Full waveform inversion (FWI), recognized for its high precision, has garnered widespread application in velocity modeling. However, the computational cost of FWI continues to be a significant hurdle in its practical application. The FWI method employing multi-source simultaneous excitation, based on source-encoding, is a pivotal strategy for enhancing efficiency. However, conventional encoding methods suffer from crosstalk noise and are not adaptable to geometry where the positions of receivers are variable. To augment efficiency and ensure inversion accuracy, this paper introduces a time-frequency hybrid domain FWI algorithm that is founded on the resampled time-domain phase-sensitive detection (TD-PSD) method. This technique leverages the orthogonality of trigonometric integrals over complete periods to effectuate the decoupling of blended wavefields in the frequency domain and enables the computation of cross-correlation gradients without crosstalk. Concurrently, we propose a lossless sampling algorithm that markedly improves the efficiency of traditional TD-PSD integration calculations. Ultimately, the reliability of our proposed method is substantiated by numerical simulations on the two-dimensional (2D) overthrust model and three-dimensional (3D) field data, demonstrating that the inversion efficiency of our method is 4-6 times higher than that of traditional TD-PSD approach.

When using neural networks to predict low-frequency data of land seismic exploration, the biggest challenge is the difficulty in obtaining the wavelets of real seismic data. Training data that replicates the waveform structure of real seismic data are not easy to be generated. When a network trained with seismic records generated with other wavelets is directly applied to real data, it often can only approximate the prediction or, in some cases, cannot predict low-frequency data of land seismic data at all. Therefore, we propose a new approach that utilizes cross-convolutions to approximately unify the wavelets of real and simulated training seismic data. This allows the trained network to be used for predicting low-frequency data. Furthermore, the wavelets of the predicted low-frequency data are also known, which is more advantageous for subsequent full-waveform inversion. We employ two different wavelets to simulate real-world application scenarios. A simple layered model is utilized to validate the feasibility of the proposed method, and its generalization is tested on more complex Marmousi and overthrust models. For field data, better results can also be obtained using our methods.

Deconvolution technology, as an effective means to enhance the resolution of seismic data, has emerged as a prominent research area in the field of seismic exploration. However, due to its inherent ill-posed nature, seismic deconvolution poses significant challenges. The conventional approach for deconvolution employs a sparse inversion strategy to reconstruct underground reflection coefficients but overlooks the spatial relationship between adjacent seismic traces, resulting in inadequate spatial continuity of the deconvolved results. In this letter, we propose an adaptive seismic deconvolution strategy that incorporates spatial continuity regularization based on local seismic similarity. The adaptive multitrace deconvolution process consists of three parts: first, we consider spatial continuity by introducing a spatial regularization term derived from local seismic similarity; second, we formulate an objective function by combining$L1$-2 norm for sparse regularization with terms accounting for misfit and spatial regularization to reconstruct reflectivity; finally, we solve the objective function using alternative direction method of multipliers (ADMMs) and L-BFGS algorithms. Our proposed method effectively preserves weak effective signals while providing a clearer depiction of geological body distribution and ensuring superior spatial continuity in complex geological structures. Synthetic and field data tests demonstrate that our proposed method yields high-resolution deconvolved results with strong spatial continuity.

The complex-frequency-shifted convolutional perfectly matched layer (CFS-CPML) has been widely used in numerical simulations of the first-order seismic wave equations to avoid artifical reflections caused by truncated boundaries. However, numerically solving the second-order seismic wave equation is preferred for some large-scale geophysical algorithms, such as reverse-time migration (RTM). Extension of CFS-CPML to the second-order seismic wave equations has been realized by using a strict derivation approach in many literatures. However, the derived CFS-CPML formulations are complex, due to introduction of a large number of intermediate variables, which decreases overall computational efficiency, visibly. We present an efficient strategy to implement CFS-CPML in finite-difference time-domain (FDTD) modeling of the second-order seismic wave equations. We do not derive any continuous CFS-CPML formulations. Instead, we split the high-order centered-grid finite-difference (CGFD) stencil applied to approximate the second-order derivatives into two-level CGFD stencils. The inner stencils are viewed as the second-order CGFD approximations of the first-order derivatives at symmetric grid points. With this viewpoint, the convolution term related to CFS-CPML can be introduced to each inner CGFD approximation, naturally. The outer CGFD stencil is also viewed as CGFD approximation of a first-order derivative and it is augmented by a convolution term as well. By this way, CFS-CPML is incorporated into the FDTD simulations, efficiently. We take three-dimensional (3D) acoustic, viscoacoustic and elastic wavefields modeling for examples to verify the feasibility of the new implementation strategy of CFS-CPML.

The irregular surface topography has significant effects on seismic wave propagation, including introduction of free surface related multiples, strong scattering, and distortion of the shapes of seismic events. Because of the high efficiency, the finite-difference time-domain (FDTD) method is the most widely used numerical approach to solve three-dimensional (3D) seismic wave equations. However, FDTD at a uniform rectangular mesh could suffer from strong spurious diffractions, due to the stairgrid approximation to irregular surfaces. To resolve this problem, we immerse the irregular surfaces into fractional points under a uniform rectangular mesh and impose the free surface boundary condition along the normal direction of the surfaces. The implementation process of the free surface boundary condition in presence of irregular surfaces can be viewed a generalization of the traditional mirror image method designed for a planar free surface. The developed finite-difference (FD) scheme with the immersed boundary method (IBM), denoted as IBM-FD, needs to update the wavefields at a small number of ghost nodes, implicitly. We adopt Seidel iterations plus a third-degree Lagrange polynomial interpolation to realize this purpose. Compared with the FD scheme without IBM, the IBM-FD scheme only increases the computational cost slightly, due to the cost to determine the ghost nodes and update the wavefields at the ghost nodes. We present 3D scalar wavefield simulation examples with irregular surface topography to confirm the accuracy and stability of IBM-FD.

