Second-order Elastic Wave Research Articles

3D numerical simulation of frequency-domain elastic wave equation with a compact second-order scheme

Finite-difference frequency-domain (FDFD) modeling has gained popularity in recent years. However, its main drawback lies in the need to solve large sparse linear systems, which is computationally expensive, especially for 3D elastic wave simulations. The existing elastic wave optimal schemes improve the simulation accuracy at the cost of using large-sized stencils or high-order finite-difference (FD) operators, and most of them only consider equal grid spacing and solid media. To address these issues, we have proposed a compact second-order scheme for simulating 3D elastic wave propagation. Starting from the second-order FD discretization of the first-order velocity-stress expression, the compact discrete scheme of the second-order elastic wave equation is obtained by using a parsimonious staggered-grid strategy to eliminate the stress variables. In this way, the scheme preserves the ability to handle high-contrast velocity and liquid-solid media, and has a more compact and sparse impedance matrix than existing schemes, which helps save computational costs. We also use an antilumped mass strategy to further improve the modeling accuracy. Dispersion analysis indicates that our compact second-order scheme has better accuracy than the classic second-order scheme, and even better accuracy than the average derivative 27-point scheme at large Poisson's ratio. Compared with the existing 3D elastic wave schemes, the compact second-order scheme has advantages in computational cost for the same grid. Finally, we employ these schemes to implement 3D numerical simulations, validating the effectiveness of our proposed scheme.

A high-order perfectly matched layer scheme for second-order spectral-element time-domain elastic wave modelling

In this study, we develop a new high-order perfectly matched layer (PML) formulation for the 3D spectral-element time-domain (SETD) method to solve the second-order elastic wave equations in terms of displacements. The high-order PML consists of multiple well-known frequency-dependent complex stretching functions, leading to the complicated temporal convolution which requires large computational resources. To avoid the evaluation of convolution, we deal with the high-order stretching functions by introducing the recursive auxiliary first-order differential equations which can be easily solved in time domain. In addition, an equivalent concept of PML is adopted for absorbing waves, preserving the original form of wave equations in physical domain. Finally, we obtain a high-order, non-convolutional and unchanged-form PML formulation, which can be easily incorporated into the SETD codes designed for the physical domain, to make them applicable for the second-order wave equations in the unbounded media. Numerical experiments demonstrate the accuracy and stability of the high-order PML through the comparison with the reference solution. It is found that the high-order PML provides a much better absorption performance than the first-order PML, particularly for the grazing-incidence waves and surface waves in the large-scale domain. We also give a fluid-filled borehole model to illustrate that the high-order PML works well for absorbing the complex guided waves in the presence of fluid-solid interaction.

A new implementation of convolutional PML for second-order elastic wave equation

The perfectly matched layer (PML) boundary condition has been widely used as a very effective absorbing boundary condition for seismic wavefield simulations. Convolutional PML (CPML) achieved by using a complex frequency-shifted stretch function was the latest development to further improve PML’s absorption performance for near-grazing angle incident waves as well as for low-frequency incident waves. However, the mathematical theory of the PML method is derived from the first-order equation, all PML implementations of second-order equations are to introduce auxiliary equations or variables to rewrite original PML equations, which will complicate the implementation. In this article, we propose a simple and efficient CPML implementation method for the second-order elastic wave equation, which directly simulates the second-order CPML equation. The main advantage of this method is that there is no need to introduce auxiliary variables or auxiliary equations to convert the second-order PML equation from the complex coordinate space to the real axis. Compared with the conventional CPML method for the second-order elastic wave equation, it introduces only eight convolution variables. We demonstrate the validity and absorption performance through extensive numerical experiments.

An Efficient Unsplit Perfectly Matched Layer for Finite-Element Time-Domain Modeling of Elastodynamics in Cylindrical Coordinates

The truncation of an unbounded medium into a finite domain and the elimination of reflections from artificial medium boundaries is important in numerical simulations of elastodynamic equations. The perfectly matched layer (PML) technique as a novel absorbing boundary has demonstrated very high efficiency for elastic wave equation models. Based on the stretched coordinate concept, an efficient and unsplit-field perfectly matched layer equation for elastic waves is formulated in a cylindrical coordinate system. By introducing integrated complex variables for radial direction and auxiliary functions, the PML formulation is extended in cylindrical coordinates based on the second-order elastic wave equation with displacements as basic unknowns. Finite-element time-domain modeling of the second-order unsplit PML for the elastodynamic equation is presented, which is a standard displacement-based formulation for the computational domain including a PML region. The formulas for the special cases with 2-D axisymmetric coordinates and polar coordinates are also given. The efficiency of the proposed formulations is illustrated by a numerical example using the finite-element method.

