Introduction to GNSS Positioning

GNSS: Global Navigation Satellite Systems Today the American GPS (Global Positioning Service), http://gps.losangeles.af.mil/index.html the Russian GLONASS, http://www.glonass-center.ru/frame_e.html In future Chinese Compass, European GALILEO

GLONASS, born almost at the same time as GPS, due to the USSR crisis in the ’90, seemed to be pushed aside. Only in 2001 the Russian government has officially declared they are going to make the system fully operational: now the system is composed of 24 operational satellites. Therefore there has not yet been a systematic and widespread development of the GLONASS signal treatment devoted to geodetic applications. GLONASS data are to be used together with the GPS ones and hopefully with those of the European system GALILEO, which should be fully operational in ?TBD?

Global Positioning System Absolute positioning/navigation A GPS receiver observes signal travel times (ranges) from several GPS satellites: ≈ 1-10 m accurate

Instrumental cost: ≅ 50-500 €

Real time GPS navigation applications people, cars, boats, trains, planes,...: guidance and control Applications Industrial: route optimization, real time overview,.. Public security: critical transports, car/trucks theft,... Assistive technologies Recreational: hiking, sailing,...

Relative kinematic and fast static positioning A reference and a rover GPS stations observe range differences: the rover position can be determined with accuracies in the range 2-5 cm – 1-2 m.

Instrumental cost: ≅ 500 - 10000 € (for the rover)

Applications High precision navigation, for example: airports, railways stations and harbours control, blinds navigation,... Cadastral and cartographic surveying

Cadastral and cartographic surveying required accuracies: 2-3 cm

Static surveys: benchmarks networks Kinematic surveys: aerial photogrammetry

Post-processed static surveys: local control

The GPS network for the monitoring of post-seismic displacements in Umbria region Required accuracies: less than 1 cm Instrumental cost: ≅ 10000 € for each receiver

Permanent Networks A network of GPS stations continuously operating and observing range differences: accuracies up to some mm.

Instrumental cost: ≅ 15000 - 20000 € for each station

Post-processed time series: movements ...

A four year time series for a (Lombardia) station...

The global GNSS network: International GNSS Service (about 380 GPS (GNSS) stations) http://igscb.jpl.nasa.gov/

Global network for the determination of the Earth shape and its movements in time

Estimated velocities of ITRF permanent stations

Local permanent networks for the determination of local movements and deformations in time Example: permanent network in Japan 1200 stations continuously operating from 2004 Estimated velocities

Survey type Notes Accuracy Absolute kinematic in real time

According to the receiver types and new signals availability

From 10 m to 1 m

Absolute static in real time

Still a research topic 10 cm

Relative kinematic in real time or post processing

According to the receiver type and the distance between reference and rover

From 1 m to 2-5 cm

Static relative and post processing

By very long surveys Better than 1 cm

The GPS satellites and milestones Satellites and milestones are grouped as follows 1978: first satellite launch Block I: experimental satellites launched from 1978 to 1985, today out of work...Block II... ...1995: fully operational system ...2010: a new signal: Block IIF, now: 6 satellites

Satellite weight ≅ 800 kg Previous satellites have 2 cesium oscillators and 2 rubidium oscillators Block IIF have 3 rubidium oscillators. The satellites use solar panels to supply themselves and retro-rockets to correct their orbits. The expected life of a satellite is 7.5 years.

GPS satellites requirements

Communicate their position and

GPS time with the highest possible accuracy.

(Block IIR satellites are able, even with a dilution of precision, to mutually position the other satellites of the same Block)

The GPS satellites system At present, 31 operational satellites, displaced on 6 different orbital planes. 55i = °,

e 0, a 26000km; 12T h=

Almost circular orbits. Each satellite moves at about 4 km/sec.

Satellite ephemerides The set of parameters necessary and sufficient to compute the satellite position in time. For the GPS satellites they can be Broadcast ephemerides (predicted by US National Geospatial Intelligence Agency (NGA, https://www1.nga.mil/ , formerly NIMA) and transmitted by the satellites by quasi-keplerian parameters). Precise ephemerides (a posteriori estimated ephemerides, delivered via web by IGS, in a tabular format).

