# Performance of Precise Point Positioning with Ambiguity Resolution for 1- to 4-Hour Observation Periods

## Summary (3 min read)

### INTRODUCTION

- During the past three decades, the development in the processing strategy for Global Positioning System (GPS) measurements has led to the products of highly accurate satellite orbits, satellite clocks, and Earth rotation parameters (ERP).
- During the PPP data processing, these products are employed to implement absolute and accurate positioning at only one single station without any synchronous GPS observations from the base stations.
- The static positioning accuracy within such a short observation period can hardly achieve millimetre level if PPP has to be applied to this field survey.
- Fortunately, recent studies have revealed that integer ambiguity resolution at a single station is possible if these UPD can be precisely determined in advance with a network of base stations ([3], [9], [16]). [9] suggested that the fractional parts of the UPD between satellites could be computed by averaging the fractional parts of all involved real-valued ambiguity estimates.
- This paper aims at comparing the performance of ambiguity-fixed static PPP with different short observation periods (i.e. one, two, three and four hours), including the efficiency and reliability of ambiguity resolution, the improvement of positioning accuracy and the occurrence of degraded solutions.

### DETERMINATION OF UNCALIBRATED PHASE DELAYS

- For briefness, multipath effects and measurement errors are not written explicitly in Equation 1.
- To remove the receiver-dependent UPD, between-satellite differences (BSD) are applied to the one-way ambiguity estimates at receiver k .
- In the following, ‘NL UPD’ denotes ‘BSD NL UPD’ and ‘WL UPD’ denotes ‘BSD WL UPD’ for briefness.
- It has been confirmed that daily mean WL UPD are quite stable over a rather long period (e.g. at least several days to even several months) ([8], [9], [16]).
- This strategy is also adopted in this study.

### AMBIGUITY RESOLUTION IN PPP

- For the ambiguity validation, two statistical tests are used, and the integer candidates are accepted only when both tests are passed in this study.
- Implying that the right side of Equation 10 might be too conservative.
- To solve this problem, [25] suggested using more observations, but this is not feasible for short-period static PPP.
- On the other hand, the authors also used the well-known ratio test which is generally defined as the ratio of the second minimum quadratic form of the residuals to the minimum quadratic form of the residuals for the fixed solution.
- It is used to discriminate between the second optimum set of integer candidates and the optimum one.

### DATA AND MODELS

- The PANDA (Positioning And Navigation Data Analyst) software [10], originally developed at Wuhan University in China, is utilized to test short-period static PPP with ambiguity resolution.
- The elevation cut-off angle for usable data was seven degrees.
- The estimated parameters included the positions, the receiver clocks, the zenith tropospheric delays, the horizontal troposphere gradients and the ambiguities.
- The 12 stations were distributed evenly within the coverage of the EPN.

### Efficiency of ambiguity resolution

- Table 2 shows the number of all solutions, the number of the solutions with successful ambiguity resolution, the number of the solutions without any ambiguities fixed, the number of the solutions with incorrect ambiguity resolution and the number of the outlier solutions in PPP for different short observation periods at all test stations.
- This 10-centimetre threshold is chosen in terms of the normal 3D position accuracy derived in hourly PPP ([13], [23]).
- The number of the failed solutions, shown in the last three columns of Table 2, decreases to zero when the observation period is over three hours.
- Hence, it can be concluded that longer observation periods are conducive to higher success rates of ambiguity resolution in short-period static PPP, and a 100.0% success rate is achievable when over three hours of observations are used.

### Reliability of ambiguity resolution

- Figure 2 shows the mean values of the test statistics for ambiguity validation in all fixed solutions at all test stations when different short observation periods are used.
- The black bars denote the 2 2 fixed float σ σ test statistic whilst the grey bars denote the 2 1 opt opt R R test statistic.
- Apparently, it appears that ambiguity resolution with longer observation period is even less reliable than that with only two hours of observations, which is not reasonable.
- Actually, this issue can be explained in terms of the different degrees of freedom for the 2 1 opt opt R R test statistics when different periods of observations are used.

### Positioning accuracy

- Table 3 shows the mean positioning accuracy of all PPP solutions at all test stations for the different short observation periods.
- As was demonstrated by [12], the positioning accuracy can be significantly improved through ambiguity resolution in hourly PPP.
- Table 3 further shows that, even in 4-hourly PPP, the accuracy of the East component is still improved significantly by approximately 66.7% and the 3D improvement achieves 35.7%.
- When the observation period increases from one hour to two, three and four hours, the 3D positioning accuracy is improved markedly from 5.0 cm to 2.9 cm, 1.9 cm and 1.4 cm, respectively.
- This pattern is not evident in the fixed solutions.

