## 1 Introduction  
In order to implement precise point positioning (PPP), satellite clock offsets need to be estimated and disseminated to users [1, 2]. Normally, there are two kinds of methods for estimating satellite clocks, one is based on undifferenced observations and the other is based on epoch-differenced [3, 4]. Obviously, the latter one is more efficient; therefore, it is more popular to use differenced observations to estimate clocks [5, 6, 7]. The difference between the two methods is whether to use an epoch-differenced strategy to eliminate ambiguity parameters in order to reduce the number of parameters to be estimated and improve the efficiency of the calculation. Based on the idea of difference, a mixed-differenced method was developed in study [6], which makes proper use of the combination of epoch-differenced phases and undifferenced codes instead of processing them separately. Different from reducing the parameters to be estimated by difference, an undifferenced estimation method that combines full-parameter and high-rate models to reduce the number of ambiguities to be estimated is proposed [8]. In addition to estimating high-rate satellite clocks, many studies have focused on multi-global navigation satellite system (GNSS) clock estimation [9, 10, 11] and the evaluation of satellite clock offset accuracy and reliability [12, 13, 14, 15, 16].  
It is a time-consuming process for initialization of the filter in estimation, and research shows that it usually takes 1 to 1.5 hours for initialization to generate stable satellite clocks [6, 7]. Even if the initial convergence process is completed, the time-varying parameters still disturb the clocks within a few hours, causing the instability of clocks and the reduction in the positioning accuracy at the user end. At present, many companies and research institutions have begun to build real-time clock estimation platforms in order to meet the needs of real-time PPP applications. However, due to the influence of network anomalies and other factors, the estimation interruption often occurs, which takes a long time to converge again. Unfortunately, we have not found a solution to the above problem in the current literature. In this context, it is of great significance to study how to accelerate the convergence of estimation and improve the stability of clocks.  
In the implementation of PPP, we usually assume that satellite code biases (SCBs) and receiver code biases (RCBs) are constants and assimilated by other parameters to obtain a full rank model [17]. This assumption is also used in clock estimation to ensure consistency. However, more and more studies show that code biases are not stable constants, and their time-varying part leads to the deviation of other parameters [18; 19; 20; 21; 22; 23; 24; 25]. In this study, we proposed a new satellite clock estimation method based on undifferenced observations, which takes into account the time-varying code biases. This paper proceeds as follows. Section 2 introduces the models of satellite clock estimation considering code biases and the strategies of eliminating rank deficiency and data processing. Section 3 exhibits the results of clock estimation and PPP to investigate the validity of the proposed method. Section 4 discusses the necessity of the proposed method. Finally, it comes to the conclusions and outlook of this study in Section 5.