Hardware delays, or biases, affect GNSS carrier-phase and code measurements and must be properly accounted for in high-accuracy positioning. Several models were proposed to handle biases in precise point positioning with ambiguity resolution (PPP-AR), all of which can be cast in an uncombined representation. In this post, I explain the unified processing scheme that I am using in my software to deal with common PPP-AR products.
Case 1: IGS clocks
Let us start with standard “code” or “float” clocks, such as the ones provided by the IGS combination. With IGS clocks, ambiguity resolution is currently not enabled and phase biases can be set to 0. Since the clocks are estimated using ionosphere-free (IF) observables, ionosphere-free biases are absorbed into the clock parameters and it is possible to set the ionosphere-free code biases equal to 0. With such constraints and a value for the differential code bias (DCB) obtained from ionospheric maps, it is possible to derive uncombined biases for the P1 and P2 observables:
where alpha and beta are frequency-dependent coefficients.
Case 2: Integer clocks (CNES - GRG)
The integer satellite clock corrections provided by CNES (with the prefix GRG) are accompanied by widelane satellite biases derived from the Melbourne-Wübbena (MW) observables, a linear combination of the widelane (WL) carrier-phase and narrowlane (NL) code observables. Since no additional correction terms are required for PPP processing, we can assume that satellite clocks are aligned to “code” clocks. This procedure was most likely done internally at CNES by shifting the phase clocks by an integer number of narrowlane wavelengths. As a result, ionosphere-free code and phase biases can be assumed to be equal, although they could differ by at most 5 cm (half of the narrowlane wavelength). The resulting system becomes:
From the previous system, we see that the uncombined code biases obtained from Case 1 and Case 2 are identical. Phase biases become a combination of the MW and DCB estimates. When these quantities are modeled as constants over a day, phase biases are also constant and do not add additional noise to the phase observables.
Case 3: Uncalibrated phase delays (UPD)
In the UPD approach, standard clocks are used as in Case 1, and the fractional part of ionosphere-free carrier-phase ambiguities are used to define uncalibrated phase delays. By combining these estimates with the MW and DCB values, we obtain:
Case 4: Decoupled clocks (NRCan – DCM)
With the decoupled-clock model (DCM) of NRCan, different clock parameters are estimated for carrier-phase and code observables. Due to the definition of the ambiguity datum needed to remove the phase-clock singularities, the phase and code clocks and not necessarily aligned and a code-phase offset (CPO) can be determined (e.g., it does not have a null value as in the CNES clocks). Furthermore, code clocks are also contaminated by noise, and the transformation above (Case 2) would introduce code noise into phase biases. For this reason, to transform the DCM biases into uncombined biases, it is convenient to fix phase biases to 0, leading to:
It should be noted that, with the DCM products, it is not possible to compute a standard float PPP solution using only the satellite clock corrections. It becomes mandatory to use both the clock corrections and the code biases because otherwise, code observations could contain (large) biases.
Case 5: Extended DCM (NRCan – EDCM)
In the above transformation, it can also be noticed that the DCB information was not used as a constraint in the system since doing so would introduce noise into phase biases. As a result, it is not possible to apply external ionospheric constraints from a global ionospheric map (GIM) using DCM products. To use such constraints, it is possible to compute differential phase biases (DPB), obtained from integer-leveled observations, see Banville et al. (2014).
By combining the MW, CPO, DPB and an additional constraint specifying that ionosphere-free phase biases are null, we obtain the extended decoupled clock model (EDCM):
Using the EDCM model with DCB instead of DPB would result in the following model:
As seen from the above system, with noisy estimates of the CPO and MW biases, the L1 and L2 phase biases would also be noisy. This is not a problem for dual-frequency users since the ionosphere-free phase biases are constrained to zero. However, after applying the uncombined biases, single-frequency users would have phase measurements with code noise which is obviously not practical. It would therefore be preferable to smooth the CPO and MW biases, which would result in smooth phase biases.
By transforming the various PPP-AR models into an uncombined representation, it becomes simpler to process GNSS data in a unified fashion. In a near future, this conversion should be done directly by analysis centers and PPP-AR products should be provided using this representation using the new IGS bias SINEX format.
Banville S, Collins P, Zhang W, Langley RB (2014) Global and regional ionospheric corrections for faster PPP convergence. NAVIGATION: Journal of The Institute of Navigation, Vol. 61, No, 2, Summer 2014, pp.115-124.