Generalized stiffness reduction method to remove the artificial edge-effects for seismic wave modelling in elastic anisotropic media

The uniaxial perfectly matched layer (PML) method uses a rectangular domain to define the PML problem and thus provides greater flexibility and efficiency in dealing with problems involving anisotropic scatterers. In this paper we first derive the uniaxial PML method for solving the time-domain scattering problem based on the Laplace transform and complex coordinate stretching in the frequency domain. We prove the long-time stability of the initial-boundary value problem of the uniaxial PML system for piecewise constant medium properties and show the exponential convergence of the time-domain uniaxial PML method. Our analysis shows that for fixed PML absorbing medium properties, any error of the time-domain PML method can be achieved by enlarging the thickness of the PML layer as $\ln T$ for large $T>0$. Numerical experiments are included to illustrate the efficiency of the PML method.

In this paper, a perfectly matched layer (PML) method is proposed to solve the time-domain electromagnetic scattering problems in three dimensions effectively. The PML problem is defined in a spherical layer and derived by using the Laplace transform and real coordinate stretching in the frequency domain. The well-posedness and the stability estimate of the PML problem are first proved based on the Laplace transform and the energy method. The exponential convergence of the PML method is then established in terms of the thickness of the layer and the PML absorbing parameter. As far as we know, this is the first convergence result for the time-domain PML method for the three-dimensional Maxwell equations. Our proof is mainly based on the stability estimates of solutions of the truncated PML problem and the exponential decay estimates of the stretched dyadic Green's function for the Maxwell equations in the free space.

The perfectly matched layer (PML) technique is a popular truncation method to model wave propagation in unbounded elastic media. Both numerical efficiency and high stability are important improvement areas in the field. In this study, we extend the optimal PML, previously proposed for acoustic media, to elastic and poroelastic media, which turns out to be more efficient and flexible than the classical PML. We investigate the accuracy and stability of the optimal PML by comparing it with the classical PML in several scenarios. First, the effectiveness of the optimal PML is studied using frequency-domain and time-domain simulations for isotropic and homogeneous elastic solids. The efficiency and accuracy of the optimal PML and classical PML are then compared across a wide range of Poisson's ratios of elastic media. The stability of the optimal PML and the classical PML is also compared taking into account the effect of the outer boundary conditions of the PML as well as the heterogeneity of the geological model. Moreover, the optimal PML is applied to poroelastic media to address the instability problem of the classical PML. Comprehensive analyses of numerical results show that the optimal PML can absorb the outgoing waves. This can be done using thinner layers and with higher accuracy than for classical PMLs. In addition, the long-time stability of optimal PML increases.

Based on the common equation of seismic wave fields and its Hamiltonian formulation, an implicit symplectic method (ISM) is proposed for seismic wave modelingin this paper. Combined with helix boundary conditions, the spectral factorization method is applied to give a fast way for the ISM modeling. The propagation of seismic wave can be described by a same scalar acoustic euqation for homogeneous, complex isotropic and anisotropic media. With the assumption of no dissipation, the scalar equation depicts the symplectic evolution of seismic wave fields which is an infinite dimensional Hamiltonian system. Therefore, the scalar equation can be rewritten into Hamiltonian formulation and consequently its symplectic schemes can be given. Compared with the explicit symplectic method (ESM), ISM is unconditionally stable and permits big time steps and long‐time calculations which may be impossible for ESM. But the inverse of large matrix must be calculated at each step of ISM, which is a time consuming process. Fortunately, the matrix is band‐shape with a helical boundary conditions. In addition, the nonzero units in each row of the matrix not only are located in the same relative positions but also have similar values. Therefore, spectral factorization method can be used to get the inversion of the matrix in a very quick way. Based on the spectral factorization method, an ISM scheme with second order in time is applied to model the wave fields in constant velocity, layer velocity and Marmousi velocity models. The results indicates that with the help of spectral factorization ISM is a quite good method in seismic wave modeling.

The subject of the paper is the study of several nonreflecting and reflecting boundary conditions for the evolution Galerkin (EG) methods which are applied for the two‐dimensional wave equation system. Different known tools are used to achieve this aim. Namely, the method of characteristics, the method of extrapolation, the Laplace transformation method, and the perfectly matched layer (PML) method. We show that the absorbing boundary conditions which are based on the use of the Laplace transformation lead to the Engquist‐Majda first and second order absorbing boundary conditions, see [3]. Further, following Bérenger [1] we consider the PML method. We discretize the wave equation system with the leap‐frog scheme inside the PML while the evolution Galerkin schemes are used inside the computational domain. Numerical tests demonstrate that this method produces much less unphysical reflected waves as well as the best results in comparison with other techniques studied in the paper.

An implementation of a second order unsplit perfectly matched layer (PML) absorbing boundary condition is presented. The formulation is given in the context of seismic modelling and simple models are used to showcase the increased performance. The formulation is presented as a correction procedure to the unchanged governing equations. This approach makes the implementation especially suited for existing FDTD modelling codes. Although, a second order PML might perform better in some cases than the classic or CFS-PML the increased computational requirements and the difficulty in optimising a greater number of PML parameters must be taken into account in deciding the most appropriate PML for a given seismic modelling problem.

