SVY 207: Lecture 13 Ambiguity 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

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

Relative positioning - Review Recall ‘one-way’ carrier phase model (in metres) LAj  Aj Aj c tA c t j  Baj Use differencing techniques to solve for carrier phase bias BABjk  BABj  BABk  (BAj  BBj BAk BBk  BAj  BBj BAk BBk Remember: BAj (Nj jA) Each bias BABjk has an integer ambiguity Double difference carrier phase model becomes:  LABjk  ABjk ABjk NABjk

Relative positioning - Review Review cont’d Step 1: Least squares “float” solution LABjk  ABjk ABjk NABjk estimate station coordinates, atmospheric delay, and carrier phase ambiguity NABjk Step 2: Ambiguity resolution example: fix NABjk to nearest integer: Step 3: Least squares “fixed” solution LABjk NABjk   ABjk ABjk left side is known: ambiguity-resolved carrier phase

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...

Ambiguity Resolution Motivation cont’d

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

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

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

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

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

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

Ambiguity Resolution Computational Aspects Geometrical 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

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.

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