GPS World, April 2015

Algorithms Methods INNOVATION μ h Dual frequency bias estimation b L 1 b L 2 b L 1 b L 2 μ h Unambiguous transformation Network side Inverse transformation Integer nature of phase ambiguities User side FIGURE 1 Phase biases estimation in the dual frequency case measurements P 1 and P 2 are expressed in meters while phase measurements L 1 and L 2 are expressed in cycles In the following we use the word clock to mean a time offset between a receiver or satellite clock and GPS System Time as determined from either code or phase measurements on different frequencies or some combination of them The code and phase measurements are modeled as 1 where D 1 and D 2 are the geometrical propagation distances between the emitter and receiver antenna phase centers at f 1 and f 2 including troposphere elongation relativistic effects and so on W is the contribution of the wind up effect in cycles e is the code ionosphere elongation in meters at f 1 This elongation varies with the inverse of the square of the carrier frequency and is applied with the opposite sign for phase Δh h i hj is the difference between receiver i and emitter j ionosphere free phase clocks Δh p is the corresponding term for code clocks Δτ τ i τj is the difference between receiver i and emitter j offsets between the phase clocks at f 1 and the ionospherefree phase clocks By construction the corresponding quantity at f 2 is γΔτ Similarly the corresponding quantity for the code is Δτ p time group delay N 1 and N 2 are the two carrier phase ambiguities By de nition these ambiguities are integers Unambiguous phase measurements are therefore L 1 N 1 and L 2 N 2 Equations 1 take into account all the biases related to delays and clock offsets The four independent parameters Δh Δτ Δh p and Δτ p are equivalent to the de nition of one clock per observable However our choice of parameters emphasizes the speci c nature of the problem by identifying reference clocks for code and phase Δh p and Δh and the corresponding hardware offsets Δτ p and Δτ These offsets are assumed to vary slowly with time with limited amplitudes Triple frequency bias estimation μ12 h 12 μ15 h 15 b L 1 b L 2 b L 5 Many possible combinations How to compute biases Network side b L 1 b L 2 b L 5 μ12 h 12 μ15 h 15 Integer nature of phase ambiguities must be kept on all viable phase combinations User side FIGURE 2 Phase biases estimation in the triple frequency case The measured widelane ambiguity also called the Melbourne Wübbena widelane can be written as 2 where is the integer widelane ambiguity is the constant widelane delay for satellite j and is the widelane delay for receiver i which is fairly stable for good quality geodetic receivers The symbol means that all quantities have been averaged over a satellite pass Integer widelane ambiguities are then easily identi ed from averaged measured widelanes corrected for satellite widelane delays Once integer widelane ambiguities are known the ionosphere free phase combination can be expressed as 3 where is the ionospherefree phase combination computed using the known ambiguity is the propagation distance is the receiver clock and is the satellite clock is the remaining ambiguity associated to the ionosphere free wavelength 107 centimeters The complete problem is thus transformed into a singlefrequency problem with wavelength and without any ionosphere contribution Many algorithms can be used to solve Equation 3 using data from a network of stations If is known with suf cient accuracy typically a few centimeters which can be achieved using a good Àoating point or realvalued ambiguity solution it is possible to simultaneously solve for and The properties of such a solution have been studied in detail A very interesting property of the satellite clocks is in particular the capability to directly x to the correct integer value the values of a receiver that was not part of the initial network The majority of the precise point positioning ambiguityresolution PPP AR implementations are based on the identi cation and use of the two quantities and These quantities may be called widelane biases and integer phase clocks a decoupled clock model or uncalibrated phase delays but they are all of the same nature www gpsworld com April 2015 GPS World 43