## 2 Materials and Methods
### Literature Review and Significance of Research
#### 2.1.3 Review of Pseudorange Multipath Estimation  
Pseudorange multipath is defined as the multipath errors in the pseudorange measurements (i.e., after signal processing in receiver hardware). The authors of [32] performed the pseudorange multipath estimation using triple-frequency pseudorange measurements and concluded that the triple-frequency approach can give an absolute estimate of multipath with the complete elimination of other errors. The estimation formula is:  
\[\mathrm{M}_{\Psi_{\Psi}}=\lambda_{k}^{2}\big{(}P_{i}-P_{j}\big{)}+\lambda_{j}^ {2}\big{(}P_{k}-P_{i}\big{)}+\lambda_{i}^{2}\big{(}P_{j}-P_{k}\big{)} \tag{1}\]  
where the subscripts \(i,j\), and \(k\) denote the three frequencies of \(L_{i}\), \(L_{j}\), and \(L_{k}\). \(M_{P_{j}|i}\) denotes the _ijk_ triple-frequency combined pseudorange multipath error, \(P\) denotes the pseudorange measurement at a frequency, and \(\lambda\) denotes the wavelength of a frequency. However, multipath errors and measurement noise at different frequencies are different if the signal designs (e.g., chipping rate and modulation) of frequencies are different. According to the error propagation law, the variance of the triple-frequency combined pseudorange multipath error contains the amplified measurement noise, and the noise level may be comparable to the pseudorange multipath errors. Moreover, triple-frequency tracking is still not common at CORS stations, which limits the availability of the estimation. Another weakness is that the pseudorange measurement quality of individual frequencies is not available for low-cost/single-frequency GNSS users. In contrast to the above method, single-frequency and dual-frequency-based approaches are more commonly used. The former approach is called code-minus-carrier (CMC) [33]. _CMC_ is the difference between pseudorange measurement \(P\) and carrier phase measurement (in the unit of meters) \(\Phi\):  
\[CMC=P-\Phi \tag{2}\]  
This method assumes the carrier-phase multipath errors are negligible when compared with the pseudorange multipath errors. Common GNSS biases and errors in pseudorange and carrier-phase measurements, such as tropospheric delay, satellite and receiver clock offsets, and satellite orbit errors, are eliminated by the differencing. However, the ionospheric effect is doubled and absorbed in the computed pseudorange multipath error (i.e., CMC). The authors of [34] used this method to test the pseudorange multipath mitigation performance of the simple-stroke and synthesized-strobe delay-locked loops (DLLs). To remedy the weakness of CMC and the insufficient observed frequencies problem, researchers usually use the linear multipath combination (i.e., _MP_) adopted in the TEQC to analyze the pseudorange multipath errors; examples are [24; 30; 32; 35]. The MP can be computed as follows:  
\[MP_{i}-B_{i}=P_{i}-\Phi_{i}-\frac{2}{\alpha-1}\big{(}\Phi_{i}-\Phi_{j}\big{)}- 2\frac{B_{j}-B_{i}}{\alpha-1} \tag{3}\]  
where the subscripts \(i\) and \(j\) denote the frequencies of \(L_{i}\) and \(L_{j}\), \(MP_{i}\) denotes the pseudorange multipath errors plus the measurement noises in \(L_{i}\), \(P\) denotes the pseudorangemeasurement, and \(B\) denotes the carrier-phase integer ambiguity, \(\alpha=\left(\frac{f_{i}}{f_{j}}\right)^{2}\). If we rear-range Equation (3), we obtain:  
\[MP_{i}-P_{i}+\left(\frac{2}{\alpha-1}+1\right)\Phi_{i}-2\frac{\Phi_{j}}{\alpha- 1}=B_{i}-2\frac{B_{j}-B_{i}}{\alpha-1} \tag{4}\]  
of which, \(\left(B_{i}-2\frac{B_{j}-B_{i}}{\alpha-1}\right)\) is constant for each given orbit arc (when no cycle slip), it can be subtracted by averaging it out in the final solution. Consequently, the pseudorange multipath plus noise can be written as follows:  
\[MP_{i}=P_{i}-\left(\frac{2}{\alpha-1}+1\right)\Phi_{i}+2\frac{\Phi_{j}}{\alpha -1} \tag{5}\]  
Equation (5) is geometry-free and ionospheric-free so it can largely reduce the error that affects the geometry-free CMC approach.