GPS + Galileo tightly combined RTK positioning for medium-to-long baselines based on partial ambiguity resolution

Multi-GNSS observation models

The above observation equations for multi-GNSS DD operations can be generalized to inter-system mixed DD which can be further categorized into those between the same frequencies or the diverse frequencies of observations (Li et al., 2017).

Because the frequencies are the same, the ambiguities \( {N}_{r_1{r}_2,i}^{s_1{s}_2} \) still have integer characteristics. However, the receiver UPDs which are related to the initial phase and hardware delay are consequently contained in the inter-system bias (ISB) and therefore cannot be eliminated, i.e. \( {ucd}_{r_{\boldsymbol{1}}{r}_{\boldsymbol{2}},i}^{A_{\boldsymbol{1}}{A}_{\boldsymbol{2}}}\ne 0 \) and \( {upd}_{r_1{r}_2,i}^{A_1{A}_2}\ne 0 \).

Ambiguity resolution in the DD model

If the validation fails, the current epoch will keep the ambiguities float instead.

Partial ambiguity resolution strategy

Then the LAMBDA method is used to fix \( {\tilde{\boldsymbol{N}}}_b \), and if \( {\tilde{\boldsymbol{N}}}_b \) can be fixed, \( {\overset{\smile }{\mathbf{N}}}_a \) and \( {\tilde{\boldsymbol{N}}}_b \) are used to update \( \overset{\smile }{\mathbf{x}} \) and \( {\mathbf{Q}}_{\overset{\smile }{\mathbf{x}}} \). Otherwise, only \( {\overset{\smile }{\mathbf{N}}}_a \) is used to update \( \overset{\smile }{\mathbf{x}} \) and \( {\mathbf{Q}}_{\overset{\smile }{\mathbf{x}}} \).

In this paper, a PAR procedure is used to determine the subset of ambiguities which in terms of the success rate and the ratio test (Wang and Feng, 2012). Figure 1 presents the flowchart of this procedure.

First, the PAR process starts with the decorrelation of the ambiguities and the diagonal elements of the decorrelated matrix are sorted in the ascending order. We can get the diagonal element set \( D=\left\{{d}_1,{d}_2,\cdots {d}_i,\cdots, {d}_n\left|{d}_1<{d}_2<\cdots {d}_i<\cdots <{d}_n\right.\right\} \) after sorting by conditional variance. Then, by updating the traversal of i = n to the minimum threshold i = n₀ and pick the subset \( D=\left\{{d}_1,{d}_2,\cdots {d}_i\right\} \) and the corresponding ambiguity subset \( {\widehat{\boldsymbol{N}}}_a\left({D}_i\right) \) and variance-covariance matrices \( {\mathrm{Q}}_{{\widehat{\boldsymbol{N}}}_a}\left({D}_i\right) \). The minimum threshold n₀ is typically 6 to ensure the selected satellites are still sufficient to get reliable positioning results. Then, the LAMBDA method is applied in the ambiguity search process. If \( {P}_S\ge {P}_{S0} \) and \( R>{R}_{\mathtt{thres}} \), the fixed ambiguities can be considered to pass the acceptance test and be used into the following position calculation. Otherwise, we will update the subset and repeat the ambiguity search and test. If the number of selected ambiguities is less than \( {n}_0 \), the iteration will stop and only the float solutions are made available.

Carrier phase and code ISB estimation

The fractional phase and code ISBs are calculated according to Eq. (6) and the results are shown in Figs. 3 and 4 Both figures show that the phase and code ISBs were stable during the period of daily experiments. However, there are differences in the ISBs of two groups of experiments over different periods. This is mainly due to the upgrade of the receiver firmware version, such as the CUT0 station’s receiver firmware version from Trimble NETR9 (4.85) to Trimble NETR9 (5.20), which is similar to the results of (Paziewski and Wielgosz, 2015). Therefore, when estimating ISB, it is necessary to note not only the receiver type but also the receiver firmware version.

We selected the ISB corrections from the experiment in 2017 to correct the ISBs in the long-baseline experiment because the receiver’s brands and firmware versions used in this experiment were the same as that used in the long-baseline experiment. The receiver brands and firmware versions of the long-baseline experiment are shown in Fig. 2.

Results of ambiguity resolution

The AR results of different combination strategy are shown for 500 min of data in Fig. 5 for the full ambiguity resolution (FAR) method and in Fig. 6 for the PAR method, including the AR success rate, number of ambiguities and ratio values. Note that in the ratio statistics, any ratio greater than 4 is assigned a value of 4, which is intended to facilitate the distribution of smaller ratio values.

