By contrasting Equation 2.36 and 2.40, it can be found that their design matrices and observed minus computed measurements are exactly the same, which demonstrates that the resulting estimates for ∆xi from the FCB- and IRC-based methods should also be identical. This
equivalence implies that a systematic difference between the ∆xiestimates in practice, if existing,
should not be caused by the differences of the two methods themselves. Ideally, the difference between the actual ∆xiestimates should be minimal and random in nature. In addition, in terms
of Equation 2.29, ∆xi also contains ZTD. Hence, identical ZTD estimates can also be achieved
using these two methods. Note that this theoretical equivalence is derived from Equation 2.30, namely the linearized measurement equation, thereby implying that identical models should be employed to reduce the raw measurements in order to achieve identical ∆xi estimates.
Nevertheless, this equivalence is largely based on the assumption that only the hardware- dependent FCBs are assimilated into the clock and undifferenced ambiguity estimates, which is not true in practice. Ge et al. (2008) illustrated that satellite-pair FCB estimates change temporally and spatially, and the fluctuation magnitude can reach up to 0.4 cycles, showing that these FCB estimates are contaminated by unknown temporally- and spatially-correlated errors, such as the inaccurate modeling of tropospheric delays. This explains why FCB estimates are not constant values in practice and thus Ge et al. (2008) proposed a 15-minute mean to achieve high-precision FCB estimates. Likewise, actual IRC estimates are likely to absorb not only the hardware-dependent FCBs, but also some unknown common errors among a network of stations. It should be stressed that these unknown redundant errors are not hardware-dependent, and thus they are likely to change under different distributions of reference stations. In the subsequent part of this thesis, “FCB estimates” thus also contain these redundant errors.
Furthermore, from the observed minus computed measurements in Equation 2.36 and 2.40, ∆tk,1+λn
c ℓB
k,1+λn
c δB
k,1 (k = 2, · · · m) can actually be taken as the satellite clock that can
assist retrieving integer ambiguities. In the FCB-based method, this clock is finally achieved by combining the satellite clock estimate and the FCB estimate. Comparatively, this clock is exactly the IRC in the IRC-based method. This difference is attributed to the different strategies of separating FCBs from integer ambiguities in these two methods. Specifically, this separation is performed at the ambiguity-estimate level in the FCB-based method, whereas at the measurement-modeling level in the IRC-based method. As a result, the composition of redundant errors can be significantly different between the actual IRC estimates and the actual FCB estimates plus their corresponding satellite clock estimates, finally leading to different ∆xi
estimates.
2.4
Method improvements made in this thesis
The FCB-based method by Ge et al. (2008) is adopted and improved for the PPP data analysis presented in this thesis. The IRC-based method by Laurichesse et al. (2009c) is also implemented and its results are compared with those by the FCB-based method. Method implementations will be addressed in Chapter 4. This section will introduce four improvements or developments that have been achieved in this thesis for the FCB-based method.
2.4.1
Derivation of undifferenced FCBs
Undifferenced FCBs are preferred to satellite-pair FCBs because the number of FCBs can be significantly reduced. For example, if m satellites are observed, m(m − 1)/2 satellite-pair FCBs will be generated. By contrast, only m undifferenced FCBs are needed to express all these satellite-pair FCBs. A large number of FCBs cannot be conveniently transmitted, especially in real-time applications. Furthermore, because undifferenced FCBs are satellite-dependent, they can actually be combined with the epoch-wise satellite clocks (Mervart et al. 2008). Such combination can further reduce the data transmission bandwidth, hence potentially enhancing the competitiveness of the FCB-based method against the IRC-based method where only satellite orbits and IRCs are transmitted to users.
In fact, the derivation of undifferenced FCBs is only a trivial task. A satellite-pair FCB φkl
can be expressed as a pseudo measurement like φkl = φk
− φl. All these pseudo measurements can be superimposed to a normal equation in order to estimate undifferenced FCBs. Note that an undifferenced FCB has to be fixed to an arbitrary value to avoid the rank deficiency of the normal matrix. In addition, the resulting residuals after this adjustment can be used to identify any possible blunders in the satellite-pair FCB estimates.
2.4.2
One FCB per satellite-pair pass over a regional area
Daily mean wide-lane FCBs can be precisely determined even on a global scale (Gabor and Nerem 1999; Ge et al. 2008; Laurichesse and Mercier 2007). However, due to the low precisions of daily mean narrow-lane FCBs, Ge et al. (2008) instead suggested a 15-minute mean.
In this thesis, it is proposed that the narrow-lane FCBs for a satellite pair can be precisely determined within each continuous tracking period by a regional network, namely each full pass by this satellite pair over a regional network. Therefore, the fractional parts of all involved narrow-lane ambiguity estimates for a satellite pair during one full pass are averaged to estimate one narrow-lane FCB. The key difference of this strategy from the 15-minute mean by Ge et al. (2008) consists in the satellite orbit arc length for one narrow-lane FCB estimate. Empirically, there are usually two narrow-lane FCB estimates per 24 hours which actually correspond to two full passes of a satellite pair over a regional network. Compared with the large number of 15-minute mean narrow-lane FCBs by Ge et al. (2008), this strategy can significantly reduce the dissemination burden of the FCB products, which is well suited for practical use in a regional network. The performance of these FCBs will be assessed in Section 5.2.
