Unbiased vs biased estimation of GPS phase ambiguities from dual-frequencycode and phase observables

With access to dual-frequency pseudorange and phase Global Positioning Syst em (GPS) data, the wide-lane ambiguity can easily be fixed. Advantage is ta ken of this information in the linear combination of the above four observa bles for base ambiguity estimation (i.e. of N-1 and N-2). Starting points f or our analysis are the Best Linear Unbiased Estimators BLUE1 and BLUE2. BL UE1 is the best one (with minimum mean square error, MSE) if the ionosphere effect is negligible. If this is not the case, BLUE2 has the smallest vari ance, but not necessarily the least mean square error. Hence, both estimato rs may suffer from a non-optimal treatment of the ionosphere bias. BLUE1 ig nores possible ionosphere bias, while BLUE2 compensates for this bias in a less favourable way by eliminating it at the price of increased noise. As a n alternative, linear estimators are derived, which make a compromise betwe en the ionosphere bias and the random observation errors. This leads to the derivation of the Best Linear Estimator (BLE) and the Restricted Best Line ar Estimator (RBLE) with minimum MSE. The former is generally not very usef ul, while the RBLE is recommended for practical use. It is shown that the M SE of the RBLE is limited by the variances of BLUE1 and BLUE2, i.e. Var(BLUE1) less than or equal to MSE(RBLE) less than or equal to Var(BLUE2) However, as is always the case with a BLE, it cannot be used strictly: some parameter tin this case the ionosphere bias) must be approximately known.