivity of water vapor can be expressed as (Smith et al., 1953).

where k = 64.8 Kmb-1 , k = 3.776x105 K2 mb-1 ,Pv is the partial pressure of water vapor (e) in mb and T in Kelvins. So, Tropospheric refractivity (N Trop ) will be

Actually the values of k , k & k are empirically determined and dependent of local situations

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Tropospheric Delay

Since the tropospheric refractive index is higher than unity varying as a function of altitude, a wave propagating between the ground and a satellite has a radio path length exceeding the geometrical path length. The difference in length can be obtained by the following integral:

OR

Where N Trop is the tropospheric refractivity.

Delay Structure of Radio Waves

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The tropospheric delay can be separated into two main components: Hydrostatic delay Wet delay.

Hydrostatic delay is caused by the dry part of the atmospheric constituents and can be estimated precisely using surface temperature and pressure measurements. By removing the hydrostatic delay from the total tropospheric delay, the remaining signal delay is called the wet delay, About 90% of the tropospheric refraction arises from the dry and about 10% from the wet component (Hofmann, 2001).

The slant tropospheric delays at arbitrary elevation angles can be expressed in terms of the zenith delays and mapping functions. This representation allows the use of separate mapping functions for the hydrostatic and wet delay components:

is elevation Angle of satellite

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Principle

Total Delay = Ionosphere Delay + Neutral Delay

Neutral Delay = Total Delay - Ionosphere Delay

Zenith Neutral Delay = Zenith Hydrostatic Delay + Zenith Wet Delay

Zenith Wet Delay = Zenith Neutral Delay - Zenith Hydrostatic Delay

Ground Based Method

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GPS signals are slowed and refracted as they passthrough the Earths atmosphere.

Measurement of the bending angle produced byatmospheric refraction provides the observable that isthe basis of space-based GPS meteorology.

Resolving the delay of GPS signals caused by the atmosphere using the most accurate geodetic receivers provides the observable that is the basis for ground-based GPS meteorology.

Noise for Some is the Signal to Others

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GPS Meteorology(Estimation of PRECIPITABLE WATER Vapor in Atmosphere from the observed delay from to troposphere

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Modelling of IONOSPHERIC Delay

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Atmospheric Layers

Ionospheric:10 100 m delay in Pseudo range

Tropospheric:

2.3 to 30 m depending on elevation and atmospheric conditions

Clock Sensitivity:

Typically GPS satellite orbital velocity is 1 Km/Sec, thus it moves by 66 meters in the position by the time the GPS signal reaches earth surface (20000Km away) in 66 msec.

An error of 1 msec would result in pseudorange error of 300 Km

For Phase processing ( 1 microSec) data is needed so that 300 m errors in pseudorange is tolerated by the solution.

Normal Ranges for Pseudorange Errors

In conventional applications these are modelled as functions of atmospheric parameters such as TEC in Ionosphere, Temperature, Pressure and Humidity of the atmosphere etc.

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IONOSPHERE

Ionosphere has a significant influence on advanced communication and navigation systems like GPS

0 50 100 150 200 250 300 350

Geographic Longitude ( oE)

-80

-60

-40

-20

0

20

40

60

80

Geographic Latitude (oN)

TEC in TECU

37+

34 to 37

31 to 34

27 to 31

24 to 27

21 to 24

18 to 21

15 to 18

12 to 15

9 to 12

6 to 9

2 to 6

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

20

40

60

80

(e)

0 50 100 150 200 250 300 350

Geographic Longitude ( oE)

-80

-60

-40

-20

0

20

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60

80

Geographic Latitude (oN)

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

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80

(a)0 50 100 150 200 250 300 350

Geographic Longitude ( oE)

-80

-60

-40

-20

0

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80

Geographic Latitude (oN)

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

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(b)

0 50 100 150 200 250 300 350

Geographic Longitude ( oE)

-80

-60

-40

-20

0

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Geographic Latitude (oN)

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

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(c)0 50 100 150 200 250 300 350

Geographic Longitude ( oE)

-80

-60

-40

-20

0

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Geographic Latitude (oN)

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

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(d)

LT: 9:30 AM LT: 11:30 AM

LT: 1:30 PM LT: 3:30 PM

LT: 5:30 PM

SPACE WEATHER

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IONOSPHERIC ERRORS: 1) TIME DELAYS

Radio signals are bent and therefore delayed by the F layer and absorbed by the D-layer

1 TECU = 10+16 e-/m2

1 ns = 0.3 m = 2.852 TECU40 TECU = 4.2 m200 TECU = 570.04 ns = 171.01 m

Receiver

400 km

Center of Earth

Sub-Ionospheric Point

x

Ionosphere

Plasmasphere

9.5 10 10.5 11 11.5 12 12.5 13 13.5

Universal Time (Hr)

-1

-0.5

0

0.5

1

dTEC/min (TECU/min)Radio signals experience

scintillations due to electron density irregularities

2) SCINTILLATIONS

Plasma bubble

9 9.5 10 10.5 11 11.5 12 12.5 13 13.5

Universal Time (Hr)

25

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Vertical TEC (TECU)

equatorial

plasma bubble

11.5 UT

21.5 LT

TEC fluctuations

GPS phase scintillation

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1 4 7 10 13 16 19 22 25

Universal Time (Hr)

150

200

250

300

350

400

h'F (km)

11 UT, 21 LT

352 km

225 km

HEIGHT VARIATIONS

Radio signals experience scintillations due to

ionospheric height variations

IONOSPHERIC MODELLING Based on TEC count

The TEC is expressed in units of electrons/m2 or occasionally TEC units (TECU)

1 TECU is defined as 1016 electrons/m2. The TEC is a function of time of day, user location, satellite

elevation angle, season, ionizing flux, magnetic activity, sunspot cycle, and scintillation.

Nominally ranges between 1016 and 1019, with the twoextremes occurring around midnight and mid-afternoon

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where = (fL1/fL2)2. Here ionospheric delay errors are removed, Drawback : measurement errors are significantly magnified through thecombination.

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A preferred approach is to use the L1 and L2 pseudorangemeasurements to estimate the ionospheric error on L1 using the following expression:

? Single frequency Receivers

In case of a single-frequency receivers, models such as the Klobuchar

model based on observed ionospheric variatios with Latitude and local

time of the Day is used

Removes (on average) about 50% of the ionospheric delay at midlatitudes

through a set of coefficients included in the GPS navigation message.

Assumption is that the vertical ionospheric delay can be approximated by

half a cosine function of the local time during daytime and by a constant

level during nigh time

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END

Almost three times as much delay is incurred when viewing satellites at low elevationthan at the zenith. For a signal arriving at vertical incidence, the delay rangesfrom about 10 ns (3m) at night to as much as 50 ns (15m) during the day. At low satelliteviewing angles (0 through 10), the delay can range from 30 ns (9m) at nightup to 150 ns (45m) during the day [15]. A typical 1-sigma value for residual ionosphericdelays, averaged over the globe and over elevation angles, is 7m [17].

Almost three times as much delay is incurred when viewing satellites at low elevation than at the zenith. For a signal arriving at vertical incidence, the delay rangesfrom about 10 ns (3m) at night to as much as 50 ns (15m) during the day. At low satellite viewing angles (0 through 10), the delay can range from 30 ns (9m) at night up to 150 ns (45m) during the day A typical 1-sigma value for residual ionosphericdelays, averaged over the globe and over elevation angles, is 7m

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