## 1 Introduction  
Real-time kinematic (RTK) positioning is one of the most widely used surveying techniques given its high reliability and preferable precision. For medium-long baseline RTK, the ionosphere and the troposphere are considered two key dominant error sources that limit the capability of carrier phase ambiguity resolution (AR) and positional precision. Numerous efforts have been exerted to improve medium-long baseline RTK positioning by considering atmospheric delay [1; 2; 3; 4]. [PERSON] [1] proposed an ionosphere-weighted model using uncombined double-differenced (DD) observation equations, and the model was proven effective for improving the precision of float ambiguity and shortening the initialization time of RTK [3; 5; 6; 7]. Network RTK is also an alternative approach for mitigating atmospheric errors [8; 9]. In addition, a number of AR strategies have been proposed for medium-long baseline RTK [10; 11; 12; 13; 14]. However, the position error of the height component for medium-long baselines can reach a few decimeters, even after applying ambiguity resolution [15; 16; 17; 18]. The troposphere error will be the remaining dominant factor for this effect. Even for network RTK, interpolation troposphere errors will increase when the distance between the master reference station and the user increases. The interpolated corrections for a rover also become vulnerable to localized anomalous errors under unfavorable atmospheric conditions [4; 19]. For a single-reference medium-long baseline RTK, the relative zenith hydrostatic delay can be corrected in advance, and the relative zenith wet delay is usually regarded as an unknown parameter [2; 10; 15; 19]. This method can overcome distance limitations; however, evaluated through condition numbers [18; 19], researchers have pointed out that the model is ill-conditioned because the troposphere parameters are highly correlated with the height component. The regularization method can overcome the ill-conditioned problem [18], but a reasonable regularization parameter is difficult to compute. In general, the tropospheric delays are assumed as a random walk process, and thus a filtering technique is used [20; 21]. The filter enables us to distinguish the troposphere zenith delay from the height component with changes in satellite geometry and mapping functions [21]. However, a relatively long period is required to initialize the troposphere parameter, and although it has been alleviated, the ill-conditioned problem remains. Additional difficulties will occur in terms of the troposphere and height solution when BeiDou Navigation Satellite System (BDS) observations are used in medium-long baseline RTK. BDS can provide reliable positioning results by itself [22; 23; 24]. Together with GPS, the ambiguity fixed rate and the precision of single-epoch RTK are increased [14; 25; 26]. However, BDS has three different orbit types: geostationary Earth orbit (GEO), inclined geosynchronous orbit (IGSO), and medium Earth orbit (MEO); therefore, its observation time, geometric structure, and change rate of satellite geometry differ from those of the GPS system. When considering the strict relationship between geometric structure and the troposphere parameter, different characteristics will be observed in BDS medium-long baseline RTK. Meanwhile, troposphere corrections derived from global zenith troposphere delay (ZTD) models now have good precision. The accuracy of the widely-used troposphere-corrected model UNB3 can reach 5.2 cm, whereas that of the modified model UNB3 m is approximately 4.8 cm [27]. The first global pressure and temperature (GPT) model was also proposed based on the ERA-40 of the European Center for Medium-Range Weather Forecasts [28]. Subsequently, improved versions such as GPT2 [29] were developed. [PERSON] [30] proposed a ZTD-constrained precise point positioning (PPP) model using virtual ZTD observations to accelerate PPP convergence. The effect of troposphere-enhanced PPP has been validated by a number of scholars [31; 32; 33].