### Degraded solutions

- [12] showed that correct ambiguity resolution may lead to degraded, rather than improved, positioning accuracy in the fixed solutions compared with the float ones.
- These solutions are defined as degraded solutions and are identified when the 3D positioning accuracy degradation exceeds 1.0 cm.
- When the observation period is increased to three hours, however, the percentage is decreased to 0.7% and the maximum accuracy degradation is reduced to 1.7 cm.
- As was discussed in [12], degraded solutions are closely related to the estimation of zenith tropospheric delays.
- Moreover, Table 5 shows the percentages of the solutions with accuracy degradation in the East, North or Up directions and the corresponding mean degradation in each direction.

### CONCLUSIONS

- PPP using short periods of observations, such as one to four hours, can hardly achieve such high accuracy when the real-valued ambiguity estimates are kept in the final solutions.
- If the non-integer uncalibrated phase delays are separated from the real-valued ambiguity estimates, integer ambiguity resolution becomes possible, thus leading to improved positioning accuracy.
- When the observation period is over three hours, the success rate of PPP ambiguity resolution can even reach 100.0%.
- In the fixed solutions of short-period static PPP, over one hour of data are sufficient for the horizontal position components to achieve an accuracy of better than 1.0 cm while over three hours of data are still required for the vertical component to achieve such accuracy.
- In addition, for degraded solutions, the accuracy degradation mainly and markedly occurred in the Up component.

Did you find this useful? Give us your feedback

##### Citations

237 citations

### Cites methods from "Performance of Precise Point Positi..."

...Note that we used to name these satellite-pair FCBs as uncalibrated hardware or phase delays in previous publications (Geng et al. 2009, 2010a,b,c)....

[...]

73 citations

51 citations

### Cites result from "Performance of Precise Point Positi..."

...Our results with four or more stations are a factor of two better than the baseline results obtained either by us or Eckl et al. Geng et al. (2010) conducted tests with 1- to 4-h sessions using precise point position of 12 stations in Europe....

[...]

...These results are consistent with those of Eckl et al. (2001), who found little baseline-length dependence up to 300 km for sessions as short as 4 h....

[...]

...We compared our results to similar tests conducted by Eckl et al. (2001) for sessions of 4–24 h and Soler et al. (2006) using sessions of 1–4 h in North America....

[...]

...Similar studies have been carried out by Eckl et al. (2001), Soler et al. (2006), and Geng et al. (2010) but with different approaches....

[...]

42 citations

41 citations

### Additional excerpts

...…GPS landslide monitoring in Puerto Rico (Wang 2010) and other empirical studies about the precision of GPS (Eckl et al., 2001; Soler et al., 2006; Geng et al., 2010; Firuzabadi and King, 2010), both 8-hour session static and 1-hour session rapid static GPS can provide sub-centimeter precision…...

[...]

##### References

3,013 citations

### "Performance of Precise Point Positi..." refers background in this paper

...The International GNSS Service (IGS) and its Analysis Centres routinely generate these products which are the basis for the development of precise point positioning (PPP) [28]....

[...]

1,499 citations

### "Performance of Precise Point Positi..." refers background in this paper

...Parameters Models & a priori constraints Static position 1 metre for each component Receiver clock White noise, 9000 metres Zenith tropospheric delay Constant within each hour, 20 cm, Niell mapping function [20] Ambiguity 10000 cycles...

[...]

1,200 citations

### "Performance of Precise Point Positi..." refers methods in this paper

...It is well-known that PPP is capable of providing millimetre positioning accuracy in the static mode using 24 hours of observations [15]....

[...]

^{1}

741 citations

### "Performance of Precise Point Positi..." refers background or methods in this paper

...Unlike WL UPD, NL UPD are normally estimated within a short observation period, such as 15 minutes, to obtain high estimation precisions [9]....

[...]

...[9] suggested that the fractional parts of the UPD between satellites could be computed by averaging the fractional parts of all involved real-valued ambiguity estimates....

[...]

...156 ambiguity estimates, thus destroying the integer properties of the ambiguities ([8], [9])....

[...]

...All stations that use cross-correlation receivers were excluded due to their relatively poor pseudo-range quality [9]....

[...]

...at least several days to even several months) ([8], [9], [16])....

[...]

578 citations

### "Performance of Precise Point Positi..." refers background in this paper

...However, to date, integer ambiguity resolution in PPP is largely ignored due to the fact that the non-integer receiver- and satellite-dependent uncalibrated phase delays (UPD) [1], which are related to hardware [8], are absorbed by the real-valued...

[...]