1. 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

2. 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

3. 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

4. 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

5. 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...

6. Ambiguity Resolution
Motivation cont’d

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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.

15. 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