## 1 Introduction  
Precise navigation and positioning with the Global Navigation Satellite System (GNSS) depends on high-precision carrier phase observations, which require accurate estimations of ambiguity. The causes of cycle slip include blocked signals, low signal-to-noise ratios (SNR), and errors in signal processing. They will interrupt the continuous tracking of carrier phase observations, result in the occurrence of cycle slips and even outliers in severe cases. The impact of cycle slips must be considered when obtaining the ambiguity [1]. Cycle slip detection and repair can avoid re-determining the ambiguity and improve the continuity of positioning, which has been studied for many years. Existing methods for fixing cycle slip methods are primarily divided into two types. The first one treats each satellite separately, using combinations of observations to detect cycle slips. For example, some of them use double difference observations [2; 3; 4; 5] to detect cycle slips, which are applied in multi-station situations. And they are not suitable for a single station; The methods based on Global Position System (GPS) data and other sensor observations [5; 6; 7] require supplementary information from the inertial navigation system (INS), which significantly constrains their feasibility in many applications, due to the cost and complexity of adding an INS system to GPS [8]; Methods based on combinations of phase, code and Doppler observations [8; 9; 10; 11; 12] are the main cycle slip fixing methods for single station, which have their own shortcomings: Methods based on combinations of carrier phase observations are supported by multiple frequencies, they are not available for single frequency; Methods based on combining observations with pseudo-range is limited by the noise level. Doppler is the instantaneous shift in the measured frequency and suitable to detect and correct cycle-slips. But the deviation in the receiver's oscillator clock may result in Doppler measurement error [13]. The methods treat each satellite separately, thereby cannot take advantage of the current multi-constellation. The second type of method is geometry-based, which treat the satellite constellation as a whole. Most of these methods utilizes the time-differenced model. [PERSON] and [PERSON] [14] predicted the atmospheric delays and fixed cycle slip in dual-frequency PPP with a time-differenced model, and [PERSON] et al. [15] expanded the method to GPS + Global Navigation Satellite System (GLONASS) observations. [PERSON] [16] utilized the time-differenced model to fix cycle slip for single frequency GPS + GLONASS observations. These methods utilize pseudo-range or Doppler observations to calculate a float solution of time-differenced changes in the ambiguity, searching for integer solutions with the least-squares ambiguity decorrelation adjustment (LAMBDA) to obtain the integer cycle slips. Some methods utilize outlier detection, such as quasi-accurate detection of outliers (QUAD) [17], the statistical hypothesis tests [18; 19], robust estimation with single frequency data [20] and so on, which adopt the residuals after the least squares adjustment to detect cycle slips and a rounding algorithm to obtain the integer cycle slips. Geometry-based methods are the current trend in multi-constellation development and more flexible and suitable for single-frequency receivers, but most of the existing geometry-based methods do not take the non-integer outliers into account. When non-integer outliers exist, the integer search with LAMBDA may fail, and the rounding algorithm have not applied validation process to detect them. In addition, the methods based on LAMBDA are influenced by the precision of the pseudo-range or Doppler observations. Moreover, the methods focused on GPS and GLONASS, while little has combined BeiDou Navigation Satellite System (BDS).