# Time-Frequency Attribute Analysis of Channel 1 Data of Lunar Penetrating Radar
## 2 Materials and Methods
### CEEMD  
CEEMD comes from EMD and EEMD. The EMD method resolves a given signal into a series of IMFs. These IMFs are components that satisfy certain conditions. The reconstructed original signal is [22]  
\[x=\sum_{i=1}^{n}c_{i}+r_{n}, \tag{1}\]  
where \(c_{i}\) is the \(i\)th IMF and \(r_{n}\) is the residue of data x after n IMFs are extracted. The decomposition flowchart of EMD is shown in Figure 3. The termination conditions of the EMD in the figure are: (1) The number of extreme points is the same as or, at most, one of the zero crossings; (2) the average value of the envelope defined by the local extremum at any point is zero. As mentioned above, the main disadvantage of EMD is the frequent occurrence of mode overlap, which is defined as a single IMF composed of signals of different scales, or a signal of likeness scale in varying IMF components. In order to overcome the frequent appearance of mode overlap, several new noise-assisted data analysis methods are proposed, such as variational mode decomposition (VMD) [25] and ensemble EMD (EEMD) [22]. The EEMD method can be expressed as follows [23]:  
1. Generate \[x^{i}[n]=x[n]+w^{i}[n],\] (2) where \(w^{i}[n]\) (\(i=1,\ldots,I\)) are different implementations of white Gaussian noise,  
Figure 2: Lunar Penetrating Radar (LPR) CH-1 data from N104 to N207 after preprocessing. (**a**) All traces from N104 to N207. (**b**) The traces after deleting the repeated traces. (**c**) The stagnation traces at each recording and navigation point from N104 to N207. Each \(x^{i}[n]\) (\(i\) = 1,..., \(I\)) is fully resolved by EMD to get their modes \(IMF_{k}^{i}[n]\), where \(k\) = 1, \(K\) represents the modes,
3. Set up \(\overline{IMF}_{k}\) to be the \(k\)-th pattern of \(x[n]\), obtained as the corresponding average \(IMF_{k}^{i}\)[23]: \[\overline{IMF}_{k}[n]=\frac{1}{l}\sum\limits_{i=1}^{I}IMF_{k}^{i}[n].\] (3)  
The resulting decomposition of EEMD solves the problem of mode mixing; however, it introduces new ones. In the technique of CEEMD, the decomposition mode will be recorded as \(\widetilde{IMF}_{k}\) and be proposed to count a unique first residue as  
\[r_{1}[n]=x[n]-\widetilde{IMF}_{1}[n]. \tag{4}\]  
\(\widetilde{IMF}_{1}[n]\) is obtained in the same way as EEMD. Then, calculate the first EMD mode over aggregate \(r_{1}[n]\) plus various realizations of a given noise, obtaining \(\widetilde{IMF}_{2}\) through averaging. The next residue can be defined as \(r_{2}[n]=r_{1}[n]-\widetilde{IMF}_{2}[n]\). The rest of the mode continues working until the requirements are met. First, we define an operator \(E_{j}(\cdot)\), which, given a signal, produces the \(j\)-th mode from EMD [23]\(w^{i}\) is the white noise with \(N(0,1)\). If \(x[n]\) is the target datastore, the method can be described by the following algorithm [23]:  
1. Disintegrate by EMD \(I\) realizations \(x[n]+\varepsilon_{0}w^{i}[n]\) to obtain their first modes and calculate \[\widetilde{IMF}_{1}[n]=\frac{1}{l}\sum\limits_{i=1}^{I}IMF_{1}^{i}[n]= \widetilde{IMF}_{1}[n].\] (5)
2. In the first phase (\(k\) = 1), obtain the first residue as in Equation (4). 3. Decompose realizations \(r_{1}[n]+\varepsilon_{1}E_{1}\Big{(}w^{i}[n]\Big{)}\), \(i\) = 1,..., \(I\), until their first EMD mode and define the second mode [23]: \[\widetilde{IMF}_{2}[n]=\frac{1}{l}\sum\limits_{i=1}^{I}E_{1}\Big{(}r_{1}[n]+ \varepsilon_{1}E_{1}\Big{(}w^{i}[n]\Big{)}\Big{)}.\] (6)
4. For \(k\) = 2,..., \(K\), calculate the \(k\)-th residue [23]: \[r_{k}[n]=r_{(k-1)}[n]-\widetilde{IMF}_{k}[n].\] (7)  
Figure 3: The decomposition flowchart of empirical mode decomposition (EMD). 5. Decompose realizations \(r_{k}[n]+\varepsilon_{k}E_{k}\Big{(}w^{i}[n]\Big{)}\), \(i=1,\ldots,I\), until their first EMD mode and define the \((k+1)\)-th mode as [23] Figure 3.0 (8)
6. Go to step 4 to continue to get the next \(k\). Repeat steps 4-6. The direct residual part cannot be further decomposed, and the final residue satisfies [23] Figure 3.1 (9)  
\(K\) is the total number of modes.