## 3 Results
### Investigation in Simple Scenarios
However, unexpected visible noises appear in the deepest 10% portion, which may arise from the end effects of EEMD [29], and GPR time windows should be at least 10% larger than targeted depths in further applications. Figure 4: (**a**) Intrinsic mode functions (IMFs) and residue of Line 1 signals from pipeline detection decomposed by EEMD. (**b**) IMFs and residue of Line 2 signals from pipeline detection decomposed by EEMD. Secondly, a numerically simulated radargram of cracked rock areas was generated by the software 'gprMax' [30]. 'gprMax' is an open source software to simulate GPR signals based on propagation physics of electromagnetic waves and the Finite-Difference Time-Domain method, widely applied in investigating signal processing approaches (e.g., [31; 32]). Numerical simulation can determine the isotropy and homogeneity of the underground medium, thus ensuring object signals are not interfered by other unknown structures. Figure 6a illustrates the simulated rock profile, with the scanning length \(L\) (\(0\leq L\leq 2\) m) and detection depth \(Z\) (\(0\leq Z\leq 1\) m). The entire section is divided into mesh grids with resolution \(0.005\times 0.0025\) m. Certain attributes have been assigned to the rock section, with the relative permittivity of 8, the conductivity of 0.02 S/m, the relative permeability of 1 and the magnetic loss of 0. There are three predefined cracked portions (sizes seen in the caption of Figure 6), each containing multiple randomly distributed cracks with widths of 2.5 mm, lengths and intervals ranging from 1 cm to 5 cm. These cracks are simulated as perfect electric conductors. The GPR antenna is simulated by the Ricker waveform with the central frequency of 800 MHz, while the detection time window is \(4\times 10^{-8}\) s. Figure 6b presents the corresponding pre-processed radargram. The pre-processing parameters are \(\alpha=0.13\), \(\varphi=0.2\) and \(Z_{0}=0.1\) m. The predefined cracked areas generate not only intensive responses at their locations but also interference signals affecting other locations. Over half of the profile is occupied by GPR signals, which increases difficulties in cracked rock identification. Figure 5: (**a**) IMF3s amplitude profile of 2 pre-buried pipelines. (**b**) IMF4s amplitude profile of 2 pre-buried pipelines. Signals with amplitudes less than 500 are removed. Two representative GPR traces on Lines 1 and 2 with cracks at different depths are compared to analyze the mode difference, as shown in Figure 7. The shallow cracked portion on Line 1 generates distinct local modes between 0.1 and 0.4 m depths (marked in the dotted circle). EEMD results of both traces are illustrated in Figure 8. Only mode functions after IMF5s are presented as IMF1s to IMF4s are entirely occupied by added white noises. IMF6s and IMF7s have captured the modal characteristics of both signal traces, as marked in dotted circles. However, the oscillations in IMF7s have extended beyond the actual crack depths. Therefore, IMF6s are chosen to reconstruct the GPR profile. Figure 6: (**a**) Simulated rock sections with three cracked areas (\(0\leq L\leq 0.5\) m and \(0.1\leq Z\leq 0.4\) m, \(0.75\leq L\leq 1.15\) m and \(0.4\leq Z\leq 0.6\) m, \(1.5\leq L\leq 2\) m and \(0.8\leq Z\leq 0.9\) m). (**b**) The corresponding radargram after pre-processing. Figure 7: GPR signals of Line 1 and 2 from numerical simulation with initialized duration. Figure 9 presents the reconstructed profile by IMF6s. Signals with amplitude weaker than 3 are eliminated since noises would affect our identification results. In the reconstructed radargram, responses are concentrated inside the cracked portions (marked as dashed rectangles), and interference signals have been almost eliminated. However, some weak signals surrounding the cracked portions are preserved (inside portions B, C, D and E) since eliminating all interference signals is difficult. Nonetheless, the intact portion F, which was covered by intensive signals in Figure 6b, has been revealed by the proposed approach.