## 3 Methodology
### EMD for Separating the Modulation Terms  
Based on the previous section, the accurate separation of the SNR modulation terms is crucial to solving the multiple phases. Currently, the modulation and trend terms in SNR are extracted mainly by the low-order polynomial or the wavelet transform [38; 41; 53]. Although these methods can obtain the modulation terms well, they need some prior knowledge. The low-order polynomial needs to predict the type of signal trend term in advance, and the wavelet transform needs to determine the best wavelet basis and the decomposition level. These requirements lead to the experiment process for signal separation becoming complex and less flexible. Compared with these methods, EMD is a better adaptive analysis method. Since it can decompose the signal directly according to the scale characteristics of the signal itself without prior knowledge, it is very suitable for extracting the characteristics of non-stationary signals [60; 61]. Moreover, using EMD in signal decomposition can significantly enhance the stability and accuracy of GNSS-IR retrieval, as demonstrated for tidal levels and  
Figure 3: Geometric principle illustrating the relationship between multipath signals of SNR and the ground environment. In the figure, \(h\) is the vertical distance from the GNSS antenna center to the horizontal ground surface, which is called vertical reflection distance; \(\theta\) represents the angle between the satellite signal and the slope. vegetation effects [19; 54]. Hence, this paper used EMD to decompose the SNR signal of each satellite. The SNR signal can be expressed as:  
\[f(t)=[a_{1},a_{2},\ldots,a_{t}],(t=1,2,\ldots,l) \tag{3}\]  
where \(a\) represents the GNSS satellite, and \(t\) and \(l\) are the observation epoch and its length, respectively. The basic idea of EMD is to decompose \(f(t)\) into an \(m\) intrinsic modal function (IMF) of different time scales and a residual term. The following two conditions must be met for each IMF. First, the number of extreme value points must be equal to or at most one unit different from the number of crossing zero points in the whole sequence; second, the average values of the envelope formed by the local maximum and the envelope formed by the local minimum must be zero [60]. The decomposition process of \(f(t)\) can be expressed as:  
\[f(t)=\sum_{k=1}^{m}IMF_{k}(t)+T_{m}(t) \tag{4}\]  
where \(T_{m}\) is the residual term of the sequence; \(m\) is the number of IMF components; \(IMF_{k}(t)\) is the intrinsic modal function, which reflects the inherent and intrinsic characteristics of the signal itself. It can be seen that the \(f(t)\) is decomposed into a limited number of IMF that contain the local feature signals of the original signal. The first IMF component is the highest-frequency component of \(f(t)\). As the order of the IMF increases, the frequency of its corresponding components gradually decreases. Ideally, \(T_{m}(t)\) is used as the low-frequency trend term of the original signal. In practice, due to the comprehensive influence of the satellite orbit and the surrounding environment of the site, the SNR of different satellites varies greatly in different periods, and the frequency change of direct signals may exceed the frequency range of a single IMF component. If the IMF component of the last layer is directly fixed as a trend term, it will cause a big error. Considering the residual component is set as the last IMF component and the combination from the boundary IMF component (\(IMF_{T}\)) to the last IMF component is used as the trend term of the original signal, it is possible to obtain the high-quality SNR modulation term required for subsequent LLS analysis. Therefore, reasonably determining the \(IMF_{T}\) of all IMF components is the key to effectively extracting the trend term and the fluctuation term of SNR. The trend term can be expressed as:  
\[\text{SNR}_{\text{d}}(t)=\sum_{k=T}^{m}IMF_{k}(t) \tag{5}\]  
In this paper, an automatic determination criterion for EMD trend terms is proposed.