Integer Least-Squares using Constant ratio test threshold (ILSC)

The ILS method as discussed in Section 4.4.3 and ratio test ambiguity validation using a constant test threshold (Section 4.4.4) are employed. The ambiguity validation and other parameter values are similar to those used in Geng et al. (2009, 2010d). The ratio test acceptance threshold is 3.0, the satellite elevation mask is 7.0 degrees, ambiguity resolution is only attempted for satellites above 15 degree elevation angle and a minimum of 1200 s carrier-phase lock time is required before attempting initial wide-lane or narrow-lane ambiguity resolution. Partial narrow-lane ambiguity resolution is used in this test: ambiguity

31 32.1

136 resolution is tested for all possible narrow-lane ambiguity combinations (Geng et al., 2009).

The partial ambiguity fixing method was initially developed as the Minimum Constellation Method (MCM) in Schuster et al. (2012).

From the results, the average time required to obtain an initial ambiguity resolution is 1400 s and its SD is 460 s. The ambiguity resolution rate is 94%. The average position error at the initial ambiguity resolution epoch is shown in Figure 5.15. The average 3D, horizontal and vertical position errors are 8.2 (SD: 10.5 cm), 5.1 (SD: 8.9 cm) and 5.4 cm (SD: 6.6 cm), respectively.

Figure 5.15 The average position error at the initial ambiguity resolution epoch using the ILSC method (one sigma standard deviation as error bars)

The distribution of the 3D position error at the initial ambiguity resolution epoch is divided into different categories in Figure 5.16. The 3D position error is larger than 10.7 cm in 20.8%

of the cases where ambiguities are fixed.

0 2 4 6 8 10 12 14 16 18 20

Error (cm)

3D Horizontal Vertical

137 Figure 5.16 The distribution of 3D position error at the initial ambiguity resolution epoch

using the ILSC method

Based on the results, the ILSC method provides better performance than the integer bootstrapping method in all aspects: the time needed to obtain an initial ambiguity resolution is shorter, ambiguity fixing rate is higher and average position error at the initial ambiguity resolution epoch and its SD are smaller. For example, the average 3D position error is 12.4 cm (SD: 13.7 cm) when employing the integer bootstrapping method and 8.2 cm (SD: 10.5 cm) when employing the ILSC method. There is a significant difference when comparing the distribution of the 3D position error at the initial ambiguity resolution epoch between the integer bootstrapping and ILSC methods, which can be seen in Figures 5.14 and 5.16. The rate of incorrect ambiguity resolution is significantly lower when using the ILSC method compared to the integer bootstrapping method.

Nevertheless, the ILSC method still suffers from various weaknesses: the rate of incorrect ambiguity resolution is high at 20.8 %. This is too large for real-life applications. In addition, there is no theoretical justification for using a constant ratio test threshold value of 3.0, chosen based on empirical experience. The lack of theoretical justification makes it is difficult to prove that the ILSC method is suitable for all possible situations such as the varying number of visible satellites or allowed failure rate.

57.1

138 5.2.4.3 Integer Least-Squares using Doubly Non-Central F-distribution based ratio

test threshold (ILSDNCF)

The ILS method presented in Section 4.4.3 is employed. The ratio test is used for narrow-lane ambiguity validation and its acceptance threshold is calculated based on the assumed Doubly Non-Central F-distribution discussed in Section 4.4.4.1. This validation is carried out as described in Feng et al. (2012). Wide-lane ambiguity validation is done using the DBT method (Section 4.3.1.1). The elevation mask used is 5 degrees and ambiguity fixing is attempted for satellites with elevation angles higher than 10 degrees.

The required confidence level (1 - failure rate) for an initial narrow-lane ambiguity resolution is 99.9%. The MCM is employed and it is required that at least four narrow-lane ambiguities can be fixed initially.

Based on the results, an initial narrow-lane ambiguity resolution can be obtained on average in 194 s and the ambiguity fixing rate is 99.9%. However, the ambiguity resolution is incorrect in most cases as shown in Figure 5.17. The percentage of the cases where the 3D position error at the initial ambiguity resolution is larger than 10.7 cm is 95.0%.

Figure 5.17 The distribution of 3D position error at the initial ambiguity resolution epoch using the ILSDNCF method

The issue is that it is too risky to fix narrow-lane ambiguities while the float position solution is still converging. It is often possible that an integer ambiguity candidate vector is sufficiently close to the float ambiguity vector, which causes the ratio test to be accepted

3 2 1.9 4.5

139 with a high confidence level. Nevertheless, the float ambiguities are far from the correct values. This leads to wrong ambiguity resolution.

To reduce the likelihood of incorrect ambiguity resolution, a minimum of 1200 s carrier-phase lock time is imposed before attempting initial ambiguity resolution. The requirement is similar to the ILSC method and is used also in Geng et al. (2010d). The lock-time requirement was not introduced when the ILSDNCF method was originally presented in Feng et al. (2010, 2012).

The average time required to obtain an initial ambiguity resolution is now 1560 s and its SD is 570 s. The ambiguity resolution rate is 87%. The average position error at the initial ambiguity resolution epoch is shown in Figure 5.18. The average 3D, horizontal and vertical errors are 6.3 (SD: 7.8 cm), 4.0 (SD: 6.7 cm) and 4.2 cm (SD: 4.7 cm), respectively. Compared to the integer bootstrapping and ILSC methods, the ILSDNCF method with the 1200 s lock time requirement provides the smaller magnitude of the position errors and their SDs.

Figure 5.18 The average position error at the initial ambiguity resolution epoch using the ILSDNCF method with the 1200 s lock time requirement (one sigma standard deviation as

error bars)

The distribution of the 3D position error at the initial ambiguity resolution epoch is shown in Figure 5.19. The position error at the initial ambiguity resolution epoch is larger than 10.7 cm in 14.6 % of the cases where ambiguities were fixed.

0 2 4 6 8 10 12 14 16 18 20

Error (cm)

3D Horizontal Vertical

140 Figure 5.19 The distribution of 3D position error at the initial ambiguity resolution epoch

using the ILSDNCF method with the 1200 s lock time requirement

Compared to the ILSC method, using the ILSDNCF method with the 1200 s lock time requirement reduced the average 3D position error at the initial ambiguity resolution epoch by 23.2% and reduced the percentage of cases with larger than 10.7 cm position error from 20.8% to 14.6%. The average time required to fix ambiguities increased by 11.4% and the ambiguity fixing rate decreased from 94% to 87%. The decrease in the overall ambiguity fixing rate reflects the smaller percentage of cases where ambiguities were fixed incorrectly.

5.2.4.4 Integer Least-Squares using Fixed Failure rate Simulation based method