## 1 Introduction  
High-rate global navigation satellite systems (GNSSs) have shown great potential in observing both ground static and dynamic motions, which is more than a favorable complement to traditional seismometers [1; 2]. Retrieving high-precision coseismic displacement in real time contributessignificantly to rapid source and rupture inversion [3; 4], rapid hazard assessment [5], and early warning of earthquake [6]. Typically, there are two primary techniques for real-time GNSS data processing. One is the relative positioning technique, which is able to achieve a positioning accuracy better than 1 cm when satellite orbits and clocks, as well as atmosphere errors, are basically eliminated through observation differences between two nearby stations [7]. The limitation is, however, that only relative displacement can be obtained with respect to the reference station, which might itself be subject to shaking in the case of a large earthquake. On the contrary, precise point positioning (PPP) can provide absolute precise position with respect to a global reference frame using a single Global Positioning System (GPS) receiver [8], and has been therefore widely used for seismo-geodesy in recent years [9; 10]. Compared with the traditional ionosphere-free combination PPP, the uncombined and undifferenced (UCUD) PPP using raw observations can retain all meaningful information and be easily extended to multi-frequency processing. Thus, it is a popularly adopted PPP model [11; 12]. In some cases, the high-rate PPP has the capability of generating kinematic position estimates at millimeter level if it is only quantifying a short period of data under favorable observation circumstances [13; 14]. In general, the 3-D displacement estimate is believed to have an accuracy of a few to ten centimeters, which hinders deep insights into rupture processes as well as possible geophysics findings. Previous studies have indicated that one of the major errors causing positioning precision deterioration is the multipath, which can transform into other parameters such as positions, tropospheric parameters, and float ambiguity terms, and primarily dominates on the low-frequency band over a few tens of seconds to minutes [15; 16]. Multipath effects cannot simply average out over a limited period of data, and probably induce spurious seismic signals as well as bias the coseismic displacement estimations. According to the spatiotemporal repeatability of multipath under static environments, two prevalent approaches were proposed to mitigate the multipath impacts on high-rate GNSS in the observation domain. For the multipath hemispherical map (MHM), the multipath is assumed to be dependent on the specific elevation and azimuth angle of the satellite [17], and it can be used for real-time application [18]. The sidereal filtering (SF) approach calculates the multipath corrections from neighboring days, and then subtracts the time-shift corrections from the observations on the day of interest based on satellite orbital repeat time [19]. This method is capable of capturing higher-frequency multipath and is more suitable for seismic waveform detection. As the multipath is frequency dependent and has a time lag for different frequencies, it is necessary to build a correction model for each frequency [20]. Therefore, it complicates the implementation of data processing for multi-frequency, especially for GLObalnaya NAvigatsionnaya Sputnikovaya Sistema (GLONASS), which has inter-frequency bias (IFB) among satellites. In addition to the multipath, the errors of satellite orbits and clocks and the atmospheric delays still remain in the phase residuals [21; 22], which could affect the performance of the SF approach to some extent. In spite of numerous studies about the performance of multi-GNSS PPP, only a few have investigated the multipath effects on PPP [16; 22]. In addition, the new BeiDou navigation satellite system (BDS) performs comparably to GPS in the relative positioning aspect but is inferior for PPP [23; 24; 25; 26].