Solving the Helmholtz equation has important applications in various areas, such as acoustics and electromagnetics. Using an iterative solver together with a proper preconditioner is key for solving large 3D Helmholtz equation. The performance of existing Helmholtz preconditioners usually deteriorates when the minimum spatial sampling density is small (approximately four points per wavelength [PPW]). To improve the efficiency of the Helmholtz preconditioner at a small minimum spatial sampling density, we have adopted a new preconditioner. In our scheme, the preconditioning matrix is constructed based on an adaptive complex frequency that varies with the minimum spatial sampling density in terms of the number of PPWs. Furthermore, the multigrid V-cycle with a GMRES smoother is adopted to effectively solve the corresponding preconditioning linear system. The adaptive complex frequency together with a GMRES smoother can work stably and efficiently at different minimum spatial sampling densities. Numerical results of four typical 3D models show that our scheme is more efficient than the multilevel GMRES method and shifted Laplacian with multigrid full V-cycle and a symmetric Gauss-Seidel smoother for preconditioning the 3D Helmholtz linear system, especially when the minimum spatial sampling density is large (approximately 120 PPW) or small (approximately 4 PPW).

The finite-difference frequency-domain (FDFD) method has important applications in the wave simulation of various wave equations. To promote the accuracy and efficiency for wave simulation with the FDFD method, we have developed a new 27-point FDFD stencil for 3D acoustic wave equations. In the developed stencil, the FDFD coefficients depend not only on the ratios of cell sizes in the x-, y-, and z-directions but also depend on the spatial sampling density (SD) in terms of the number of wavelengths per grid. The corresponding FDFD coefficients can be determined efficiently by making use of the plane-wave expression and the lookup table technique. We also develop a new way for designing an adaptive FDFD stencil by directly adding some correction terms to the conventional second-order FDFD stencil, which is simpler to use and easier to generalize. Corresponding dispersion analysis indicates that, compared to the optimal 27-point stencil derived from the average-derivative method (ADM), the developed adaptive 27-point stencil can reduce the required SD from approximately 4 to 2.2 points per wavelength (PPW) for a cubic mesh and to 2.7 PPW for a general cuboid mesh. Numerical examples of a 3D homogeneous model and SEG/EAGE salt-dome model indicate that the developed stencil is more accurate than the ADM 27-point stencil for cubic and general cuboid meshes, while requiring similar CPU time and computational memory as the ADM 27-point stencil for direct and iterative solvers.

Seismic numerical modelling involved in each stage of seismic exploration process is aimed at predicting seismograms in an assumed subsurface medium. The energies and phases of waves vary with the presence of fluids, while the frequency-dependent seismic reflections in a reservoir with hydrocarbons was explained using the diffusive-viscous wave (DVW) equation. We proposed a Galerkin finite-element method (FEM) to numerically study the propagation properties of the DVW in fluid-filled media to ascertain the influences of saturated fluids on characteristics of the wavefield. We also theoretically analysed the numerical dispersion and stability condition of the FEM algorithm, which indicated that a minimum of six nodes per wavelength is recommended to achieve more accurate results. In numerical simulation, we presented a non-split perfectly matched layer (NPML) boundary condition for the DVW equation to absorb the artificial reflections in finite-element modelling, using a homogeneous model to demonstrate the effectiveness of the NPML boundary condition through comparisons with the results without the PML condition. Moreover, we modelled the DVW propagation in a fluid-saturated (gas, oil and water) medium with sharp edges and curved interfaces using the proposed method and compared the results with those pertaining to acoustic waves. The numerical results indicated significant amplitude damping and phase variation in the DVW when it propagates across the fluid-saturated layers compared with those of acoustic waves. Furthermore, we compared the numerical results in the fluid-saturated model calculated via the FEM with those calculated via FCT-FDM (flux corrected transport-finite-difference method) to demonstrate the validity of the former.

Wavelet and reflectivity inversion (WRI) is an important issue in seismic data processing. To overcome the ill-posedness of WRI inversion with more efficient parameter selection and better lateral continuity of reflectivities, we propose a new WRI algorithm named compact smoothness and relative sparsity (CSRS) algorithm, where a normalized compact constraint and a normalized smooth regularization is proposed for the wavelet inversion, and a relative sparsity constraint is proposed for the reflectivity inversion. The proposed constraints and regularization make the parameters of WRI easy to be selected. The proposed relative sparsity constraint can lead to a reflectivity profile with good lateral continuity as it can be suitable for various seismic data with a fixed sparsity parameter. We also propose an efficient algorithm for solving corresponding WRI optimization problem. The whole WRI problem is divided into reflectivity inversion subproblem and wavelet inversion subproblem by using alternating iterative method, where the initial wavelet is estimated by smoothing the absolute amplitude spectrum of averaged seismic data. The proximal algorithm is applied to solve both reflectivity inversion subproblem and wavelet inversion subproblem. By replacing Toeplitz matrix multiplication with fast Fourier transform (FFT) and using compact wavelet, our algorithm can be efficient for 3D seismic data. The numerical examples on 2D synthetic data, 2D offshore field data and 3D onshore field data demonstrate that, compared to Toeplitz-sparse matrix factorization (TSMF) algorithm, the CSRS algorithm with fixed default parameters can get high-resolution reflectivities with better lateral continuity, and requires much less computational time.

Seismic wave forward simulation is the basis of seismic inversion and imaging. In recent years, deep learning methods represented by physics-informed neural networks (PINN) have been widely applied in the field of scientific computing, and have obtained significant attention as mesh-free methods in seismic inversion. Currently, the training of PINN is usually constructed by fixed points in seismic wavefield simulation, and the large number of training points can result in high computational costs. This paper proposes an adaptive strategy for PINN training points selection based on the velocity distribution and residuals of viscoacoustic wave equations. This strategy initializes the positions and quantities of training points based on the velocity distribution of the model, and dynamically adjusts the training points based on the values of loss function during the calculation process. The proposed approach reduces the number of training points, thereby improving the training efficiency of PINN. Numerical examples including the homogeneous model, the layered model with irregular topography, and the Marmousi2 model show that the proposed sampling strategy effectively reduces the calculation time by 54.37% while ensuring accuracy.

Seismic deconvolution is a useful tool in seismic data processing. Classical non-machine learning deconvolution methods usually apply quite a few constraints to both wavelet inversion and reflectivity inversion. Supervised machine learning deconvolution methods often require appropriate training labels. The existing self-supervised machine learning deconvolution methods need a given wavelet, which is a non-blind process. To overcome these issues, we propose a blind deconvolution method based on self-supervised machine learning. This method first estimates an initial zero-phase wavelet by smoothing the amplitude spectrum of averaged seismic data. Then, the loss function of self-supervised machine learning is taken as the error between the observed seismic data and the reconstructed seismic data that come from the convolution of phase-rotated wavelet and reflectivity generated by the network. We utilize a residual neural network with long skip connections as the reflectivity inversion network and a fully connected convolutional neural network as the wavelet phase inversion network. Numerical experiments on synthetic data and field data show that the proposed method can obtain reflectivity inversion results with higher resolution than the existing self-supervised machine learning method without given wavelet.