Linear Combinations

Common linear combinations of GPS signals

Linear combinations are combinations of measurements belonging to the same receiver and to the same epoch. Linear combinations are designed to optimize the observation characteristics for specific tasks.

Two main objectives for the formation:

1. Mitigation of errors, such as the ionospheric delay, or highlighting of selected effects, such as the ionospheric delay or multipath.
2. Generation of observations with different wavelengths, required to facilitate the recovery of the integer ambiguities.

It is possible to generate observations with different wavelength (λ) or different sensitivities to atmospheric and environmental disturbances. However, the advantages of linear combinations are accompanied by increased noise and a loss of the physical observation characteristics.

We’ll begin by re-introducing the full observation model for the pseudorange and carrier-phase observables, followed by the steps to simplify the models to what is typically presented, then we’ll go into the derivations of some of the common linear combinations of GPS signals.

Pseudorange observable
Carrier phase observable
Comparison of the carrier phase and pseudorange observables
Simplified pseudorange and carrier phase observable
Inter-frequency carrier phase combinations
– Widelane criterion and derivation
– Narrowlane criterion and derivation
– Iono-free derivation
– Geometry free derivation
– Multipath derivation

The literature presented was a synthesis of the material presented by Kleusberg and Teunissen (1996) and Collins (1999). For the results presented in Collins (1999), I have recomputed the calculations which can be downloaded from the utilities section. For all the derivations, I tried to complete all the steps that would be typically “left to the reader”, if you find any mistakes please let me know and I will update the blog post.

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