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SVY 207: Lecture 11GPS Error Sources: Review

• Where can errors occur?

– Satellite: ephemeris, clock, S/A (history) A/S

– Propagation: ionosphere, troposphere, multipath

– Receiver: antenna, clock,measurement eror

– Earth: Earth surface kinematics

– Data Processing

» Algorithmic: cycle slips, ambiguity

» Stochastic: errors in the error model

» A general least squares perspective

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Ephemeris Errors (1)

– To undertake precise positioning, the position of the satellite must be known at transmit time.

– Navigation Message contains Keplerian-type parameters which are valid for approx. 1 hour

» orbit position in WGS-84 using “ephemeris algorithm”

» “old” ephemeris will degrade rapidly with time due to unmodelled forces on the satellite

– A number of civilian agencies tabulate precise orbit positions in the Earth fixed frame

» NGS (National Geodetic Survey, USA)

» IGS (International GPS Service for Geodynamics)• up to factor of 100 improvement over Navigation Message

• but not in real time (few days delay)

» Up to factor of 10 for real time predictions

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Ephemeris Errors (2) • Reference Frame

– Broadcast orbit frame

» Positions in WGS-84 reference frame

» WGS-84 is inherently accurate to decimetres (globally)

» Suitable for most surveying, engineering and precise navigation applications

» Broadcast orbits continue to improve

– Precise orbit frame

» Positions in ITRF,• IERS Terrestrial Reference Frame

• IERS = International Earth Rotation Service

» ITRF is inherently accurate to a centimetre (globally)

» WGS-84 is consistent with ITRF

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Ephemeris Errors (3)• Error in relative coordinates r between two stations separated

by distance L: – “rule of thumb” r Ls A– where s is orbit error, A is orbit altitude (20,000 km)

– depends on DOP, estimation strategy, orbit improvement

• Accuracy:Source Delay Orbit Error Position Error

(L=distance)

Broadcast Instant 5 m <10-6 L

IGS (predicts) Instant 1 m <10-7 L

IGS (rapid) few days 0.2 m 10-8 L

IGS (final) 1 week 0.05 m < 10-8 L

– NOTE: position errors assume that processing softwareuses a function model appropriate to distance L.

• JPL now produce rapid orbit products–

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Satellite Clock Error• For precise relative positioning, the satellite

clock is– Either

» estimated (at every epoch).

» OR removed by differencing data between receivers• differencing introduces mathematical correlations into the data

which must be modelled correctly.

• An alternative method– JPL provides precise clock solutions with their precise

ephemerides to allow process known as precise point positioning (1 cm accuracy).

– Use of this currently requires proprietary software (GIPSY).

» Again, a delay in the provision of the product.

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Selective Availability (S/A)(Historical note)

• Intentional errors imposed on the GPS signal– Epsilon

» Navigation message includes errors in ephemeris i.e. orbits

» Apparently not used (according to daily comparisons between broadcast and IGS orbit solutions)

» Even if it were used, it would have no effect when using precise orbits from another source

– Dither

» The satellite reference frequency is dithered

» Special military receivers can remove this dither

» Looks exactly like a satellite clock error

– S/A is mitigated by relative positioning

– S/A switched off on May 2nd 2000 - but could be tuned on again in the future.

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Anti-Spoofing (A/S)• Encryption of the P-code

– Still used to deny civilian users access to precise codes on L1 and L2

– Recall, modern geodetic receivers can nevertheless form 2 precise pseudorange observables e.g. cross correlation.

– Pseudorange noise is increased• However, we rely on phase for precise work

• May degrade cycle-slip detection and ambiguity resolution algorithms

– L2 phase noise is increased• from ~1 mm to ~ 1 cm

• No serious effect on long sessions (static positioning)

• Degrades short session positions (rapid static, kinematic)

• Larger degradation at low elevations (up to 2 cm)

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Ionospheric Errors (1)• Ionosphere delays radio signals

– Upper atmosphere: km ionosphere contains charged particles (electron positive ions) caused by the Sun’s UV rays

– Electrons delay the GPS coded signal (units of distance):

ion k. TECf 2– k varies unpredictably with time and direction

– Peak TEC occurs around km altitude

• The ionosphere is dispersive for microwaves– The carrier phase is delayed (satellites appear further

away)

– The code signal is advanced (satellites appear closer)

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Ionospheric Errors (2)• Ionspheric Delay

– Important for single frequency receivers

– Delay of 0.1 (night) - 100 metres (day)

– Maximum effect:

» 14:00 hours local time

» Spring Equinox

» Peak of 11-year solar sunspot cycle

» 20° above and below magnetic equator

» up to 30 m error at zenith - can be 3 times larger near to horizon (increased slant depth)

• Rapid Variations– Can cause receiver problems (cycle slips),

– Especially older, cheaper receivers

– Magnetic activity around polar regions

– Sometimes high levels of atmospheric turbulence near equator

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Tropospheric Error (1)• Lower atmosphere

– 0 - 20km

– Very thin and local

– Temperature decreases rapidly with altitude

– Most delay from 0 - 7km

» Contains 80% of atmosphere by mass

– Non dispersivee medium for radio waves - affects L1 and L2 carriers by the same amount

• Tropospheric error has Two components: – Dry: mainly nitrogen and oxygen

– Wet: water vapour (very small effect from clouds and rain)

• “Dry Delay” affects radio + optical» Zenith delay 1.9 to 2.3 m

» Function of pressure and temperature profile

» Therefore, depends on height above sea-level

» Predictable and easily calibrated

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Tropospheric Error (2)• “Wet Delay” strongly affects radio

– Zenith delay of 0 to 20 cm

– Function of humidity profile

– Therefore, depends on climate (e.g., tropical, desert, etc.)

