Solution convergence time

Based on the results from the float PPP tests in 5.2.3, it takes on average 1570 s to obtain 3D position error smaller than 10 cm and on average 2400 s to obtain 3D error smaller than 5 cm. The success rates of obtaining better than 10 or 5 cm 3D position errors are 88.0% and 55.4%, respectively, for the one hour datasets used.

Based on the fixed ambiguity PPP tests in Section 5.2.4, the ILSDNCF method with a 1200 s minimum lock time requirement was found to be the most suitable. The average time required to obtain an initial ambiguity resolution was found to be 1560 s, rate of correct ambiguity resolution 74.2% and rate of incorrect ambiguity resolution 12.7%. The time required to obtain 3D position errors better than 10 and 5 cm is 1310 and 1780 s, respectively. The success rates of obtaining 3D position errors better than 10 and 5 cm are 83.0% and 66.3%, respectively.

The results show that typical PPP solution convergence times are unacceptably long compared to the requirements of most applications. For example, many agricultural, aviation and LSB applications require immediate solution convergence to centimetre level accuracy. The long convergence time is the most important limiting factor which prevents the wider use of PPP.

The ionosphere-free combination is used by the PPP methods tested. The combination eliminates the first order ionospheric error, but magnifies measurement noise. The large magnitude of the noise makes convergence slow. In addition, the current fixed ambiguity PPP methods discussed in 4.3 employ GPS alone. In terms of the solution convergence, it would be beneficial to use also other GNSS systems such as GLONASS, to improve satellite geometry and increase measurement redundancy.

The insufficient quality of the error correction products such as satellite orbit and clocks and error models such as tropospheric models also impacts convergence in a negative way,

150 because it causes the position estimation to be sub-optimal. It is possible that the insufficient the quality of the error correction products is the reason why 10 cm position accuracy is not achieved in 12.0% of the tested cases in the float PPP tests in Section 5.2.3.

The PPP performance issues caused, for example, by the insufficient accuracy of satellite orbit corrections are discussed in Laurichesse (2011). To achieve the best possible convergence, all error sources should be eliminated. However, this is impossible in practice.

Solution convergence time is also linked to the speed and reliability of initial ambiguity resolution. When employing PPP, correct narrow-lane ambiguity resolution can improve convergence to the 5 cm accuracy, because resolved ambiguities constrain the position error. However, as shown in Section 5.2.4, the reliability of the current fixed ambiguity PPP methods is not sufficient. The limitations of the current PPP ambiguity resolution methods are discussed in detail in the next Section.

5.2.6.2 Carrier-phase ambiguity resolution and validation

Ambiguity resolution and validation are the main challenges when employing fixed ambiguity PPP. Based on the tests in Section 5.2.4, the ILSDNCF method with a 1200 s lock time requirement is the most reliable method. However, the rate of incorrect ambiguity resolution is still 12.7%, unacceptably high for real-life applications.

The ratio test based on the ILSC, ILSDNCF and ILSFFS methods (Section 4.4.4) do not test the absolute correctness of the fixed ambiguity candidate vector. Instead they provide information on the closeness of the integer ambiguity candidate vector to the float ambiguity vector (Teunissen and Verhagen, 2008). Therefore, the integer ambiguity candidate vectors accepted could be incorrect, if the float ambiguity vector is incorrect, for example, as a result of multipath, insufficiently accurate correction products or other error sources.

In reality, it is often possible that a float position solution has large errors. For example, in the float PPP test in Section 5.2.3, 10 cm position accuracy is not achieved in 12.0 % of the cases. Therefore, the possibility of wrong float solution must be taken into account when validating carrier-phase ambiguities.

151 For example, when employing the ILSDNCF method with a 1200 s lock time requirement and using data recorded at the BDOS NOAA station on 12 February 2013 between 04:00 and 05:00, wrong ambiguity resolution occurs at the 1200 s epoch, as shown in Figure 5.26. The values of fixed and float ambiguities at the initial ambiguity resolution epoch are shown in Table 5.6. It can be seen that the float ambiguity values are close to integers. Thus, the ratio test accepts the fixed ambiguity candidate vector with the test statistic value of 19.47, when the threshold is 10.71. However, the float ambiguity values are far from correct, because the float position solution is still converging and the magnitude of the 3D position error is approximately 0.3 m. The correct float ambiguity values correspond to the case where the magnitude of the position error is zero. However, it is not possible to know the exact correct float ambiguity values when using real data. Thus, it is assumed that float ambiguities are far from correct when the position error is large (3D error more than 10.7 cm). In summary, this case shows that wrong ambiguity resolution may occur when using the ratio test alone and the float ambiguities are far from the correct values.