Frequency-Domain Finite-Difference Elastic Wave Modeling in the Presence of Surface Topography

Seismic numerical modeling in the presence of surface topography has become a valuable tool to characterize seismic wave propagation in basin or mountain areas. Regarding advantages of frequency-domain seismic wavefield simulations (e.g., easy implementation of multiple sources and straightforward extension of adding attenuation factors), we propose a frequency-domain finite-difference seismic wavefield simulation in 2D elastic media with an irregular free surface. In the frequency domain, we first transform second-order elastic wave equations and first-order free surface boundary conditions from the Cartesian coordinate system to the curvilinear coordinate system. Then we apply complex frequency-shifted perfectly matched layer (CFS-PML) absorbing boundary condition to second-order elastic wave equations in the curvilinear coordinate. To better couple free surface boundary conditions and CFS-PML absorbing boundary condition, we also apply the complex coordinate stretching method used in CFS-PML to free surface boundary conditions in the curvilinear coordinate. In the first numerical test, the comparison of the seismograms calculated by our algorithm with an analytical solution indicates that our algorithm can accurately simulate seismic wavefield in the frequency domain. Finally, we choose three more elastic models with different types of surface topographies to further characterize seismic wave propagation.

A simple implementation of PML for second-order elastic wave equations

When modeling time-domain elastic wave propagation in an unbound space, the standard perfectly matched layer (PML) is straightforward for the first-order partial differential equations (PDEs); by contrast, the PML requires tremendous re-constructions of the governing equations in the second-order PDE form, which is however preferable, because of much less memory and time consumption. Therefore, it is imperative to explore a simple implementation of PML for the second-order system. In this work, we first systematically extend the first-order Nearly PML (NPML) technique into second-order systems, implemented by the spectral element and finite difference time-domain algorithms. It merits the following advantages: the simplicity in implementation, by keeping the second-order PDE-based governing equations exactly the same; the efficiency in computation, by introducing a set of auxiliary ordinary differential equations (ODEs). Mathematically, this PML technique effectively hybridizes the second-order PDEs and first-order ODEs, and locally attenuates outgoing waves, thus efficiently avoid either spatial or temporal global convolutions. Numerical experiments demonstrate that the NPML for the second-order PDE has an excellent absorbing performance for elastic, anelastic and anisotropic media in terms of the absorption accuracy, implementation complexity, and computation efficiency.

A stable auxiliary differential equation perfectly matched layer condition combined with low-dispersive symplectic methods for solving second-order elastic wave equations

The stable implementation of the perfectly matched layer (PML), one of the most effective and popular artificial boundary conditions, has attracted much attention these years. As a type of low-dispersive and symplectic method for solving seismic wave equations, the nearly-analytic symplectic partitioned Runge-Kutta (NSPRK) method has been combined with split-field PML (SPML) and convolutional complex-frequency shifted PML (C-CFS-PML) previously to model acoustic and short-time elastic wave modelings, not yet successfully applied to long-time elastic wave propagation. In order to broaden the application of NSPRK and more general symplectic methods for second-order seismic models, we formulate an auxiliary differential equation (ADE)-CFS-PML with a stabilizing grid compression parameter. This includes deriving the ADE-CFS-PML equations and formulating an adequate time integrator to properly embed their numerical discretizations in the main symplectic numerical methods. The resulting (N)SPRK+ADE-CFS-PML algorithm can help break through the constraint of at most second-order temporal accuracy that used to be imposed on SPML and C-CFS-PML. Especially for NSPRK, we implement the strategy of neglecting the treatment of third-order spatial derivatives in the PML domain and obtain an efficient absorption effect. Related acoustic and elastic wave simulations illustrate the enhanced numerical accuracy of our ADE-CFS-PML compared with SPML and C-CFS-PML. The elastic wave simulation in a homogeneous isotropic medium shows that compared to NSPRK+C-CFS-PML, the NSPRK+ADE-CFS-PML is numerically stable throughout a simulation time of 2 s. The synthetic seismograms of the 2D acoustic SEG salt model and the two-layer elastic model demonstrate the effectiveness of NSPRK+ADE-CFS-PML for complex elastic models. The stabilization effect of the grid compression parameter is verified in the final homogeneous isotropic elastic model with free-surface boundary.