Figure 5 shows that the AR success rate and number of ambiguities of the three baselines are similar. However, the number of AR changes more frequently as the baseline length increases. When there is a newly-rising satellite, the success rate will drop dramatically. Fortunately, the TC strategy can provide more satellite observations, at least one more than the LC strategy. In this way, the TC strategy can be faster to reach the success rate at 99%, and has the ability to eliminate the sudden change in the number of satellites.

It is worth noting that the ratio values of the medium-long-baseline and the long-baseline are very different. The ratio of the medium-long-baseline ratio values greater than 3 is about 90% and the long-baseline is just about 5%. This reflects the fact that the long-baseline is affected by the atmospheric delay error and is difficult to satisfy the condition of correct AR. In addition, TC did not improve the ratio values, but deteriorate. For example, the ratio of the NNOR-CUT0 long-baseline to the ratio values greater than 3 under the three different combinations of LC, TC, and tight combination with ISB corrections (TC + ISB) is 4.83, 4.16 and 4.34%, respectively. This indicates that more observations do not increase the ambiguity ratio values, but rather deteriorate.

Figure 6 shows that the application of the PAR method can significantly shorten the convergence time. The three baselines under different combinations can all reach the success rate of 99% within two or three epochs.

For medium-long-baselines, the number of resolved ambiguity is relatively stable and the problem of newly-rising or falling satellites is effectively suppressed. Because the ambiguity subset selection process can remove these satellites according to the variance of their ambiguity, and the ambiguity variance of such satellites is generally large. In the following epochs, their ambiguity precision improves, and their corresponding ratio values have also been improved, the ratio values also drops a lot only in the number of ambiguity changes epoch, but it still meets the threshold.

However, for long-baselines, the number of ambiguities to be fixed changes frequently. Because of the influence of atmospheric delay and so on, the ambiguity subset can’t be fixed, even when the number of ambiguity reaches the threshold. Moreover, the frequent occurrence of falling satellites also affects the ambiguity subset fixed. Since the ambiguity subset can’t be fixed, only the float solutions are available and then the ambiguity subset selection is tried again in the next epoch. However, the PAR method is still able to greatly improve the ambiguity fixed rate. For example, the ratio of the NNOR-CUT0 long-baseline to the ratio values greater than 3 under the three different combinations of LC, TC, and TC + ISB for the PAR method is 27.36, 27.23 and 27.30%, respectively. Compared to the FAR strategy, it was increased by 22.53, 23.07 and 22.96%, respectively.

Results of positioning

The positioning capabilities of the two combination strategies and the two AR strategies are now verified with real GNSS data for single-frequency combined GPS + Galileo. The baseline errors of different combination strategies are shown in Figs. 7 and 8. Baseline errors are the difference between the estimated baseline length and precise reference baseline length.

Figure 7 shows that the medium-long-baseline error (1.5 cm) is smaller than that of the long-baseline (10 cm) static relative positioning for the FAR strategy. Affected by the residual atmospheric errors, such as residual tropospheric and ionospheric delays, the positioning results reflect some systematic biases especially for the longer baseline. The positioning accuracy based on the TC strategy is improved, especially for the medium length baseline. The accuracy of positioning based on TC + ISB is equivalent to that of TC, because the ISBs of this experiment are so small that it can be ignored. The positioning accuracy based on PAR method has improved, especially for the long-baseline. Because of after fixing the partial ambiguities by the PAR method, the fixed integer ambiguities can be back tracked into the observations to update the troposphere and the ionospheric parameters, a more accurate atmospheric delay correction is obtained, and the positioning parameters are updated by fixing the remaining ambiguities. The proportions of fixed solution of the NNOR-CUT0 long-baseline static relative positioning under the three different combinations of LC, TC, and TC + ISB for the FAR method is 11.98, 17.33 and 17.40%, respectively. For the PAR strategy, that is 75.80, 76.70 and 76.96%, respectively.

Figure 8 shows that the baseline errors of different combination strategies RTK positioning for the FAR method and the PAR method. Introducing the PAR method into the TC strategy can not only fast converge, but also effectively reduce the baseline error. The results are similar to the static mode, however, the positioning accuracy and the proportion of the fixed solution decrease. For example, the proportions of fixed solution of the NNOR-CUT0 long-baseline RTK positioning under the three different combinations of LC, TC, and TC + ISB for the PAR method is 17.38, 18.23 and 18.69%, respectively.