Perfectly matched layer-absorbing boundary condition for finite-element time-domain modeling of elastic wave equations

The perfectly matched layer （PML） is a highly efficient absorbing boundary condition used for the numerical modeling of seismic wave equation. The article focuses on the application of this technique to finite-element time-domain numerical modeling of elastic wave equation. However, the finite-element time-domain scheme is based on the second- order wave equation in displacement formulation. Thus, the first-order PML in velocity-stress formulation cannot be directly applied to this scheme. In this article, we derive the finite- element matrix equations of second-order PML in displacement formulation, and accomplish the implementation of PML in finite-element time-domain modeling of elastic wave equation. The PML has an approximate zero reflection coefficients for bulk and surface waves in the finite-element modeling of P-SV and SH wave propagation in the 2D homogeneous elastic media. The numerical experiments using a two-layer model with irregular topography validate the efficiency of PML in the modeling of seismic wave propagation in geological models with complex structures and heterogeneous media.

PML and CFS-PML boundary conditions for a mesh-free finite difference solution of the elastic wave equation

Mesh-free finite difference(FD)methods can improve the geometric flexibility of modeling without the need for lattice mapping or complex meshing process.Radial-basisfunction-generated FD is among the most commonly used mesh-free FD methods and can accurately simulate seismic wave propagation in the non-rectangular computational domain.In this paper,we propose a perfectly matched layer(PML)boundary condition for a meshfree FD solution of the elastic wave equation,which can be applied to the boundaries of the non-rectangular velocity model.The performance of the PML is,however,severely reduced for near-grazing incident waves and low-frequency waves.We thus also propose the complexfrequency-shifted PML(CFS-PML)boundary condition for a mesh-free FD solution of the elastic wave equation.For two PML boundary conditions,we derive unsplit time-domain expressions by constructing auxiliary differential equations,both of which require less memory and are easy for programming.Numerical experiments demonstrate that these two PML boundary conditions effectively eliminate artificial boundary reflections in mesh-free FD simulations.When compared with the PML boundary condition,the CFS-PML boundary condition results in better absorption for near-grazing incident waves and evanescent waves.

THE PARAMETER PERTURBATION METHOD ON ELASTIC WAVE EQUATION IN INHOMOGENEOUS MEDIUM

In this paper, the medium parameters of the elastic wave equation in inhomogeneous medium are rewritten by introducing the referential variables and the perturbational variables, and the wave equation whose sources are the medium parameter perturbational term in homogeneous medium is obtained. By using the Green function theory, the integral equation of the perturbational parameters is obtained. Then the displacement field in homogeneous medium is considered the result of the first iteration, and the displacement field is solved by this integral equation. When the perturbations of medium parameters are about 50 percent, this method can solve the displacement field effectively. from the analysis of the numerical results, the characteristics of wave field in inhomogeneous medium are obtained. The results conform with the local principles of wave function in inhomogeneous medium.

A Stable Finite Difference Method for the Elastic Wave Equation on Complex Geometries with Free Surfaces

A stable and explicit second order accurate finite difference method for the elastic wave equation in curvilinear coordinates is presented.The discretization of the spatial operators in the method is shown to be self-adjoint for free-surface,Dirichlet and periodic boundary conditions.The fully discrete version of the method conserves a discrete energy to machine precision.

An improved convolution perfectly matched layer for elastic second-order wave equation

A convolution perfectly matched layer(CPML)can efficiently absorb boundary reflection in numerical simulation.However,the CPML is suitable for the first-order elastic wave equation and is difficult to apply directly to the second-order elastic wave equation.In view of this,based on the first-order CPML absorbing boundary condition,we propose a new CPML(NCPML)boundary which can be directly applied to the second-order wave equation.We first systematically extend the first-order CPML technique into second-order wave equations,neglecting the space-varying characteristics of the partial damping coefficient in the complex-frequency domain,avoiding the generation of convolution in the time domain.We then transform the technique back to the time domain through the inverse Fourier transform.Numerical simulation indicates that the space-varying characteristics of the attenuation factor have little influence on the absorption effect and increase the memory at the same time.A number of numerical examples show that the NCPML proposed in this study is effective in simulating elastic wave propagation,and this algorithm is more efficient and requires less memory allocation than the conventional PML absorbing boundary.

