## 3 Research Status of the Multipath Error Data Processing Algorithm
### Multipath Error Mitigation Method Based on the Function Model
#### 3.2.1 Multipath Error Mitigation Method Based on the Coordinate Domain  
The coordinate sequence of the previous repeat period is analyzed using the filtering method to extract the multipath error time series in the coordinate domain, and then the multipath error correction model is constructed. Based on the repetitive characteristic of multipath effects, multipath error correction is performed on the sequence of coordinates for the next repeat period. The GPS/BDS/GLONASS/Galileo repeat period can be observed in Table 3. It is a key point to use various filtering methods to denoise and isolate the multipath correction model with repeatability. Such methods can only be used for data post-processing. [PERSON] et al. [42] proposed a data filtering method called cross-validation Vondrak filter (CVVF) based on the Vondrak filter and cross-validation technology, which is used for signal and noise separation in data sequences and is applied to mitigate GPS multipath effects for deformation monitoring. [PERSON] et al. [43] established a new denoising filtering method based on empirical mode decomposition (EMD) and applied this method to GPS multipath error correction. EMD entails the separation of several basic signals from complex signals, from high frequency to low frequency, and the separated signal is called the intrinsic mode function (IMF). For a signal mixed with random noise, the decomposed high-frequency IMF components are usually the noise of the signal and the low-frequency components are multipath signals. However, EMD presents problems that need to be addressed. [PERSON] and [PERSON] proposed two methods based on EMD improvements, ensemble empirical mode decomposition (EEMD) [44] and complete ensemble empirical mode decomposition (CEEMD) [45], to filter and denoise the signal. EEMD and CEEMD are able to mitigate the modal aliasing problem in EMD. [PERSON] [46] proposed a method for extracting GPS/BDS-3 multipath errors based on complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN)-wavelet transform (WT). The positioning results of this method were better than those of EEMD and CEEMD. [PERSON] et al. [47] used the IMF obtained by EMD as the input and applied the generalized eigenvalue decomposition (GED) blind source separation algorithm to extract the multipath error model of the repeat period. [PERSON] et al. [48] first used EMD to decompose the original signal to obtain three IMFs, and then used the recursive least-squares algorithm (RLS) to filter the mixed IMFs. The obtained new IMF and information IMF were reconstructed to extract the multipath error correction model. Singular spectrum analysis is an effective method for extracting information from short noise time series. [PERSON] et al. [49] proposed a new SSA filtering method based on the significant difference between noise and signal Hurst exponents, and verified its effectiveness using GPS data. [PERSON] et al. [50] proposed a wavelet signal-layer automatic identification method based on cross-validation technology, which can separate the noise whose amplitude level is lower than about half of the signal amplitude, and applied it to the mitigation of GPS multipath effects. The repeatability of multipath effects is the highest for two consecutive periods; however, repeatability decreases as the time between the first period and subsequent periods increases. In order to solve this problem, [PERSON] et al. [51] developed a combined method of principal component analysis and empirical mode decomposition and independent component analysis with reference (PCA-EMD-ICA-R), which analyzes the coordinate time series of the first period to obtain the reference multipath model for the second period and has a significant multipath error mitigation effect in both static and dynamic environments. [PERSON] et al.