Empirical mode decomposition (EMD) lacks a strong theoretical support although extensively applied. We propose a theoretical framework for a succinct EMD in this work, with the assumption of invariant extrema locations for one IMF extraction. We define the envelope mean filter (EMF) and prove that the filter matrix satisfies five properties. The sifting matrix is convergent to an idempotent matrix. An IMF is the projection of the input signal on the generalized eigenspace of the EMF matrix. An IMF is orthogonal to the residual signal, but different IMFs have no orthogonality. With numerical experiments on different signals, our framework achieves similar results to the classic EMD.

We propose two strategies for eliminating the speckle noise in a laser Doppler vibrometer according to Fourier analysis. Fourier transform is theoretically conducted on the speckle pattern phases, whose variation dominantly contributes to the speckle noise. The calculated and experimental frequency spectra of speckle noise both present oscillations of the frequency series (frequency peaks have constant intervals). (1) A low-pass filter can remove the noise if the vibration frequency is far lower than the first frequency peak of the noise. (2) The vibration energy can be revealed by removing the oscillating frequency trend. Physical experiments demonstrate the effectiveness of both despeckling strategies.

Laser Doppler Vibrometer (LDV) is extensively applied in remote and precise vibration measurements for structural monitoring. Speckle noise is a severe signal issue restricting LDV applications, mainly when an LDV scans from moving platforms. Realistic simulations and thorough characterizations of speckle noise can support the despeckle procedure. A novel approach to numerically simulate speckle noise is proposed based on the statistical properties of speckle patterns. Surface roughness and other affecting factors are thoroughly studied. The simulated distributions agree well with the literature when investigating speckle properties. Single-point and continuously scanning speckle noise are both numerically generated and experimentally acquired. Their corresponding time-series and fast Fourier spectra present good agreement. In addition, similar amplitude distributions, approximating a Gaussian distribution, are achieved. Speckle noise is different from Gaussian white noise because of the varying frequency distribution. The speckle noise grows with increasing surface roughness to a critical value. When simulating and acquiring the scanning speckle noise, the noise energy increases with the scanning speed, but the signal drop-outs decrease in intensity and density. These promising results demonstrate the simulation accuracy and can further support despeckle procedures.

With increasing requirements for structural stability and durability, effective monitoring strategies for existing and potential damage are necessary. A laser Doppler vibrometer on moving platforms (LDVom) can remotely capture large-scale structural vibrations, but speckle noise, a significant signal issue mainly when one-way continuously scanning from moving platforms, restricts its applications. A novel approach based on ensemble empirical mode decomposition (EEMD) is proposed to eliminate speckle noise. Moving root-mean-square thresholds are used to cut off signal drop-outs. With both numerically simulated and experimentally acquired signals, the proposed EEMD-based approach reveals the true vibrations despite the low initial signal-to-noise ratio. Other methods fail to eliminate the speckle noise. In physical experiments, the despeckled signal energy is concentrated at defect locations in the Hilbert-Huang spectrum. The identified damage locations agree well with the actual damage locations. Therefore, the developed approach demonstrates advantages and robustness of eliminating speckle noise in LDVom signals for damage inspection.

As the durability and stability of structures in operation are required, effective technologies are necessary to monitor the structural health. Laser Doppler vibrometer (LDV) is a non-contact and non-destructive vibration detector suitable for acquire broadband signals remotely and continuously. The significant signal issue is speckle noise mainly by LDV scanning from moving platforms (LDVom). This paper presents a novel approach based on ensemble empirical mode decomposition for eliminating speckle noise. The instantaneous frequency in intrinsic mode functions (IMFs) corresponds to the instantaneous vibration that is acquired by LDVom, and the signal is continuous in a sole IMF. In numerical simulations, the EEMD-based approach can effectively reveal the true vibration despite intensive noise. The correlation coefficient remains over 0.9 when the initial signal-to-noise ratio decreases to -15 db. In experiments, the 500 Hz vibration is almost revealed by EEMD regardless of the noise intensity and the vibration strength. Therefore, our approach is applicable in eliminating speckle noise.