: We perform a detailed stability analysis based on the unsplit-field uniaxial perfectly matched layer (PML) formulation. Our finding shows that it is essential to have transverse field gradients present at all times to stabilize PML calculations. In the absence of transverse field gradients, the PML method becomes unstable with the axial field components growing linearly in time.

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving acoustic seismic wave equations in geophysical modeling. However, they often encounter challenges such as spectral bias, gradient instability, and slow convergence when applied to complex media, high-frequency wavefields, and singular source terms. To address these limitations, we propose Adaptive SIREN-PINNs, an enhanced framework that integrates learnable amplitude and frequency modulation parameters into the Sinusoidal Representation Networks (SIRENs), which employs periodic activation functions to naturally represent continuous signals and their derivatives, along with a principled weight initialization strategy. This enhancement improves the model’s frequency-aware representation capability, enabling it to better capture both low- and high-frequency components while enhancing gradient propagation stability during training. Furthermore, to mitigate the singularity associated with point sources, we adopt a Gaussian approximation of the Dirac function and reformulate the wave equation in terms of the scattered wavefield, following established regularization practices in seismic wave modeling. Additionally, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) to gain insights into the model’s learning dynamics across different frequency components during training. Numerical experiments conducted on isotropic homogeneous medium, the Marmousi model, and anisotropic TTI/VTI medium demonstrate that the proposed method significantly outperforms conventional PINNs and baseline SIREN-PINNs in terms of wavefield reconstruction accuracy and convergence efficiency. The results validate the effectiveness of the adaptive SIREN mechanism in improving both the expressive power and training stability of PINNs, offering a mesh-free and solver-free alternative to traditional numerical methods.

Theoretical and quantitative evaluation of hybrid PML-ABCs for seismic wave simulation

Chapter 3 - Seismic wave modeling and high-resolution imaging

A rotated staggered grid finite-difference with the absorbing boundary condition of a perfectly matched layer

Applying an advanced temporal and spatial high-order finite-difference stencil to 3D seismic wave modeling

Seismic imaging necessitates precisely separating P and S waves to mitigate undesirable crosstalk between them. Failure to properly handle this crosstalk can lead to distorted or misinterpreted images. In an elastic anisotropic medium, the polarization of a P wave is not necessarily parallel to its propagating direction, making the separation of P and S waves more difficult than in the case of an elastic isotropic medium. A pure acoustic transversely isotropic (TI) wave equation involves a mixed-domain problem. This usually means the space dependencies and wavenumbers are coupled in the wavefield extrapolator. An efficient solution to the wavefield requires a separation between the space dependencies and wavenumbers in the extrapolator. There are two primary approaches: (1) an analytical separation of the mixed-domain term; and (2) a numerical solution to the mixed-domain term, such as low-rank and Poynting vector methods. In this study, the acoustic approximation is derived by setting the S-wave phase velocity to zero in all phase directions. An analytical separation is developed in this study by employing a Taylor series to linearize the anellipticity term, thereby decoupling the space-dependent and wavenumber-dependent variables within the wavefield extrapolator. The kinematics analysis demonstrates that our method has competitive phase velocity accuracy compared with existing methods. Moreover, the corresponding TI acoustic wave equation can be efficiently solved by the hybrid finite-difference/pseudospectral method. In a 2D case, it only requires one fast Fourier transform, two inverse Fourier transforms, and a few additional spatial differentiations at each time step. Remarkably, our wavefield solution remains independent of the wavefield heterogeneity (as observed in the low-rank approach) and is immune to wavefield interference (as encountered in the Poynting vector method). The developed acoustic approximation ensures the absence of any S-wave artifacts with relatively cheap computation. Numerical examples indicate our acoustic tilted transversely isotropic wave equation is reliable for accurate wavefield modeling and reverse time migration.

The paper deals with finite–difference (f‐d) approach to simulation of elastic waves' propagation in anisotropic elastic media with general symmetry. Any implementation of this approach claims resolution of two key problems: construction of an effective f‐d scheme itself; we propose to use the Lebedev's scheme (LS) being a natural generalization of Virieux staggered grid scheme (VS) widely used for isotropy; we prove that LS possesses better dispersion properties in comparison with well known Rotated Staggered Grids Scheme (RSGS). stable domain distension. The Perfectly Matched Layers(PML) useful for isotropic problems can be unstable in the case of anisotropy. Lebedev scheme allows one to use Optimal Grids (OG) which gives a possibility to implement efficient and low cost domain distension. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Seismic forward modeling is a computationally and data-intensive stage in the seismic processing workflow. By profiling the kernels of seismic forward modeling algorithms, it was observed that they need to access a wide variety of memory locations, in addition to the computational cost of performing floating-point operations for the numerical solution of wave equations. In this context, the Roofline model was used to analyze six representative computing kernels in seismic modeling on GPU environment to indicate bottlenecks in the performance and suggest improvements of these wave equation propagators. Based on this, six viscoacoustic equations were implemented using the Devito tool. Experimental data have shown that optimizations in increasing data reuse and decreasing off-chip memory traffic can significantly improve performance.