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Acoustic impedance (AI) is an important parameter for seismic reservoir characterization. Traditional algorithms can obtain AI whereas the resolution is open to improvement. Single-channel supervised algorithms can characterize seismic data accurately to achieve high-resolution AI at the cost of a large volume of labels. In the case of limited labels, the single-channel closed-loop (SCCL) algorithm based on linear convolution forward modeling can provide some help while the horizontal continuity can be enhanced. To further improve the horizontal continuity, we propose a multichannel closed-loop (MCCL) AI estimation algorithm for seismic data with limited single-channel labels. By masking multichannel output with ones at the well locations and zeros at nonwell locations, we properly use available single-channel labels for network training. A patching strategy is also used to enlarge labeled data. Besides, the used linear convolution operator cannot fully characterize complicated features in post-stack profiles, so we design a nonlinear correction procedure to capture refined features. Nonstationary examples of the Marmousi-II model with five single-channel labels quantitatively demonstrate the effectiveness of the proposed MCCL algorithm with nonlinear correction. It achieves superiority in terms of the recovered signal-to-noise ratio (S/N) when compared to SCCL with/without nonlinear correction and MCCL. The obtained AI profile of field data using the MCCL algorithm with nonlinear correction is of high resolution and horizontal continuity visually. Quantitative assessments at the well locations underscore the proposed algorithm’s ability to improve AI estimation performance.

The enhancement of seismic migration and inversion processes critically depends on the precise interpolation of seismic data. In recent years, the rapid advancements in deep learning (DL) have led to the widespread adoption of convolutional neural networks (CNNs) in seismic interpolation applications. Nonetheless, the inherent limitations of traditional CNNs, due to the fixed structure of their convolution kernels, impede the extraction of high-level features of seismic data. The accuracy of CNN-based seismic data characterization and interpolation is open to improvement. Therefore, we introduce deformable convolution kernels and design a novel deformable convolution residual-U-Net (DRU-Net) for a more nuanced characterization and interpolation of seismic data. The proposed DRU-Net allows a deformable convolution kernel with adaptive shape adjustments for the receptive field by using learnable offsets to effectively extract advanced features from the training data. In conjunction with a residual learning approach, the DRU-Net significantly refines the network’s learning process and boosts interpolation precision. Through numerical experiments on synthetic and field data with irregular traces missing, the proposed DRU-Net achieves superior precision in seismic data interpolation compared with traditional U-Net and U-Net with the squeeze and excitation block (SE-block).

Antialiasing seismic data interpolation algorithms can reconstruct sparse seismic data into dense data, which helps obtaining high-precision migration images and accurately locate reservoirs. Nonlinear seismic interpolation techniques based on deep learning (DL) have become popular in recent years. The majority of supervised techniques, however, depend on large volumes of labeled datasets, which are rarely available for field data interpolation. To overcome the limitations of the labeled data requirement in supervised learning, some unsupervised interpolation techniques, such as the well-known deep image prior (DIP), have been developed. However, these unsupervised techniques often have poor generalization. In order to avoid the complete labeled data requirement and achieve an accurate interpolation result efficiently, we propose a novel self-supervised interpolation via frequency extrapolation (SIFE) algorithm for regularly missing seismic data. The proposed SIFE mainly contains two steps: aliasing-free low-frequency complete data reconstruction via the Nyquist sampling theorem and high-frequency data recovery via self-supervised frequency extrapolation. In the first step, a low-frequency filter is adopted to obtain aliasing-free sparse data, which can be interpolated into dense low-frequency data via the Nyquist sampling theorem. In the second step, the low-frequency filtered observation data is mapped to its original full-band observed data via self-supervised learning for frequency extrapolation. After the training convergence, we can obtain an optimized network that can map the reconstructed low-frequency data at the missing locations to the corresponding full-band seismic data via frequency extrapolation, i.e., reconstructing the missing seismic traces. Numerical examples with synthetic and field data show the superiority of the proposed SIFE when compared with DIP.

Full waveform inversion (FWI) has the potential to recover high-resolution physical parameters of the Earth from seismic data. Considering the attenuation effects of the subsurface, we develop a new visco-acoustic FWI method in the time domain with a known Q model. The seismic wavefield is modelled by solving a decoupled fractional Laplacian visco-acoustic wave equation, which can better describe amplitude attenuation and phase distortion during wave propagation. To weaken the initial model dependence of visco-acoustic FWI, we introduce the optimal transport-based misfit functional which can measure the data residual globally. Numerical examples on the 2D Marmousi model demonstrate the superiority of the proposed method compared with the L2norm-based method which measures the misfit of observed and synthetic seismic data locally, generating cycle-skipping issue. The inversion results of seismic data without low-frequency components further demonstrate the validity and effectiveness of the proposed method, which is insensitive towards low-frequency components, and may achieve reasonable inversion results for field data application.

Because of the environment limitations, irregularity appears in the observed seismic data. In addition, the observed seismic data contains random noise from the acquisition equipment and surrounding environment, which affects the performances of multi-channel techniques, such as surface related multiple elimination (SRME) and amplitude variation with offset (AVO) analysis. The projection onto convex sets (POCS) method, known as an efficient interpolation method, is suitable for high signal-to-noise ratio (SNR) situations; however, the existing random noise may affect its final performance. In our previously published paper, the POCS formula was deduced in the view of iterative hard thresholding (IHT) method using a projection operator. In this paper, more physical illustrations about its detailed deduction are provided to show the differences between IHT and POCS in noise-free and noisy situations with easy understanding for readers. Then, performances of the POCS and IHT methods are compared in both noise-free and noisy situations, in terms of seismograms, frequency wavenumber (FK) spectra and single traces. For noise-free data, both the POCS and IHT methods can achieve good interpolation results. For noisy data, the POCS method is unsuitable because of the observed noisy data insertion, while the IHT performance is satisfactory because it uses a thresholding operator to eliminate random noise. Numerical examples on noise-free datasets demonstrate the validities of the POCS and IHT methods for interpolation. Tests on noisy data contaminated with additive white Gaussian noise prove the ability and superiority of the IHT method with anti-noise property compared with the POCS method.

Seismic interpolation, as an efficient strategy of providing reliable wavefields, belongs to large-scale computing problems. The rapid increase of data volume in high dimensional interpolation requires highly efficient methods to relieve computational burden. Most methods adopt the L1norm as a sparsity constraint of solutions in some transformed domain; however, the L1norm is non-differentiable and gradient-type methods cannot be applied directly. On the other hand, methods for unconstrained L1norm optimisation always depend on the regularisation parameter which needs to be chosen carefully. In this paper, a fast gradient projection method for the smooth L1problem is proposed based on the tight frame property of the curvelet transform that can overcome these shortcomings. Some smooth L1norm functions are discussed and their properties are analysed, then the Huber function is chosen to replace the L1norm. The novelty of the proposed method is that the tight frame property of the curvelet transform is utilised to improve the computational efficiency. Numerical experiments on synthetic and real data demonstrate the validity of the proposed method which can be used in large-scale computing.