Audio-frequency magnetotellurics (AMT) is one of the commonly used methods in geophysical exploration; however, its signal energy is relatively weak and easily submerged by various cultural noises, making denoising a critical step in AMT data processing. Currently, deep learning-based neural networks have achieved superior denoising performance compared to traditional methods in many fields, but in AMT denoising, the neglect of the sparsity characteristic of cultural noise results in degraded denoising performance. To enable the neural network to consider sparsity features during the denoising process and thereby enhance denoising accuracy, we adopt a convolutional neural network (CNN) as the network backbone and design a multilevel wavelet convolutional neural network (MWCNN) from the perspective of sparse representation. This network improves CNN blocks via shortcut connections and enhances feature transmission efficiency by replacing pooling layers and interpolation with wavelet transforms, thereby enabling the network to account for the sparsity of cultural noise, capture underlying noise spectral information and improve denoising performance. Furthermore, we discuss the influence of various network parameters on denoising performance. Finally, we validate the effectiveness of MWCNN in AMT denoising through comparative experiments on both synthetic and field AMT datasets against wavelet transform, bounded influence remote reference processing, data-driven tight frame, CNN and residual networks. Comprehensive evaluations based on signal-to-noise ratio, wavelet time-frequency spectra, denoised results and residuals, apparent resistivity and phase curves, error analysis, one-dimensional inversion results and Nyquist diagrams confirm the superiority of MWCNN for AMT denoising.

Bearing vibration data are often contaminated with noise, which is detrimental to equipment fault diagnosis and predictive maintenance. Denoising bearing vibration data is crucial. Traditional denoising methods have certain limitations. For instance, when employing wavelet denoising, fixed basis functions may fail to perfectly match all signal structures, potentially compromising denoising accuracy. Similarly, when utilizing data-driven tight frame (DDTF) denoising, the learned basis, due to the lack of noise constraints, may incur a risk of overfitting. To optimize denoising performance in both scenarios, this paper proposes a method that combines wavelet transform and DDTF dictionary learning to extract noise based on a doubly sparse dictionary. The specific approach involves mutually cascading the wavelet transform and DDTF. After applying the wavelet transform to the noisy signal, multi-layer wavelet sparse coefficients are obtained. DDTF processing is then applied to each layer of wavelet sparse coefficients. Subsequent inverse transformation achieves noise suppression. This method integrates the structural constraint capability of wavelet decomposition with the learning capability of DDTF, thereby mitigating their respective limitations to some extent. The denoised data are fed into a residual network model, and training results confirm that the proposed method achieves the best classification performance. Experimental results from both data denoising and deep learning classification demonstrate that the proposed method exhibits superior denoising performance. Although the algorithm structure of this method is more complex compared to other approaches, it is meaningful in scenarios where high-precision denoising is required.