2.4.3
Implementation of real-time ambiguity resolution
To date, FCB-based methods are assessed primarily for post-processing applications. Mervart et al. (2008) reported some real-time results but few technical details are released due to commercial reasons. Hence, it is still necessary to propose a PPP-RTK model where PPP provides rapid convergence to a reliable centimeter-level positioning accuracy based on an RTK reference network (e.g. Bisnath and Gao 2007; W¨ubbena et al. 2005), which can improve on current real-time ambiguity-float PPP services (e.g. Dixon 2006).
Overall, a PPP-RTK model consists of a server module providing FCB products and a user module achieving ambiguity-fixed solutions. For the server module, wide-lane FCBs are generated according to the method by Ge et al. (2008) as addressed in Section 2.2.2. Ge et al.
2.4 Method improvements made in this thesis 31
Figure 2.5: Data flowchart of a PPP-RTK model based on ambiguity resolution. Cylinders denote data storage and squares denote data-processing modules; dashed arrows denote data input or output; and solid arrows denote processing sequences
(2008) showed that wide-lane FCB estimates change within ±0.05 cycles within 2 weeks. Hence, wide-lane FCB products can be precisely predicted at an update rate of several days for real-time applications. However, narrow-lane FCBs have to be estimated at a sub-daily frequency in post- processing applications. Likewise, we have to frequently re-estimate and update the narrow-lane FCBs at the PPP-RTK server module. For the user module, wide-lane ambiguity resolution relies on the noisy Melbourne-W¨ubbena combination measurements. Hence, a sufficiently long time has to be spent before a wide-lane resolution can be reliably achieved. This time span is mainly subject to the pseudorange measurement noise, multipath effects and atmospheric delays. Afterwards, narrow-lane ambiguity resolution can be attempted according to the theories in Section 2.2.2. Note that the narrow-lane FCBs that are contributing to the achieved ambiguity- fixed solutions should be frequently updated because they are frequently re-estimated and re- disseminated at the server module (see Section 5.6.3 for detailed explanations).
Accordingly, Figure 2.5 presents the data flowchart at both the server and user modules for a conceptual PPP-RTK model. Compared with current real-time ambiguity-float PPP models, this model generates and disseminates not only the satellite orbits and clocks, but also the FCB products for users to achieve ambiguity-fixed solutions at a single receiver. Moreover, this model has to keep consistency among orbits, clocks and FCB products, rather than only between orbits and clocks (Geng et al. 2010c). In addition, in Figure 2.5, except the square denoting “orbit & clock determination”, the other four squares exhibit three modules which are crucial to ambiguity resolution. It is presumed that wide-lane FCB products are easily known, and thus their determination module is ignored in Figure 2.5 for brevity. From the solid arrows in Figure 2.5, wide-lane ambiguity resolution is the prerequisite for both narrow-lane FCB determination and narrow-lane ambiguity resolution.
2.4.4
Constraints from integer double-difference ambiguities
As demonstrated in Section 2.2.2, FCBs are estimated by averaging the fractional parts of all involved ambiguity estimates at the reference stations. Nonetheless, due largely to the possible biases that are absorbed into the float ambiguity estimates, a sophisticated averaging technique, such as Equation 2.12, has to be adopted to achieve correct FCB estimates (see also Ge et al.
2008). Moreover, the temporal instability of narrow-lane FCBs (see Section 2.3.4) is likely to further complicate such averaging operations.
Fortunately, such averaging for narrow-lane FCBs can be avoided by applying tight con- straints from integer double-difference ambiguities. In the FCB-based method by Bertiger et al. (2010), undifferenced ambiguity estimates can directly form double-difference ambiguities and integer resolutions can subsequently be attempted. This strategy was also discussed by Ge et al. (2005a, 2006). Successful ambiguity resolution will then improve the accuracy of undifferenced ambiguity estimates (Feng et al. 2007). Suppose we have two single-difference narrow-lane ambiguity estimates between satellites k and l at receiver i and j, namely ˆNkl
i and ˆN kl j , their
difference ˆNkl
i − ˆNjkl should not be an integer after an initial PPP data processing. However, if
double-difference ambiguity resolution is successfully performed on ˆNkl
i − ˆNjkl, this difference can
then be tightly constrained to an integer in a second PPP data processing. Hence, the fractional parts of the resulting estimates for ˆNkl
i and ˆN kl
j are identical. In this way, most fractional
parts of single-difference ambiguity estimates for a satellite pair are identical if the fixing rate of double-difference ambiguities is sufficiently high. Fortunately, Ge et al. (2005a) suggested an innovative strategy and showed that a fixing rate of higher than 97% can be routinely achieved using daily GPS measurements on a global scale. Therefore, narrow-lane FCBs can then be easily identified from the identical fractional parts of narrow-lane ambiguity estimates for a satellite pair.
Finally, it is worth highlighting that narrow-lane FCBs determined through these integer double-difference constraints theoretically have higher accuracy than that determined without these constraints. This point will be empirically verified in Section 5.7.