## 4 Discussion  
The external constraint can help alleviate the ill-conditioned problem and improve the accuracy of estimated parameters in medium-long baseline RTK; however, the external constraint may also  
Figure 12: Improvement of (**a**) STD and (**b**) RMS on DOY016, 92, 190, and 289 when comparing the GPT2-constrained model with the traditional model.  
result in a bias for the other parameters when the corresponding variance \(\sigma^{2}\) is too small. In contrast, if \(\sigma^{2}\) is excessively large, then the contribution of constraints will fade. Possible introduced bias and the positive effect on height precision of external tropospheric corrections should be balanced. The improvement rates of STD indicators on a summer day were lower than on a winter day based on the analysis in the previous section (Table 3). Negative values even appeared for the improvement rates of some baselines. This indicates that position bias may have been introduced because of external constraints. In addition, the precision of the a priori tropospheric corrections derived from the GPT2 model may also vary at different times and areas. Consequently, an appropriate \(\sigma\) should be set for external constraints.  
The data collected on DOY016 and 190 for all 15 baselines were processed using the GPT2-constrained RTK model with different STD \(\sigma\) for the a priori GPT2 tropospheric corrections. Figure 13 shows the vertical STDs of the fixed RTK solution for BDS-only and GPS/BDS-combined scenarios when \(\sigma\) for external RZTD information in Equation (5) was set as 1.5, 4.0, and 15 cm, respectively. The statistical results only included the correctly fixed solution. The traditional RTK model can be regarded as the RZTD-constrained RTK model when \(\sigma\) is set as \(\infty\) for GPT2 troposphere information. In addition, without estimating the RZTD parameter, the method correcting the troposphere error in phase and pseudo-range observations with the GPT2 model was also used for comparison. This method is also called the ZTD-corrected model, which can be regarded as the RZTD-constrained model when \(\sigma\) is set infinitely close to zero. Compared to the BDS-only situation, GPS/BDS-combined RTK seems to be more sensitive to the value of \(\sigma\). In two different tropospheric conditions, compared to traditional RTK, the STDs of the BDS-constrained RTK solution greatly decreased when \(\sigma\) was set between 1.5 cm and 15 cm; however, the performance of the GPS/BDS-combined GPT2-constrained RTK varied greatly with different \(\sigma\). Additionally, it can be seen that on DOY016, for most baselines, the STD of the positioning results increased for both BDS-only and GPS/BDS-combined scenarios when the value of \(\sigma\) increased, and the RZTD-corrected model obtained a preferable effect on this day. However, on DOY190, the STD of the BDS-only fixed solution for Baselines 11, 12, 14, and 15 did not increase as the value of \(\sigma\) increased; the STD of the GPS/BDS-combined RTK for most baselines greatly decreased when the value of \(\sigma\) varied from 1.5 cm to 15 cm, and the performance of the RZTD-corrected model was the worst for the GPS/BDS-combined situation on this day. This means that for relatively stable tropospheric conditions on a winter day, a smaller \(\sigma\) value seems preferable. However, on a summer day, a higher \(\sigma\) value is helpful to decrease possible introduced bias. In general, to ensure the reliability of the RZTD constrained model, a relatively larger variance is usually proposed.  
Figure 13: STD of the fixed RTK solution on (**a**) DOY016 and (**b**) DOY190 with different STD \(\sigma\) for the a priori GPT2 tropospheric corrections.