– Unpredictable and difficult to calibrate

• Delay effect of Toposphere– Total effect (dry + wet) at zenith: 1.9 - 2.5 m

– Total effect near horizon: 20 m

– Very little variation with azimuth

• Variation with time– Zenith delay typically varies only 1 cm in 1 hour

– Strongly correlated between stations < 30 km apart

• GPS Data Processing – Mitigating Tropospheric affects is very important for

precise positioning - (cf. Lecture 14)

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Multipath Error

• Multiple paths between satellite and receiver– Caused by reflection of signal from conducting surfaces

in receiver environment

» Chain link fence, vehicles, wet ground, sea surface, etc.

» Beware of large, regularly-shaped surfaces nearby

– Secondary path is weaker, and error is often difficult to detect

– Tends to produce oscillating errors

» tends to average down very quickly (few minutes)

» but oscillation is slower (and stronger) for closer objects

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Antenna• Phase Centre Offset

– Electrical centre doesn’t coincide with geometrical centre

– Each antenna has a phase centre offset

– Same type of antenna has approximately the same offset(especially if both are oriented North)

– Hence this approximately cancels for relative positions

– But offset values must be used if antennas are different

– Most important for very precise work

– Avoid mixing antenna types on same baseline

• Antenna structure also produces multipath– causes an apparent “phase centre variation”

– electrical centre moves depending on satellite direction

– can be at the 1 centimetre level

– but approximately cancels between similar antennas

– antennas can be calibrated as function of sky direction

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Receiver Clock Error• Receiver clock is used to stamp the time

associated with each measurement– Effect on observation is cancelled by double differencing

• But it is also used as the epoch time to compute the geometrical range (in b vector)

– which changes as the satellites move and Earth rotates

• If clock is synchronised to GPS time to s – satellite position error 10-6 s4 km/s mm

– Earth rotation error 10-6 s5 km/s mm

– computed range error 10-6 s1 km/s mm

• If not synchronised, then:– first use pseudorange point position method to estimate

receiver clock offset.

– then use carrier phase relative positioning, where double difference geometry is computed using corrected time

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Error in Fixed Receiver Position• In precise relative positioning using double

differencing, one receiver is held fixed, while the other is estimated

– Fixing one receiver to an incorrect coordinate, is equivalent to moving the orbits in the opposite direction to the error (a functional model error)

– Sometimes fixed receiver coordinate is obtained from a poor point position

– Hence, we can estimate the effect of fixed receiver position error

• Rule of thumb– 10 meter error in fixed station coordinate can produce

» 1 cm relative position error over 20 km

» 5 cm relative position error over 100 km, or 10-6 L

– a 50 meter error can produce

» 5 cm relative position error over 20 km, or 5 x 10-6 L

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Measurement Error• Definition

– The difference between the actual measurement, andone that would be made using perfect instrumentation

• Systematic– Caused by physical effects which have a pattern that could be

predicted if we had sufficient data and a good enough model about the processes involved.

– Receiver firmware (code correlation, phase tracking loop)

– Electronic channel biases which vary with temperature

• Random– Caused by physical effects which have no predictable pattern, but

may be treated statistically

– Electronic noise (random atomic motion)

– Sampling error (finite bandwidth smooths out signal)

– Finite clock instability (random atomic motion)

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Earth Model• In static positioning, receiver is attached to

Earth’s surface.

• But Earth is constantly changes shape, rate of spin, and spin direction.

– Solid Earth tides

– Distortion due to atmospheric pressure

– Distortion due to oceanic pressure

– Plate tectonics, seismic motion.

• If ignored, relative accuracy is limited L

• Most of this can be modelled to allow for relative position at the level L

– Will be covered in SVY307: Geophysical Geodesy

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Cycle-Slips• Discontinuities in the integer ambiguity.

• Cycle slips are more prone to happening:– obstructions (e.g., operator stands near antenna!)

– magnetic disturbance in ionosphere

– antenna undergoes significant acceleration

– receiver clock becomes unstable

– signal to noise is low (e.g., long antenna cable, low elevation satellite).

• If not accounted for– This consistutes a blunder

– Error estimates will be meaningless unless sufficient confidence can be place in removing cycle-slips.

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Errors - A general least squares perspective

– error in computing the solution:

where:

– Formal error in parameters is given by the computed covariance matrix:

– Does not account for systematic errors arising from

b L L x AL

x xW vvO C

C T

( )0

0

1

x A WA A WbT T 1

L

L x

x

A x

vv

O

C

T

observations

functional model

list of parameters

design matrix

stochastic model

( )0

0

- blunders (transcription, equipment failure...)?- inaccurate formulas or missing terms?- errors due to parameters not on the list?- bad provisional values (iteration needed)?- inaccurate assumptions on expected errors?

C xx A WAxT T

1

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Carrier Phase Ambiguity

• Formal error– If all ambiguities can be resolved (fully “fixed” solution)

» reduces number of least squares parameters

» improves redundancy (No. data No. parameters)

» therefore improved formal precision (covariance matrix)

– If not all ambiguities can be resolved (partially fixed solution)

» larger number of least squares parameters than above

» formal precision (covariance matrix) larger than above

• Blunders– If resolved to the wrong integer, this constitutes a blunder

– several centimetre error possible

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Stochastic Errors• Finally, it should be recognised that we will

introduce an error into our estimates if we have an error in our error model.

• Incorrect weight matrix for least-squares– incorrect data variances might be used

» low-elevation satellites usually more noisy (low S/N ratio)

» In a network solution, we might apply inappropriate weights to a particularly noisy receiver

– We might ignore physical correlations between measurements

» due to multipath

» due to varying atmospheric conditions, etc.

» these types of error can equivalently be considered as systematic errors in the functional model.

– Software might ignore known mathematical correlations