## 1 Introduction  
Precise point positioning (PPP) [1; 2] eliminates or weakens many errors in pseudo-range and carrier-phase to positioning solutions. However, the instrumental biases in pseudorange and carrier phase are not eliminated, and convergence time of PPP is too long-about 30 min [2]. [PERSON] [3] suggested that users can handle this with the differential code bias (DCB) in PPP to further improve errors and convergence time. In addition, Global Positioning System (GPS) satellites began to provide triple frequency signals [4; 5; 6]. The Chinese BeiDou Navigation Satellite System (BDS) is the world's first Global Navigation SatelliteSystem (GNSS) to provide three-frequency signal services. BDS had already launched a regional navigation service at the end of 2012 and continued to develop a global positioning service for 2018 [7; 8]. Triple-frequency, which can be used to decrease the convergence time of PPP, became a research focus. [PERSON] [9] estimated P1-C5 satellites differential code biases based on different criteria and assessed the performance of triple-PPP by processing the modernized L5 signal along with the legacy GPS signals. With the introduction of a third frequency signal, inter-frequency clock biases (IFBs) from satellite clocks and receiver clocks among multi-frequency signals have been found. [PERSON] et al. [10] summarized a new receiver bias named receiver IFB from P3. For the carrier phase observations, the bias can be absorbed into the ambiguity making it as a float form, resulting in a much longer convergence time and degrading the positional accuracy of PPP. [PERSON] [11] investigated a method of estimating the satellite IFB by dividing the satellite IFB into a constant and a variable part. Subsequently, the method was widely used to analyze the characteristics of the satellite IFB [12; 13] and its influence for positioning [14; 15; 16; 17]. Furthermore, [PERSON] and [PERSON] [18] proposed an alternative approach where a second satellite clock parameter dedicated to the L5 signals is estimated along with the legacy L1/L2 clock in the undifferenced uncombined GNSS model. It still lacks a uniform formula of IFB from both the receiver and satellites. In addition, the model with IFB derived from the receiver can not be used in the future triple-frequency ambiguity-fixed PPP. To overcome the problem of the long convergence time of PPP, multi-system combination (which adds observations) is also an available method. Both receiver clocks and satellite clocks in any two systems are unequal, they lead to inter-system clock bias (ISB). [PERSON] [19] reviewed various types of biases in GNSS data, including satellite and receiver hardware biases, differential code biases, differential phase biases, initial fractional phase biases, inter-system receiver time biases, and system time scale offset. [PERSON] and [PERSON] [20] analyzed the improvements of the float ambiguities with the biases correction by observing PPP convergence characteristics with or without bias correction in dual-frequency and triple-frequency scenarios during the first few minutes. The models lack conjoint analysis of ISB and IFB. To solve the above bias problems, we revised the formula of the receiver IFB on a third frequency carrier phase observation to increase the fixed success rate of the third frequency ambiguity. Such a revised IFB decreases convergence time and increases the positional accuracy of PPP, compared with the model of [PERSON] et al. [10]. Furthermore, we merged the formulae of IFB from both receiver and satellite and shed light on the mathematical meaning of ISB. Finally, we presented the observation equations of uncombined observation model with the IFB and ISB and analyzed the daily positioning performance in GPS/BDS PPP with the use of real triple-frequency data. The remainder of this paper is organized as follows.