# Towards Explainable Artificial Intelligence for GNSS Multipath LSTM Training Models
## 2 Background  
### GNSS Multipath Effect  
Multipath effects occur when GNSS signals reach a receiver through multiple paths due to reflections from surrounding objects. This phenomenon is particularly pronounced in urban environments where buildings and other structures create complex reflection patterns. The interaction between direct and reflected signals can significantly degrade positioning accuracy, making multipath detection and mitigation crucial for reliable GNSS applications. The impact of multipath can be quantified through two fundamental GNSS measurements: pseudorange (code phase) and carrier phase observations. The pseudorange measurement \(P_{i}\) for carrier frequency \(f_{Li}\) (\(i=1,2\)) includes various error components:  
\[ \begin{array}{l}P_{1}=r+I_{1}+T+M_{1}+c(dt_{r}-dt^{s})\\ P_{2}=r+I_{2}+T+M_{2}+c(dt_{r}-dt^{s})\end{array} \]  
where  
* \(P_{i}\) is code-phase observable for carrier frequency \(f_{Li}\) (\(i=1,2\));
* \(r\) is the geometric distance;
* \(I_{1}\), \(I_{2}\) are ionospheric delay errors;
* \(T\) is the tropospheric delay error;
* \(M_{1}\), \(M_{2}\) are pseudorange multipath errors;
* \(c\) is the speed of light;
* \(dt_{r}\) is the receiver clock error;
* \(dt^{s}\) is the satellite clock error. Carrier phase measurements provide more precise observations but include an unknown integer ambiguity term:  
\[ \begin{array}{l}L_{1}=r-I_{1}+T+m_{1}+n_{1}\lambda_{1}+c(dt_{r}-dt^{s})\\ L_{2}=r-I_{2}+T+m_{2}+n_{2}\lambda_{2}+c(dt_{r}-dt^{s})\end{array} \]  
where:  
* \(m_{1}\), \(m_{2}\) are carrier phase multipath errors;
* \(n_{1}\), \(n_{2}\) are the integer ambiguity terms;
* \(\lambda_{1}\), \(\lambda_{2}\) are the carrier wavelengths. The multipath effect in GNSS occurs when satellite signals reach the receiver through multiple paths due to reflection and diffraction from surrounding objects. To estimate multipath errors, we utilize both pseudorange and carrier phase measurements from dual-frequency observations. Following the methodology proposed by [PERSON] and [PERSON] [39], the multipath equations can be derived through linear combinations of these measurements. For dual-frequency observations, assuming the atmospheric path delay is almost identical for both frequencies, the carrier phase linear combinations can be expressed as:  
\[ \begin{array}{l}I_{1}+\frac{1}{\alpha-1}(n_{1}\lambda_{1}-n_{2}\lambda_{2}+ m_{1}-m_{2})=\frac{1}{\alpha-1}(L_{1}-L_{2})\end{array} \]  
or  
\[ \begin{array}{l}I_{2}+\frac{\alpha}{\alpha-1}(n_{1}\lambda_{1}-n_{2}\lambda_ {2}+m_{1}-m_{2})=\frac{\alpha}{\alpha-1}(L_{1}-L_{2})\end{array} \]where:  
* \(L_{1}\), \(L_{2}\) are the carrier phase measurements;
* \(I_{1}\), \(I_{2}\) represents the ionospheric delay errors;
* \(n_{1}\), \(n_{2}\) are the integer ambiguities;
* \(m_{1}\), \(m_{2}\) are the phase multipath errors;
* \(\alpha\) is the ratio related to dual frequencies. The ionospheric delay error ratio \(\alpha\) between dual frequencies can be expressed as:  
\[\alpha=\frac{f_{L1}^{2}}{f_{L2}^{2}}\]  
To isolate multipath effects, we can form linear combinations of these measurements. The resulting multipath combination \(mp_{i}\) can be expressed as:  
\[ \begin{array}{l}mp_{1}=P_{1}-\left(1+\frac{2}{\alpha-1}\right)L_{1}+\left( \frac{2}{\alpha-1}\right)L_{2}=M_{1}+B_{1}-\left(1+\frac{2}{\alpha-1}\right)m _{1}+\left(\frac{2}{\alpha-1}\right)m_{2}\\ mp_{2}=P_{2}-\frac{2\alpha}{\alpha-1}L_{1}+\left(\frac{2\alpha}{\alpha-1}-1 \right)L_{2}=M_{2}+B_{2}-\left(\frac{2\alpha}{\alpha-1}\right)m_{1}+\left( \frac{2\alpha}{\alpha-1}-1\right)m_{2}\end{array} \]  
where:  
* \(mp_{1}\), \(mp_{2}\) are the multipath combinations;
* \(P_{1}\), \(P_{2}\) are the pseudorange measurements;
* \(M_{1}\), \(M_{2}\) represent the multipath errors;
* \(B_{1}\), \(B_{2}\) are the bias terms related to integer ambiguities. The bias terms can be expressed as:  
\[