A Note on Optimal Linear Combinations

There are a lot of papers written about “optimal” linear combinations of observations, some of which are excellent, such as the work by Cocard et al. [2008]. The optimality of linear combinations can be defined based on several criteria such as noise reduction, ionospheric delay reduction, or wavelength amplification. This quest for optimality has however obscured what I believe is the true benefit of linear combinations: reducing the number of parameters to be estimated. It is unfortunately a widespread belief that the use of linear combinations can reduce errors in the observations and lead to better positioning results or faster ambiguity resolution than uncombined signals.


To clarify this issue, I will use an example based on the Melbourne-Wübbena (MW) combination, which “eliminates” all dispersive (geometry) and non-dispersive (ionosphere) effects, while reducing code noise and increasing the wavelength of the signal. What could be more optimal than that? Let us start with the values of four simulated double-differenced (DD) observables (code and phase measurements on both frequencies):

Now, let us compute the noisy widelane ambiguity estimate based on the MW combination:

Instead of forming the MW combination, let us try to explicitly estimate the DD range and ionospheric delay between the receivers and satellites:

Solving for those four unknowns leads to:

which gives the widelane ambiguity:

Why are we getting the same answer? Didn’t the MW combination eliminate all non-dispersive and dispersive errors? Well, not really… Eliminating (or should we say reducing) parameters through linear combinations is equivalent to estimating those quantities without any a priori constraints. If precise external information is available (e.g. on the range or the ionosphere), then linear combinations are clearly not optimal. Obviously this is no news, and was demonstrated close to 30 years ago, see e.g. the work of Wells et al. [1987].

 

So next time you are thinking of forming linear combinations, remind yourself of your true motive!

 

References

Cocard, M., S. Bourgon, O. Kamali, and P. Collins (2008). “A systematic investigation of optimal carrier-phase combinations for modernized triple-frequency GPS,” Journal of Geodesy, Vol. 82, No. 9, pp. 555-564.

 

Wells, D.E., W. Lindlohr, B. Schaffrin, and E. Grafarend (1987). “GPS design: undifferenced carrier beat phase observations and the fundamental differencing theorem,” Technical Report No. 116, Department of Geodesy and Geomatics Engineering, University of New Brunswick (PDF).



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