Accurate and automatic railhead inspection is crucial for the operational safety of railway systems. Deep learning on visual images is effective in the automatic detection of railhead defects, but either intensive data requirements or ignoring defect sizes reduce its applicability. This paper developed a machine learning framework based on wavelet scattering networks (WSNs) and neural networks (NNs) for identifying railhead defects. WSNs are functionally equivalent to deep convo-lutional neural networks while containing no parameters, thus suitable for non-intensive datasets. NNs can restore location and size information. The publicly available rail surface discrete defects (RSDD) datasets were analyzed, including 67 Type-I railhead images acquired from express tracks and 128 Type-II images captured from ordinary/heavy haul tracks. The ultimate validation accuracy reached 99.80% and 99.44%, respectively. WSNs can extract implicit signal features, and the support vector machine classifier can improve the learning accuracy of NNs by over 6%. Three criteria, namely the precision, recall, and F-measure, were calculated for comparison with the literature. At the pixel level, the developed approach achieved three criteria of around 90%, outperforming former methods. At the defect level, the recall rates reached 100%, indicating all labeled defects were identified. The precision rates were around 75%, affected by the insignificant misidentified speckles (smaller than 20 pixels). Nonetheless, the developed learning framework was effective in identifying railhead defects.

The quality of the surrounding rock is crucial to the stability of underground caverns, thereby requiring an effective monitoring technology. Ground-penetrating radar (GPR) can reconstruct the subterranean profile by electromagnetic waves, but two significant issues, called clutter and hyperbola tails, affect the signal quality. We propose an approach to identify fractured rocks using 2D Wavelet transform (WT) and F-K migration. F-K migration can handle the hyperbola using Fourier analysis. WT can mitigate clutter, distinguish signal discontinuity, and provide signals with a good time-frequency resolution for F-K migration. In the simulation, the migration result from horizontal detail coefficients highlight the crack locations and reduce the scattering signals. Noise has been separated by 2D WT. Hyperbola tails are decomposed to vertical and diagonal detail coefficients. Similar promising results have been achieved in the field measurement. Therefore, the proposed approach can process GPR signals for identifying fractured rock areas.

Automatic and efficient ground penetrating radar (GPR) data analysis remains a bottleneck, especially restricting applications in real-time monitoring systems. Deep learning approaches have good practice in automatic object identification, but their intensive data requirement has reduced their applicability. This paper developed a machine learning framework based on wavelet scattering networks to analyze GPR data for subsurface pipeline identification. Wavelet scattering network is functionally equivalent to convolutional neural networks, and its null-parameter property is intended for non-intensive datasets. A double-channel framework is designed with wavelet scattering networks followed by support vector machines to determine the existence of pipelines on vertical and horizontal traces separately. Classification accuracy rates arrive around 98% and 95% for datasets without and with noises, respectively, as well as 97% for considering surface roughness. Pipeline locations and diameters are convenient to determine from the reconstructed profiles of both simulated and practical GPR signals. However, the results of 5 cm pipelines are sensitive to noises. Nonetheless, the developed machine learning approach presents promising applicability in subsurface pipeline identification.

Surrounding rock quality of underground caverns is crucial to structural safety and stability in geological engineering. Classic measures for rock quality investigation are destructive and time consuming, and therefore technology evolution for efficiently evaluating rock quality is significantly required. In this paper, the non-destructive technology ground penetrating radar (GPR) assisted by an ensemble empirical mode decomposition (EEMD)-based signal processing approach is investigated for identifying unstable subsurface rock structures. By decomposing the pre-processed GPR signals into multiple intrinsic mode functions (IMFs) and residues, one typical IMF can preserve the distinct local modes and is considered to reconstruct the subterranean profile. Promising results have been achieved in simple scenarios and filed measurements. The reconstructed profiles can accurately illustrate the subsurface interfaces and eliminate the interference signals. Unstable rock structures have been identified in further field applications. Therefore, the developed approach is efficient in unstable rock structure identification.