The system control network and the broadcast ephemerides A network of GPS PSs of the United States Army, to predict both the satellites orbits and clocks: the stations are almost uniformly distributed on the Earth equator. It is also called control segment of the GPS system.

The GPS control network, http://www.kowoma.de

The observations of each control station to each satellite are sent to the Control center located at the Master station (Colorado Springs). By complex modeling applied to the received observations, the control center predicts the orbit of each satellite for the next 24 hours. The predicted ephemerides are transmitted to the satellite, which distribute them to the users. The broadcast ephemerides are in WGS84 realization of ITRF and have an accuracy of about 1m.

Precise ephemerides Precise ephemerides are a posteriori computed by IGS Rapid ephemerides, available within 1 day, Final ephemerides: available within 2 weeks. They consist of the a posteriori estimates of the satellites positions in the IGS reference frame and are distributed via web. Final ephemerides have an accuracy better than 2.5 cm.

... SKIPPED HEADER * 2007 9 30 0 0 0.00000000 PG01 2667.489628 14512.617647 22299.283328 155.952112 PG02 -12204.536387 -23228.393961 4878.362107 140.117083 ... PG32 -2811.094639 22575.212168 14287.586567 99999.999999 * 2007 9 30 0 15 0.00000000 PG01 173.660695 14683.334461 22339.970226 155.952820 ... * 2007 9 30 23 45 0.00000000 ... EOF

Visibility of satellites from an observer The system has been designed to always guarantee the visibility of at least 4 satellites everywhere on the Earth surface. A typical visibility table in the 24 hours: typically 6-7 satellites are visible, each satellite is visible for a period of 1- 6 hours.

Satellites elevation: η The angle between the signal direction and the horizontal plane.

Only signals coming from satellites whose elevations are η > 10° or 15° are commonly used, in order to reduce the atmospheric delays.

Skyplot For a given time span, shows the polar (azimuth and elevation) trajectory of all the satellites in view. The above skyplot, for a 24 hours period, is common at our latitudes: no tracks toward North

The signal The on-board oscillators produce a signal, whose nominal frequency 0f is equal to 10.23 MHz, very stable in time (

13 14/ 10 /10f f − −Δ = ). From 0f : two sinusoidal carrier phases (L1 and L2) four binary codes: C/A (Coarse Acquisition Code), P (Precise Code), nowadays transformed in Y (EncrYpted), L2C (Block II-R) M (Block IIR-M), navigational message D (Navigation Data).

The carrier phases for an ideal oscillator Example of a cyclic oscillating phenomenon and the corresponding behavior in time.

An oscillating phenomenon, which repeats itself cyclically in time (sinusoidal) is described by the following equation:

0 0 0( ) sin( ) sin( ( ))A t A t A tω ϕ ϕ= + = 0A : signal amplitude, ω: angular velocity (rad/s), 0ϕ : initial phase

(rad), 0tω ϕ+ instantaneous phase. ( )tϕ is the state of the phenomenon at epoch t, can be expressed

in radians, or, taking ( ) ( ) / 2t tφ ϕ π=

in part of a cycle. The period T of the signal is the time needed to complete one full revolution

2 /T π ω= indeed

0 0 0 0( ) sin( ( 2 / ) ) sin( 2 ) ( )A t T A t A t A tω π ω ϕ ω π ϕ+ = + + = + + = The frequency is defined as follows

1/ / 2f T ω π= = f can be defined also as the first derivative of the phase with respect to time: in fact, to a time interval tΔ a phase variation

f tφΔ = Δ corresponds

/f tφ= Δ Δ taking the limit as 0tΔ → , one gets

/f d dtφ= Let consider an oscillating phenomenon that propagates in space (e.g. the sea waves): it is characterized by a law which depends both on time (t) and space (x). The propagation law is given by

0 0 0 0( , ) sin( ( ) ) sin(2 ( ) )x t xA x t A t Ac T

ω ϕ π ϕλ

= − + = − +

c : signal space propagation velocity,

/cT c fλ = = : wavelength, the distance in space between two successive repeating units.