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

We develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two-and three-dimensional spatial domains.In this method,waves are slowed down and dissipated in sponge layers near the far-field boundaries.Mathematically,this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain,where the elastic wave equation is solved numerically on a regular grid.To damp out waves that become poorly resolved because of the coordinate mapping,a high order artificial dissipation operator is added in layers near the boundaries of the computational domain.We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy,which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain.Our spatial discretization is based on a fourth order accurate finite difference method,which satisfies the principle of summation by parts.We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries.Therefore,the coefficients in the finite difference stencils need only be boundary modified near the free surface.This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains.Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer.The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem,where fourth order accuracy is observed with a sixth order artificial dissipation.We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

An Energy Absorbing Far-Field Boundary Condition for the Elastic Wave Equation

We present an energy absorbing non-reflecting boundary condition of Clayton-Engquist type for the elastic wave equation together with a discretization which is stable for any ratio of compressional to shear wave speed.We prove stability for a second-order accurate finite-difference discretization of the elastic wave equation in three space dimensions together with a discretization of the proposed non-reflecting boundary condition.The stability proof is based on a discrete energy estimate and is valid for heterogeneous materials.The proof includes all six boundaries of the computational domain where special discretizations are needed at the edges and corners.The stability proof holds also when a free surface boundary condition is imposed on some sides of the computational domain.

P-and S-wavefield simulations using both the firstand second-order separated wave equations through a high-order staggered grid finite-difference method

In seismic exploration, it is common practice to separate the P-wavefield from the S-wavefield by the elastic wavefield decomposition technique, for imaging purposes. However, it is sometimes difficult to achieve this, especially when the velocity field is complex. A useful approach in multi-component analysis and modeling is to directly solve the elastic wave equations for the pure P- or S-wavefields, referred as the separate elastic wave equa- tions. In this study, we compare two kinds of such wave equations： the first-order （velocity-stress） and the second- order （displacement-stress） separate elastic wave equa- tions, with the first-order （velocity-stress） and the second- order （displacement-stress） full （or mixed） elastic wave equations using a high-order staggered grid finite-differ- ence method. Comparisons are given of wavefield snap- shots, common-source gather seismic sections, and individual synthetic seismogram. The simulation tests show that equivalent results can be obtained, regardless of whether the first-order or second-order separate elastic wave equations are used for obtaining the pure P- or S-wavefield. The stacked pure P- and S-wavefields are equal to the mixed wave fields calculated using the corre- sponding first-order or second-order full elastic wave equations. These mixed equations are computationallyslightly less expensive than solving the separate equations. The attraction of the separate equations is that they achieve separated P- and S-wavefields which can be used to test the efficacy of wave decomposition procedures in multi-com- ponent processing. The second-order separate elastic wave equations are a good choice because they offer information on the pure P-wave or S-wave displacements.

3D elastic waveform modeling with an optimized equivalent staggered-grid finite-difference method

Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coefficients on spatial derivatives,but the simulation results suffer serious numerical dispersion on a large frequency zone.We develop an optimized equivalent staggered-grid(OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3 D elastic wave equation.On the one hand,we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave,S-wave,and converted-wave(C-wave) terms.On the other hand,a novel plane wave solution for the 3 D elastic wave equation is derived from the matrix decomposition method to construct the time-space dispersion relations.FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method.Finally,we construct a new objective function to analyze P-wave,S-wave,and C-wave dispersion concerning frequencies.The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method.The synthetic examples demonstrate the effectiveness and superiority of the presented method.

Hybrid absorbing boundary condition for threedimensional elastic wave modeling

Edge reflections are inevitable in numerical modeling of seismic wavefields, and they are usually attenuated by absorbing boundary conditions. However, the commonly used perfectly matched layer （PML） boundary condition requires special treatment for the absorbing zone, and in three-dimensional （3D） modeling, it has to split each variable into three corresponding variables, which increases the computing time and memory storage. In contrast, the hybrid absorbing boundary condition （HABC） has the advantages such as ease of implementation, less computation time, and near-perfect absorption; it is thus able to enhance the computational efficiency of 3D elastic wave modeling. In this study, a HABC is developed from two-dimensional （2D） modeling into 3D modeling based on the I st Higdon one way wave equations, and a HABC is proposed that is suitable for a 3D elastic wave numerical simulation. Numerical simulation results for a homogenous model and a complex model indicate that the proposed HABC method is more effective and has better absorption than the traditional PML method.

