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SVY 207: Lecture 13Ambiguity Resolution

• Aim of this lecture:– To introduce methods of ambiguity resolution in detail

and to look at applications of these for RTK GPS.

• Overview– Relative positioning review

– Ambiguity resolution review

– Ambiguity resolution - Motivation

– Ambiguity resolution techniques

– Implications for GPS surveying

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Relative positioning - Review• Requirements - Precise engineering surveying

– 2 stations (baseline), or multiple stations (network)

– Carrier phases from 4 satellites, then double-difference

– Use broadcast orbits and clocks

– Assume values for one station and its clock time

– Estimate, using weighted least squares, station coordinates, and carrier phase ambiguities

» fix ambiguities to integer values and iterate.

– Achievable precision: < 1 cm

» over few 10 km using broadcast orbits

– Can be post-processed or real-time

• Process depends upon AMBIGUITY RESOLUTION

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Relative positioning - Review• Review

– Recall ‘one-way’ carrier phase model (in metres)

LAj A

j Aj c tA c t j Baj

– Use differencing techniques to solve for carrier phase bias

BABjk BAB

j BABk

(BAj BB

j BAk BB

k

BAj BB

j BAk BB

k

– Remember: BAj (N

j jA)

– Each bias BABjk has an integer ambiguity

– Double difference carrier phase model becomes:

LABjk AB

jk ABjk NAB

jk

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Relative positioning - Review

• Review cont’d

• Step 1: Least squares “float” solutionLAB

jk ABjk AB

jk NABjk

– estimate station coordinates, atmospheric delay, and carrier phase ambiguity NAB

jk

• Step 2: Ambiguity resolution– example: fix NAB

jk to nearest integer:

• Step 3: Least squares “fixed” solutionLAB

jk NABjk AB

jk ABjk

– left side is known: ambiguity-resolved carrier phase

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Ambiguity Resolution

• Motivation - – Resolution of initial phase ambiguity is key to sub-

centimetre position accuracy in GPS surveying

– Fewer parameters to estimate greater precision

– Time period to resolve ambiguities

– Prior to 1995 majority of GPS employed static techniques

– (Remondi 1985) ‘rapid-static’ approach

» Ambiguities could be resolved in minutes as opposed to hours

» Greater efficiency

» New applications - RTK, Machine guidance etc...

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Ambiguity ResolutionMotivation cont’d

100 experiments2.2 km baselinedual frequency phase data3D coordinate system: local North, East, Up7 satellites2 epochs of data with 5 seconds in between

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Ambiguity Resolution

• More detail– Specifically, ambiguity resolution OR “Initialisation” is

the problem of finding N, where :

N = The full cycles of double differenced N’s

– If can initialise then difference between two epochs, collected by same receiver to same satellite = change in topocentric range i.e.,

L’ (L N) – Initialisation not easy, requires

» Good station-satellite relative geometry

» Low level of observation errors

» Reliable algorithm

– Need to be able to validate if have initialised correctly

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Ambiguity Resolution

• Classical ‘static’ Techniques – Majority use TWO step approach

» First step to estimate station coordinates and real-valued ambiguities

» Second step to resolve initial ambiguities to integer values -methods include:

• Round real values to nearest integers

• Use estimated errors to evaluate if resolution to integer is feasible. Ambiguity only fixed if integer value is within an appropriate confidence interval e.g. +/- 3

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Ambiguity Resolution

• More modern techniques– Developed since the early 1990’s

» Ambiguity searching • Form all feasible combinations of integer ambiguity values

around the real-valued estimates. Test each set of ambiguities to find the most probabilistic values. This technique also known as ambiguity searching. This method also suitable for On-the-Fly (OTF) initialisation.

» Suitable for both static and kinematic applications

» For short baselines precision is comparable to traditional techniques

» All follow similar approach - c.f. lecture 12

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Ambiguity Resolution– An example - Ambiguity Function method (After Mader

[1992]) - (1)

» Determine approximate coordinates• Use pseudo-ranges to compute approximate coordinates for

unknown point

• Determine a search volume typically a cube around approximate postition

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Ambiguity Resolution– An example - Ambiguity Function method (2)

» Construct an ambiguity mapping function such that when the observed minus computed phase single difference is an integer for all observations, the function will be a maximum

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Ambiguity Resolution

– An example - Ambiguity Function method (3)

» For each integer point contained within cube ‘test points’ compute ambiguity function

» Correct ‘test position’ should emerge as a recognisable peak, with largest ambiguity function value

» Test if the position with the highest ambiguity function is correct. - Statistical testing

» If pass statistical testing then fix ambiguities if fail then re-try with next epoch of data

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Ambiguity Resolution

• Computational Aspects– Construction of optimal search space - e.g. 1m cube with

integer spacing at 1cm results in 1million test ambiguity combinations

– Fast robust algorithm

– Effective validation and rejection criteria

• Geometrical Aspects– Dependant on geometry of observation e.g. geometry of

satellite constellation w.r.t. base and rover station

– Quality of actual signals being observed

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Ambiguity Resolution

• Implications for GPS surveying– Increased efficiency resulting from reduced requirement

for long data observation sessions

» with no loss of precision over short baselines

– Kinematic surveying now true alternative to Total Station

– Addition of Radio to base and rover stations allows Real Time Kinemtic Surveying, but…

» Never with poor DOP values

» Always ensure ambiguities resolved

» Take checks during survey e.g. re-survey known points

» Follow sound survey practise at all times.

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Ambiguity Resolution

• References» GPS world ‘innovations’

• April 1993

• April 1994

• May 1995

• September 1998

• May 2000

» Teunissen P.J.G., Kleusberg A., ‘GPS for Geodesy’, 2nd ed, Springer

» Leick A., ‘Satellite Surveying’ 2nd ed, Wiley