The main characteristics of L1 and L2 carrier phases (sinusoidal signals propagating in space)

Name f (MHz) λ (cm)

L1 154f0=1575.42 ≅ 19 L2 120f0=1227.60 ≅ 24

Binary Codes A binary code is a sequence of pulses equal to +1 and -1. The pulses transmission sequence, according to a proper decoding key, represents the signal content.

Code duration: time needed to transmit the whole sequence, Code frequency: the inverse of one pulse lasting, Code wavelength: length of one pulse. Name f (MHz) λ (m) Pulses number T All satellites codes C/A 1.023 293.0 1023 1 ms P(Y) 10.230 29.3 3.2703264×1016 37 weeks Block IIR-M satellites new codes: here not discussed

C/A (Coarse acquisition) code 1023 pulses (1 ms.), one different C/A for each satellite used to identify the satellite and for pseudo-range observations. P (Precise) or Y (EncrYpted P) military code 37 weeks long, common to all the satellites. P(Y) guarantees better pseudo-range precisions than C/A code. At the beginning it was public, since August 1994 it has been encrypted in Y code, that, theoretically can be exploited only by the US Army receivers.

Navigational message D (Navigation Data) A further binary code is generated, called navigational message D (f=50 Hz): it consists of 25 blocks, each 30 s long, with a resulting duration of 12.5 min. Each block contains satellite ephemerides and clock offsets, ionospheric model, cyclic information about the state of the other orbiting satellites (almanacs).

Combination of the signals: carrier phase modulation by a binary code. Carrier phases are pure oscillating signals, codes are sequences of square pulses. By multiplying them: a signal equal to the carrier phase but for slips equal to 180° corresponding to code state transitions.

The final signal transmitted by the satellites Two versions of L1 and L2 with a difference in phase equal to 90°, are transmitted: L1 carries C/A, P(Y) L2 carries P(Y) Why a so complex signal? Two frequencies allow to compute ionospheric free observables. Originally, two separate codes (C/A and P(Y)) for civilian and military users. C/A, less precise, only one frequency, public, P(Y): more precise, two frequencies, public in experimental phase (up to the full operability), encrypted in Y and restricted to US army receivers after full operability.

In the last 40 years, changes of US strategies, design of concurrent GNSS constellations: introduction of new signals to improve from one side civilian positioning, from the other, better restrict military codes. Two carriers are a technical need to vehicle binary codes: the two carriers allow phase observations, that are the basis for geodetic GPS surveys. The navigation message is needed to communicate broacast ephemerides, clocks, ionospheric models and other constellation information.

Frequency, phase and time of a real oscillator For each epoch t, phase and frequency of an oscillator are given by

0

0( )( ) , ( ) ( )

t

t

d tf t t f ddtφ φ τ τ φ= = +∫

If i is a clock based on an atomic oscillator with nominal frequency 0f , the time ( )it t of the clock is computed according to the

0 0

0 0 0

( ) ( ) ( ) ( )( ) i i i ii

t t t tt tf f f

φ φ φ φ−= = −

00

1( ) ( )t

i it

t t f df

τ τ= ∫

The frequency of an oscillator can be written as

0( ) ( )i if t f f tδ= + ( )if tδ is the frequency fluctuation.

The clock time can be written as

0 0

00 0

1 1( ) ( )t t

i it t

t t f d f df f

τ δ τ τ= +∫ ∫

( ) ( )i it t t dt t= + where t is GPS time, ( )idt t is called the clock offset or error.

The phase at the epoch t can be written as φi (t) = f0ti (t)+φi (t0 ) φi (t) = f0[t + dti (t)]+φ(t0 ) = f0t + f0dti (t)+φi (t0 )

GPS time and satellites clock offsets The offset between satellites clocks ( St ) and GPS time ( GPSt ) is defined as follows

S SGPSdt t t= −

The offset is time-varying: satellites clocks offsets can be described (with a sufficient accuracy for positioning purposes) by a polynomial of 2nd order in time

20( )S S S S

GPS GPS GPSdt t dt a t b t= + +

Satellites transmit broadcast clock parameters More accurate clock offsets are a posteriori estimated by IGS and distributed via web Accuracies - broadcast offsets: 3 ns, corresponding to 1 m - precise estimates: better than 0.15 ns, corresponding to 5 cm.

