Advanced Ionospheric Modeling

Peter KolbTrimble Terrasat

GPSNet User SeminarMay 30, 2006

Outline

• Origin and phenomenology of the ionosphere• Effect of the ionosphere on GPS signals• Effect of the ionosphere on network solutions• Formulation of the ionospheric model • Model results

– State of the ionosphere– Ambiguity resolution

• Summary

The origin of the ionosphereschematic, not to scale

Ionization of molecules

Electron density as function of altitudeA typical output of the International Reference Ionosphere

(combination of measurement and sophisticated modeling)D. Bilitza, International Reference Ionosphere 2000, Radio Science 2 (26) 2000

http://nssdc.gsfc.nasa.gov/space/model/ionos/iri.html

Strongly peaked at a particular altitude

Integrated value: Vertical TEC (VTEC)

The origin of the ionosphere

• Gravity keeps gas molecules in the vicinity of earth's surface, forming the atmosphere

• Radiation from the sun ionizes a small fraction of the gases, forming a dispersive medium, the ionosphere

• Ionosphere experiences disturbances depending on– solar activity (11 year solar cycle)– geomagnetic disturbances, solar winds – geographic location of the sun relative to earth– air currents (winds) in the upper layers of the

atmosphere

11 year solar cycle,observed number of sunspots

http

://s

cien

ce.n

asa.

gov/

ssl/

pad/

sola

r/su

nspo

ts.h

tm

11 year solar cycle, predicted number of sunspots

http://science.nasa.gov/ssl/pad/solar/sunspots.htm

rough times are coming up soon again

http://sidc.oma.be

A global VTEC map, calm ionosphereESA Ionospheric Monitoring Facility

http://nng.esoc.esa.de/gps/ionmon.html

Vertical Total Electron Content on a global map, as derived from GPS base station observations

00:00

A global VTEC map, active ionosphere

02:0004:0006:0008:0010:0012:0014:0016:0018:0020:0022:0024:00

CODE’S GLOBAL IONOSPHERE MAPS FOR DAY 280,2005G

eogr

aphi

c la

titud

e (d

egre

es)

-60

-30

0

3

0

6

0

-180 -135 -90 -45 0 45 90 135 180Geographic longitude (degrees)

0 10 20 30 40 50 60 70 TEC (TECU)

Center for Orbit Determination in Europe (CODE)http://www.aiub.unibe.ch/ionosphere

Complex dynamical behavior also on small scales

-20 -10 0 10 20 30 40Longitude (degrees)

Latit

ude

(deg

rees

)4

0

50

6

0

70

DLR Institute for Communications and NavigationN. Jagowski, S. Schlüter, A. Jungstand, http://www.kn.nz.dlr.de

Vertical Total Electron Content over Europe Oct. 7 2005 , as derived from GPS base station observations.

0

1

0

2

0

30

TEC

(m-2

)

Effects of charged gas on radio signals

Phase signal

Code signal

Sendercharged gas Receiver

schematic, not to scale

• Between satellite and rover– the number of phase cycles decreases (phase

velocity increases)– the number of code chips increases (code velocity

decreases)– by the same amount if expressed in units of length

• Magnitude of the effect is – proportional to the density of electrons integrated

over the path of signal propagation (total electron content - TEC)

– proportional to the square of the carrier wavelength(dispersive medium): Great tool for multi-frequency signals

Effects of ionosphere on GPS signals

A case study: BLVA-network

49.2 km

74.9 km

Electron density (cm-3)

IRI -2001

λφ

Electron density, ionospheric layer, pierce points and mapping function

Enhancement of 'seen' electron density along the propagation path:mapping function m

Modeling gradients of the ionosphere

λφ

Taylor expansion of Ionospheric effect across the network projection to the ionosphere

Gradients

λφ

Taylor expansion of Ionospheric effect across the network projection to the ionosphere

Gradients

Modeling gradients of the ionosphere

Piercepoint geometry and dynamical evolution

pierce point coordinates in geocentric coordinatesduring a period of 6 hours

20 minutes between frames

Longitude (degrees)

Latit

ude

(deg

rees

)

Piercepoint geometry and dynamcial evolution

pierce point coordinates in solar centered coordinatesthroughout the course of a day

TEC

valu

es o

btai

ned

from

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tp:/

/ww

w.a

iub.

unib

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/ion

osph

ere

Longitude (degrees)

Latit

ude

(deg

rees

)

wavelength x(phase + int. amb.)

