For the successful inversion and interpretation of multicomponent seismic data, it is crucially important to map the PS-wave to the time domain of the PP-wave. The accuracy of traditional methods decreases when the data differ due to noise, amplitude, phase, and frequency perturbations. Recently, deep learning (DL) is used for seismic data registration, requiring fewer assumptions and lower computational costs. However, current methods primarily focus on direct matching through neural networks, often neglecting physical constraints. To address the above issue, we propose a seismic data registration method based on a physically constrained unsupervised framework (PCUS). Firstly, to avoid undesirable foldings in the warped PS-wave, we require a monotonic warping function that preserves the waveform of the warped PS-wave. We guarantee the monotonicity of the warping function by parameterizing it with a physical constraint of the relationship between the warping function and the velocity ratioVP/VS. Then, we consider the continuity of seismic events and add a smooth regularizer to guarantee the smoothness of the warping function. Finally, we utilize the unsupervised deep learning framework to address this issue. Experiments on synthetic and field datasets indicate the validity and flexibility of the PCUS method. In the synthetic data, the proposed PCUS method demonstrates robustness against amplitude, phase, frequency, and noise perturbations. The relative root mean square error is at least one-tenth of that produced by the traditional dynamic image warping method and a DL-based registration method. In the field data, the proposed PCUS method aligns the seismic events of the PS-wave and PP-wave under the premise of keeping the waveform of the PS-wave.

Seismic registration is a strongly nonlinear and ill-posed optimization problem in the presence of large misalignments or intense noise. Traditional cross correlation-based methods or dynamic image warping (DIW) methods may fail in such situations. From the perspective of optimization, the constraint on the shift may provide a more stable solution. We propose using a deep neural network (DNN) to constrain and solve for the shift. The continuity of the shift is well preserved across different traces with the DNN constraint. In addition, deep learning (DL) optimization is specifically efficient for solving nonlinear problems. The input of the DNN is optimized to produce a shifting image that matches two seismic images. This method is self-supervised; that is, only seismic images and warped versions are required. In our method, the signal-to-noise ratio (SNR) of the predicted shifts is improved by at least 7 dB compared with the traditional DIW method.

Deep learning has been widely applied to seismic data interpolation. However, most existing methods are based on supervised learning, suffering from limitations such as low generalization ability and the necessity for a labeled training dataset. To address these issues, we propose a novel self-supervised transfer learning framework. The backbone network used is our previously developed projection-onto-convex-sets network (POCS-Net). To our knowledge, this represents the first integration of a data- and model-driven dual approach with a self-supervised learning method. The proposed approach consists of two steps. In the first step, the network undergoes pretraining with synthetic training samples using a supervised learning framework. The parameters obtained from this pretraining are then used to initialize the following transfer training. In the second step, the training dataset is constructed by further downsampling the already corrupt data. The proposed framework is evaluated through numerical experiments on 2-D synthetic and 3-D field prestack data, demonstrating its superiority over existing methods. Compared to supervised learning using synthetic dataset, the signal-to-noise ($S/N$) ratio of 2-D synthetic and 3-D field data improves by about 9 dB.

Full-waveform inversion (FWI) is a powerful method of reconstructing subsurface properties during seismic exploration. However, it is difficult for FWI to accurately describe a subsurface model with sharp surfaces and smooth variations because of the highly nonlinear and ill-posed problems associated with FWI. We first propose a novel FWI with shearlet transform and total generalized variation (TGV) regularization on a subsurface model to alleviate this challenge. Shearlet transform is particularly well adapted to preserve the abundant geometric information of models by representing anisotropic features such as curves and edges; however, it often induces the boundary effect, leading to a resolution reduction. To address shearlet transform drawbacks, we employ TGV to reduce the artifacts by involving various order derivatives to adjust different degrees of smoothness. The proposed regularization scheme is robust to the noise of the observed data during the inversion process. Using the simple synthetic, Society of Exploration Geophysicists (SEG)/European Association of Geoscientists and Engineers (EAGE) overthrust, Marmousi, and modified 2004 BP models, we demonstrate that the proposed method reconstructs subsurface geophysical models with sharp interfaces and smooth background variations more accurately than the conventional methods without any regularization and those with only the total variation (TV) regularization, TGV regularization, and shearlet transform regularization.

Multidimensional prestack seismic data reconstruction can be viewed as a low-rank tensor completion problem. Recently, the nuclear norm has been widely used as a convex surrogate of the tensor rank function for low-rank tensor recovery and has been successfully applied to 5-D seismic data reconstruction. However, solving the nuclear norm-based relaxed convex problem typically leads to a suboptimal solution of the original rank minimization problem, often degrading the reconstruction performance. In this study, to seek solutions to the aforementioned problems, we established a nonconvex logDet function as a smooth approximation for the tensor rank instead of the convex tensor nuclear norm and applied it to solve the 5-D seismic data reconstruction problem. Thereafter, we propose solving the obtained nonconvex relaxation problem using an alternating direction method of multipliers (ADMMs) algorithm. Numerical experiments of our approach on synthetic 5-D seismic data demonstrated remarkable reconstruction performance compared with the performances of higher-order singular value decomposition (HOSVD), nuclear norm, and parallel matrix factorization (PMF) methods in terms of visual examination and numerical test. We further illustrate the performance of the proposed method using a land data survey.

The seismic data regularization problem is vital to seismic data processing. We propose a joint sparsity-promotion method based on the compressive sensing theory named the curvelet-data-driven-tight-frame-based sparsity-promoting (CDSP) method. The CDSP method regularizes the seismic data directly on the nonequispaced grid along the spatial dimension. The joint sparsity is exploited in the curvelet and data-driven tight frame transform simultaneously on the projected regular spatial grid. The projection from the nonequispaced grid to the equispaced grid is achieved by the nonequispaced discretized Fourier transform. Comparing with the curvelet-sparsity-promotion-based (CSP) regularization method, CDSP combines the advantage of the predefined curvelet transform and adaptive-learned sparse transform into one single optimization model. An alternative directional method of multipliers (ADMM) is applied to solve the optimization problem. One synthetic and two field examples show that the CDSP method performs better on the preservation of the continuity of events and produces less artifacts than the CSP method.

We have introduced a new decomposition method for seismic data, termed complex variational mode decomposition (VMD), and we have also designed a new filtering technique for random noise attenuation in seismic data by applying the VMD on constant-frequency slices in the frequency-offset (f -x) domain. The motivation behind this paper is to overcome the potential low performance of empirical mode decomposition (EMD) for energy preservation of the steeply dipping events when used for noise attenuation, and low resolution when used for signal decomposition. The VMD is proposed to decompose a signal into an ensemble of band-limited modes. For seismic data consisting of linear events, the constant-frequency slices of its f -x spectrum are exactly band-limited. The noise attenuation algorithm is summarized as follows. First, the Fourier transform is applied on the time axis of the 2-D seismic data. Next, the VMD is applied on each frequency slice of the f -x spectrum and the decomposed modes are combined to obtain the filtered frequency slice. Finally, an inverse Fourier transform is applied on the frequency axis of the f -x spectrum to obtain the denoised result. The resulting VMD-based noise attenuation method is equivalent to applying a Wiener filter on each decomposed mode, which is achieved during the decomposition progress. We also applied 2-D VMD on 3-D seismic data for denoising. Numerical results show that the proposed VMD-based method achieves a higher denoising quality than both the f -x deconvolution method and the EMD-based denoising method, especially for preserving the steep slopes.