Observation equations Code (or pseudo-range) observation

The satellite is identified by its own C/A code

After the identification the receiver performs a correlation between

its internally generated C/A code and the one received from the satellite.

These observations can be done with both C/A and P(Y) codes

The delay between the received signal

and the internal one is electronically measured. ( )S

RT tΔ

The starting epoch is recorded by the satellite clock, The receiving epoch is recorded by the receiver clock: Therefore, the epochs are related to the local time of the two clocks: they are affected by the two clocks offsets.

Let t be the observation epoch; let ( )S

RT tΔ be the delay observed between the receiver R and the satellite S signals. The observation equation is given by:

( ) ( ) ( )S S SR R RT t t t t t τΔ = − −

where: ( )Rt t is the receiving epoch recorded by the receiver R clock, ( )S S

Rt t τ− is the starting epoch recorded by the satellite S clock. τ is the signal traveling time from the satellite to the receiver;

The satellite clock is not synchronized with the GPS time; we have: ( ) ( )S S S S S

R R Rt t t dt tτ τ τ− = − + − in 66msτ ≅ clock offset doesn’t change: ( ) ( )S S S

Rdt t dt tτ− ≅ for the receiver clock ( ) ( )R Rt t t dt t= +

Therefore, we have

( ) ( ) ( ) ( ) ( )S S S S SR R R R RT t t t dt t dt t dt t dt tτΔ = − + − = + −

Multiplying by the propagation signal velocity in vacuum, the so called code (or pseudo-range) observation is obtained ( ) ( ) ( ( ) ( ))S S S S

R R R RP t c T t c c dt t dt tτ= Δ = + − Observation noise P(Y): 10-30 cm C/A: 30-200 cm (depending on the receiver quality)

Phase observations Basic concepts An observation of the difference in phase between the received carrier (L1 or L2) and a sinusoid of the same frequency internally generated by the receiver can be done.

The observation equation at the epoch t is given by ( ) ( ) ( )S S

R R S Rt t tφ φ φ τ= − − ( )S

R tφ is the phase difference observation at the epoch t ( )R tφ is the receiver internally generated phase at the epoch t ( )SS Rtφ τ− is the phase of the satellite S signal generated at the

emission epoch. taking into account the previous formulas,

0 0 0 0 0 0

0 0 0 0

( ) ( ) ( ) ( )

( ( ) ( ))

S S S S SR R R R R

S S SR R R

t f t f t f dt t f dt t

f f dt t dt t

φ τ τ φ φτ φ φ

= − − + − − + −

= + − + −

The integer ambiguity

The integer ambiguity The receiver can observe the fractional part of the incoming carrier, but not the integer number of cycles from its emission at the satellite: a integer unknown has to be added to the observation equation

0 0 0 0( ) ( ( ) ( )) ( )S S S S SR R R R Rt f f dt t dt t N tφ τ φ φ= + − + − +

( )S

RN t is the integer number of cycles passed from the satellite signal phase at the emission epoch and the satellite signal phase at the receiving epoch: N cannot be observed.

The observation can be multiplied by the wavelength of the carrier: Thus the so-called phase observation is obtained

0 0( ) ( ) ( ) ( ( ) ( )) ( ( ) )S S S S S SR R R R R RL t t c t c dt t dt t N tλφ τ λ φ φ= = + − + + −

Observation noise: about 1 mm, both for L1 and for L2.

0 0, , ( )S SR RN tφ φ represent completely different physical quantities.

0 0,

SRφ φ are fractional values, constant in time: represent the initial

phases of receiver and satellite respectively. 1( )

SRN t is an integer, time varying, value: represents the integer part

of the distance between satellite and receiver.