Bayesian inversion is capable of integrating seismic data, well-log data, and geological data to obtain a posterior probability distribution function (PPDF) of elastic parameters. Due to the absorption of strata, amplitude attenuation and phase distortion inevitably occur during the propagation of seismic wave, resulting in low resolution of seismic data. The traditional linear Bayesian inversion method is based on the stationary convolution model, and amplitude compensation and phase correction are needed to be conducted in advance. Uncertainties exist in both compensation and inversion, and the inversion results cannot account for the uncertainties in the directly observed seismic data resulting in accurate inversion results. To estimate uncertainty more accurately and improve inversion accuracy, in this article, we develop a nonstationary prestack linear Bayesian stochastic inversion (NSPLBSI) method. Through the proposed method, prior information from well-log data can be effectively introduced to constrain inversion. Compensation and inversion are integrated into one procedure, which can estimate the posterior uncertainties more accurately directly from seismic data compared with traditional two-step inversion methods, i.e., first compensating attenuation and second performing inversion. Also, more accurate inversion results are obtained. In order to verify the rationality and effectiveness of our proposed method, we conduct inversions using synthetic seismic data on the Marmousi model and a field seismic dataset. Numerical examples show that it can not only compensate amplitude well but also obtain high-precision inversion results with smaller prediction interval.

Full-waveform inversion (FWI), as a high-precision model building method, plays an invaluable role in seismic exploration. The accuracy of conventional FWI is universally reduced by the cycle skipping, which can be improved by the optimal transport distance (OTD) objective function. However, the OTD-based FWI cannot converge to meaningful results with an inaccurately estimated source wavelet. To solve this dilemma, we construct a novel convolutional Wasserstein distance (CW) objective function by applying the OTD objective function to convolved seismograms. Before the standard non-negative and normalization preprocessing of OTD, we first convolve the observed data with a reference trace selected from simulated seismograms and convolve the simulated data with a trace selected from the observed data. The both convolved data sets are naturally regarded as with an identical source, so the data difference caused by the inaccurately estimated source wavelet is eliminated. The adjoint source corresponding to the new objective function is derived. The velocity model can be updated by using a quasi-Newton method according to the gradient of the objective function generated by the adjoint-state method. We investigate the effectiveness of our objective function by one-dimensional signals and several FWI examples. Furthermore, this new objective function still delivers an excellent performance in releasing the local minimum problem and resisting noise when the wavelet used in FWI is inaccurate.

Absorption in subsurface media severely degrades seismic data quality. Seismic attenuation compensation as an important processing method can effectively improve the resolution and fidelity of seismic data. Based on sparse reflectivity model and attenuated convolution function, inversion-based compensation approaches show better stability and accuracy over traditional direct compensation schemes. However, conventional inversion-based compensation methods are conducted on single trace, which ignore the subsurface spatial continuity and make the compensated result contaminated with high-frequency noise. In this paper, we develop a structurally constrained multichannel L1-2 minimization for seismic attenuation compensation. We first estimate structure tensors from migrated seismic images. The structure tensors can be decomposed by eigenvalues and eigenvectors, which can reflect the structural orientations. Then, we introduce the estimated orientations as a regularization term to the L1-2 inversion-based compensation objective function. In this way, we can improve the stability of the compensation result and enhance the spatial continuity of the compensated seismic reflectors. The structure-guided L1-2 regularized compensation objective function can be efficiently solved via difference of convex algorithm and alternating direction method of multipliers. Synthetic and field data examples demonstrate that the proposed method possesses superior performance over conventional L1-2 regularized inversion-based compensation.

Seismic deconvolution is a typical ill-posed inverse problem. The regularization technique in terms of different prior information is used for a unique and stable solution. Due to the difference between prior information and the actual subsurface situation, it is hard to obtain a solution with satisfactory accuracy and resolution. This letter presents the dictionary learning as an efficient adaptive deconvolution method for the reflectivity reconstruction problem. Considering the curse of dimensionality of conventional dictionary learning and the suboptimal solution of the patch-based dictionary learning, we take the convolutional sparse coding (CSC) model as the dictionary learning method. In this method, the prior information can be obtained from the well-log data in the form of sparse CSC dictionary of reflectivity. On the assumption that the deposition of the subsurface layers is stable, the CSC dictionary extracted from the well-log data can also be applied in the whole work area. The CSC-based deconvolution can be seen as the adaptive deconvolution due to the independence of the assumption made about the reflectivity and seismic data. The process of the adaptive CSC-based deconvolution is divided into three parts. First, the CSC dictionary is learned from the well-log data. Then, the objective function is formulated by combining the CSC dictionary and the single-channel seismic record misfit term for the reconstruction of reflectivity. Finally, the objective function is efficiently solved with the coordinate descent approach. We illustrate the performance of our adaptive deconvolution with synthetic and field seismic data.