A New Second Order Absorbing Boundary Layer Formulation for Anisotropic-Elastic Wavefield Simulation

The hybrid perfectly matched layer (H-PML) approach to boundary absorption, a combination of convolutional and multiaxial perfectly matched layer (C-PML and M-PML) approaches, is extended to simulate second-order displacement-stress elastic wave equations. The displacement components instead of the velocity components can be directly updated, which can further be used in elastic full waveform inversion. A general stability condition for the second-order displacement-stress elastic wave equations is also proposed. The C-PML and H-PML simulation that results in isotropic and anisotropic media are compared. H-PML is capable of absorbing boundary reflections in both isotropic and anisotropic media, but C-PML suffers severely from the boundary reflections in anisotropic media. The H-PML simulation results for both first- and second-order elastic wave equations show its efficiency in boundary reflections suppression. The computational cost comparison between C-PML and H-PML also demonstrated that H-PML needs smaller computational volumes than C-PML for suppressing the same level of boundary reflections, which is more suitably applied in anisotropic full waveform inversion to reduce the computational volume.

Improved forward wave propagation and adjoint-based sensitivity kernel calculations using a numerically stable finite-element PML

In recent years, the application of time-domain adjoint methods to improve large, complex underground tomographic models at the regional scale has led to new challenges for the numerical simulation of forward or adjoint elastic wave propagation problems. An important challenge is to design an efficient infinite-domain truncation method suitable for accurately truncating an infinite domain governed by the second-order elastic wave equation written in displacement and computed based on a finite-element (FE) method. In this paper, we make several steps towards this goal. First, we make the 2-D convolution formulation of the complex-frequency-shifted unsplit-field perfectly matched layer (CFS-UPML) derived in previous work more flexible by providing a new treatment to analytically remove singular parameters in the formulation. We also extend this new formulation to 3-D. Furthermore, we derive the auxiliary differential equation (ADE) form of CFS-UPML, which allows for extension to higher order time schemes and is easier to implement. Secondly, we rigorously derive the CFS-UPML formulation for time-domain adjoint elastic wave problems, which to our knowledge has never been done before. Thirdly, in the case of classical low-order FE methods, we show numerically that we achieve long-time stability for both forward and adjoint problems both for the convolution and the ADE formulations. In the case of higher order Legendre spectral-element methods, we show that weak numerical instabilities can appear in both formulations, in particular if very small mesh elements are present inside the absorbing layer, but we explain how these instabilities can be delayed as much as needed by using a stretching factor to reach numerical stability in practice for applications. Fourthly, in the case of adjoint problems with perfectly matched absorbing layers we introduce a computationally efficient boundary storage strategy by saving information along the interface between the CFS-UPML and the main domain only, thus avoiding the need to solve a backward wave propagation problem inside the CFS-UPML, which is known to be highly ill-posed. Finally, by providing several examples we show numerically that our formulation is efficient at absorbing acoustic waves for normal to near-grazing incident body waves as well as surface waves.

A first-order k-space model for elastic wave propagation in heterogeneous media

A pseudospectral model of linear elastic wave propagation is described based on the first order stress-velocity equations of elastodynamics. k-space adjustments to the spectral gradient calculations are derived from the dyadic Green's function solution to the second-order elastic wave equation and used to (a) ensure the solution is exact for homogeneous wave propagation for timesteps of arbitrarily large size, and (b) also allows larger time steps without loss of accuracy in heterogeneous media. The formulation in k-space allows the wavefield to be split easily into compressional and shear parts. A perfectly matched layer (PML) absorbing boundary condition was developed to effectively impose a radiation condition on the wavefield. The staggered grid, which is essential for accurate simulations, is described, along with other practical details of the implementation. The model is verified through comparison with exact solutions for canonical examples and further examples are given to show the efficiency of the method for practical problems. The efficiency of the model is by virtue of the reduced point-per-wavelength requirement, the use of the fast Fourier transform (FFT) to calculate the gradients in k space, and larger time steps made possible by the k-space adjustments.