The novel enhancements to increase the reliability of the ILSDNCF method are discussed in this section. These include time window based ambiguity validation (Section 5.3.2.1) and
160 additional validations (Sections 5.3.2.2) to reduce the probability of incorrect ambiguity resolution.
5.3.2.1 Time window based ambiguity validation
The principle of time window based ambiguity validation is to require that the best ambiguity candidate vector is the same over a given number of epochs consecutively and that the ambiguities belonging to that vector can be fixed to the same integers during the chosen time-period. The theory behind this test is that the float ambiguities change when the float solution is converging. Therefore, if for the given time window the best ambiguity candidate vector is the same and the ambiguities in the vector are fixed to the same values, then it is less likely that float ambiguities have not converged to the correct values and are close to the nearest integer.
Ambiguity resolution is tested with multiple time window lengths such as 1, 50, 150, 250 and 500 s using the NOAA dataset and ILSDNCF method. The required confidence level of ambiguity validation is set to 99.9%. The results are shown in Table 5.7 and analysed in terms of the rate of ambiguity resolution, rate of correct ambiguity resolution, rate of incorrect ambiguity resolution, time required to obtain an initial ambiguity resolution and position error at the initial ambiguity resolution epoch.
Time Table 5.7 The effect of the selected time window length on ambiguity resolution results
161 The idea of using the time-window based ambiguity validation for fixed-ambiguity PPP is novel and developed in this thesis. When employing cRTK, the time window based ambiguity validation was used to complement the ILSC method based carrier-phase ambiguity validation in Wei and Schwarz (1995). However, using the time-window based ambiguity validation with the ILSDNCF method has not been attempted, even with cRTK. In addition, time-window based ambiguity validation is not used for PPP ambiguity validation in the literature. Therefore, the time-window based ambiguity validation can be defined a new PPP development, even though the idea of it is not completely novel.
Using the time-window based ambiguity validation is particularly useful for PPP, because it verifies that float narrow-lane ambiguities with 10.7 cm wavelength have stabilised sufficiently before deciding ambiguity fixing. The narrow-lane ambiguities in the PPP estimation are particularly vulnerable to errors in measurements and correction products, because of the short wavelength. The time window based ambiguity validation ensures that ambiguity resolution is not done only based on the one acceptance of the ratio test as in the case of the current methods presented in the literature. It is often possible that float ambiguities are sufficiently close to some (wrong) integers that the ratio test is accepted for a few epochs. Nevertheless, it is significantly less likely that the float ambiguities would remain close to incorrect integers for long time periods such as 150 s.
The time window lengths of 150 and 250 s are selected for further investigation, because they give a good compromise between the rate of correct and incorrect ambiguity resolution, time required to obtain an initial ambiguity resolution and average position error and its standard deviation at the initial ambiguity resolution epoch. For the time window lengths of 1 and 50 s, the magnitude of the average position error at the initial ambiguity resolution epoch is unacceptably large. On the other hand, long time window lengths such as 500 s provide a small average position error, but the obtained rate of ambiguity resolution (54.7%) is unacceptably low.
5.3.2.2 Additional validations to make incorrect ambiguity resolution less likely When employing the LAMBDA method implementation, the number of float narrow-lane ambiguities in a vector must be at least four. However, requiring that only four narrow-lane
162 ambiguities are fixed initially is not always optimal, because it is more likely that inaccurate float solution estimation can result in incorrect ambiguity resolution. The probability that all float ambiguities are close to wrong integers decreases when the number of ambiguities to test increases. Therefore, ambiguity resolution is tested by requiring that at least four, five and six ambiguities are fixed initially using the NOAA dataset. The length of the ambiguity validation time window in this test is 150 s and required confidence level is set to 99.9%.
The results are shown in Table 5.8. The requirement of fixing initially at least five narrow-lane ambiguities gives a good compromise between the reliability and ambiguity resolution rate when the time window length is 150 s. The reliability is measured in terms of the rate of incorrect ambiguity resolution and average position error and its standard deviation at the initial ambiguity resolution epoch. Table 5.8 The effect of requiring initial ambiguity resolution for more than four
narrow-lane ambiguities (150 s time window)
Table 5.9 shows the ambiguity resolution results when the length of the time window is 250 s and the required confidence level is 99.9%. The ambiguity resolution rate is already as low as 80.4% when at least four ambiguities are initially required to be fixed. Increasing the minimum number of narrow-lane ambiguities to be fixed would further decrease the fixing rate. That is shown in Table 5.9 when at least five ambiguities are initially required to be fixed.
163 Table 5.9 The effect of requiring initial ambiguity resolution for more than four
narrow-lane ambiguities (250 s time window)
When employing the ILSDNCF method, the ratio test statistic must be larger than the threshold for an integer ambiguity candidate vector to be accepted. The variable confidence level method is developed in this thesis. The variable confidence level method cannot be considered as a new development, but it can be considered as a more optimal way of employing the ILSDNCF method. The principle of the variable confidence level method is to use a confidence level threshold value of 99.99% during the float solution convergence period and 99.00% otherwise. The solution is defined to be in the convergence period when the longest carrier-phase lock time is smaller than 2000 seconds. The reason for using a higher threshold during the float solution convergence period is that the position error is typically larger during this period. Thus, it is more likely that float ambiguities are close to wrong integers.
The results of the comparison of the use of constant confidence levels of 99.00%, 99.90%
and 99.99% during the whole test and a variable confidence level are shown in Table 5.10.
The results show that the higher confidence level of 99.99% during the convergence period is beneficial in terms of reducing the rate of incorrect ambiguity resolution. On the other hand, it is beneficial to use a lower confidence level (99.00%) after the solution has converged to increase the rate of correct ambiguity resolution. When using a time window length of 250 s and requiring that at least four ambiguities are fixed initially, the variable confidence level method reduced the incorrect ambiguity resolution rate from 6.3% to 4.5%
and increased the correct ambiguity resolution rate from 74.1% to 82.0% compared to using the constant confidence level 99.9%. When the time window length is 150 s and an initial fix of five ambiguities is required, the variable confidence level method reduces the rate of incorrect ambiguity resolution from 7.9% to 6.0% and increases the rate of correct