Figure 5.26 3D position error in the case of fixing ambiguities incorrectly

152

Table 5.6 Ambiguity values in a wrong ambiguity resolution case

The reliability of ambiguity resolution is important when using PPP for real-life applications.

The large incorrect ambiguity resolution rate of 12.7% is not acceptable even for non-life-critical applications such as land surveying. To make fixed-ambiguity PPP useful for practical applications, the incorrect ambiguity resolution rate closer to the cRTK method is required.

PPP ambiguity resolution with the GPS L1 and L2 signals without external ionospheric corrections is typically significantly more difficult than cRTK ambiguity resolution. The reason is that constraining the position error by employing the geometry-dependent wide-lane combination with the 86 cm wavelength and fixing the wide-wide-lane ambiguities is not possible when employing PPP. Ionospheric errors affect the geometry-dependent wide-lane combination significantly as shown in equation (5.1), where _𝑓^𝐼1

12 and _𝑓^𝐼1

22 are the magnitudes of the first order ionospheric delays at the L1 and L2 frequencies, respectively, in metres.

Based on this, the magnitude of the ionospheric delay in the geometry-dependent wide-lane combination can be at the metre level when measurements are not differenced across receivers. Thus, the combination cannot be used in PPP. This is the reason why geometry-dependent PPP ambiguity resolution must be carried out directly using the narrow-lane combination, which has a 10.7 cm wavelength. Measurement errors and inaccurate float solutions aggravate narrow-lane ambiguity resolution significantly more than wide-lane resolution, because of the short narrow-lane wavelength.

𝐼_𝑤𝑙 = 1

𝑓₁− 𝑓₂(𝑓₁ 𝐼₁

𝑓₁²− 𝑓₂ 𝐼₁

𝑓₂² ) (5.1)

The 10.7 cm wavelength of the narrow-lane combination is a limiting factor which prevents fast ambiguity resolution compared to cRTK. As discussed in 4.3.4, using the new GNSS signals and systems adds more possible geometry-dependent signal combinations which have longer wavelength than the GPS L1/L2 narrow-lane. As discussed in Section 4.3.4.1,

153 employing longer wavelength geometry-dependent signal combinations enables faster ambiguity resolution. However, the signals and systems are not currently available and cannot be used for practical applications.

Fixing GLONASS ambiguities when employing PPP is not practical as discussed in Section 4.3.3, because of the satellite/frequency/receiver type specific inter-frequency biases in GLONASS measurements. However, it is possible to use both GPS and GLONASS for the float solution estimation and fix GPS ambiguities as discussed in Section 5.3.3.

In conclusion, the major issues with the current ambiguity validation methods are the vulnerability for inaccurate float solutions, a high rate of wrong ambiguity resolution and low rate of correct ambiguity resolution.

5.2.6.3 Accuracy

There are applications such as control surveying which require millimetre-level accuracy.

However, most applications require positioning accuracy between 1 and 10 cm. Based on the tests, fixed-ambiguity PPP can fulfil the requirement of accuracy between 1 and 10 cm in 3D, horizontal and vertical levels, if carrier-phase ambiguities are fixed correctly.

Similar to the solution convergence time, the quality of error correction products and models also impacts accuracy. If the magnitude of the residual error after applying correction products and models is significantly large compared to the total error budget shown in Table 3.4, the PPP solution may not converge to the required accuracy and correct narrow-lane carrier-phase ambiguity resolution may not be possible.

Wrong ambiguity resolution can also result insufficient positioning accuracy. Based on the example shown Section 5.2.6.2, where ambiguities were fixed incorrectly at the initial ambiguity resolution epoch, horizontal and vertical position errors and protection levels are shown in Figures 5.27 and 5.28. This shows that the ICRAIM method cannot be used to detect errors caused by wrong ambiguity resolution, because the position solution is self-consistent with the wrongly fixed ambiguities. The reason for the self-consistency is that the position solution is re-calculated based on the incorrectly fixed ambiguities. Therefore, the

154 magnitude of the measurement residuals is small and the solution can pass the ICRAIM based integrity check.

Figure 5.27 Horizontal position error and Protection Level (HPL) in the case of fixing ambiguities incorrectly

Figure 5.28 Vertical position error and Protection Level (VPL) in the case of fixing ambiguities incorrectly

155 In general, real-time PPP using the L1 and L2 signals can provide sufficient accuracy when ambiguities are fixed correctly or the solution has converged over a sufficiently long time (at least 30 min). However, the primary issue when using PPP for real-life applications is to guarantee the accuracy, i.e. to monitor the integrity.