distance traveled by signal

clock offset,user and satellite

ionospheric phase advance

tropospheric delay

multipath contribution

measurement and model error

2 frequency phase measurement and state variables

(Ambiguity, Multipath, Ionosphere)

measurement state variables

model error + system noise

using and forming difference for

Kalman filter – part 1: Observation equation

Observation, measurement

measurements from N stations

Statevector N ambiguities

N stationsN multipathsN stations

3 parameters1 ionosphere

Designmatrix

mapping function

relative distance to reference piercepoint

Kalman filter – part 2:Time update

Systemnoise

Parameters of the calculation

Same set of parameters applied for all networks around the globe

Statevector

state of current epochestimated state for next epoch

Timeupdatematrix

Motion of reference point

Components of the state vector

12:30 15:00

6.0

5.8

5.6

5.4

5.2

5.0

4.8

differences between ambiguities are constant

ambiguity (meters)ambiguity (meters)

0

12:30 15:00

54321

-1-2

-3-4

uncorrelated between stations

multipath (centimeters)multipath (centimeters)

12:30 15:00

0

-0.5

0.5

1.0

1.5

2.0

2.5

activity of ionosphere about to decline

ionosphere parametersionosphere parameters

Use double differences together with code measurement and tropo model to resolve the ambiguity (FAMCAR)

0.5

correlated between days

1.0

0.0

-0.5

-1.06000 8000 10000 12000 14000 16000

Japan, day 1Japan, day 2 (shifted by 4 minutes)

Ionospheric state parametersfrom the bavarian sub network

local time (h)

local time (h) local time (h)

Longitude (degrees)

Latit

ude

(deg

rees

) CODE TEC Map (schematic)

Obs

erva

tion

data

: B

LVA

Day

06

8,

20

02

local time (h)

local time (h) local time (h)

Longitude (degrees)

Latit

ude

(deg

rees

) CODE TEC Map (schematic)

Obs

erva

tion

data

: G

SI D

ay 0

19

, 2

00

2

Ionospheric state parametersfrom a Japanese network

Contributions of first and second order correction termsTypical orders of magnitude for first and second order derivatives

linear terms O(aλ ) ~ O(aϕ ) ~ 1 x 10-6 = 1 mm / km

second order O(aλλ ) ~ O(aλϕ ) ~ O(aϕϕ ) ~ 1 x 10-12 m-1

Network with extensions O(Δλ) ~ O(Δϕ) ~ 100 km

The corrections of the individual terms to the Ionospheric value is thus

linear terms O(aλ Δλ) ~ O(aϕ Δϕ) ~ 10 cm

second order O(1/2 aλλ Δλ2) ~ O(aλϕ Δλ Δϕ) ~ O(1/2 aϕϕ Δϕ2) ~ 1 cm

Depending on your application, higher order terms need to be taken care of, or can be neglected

Application of the model in neworking software, ambiguity resolution

Tracked satellites and fixed satellites Number of unfixed satellites

homogene

10

8

6

4

2

9:00 18:0012:00 15:00local time

9:00 18:0012:00 15:00

10

8

6

4

2

10

8

6

4

2

9:00 18:0012:00 15:00local time

10

8

6

4

2

9:00 18:0012:00 15:00

num

ber o

f sat

ellit

es

num

ber o

f sat

ellit

es

num

ber o

f sat

ellit

es

num

ber o

f sat

ellit

es

Zeroth order

First order

First order

Zeroth order

Resolving ambiguities with different orders of approximationJapanese network (GSI): homogeneous Iono: 96.68 %7 stations first order Iono: 97.73 %70 km x 60 km second order: 97.58 %

German network (ASCOS): homogeneous Iono: 98.03 %28 stations first order Iono: 99.29 %250 km x 350 km second order: 99.29 %

0

0.5

1

1.5

2

2.5

3

3.5

Japan Swissto po A SC OS B LVA

ho mo geneo us Io no

first o rder Io no

seco nd o rder

unfix

ed p

erce

ntag

e

(smaller is better)

Swiss network (Swisstopo): homogeneous Iono: 98.11 %29 stations first order Iono: 98.52 %310 km x 210 km second order: 98.57 %

Summary

Model ionospheric phase advance in terms of Taylor series to the desired order across local area network

Extract ionospheric parameters, multipath and doubly differenced ambiguities by means of a Kalman filter

Obtain increased fixing performance and reliability for small to intermediate network sizes together with a physical picture of the evolution of the ionosphere

EXTRAS

Mapping function on the curved surface

schematic, not to scale

RE

h ϕ

ϕ '

1

1.5

2

2.5

3

3.5

0 10 20 30 40 50 60 70 80 90

2

cos1

1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

−

=

ϕhR

Rm

E

E

map

ping

func

tion

m

Some trigonometryleads to

elevation angle

Illustration of PP Dynamics

RS

RE

PP

RE +h

http://sidc.oma.be

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