the ambiguity-fixed ones achieve this after about 600 s. Hence, according to the statistics above, it is demonstrated that ambiguity resolution can speed up the convergence to centimeter-level accuracy of epoch-wise position estimates within a few tens of minutes.
5.6.5
Conclusions
First, it is concluded that at least 10 minutes of observations are required for most receiver types to reliably fix about 90% of wide-lane ambiguities corresponding to elevations of over 15◦
, and over 20 minutes to fix about 90% of those corresponding to elevations of below 15◦
. Moreover, receivers of cross-correlation types or without choke-ring antennas require longer periods than those of others. Two thresholds, namely 10 minutes for the time span and 15◦
for the time-mean elevation angle, are proposed to decide when a wide-lane ambiguity can be reliably fixed.
Second, it is concluded that several tens of minutes are usually required for a regional network before a narrow-lane FCB estimate stabilizes to an accuracy of far better than 0.1 cycles. Large fluctuations of narrow-lane FCB estimates are mainly caused by satellites’ rise from a low elevation with respect to a regional network. However, we can still generate and disseminate narrow-lane FCB products once wide-lane ambiguity resolution is achieved. These products are usable for ambiguity resolution, but we have to update them, every 5 s for example, in ambiguity-fixed solutions in order to achieve highly accurate position estimates.
Finally, ambiguity-fixed solutions with centimeter-level positioning accuracy can be achieved within a few tens of minutes. For hourly measurements, ambiguity resolution significantly reduces the RMS statistics of differences between the epoch-wise and daily position estimates from 13.7, 7.1 and 11.4 cm to 0.8, 0.9 and 2.5 cm on average for the East, North and Up components, respectively.
For the PPP-RTK model in Section 2.4.3, 10 minutes for wide-lane resolution of high- elevation ambiguities, 10 minutes for narrow-lane FCB generation and 5-second update rate are all acceptable at the server end. However, 10 minutes for wide-lane ambiguity resolution and even longer period for narrow-lane ambiguity resolution constitute the technical bottleneck which prohibits many real-time users who require instantaneous precise positioning from applying this PPP-RTK model. However, this model is still useful in some remote sensing applications where the timeliness requirement on the first ambiguity-fixed solution is not as critical as that in the instantaneous positioning, such as the near-real-time GPS meteorology. For future technical development, rapid ambiguity resolution at a single receiver will be the key problem to be resolved before we can implement an instantaneous PPP-RTK model where ambiguity-fixed solutions can be achieved using a few seconds or even one second of measurements.
5.7
Impact of integer double-difference constraints
As suggested in Section 2.4.4, hard constraints from integer double-difference ambiguities in a network solution can improve the accuracy of narrow-lane FCB estimates, which is beneficial for the positioning accuracy of ambiguity-fixed PPP. This section thus aims at quantifying the impacts of such constraints.
5.7.1
Data, models and methods
One year of daily GPS data at about 350 globally-distributed reference stations from the IGS permanent network in 2008 were used (Dow et al. 2009). Data files covering less than 6 hours of measurements were removed. Moreover, CODE final satellite orbits, 30-s satellite clocks, ERPs and P1-C1 DCBs were used (Dach et al. 2009). Note that using CODE satellite products, rather than IGS ones, is to avoid the possible inhomogeneities of the IGS final products which can degrade the positioning quality of PPP (Teferle et al. 2007).
For data modeling, the absolute phase centers (Schmid et al. 2007), phase-wind up ef- fects (Wu et al. 1993) and station displacement models proposed by IERS conventions 2003 (McCarthy and Petit 2004) were applied. A cut-off angle of 7◦
was set for usable measurements and an elevation-dependent weighting strategy was applied to measurements at low elevations (Gendt et al. 2003). Moreover, ZTDs were estimated every one hour by applying the global pressure/temperature model and the global mapping function (Kouba 2009b), while horizontal tropospheric gradients every 12 hours (Bar-Sever et al. 1998).
CODE satellite products and ERPs were fixed during the data processing and about 180 globally-distributed stations were selected to estimate FCBs. According to Ge et al. (2008), narrow-lane FCBs were computed every 15 minutes. Then these FCBs were applied to all 350 stations in order to achieve ambiguity-fixed daily solutions. In this section, the narrow-lane FCBs that are derived by applying the double-difference ambiguity resolution are named as improved narrow-lane FCBs, and those otherwise as original narrow-lane FCBs.
5.7.2
Results
The original and improved narrow-lane FCB estimates are first compared. Daily RMS statistics of narrow-lane FCB differences for all satellite pairs at all 15-minute time spans are computed. All daily RMS statistics range from 0.021 to 0.089 cycles and the mean RMS of one year is about 0.035 cycles. These small differences are due to the highly-accurate daily ambiguity estimates even if double-difference ambiguity resolution is not applied.
Moreover, the daily position estimates based on the original and improved narrow-lane FCBs are also compared. Figure 5.15 shows the magnitude distribution of all these position
Figure 5.15: Magnitude distribution of all differences between the position estimates based on the original and improved narrow-lane FCBs for the East, North and Up components. The top-left corner of each subfigure shows the bias and the standard deviation (σ), whereas the top-right corner shows the percentages of deviations that are within 2σ, or larger than 3σ (σE=1.2 mm, σN=0.4 mm and
5.7 Impact of integer double-difference constraints 93 Table 5.15: Mean RMS of transformed residuals of the daily position estimates against the IGS weekly solutions in 2008
Solution types East North Up Ambiguity-float 3.6 2.3 6.4 Ambiguity-fixed by original narrow-lane FCBs 2.6 2.2 6.1 Ambiguity-fixed by improved narrow-lane FCBs 2.2 2.2 6.1
differences for the East, North and Up components. We can see that the absolute biases of these position differences are not larger than 0.1 mm for all three components. Correspondingly, the standard deviations are 1.2, 0.4 and 1.5 mm for the East, North and Up components, respectively. Furthermore, 94.1%, 94.7% and 95.2% of all position differences are within two times the standard deviations for the three components, and minimally 1.0%, 0.8% and 1.1% of all are larger than three times the standard deviations. These statistics illustrate the small difference between the daily ambiguity-fixed position estimates based on the original and improved narrow- lane FCBs.
Nevertheless, the improvement of the positioning quality based on the improved narrow- lane FCBs can be identified when these daily position estimates are compared with the IGS weekly solutions. Seven-parameter Helmert transformations are applied during these daily comparisons. Table 5.15 shows the mean RMS statistics of the transformed residuals of the daily position estimates against the IGS weekly solutions in 2008. We can see that ambiguity resolution significantly reduces the RMS statistics, especially for the East component by 27.8%, hence confirming the results by Ge et al. (2008). After double-difference ambiguity resolution is applied to the network solution, the resulting narrow-lane FCB estimates contribute to a further reduction of the RMS statistics for the East component by 0.4 mm, although the RMS statistics for the North and Up components remain the same.
In addition, Figure 5.16 shows the RMS of the transformed residuals of the daily position estimates against the IGS weekly solutions for the East component. We can clearly find the RMS reduction led to by ambiguity resolution at a single receiver. Moreover, we can also easily identify the smaller further RMS reduction due to the improved narrow-lane FCBs. Specifically, almost on each day of the year, the RMS based on the improved narrow-lane FCBs is smaller than that based on the original narrow-lane FCBs. This result clearly demonstrates the improvement
Figure 5.16: Daily RMS of transformed residuals of the position estimates against the IGS weekly solutions for the East component in 2008
of the positioning quality contributed by applying double-difference ambiguity resolution to narrow-lane FCB determination.