Elastic Full Waveform Inversion Based on the Trust Region Strategy

In this paper, we investigate the elastic wave full-waveform inversion (FWI) based on the trust region method. The FWI is an optimization problem of minimizing the misfit between the observed data and simulated data. Usually, the line search method is used to update the model parameters iteratively. The line search method generates a search direction first and then finds a suitable step length along the direction. In the trust region method, it defines a trial step length within a certain neighborhood of the current iterate point and then solves a trust region subproblem. The theoretical methods for the trust region FWI with the Newton type method are described. The algorithms for the truncated Newton method with the line search strategy and for the Gauss-Newton method with the trust region strategy are presented. Numerical computations of FWI for the Marmousi model by the L-BFGS method, the Gauss-Newton method and the truncated Newton method are completed. The comparisons between the line search strategy and the trust region strategy are given and show that the trust region method is more efficient than the line search method and both the Gauss-Newton and truncated Newton methods are more accurate than the L-BFGS method.

Research on the Seismic Wave Field of Karst Cavern Reservoirs near Deep Carbonate Weathered Crusts

Fracture and cavern hydrocarbon reservoirs in carbonates are an important pool type worldwide. The karst cavern reservoirs are easiest to identify on seismic reflection data. The prediction, exploration, and development of this type of reservoir require theoretical research on seismic wave fields reflected from complex inhomogeneous media. We compute synthetic seismic sections for fluidfilled cavern reservoirs of various heights and widths using random media models and inhomogeneous media elastic wave equations. Results indicate that even caverns significantly smaller than 1/ 4 wavelength are detectible on conventional band-width seismic sections as diffractions migrated into bead-type events. Diffraction amplitude is a function of cavern height and width. We introduce a width-amplitude factor which can be used to calculate the diffraction amplitude of a cavern with a limited width from the diffraction amplitude computed for an infinitely wide cavern.

An eigen theory of waves in piezoelectric solids

Based on the standard spaces of the physical presentation, both the quasi-static mechanical approximation and the quasi-static electromagnetic approximation of piezoelectric solids are studied here. The complete set of uncoupled elastic wave and electromagnetic wave equations are deduced. The results show that the number and propagation speed of elastic waves and electromagnetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. Based on these laws, we discuss the propagation behaviour of elastic waves and electromagnetic waves in the piezoelectric material of class 6 mm.

Mortar DG Method with Staggered Hybridization for Rayleigh Waves Simulation

The simulation of Rayleigh waves is important in a variety of geophysical applications.The computational challenge is the fact that very fine mesh is necessary as the waves are concentrated at the free surface and decay exponentially away from the free surface.To overcome this challenge and to develop a robust high order scheme for the simulation of Rayleigh waves,we develop a mortar discontinuous Galerkin method with staggered hybridization.The use of the mortar technique allows one to use fine mesh in only a local region near the free surface,and use coarse mesh in most of the domain.This approach reduces the computational cost significantly.The staggered hybridization allows the preservation of the strong symmetry of the stress tensor without complicated construction of basis functions.In particular,the basis functions are piecewise polynomial without any continuity requirement,and the coupling of the basis functions is performed by using carefully chosen hybridized variables.The resulting scheme is explicit in time,and only local saddle point system are solved for each time step.We will present several benchmark problems to demonstrate the performance of the proposed method.

Stable Grid Refinement and Singular Source Discretization for Seismic Wave Simulations

An energy conserving discretization of the elastic wave equation in second order formulation is developed for a composite grid,consisting of a set of structured rectangular component grids with hanging nodes on the grid refinement interface.Previously developed summation-by-parts properties are generalized to devise a stable second order accurate coupling of the solution across mesh refinement interfaces.The discretization of singular source terms of point force and point moment tensor type are also studied.Based on enforcing discrete moment conditions that mimic properties of the Dirac distribution and its gradient,previous single grid formulas are generalized to work in the vicinity of grid refinement interfaces.These source discretization formulas are shown to give second order accuracy in the solution,with the error being essentially independent of the distance between the source and the grid refinement boundary.Several numerical examples are given to illustrate the properties of the proposed method.

A New Approach of High OrderWell-Balanced Finite Volume WENO Schemes and Discontinuous Galerkin Methods for a Class of Hyperbolic Systems with Source Terms

Hyperbolic balance laws have steady state solutions in which the flux gradients are nonzero but are exactly balanced by the source terms.In our earlier work[31–33],we designed high order well-balanced schemes to a class of hyperbolic systems with separable source terms.In this paper,we present a different approach to the same purpose:designing high order well-balanced finite volume weighted essentially non-oscillatory(WENO)schemes and RungeKutta discontinuous Galerkin(RKDG)finite element methods.We make the observation that the traditional RKDG methods are capable of maintaining certain steady states exactly,if a small modification on either the initial condition or the flux is provided.The computational cost to obtain such a well balanced RKDG method is basically the same as the traditional RKDG method.The same idea can be applied to the finite volume WENO schemes.We will first describe the algorithms and prove the well balanced property for the shallow water equations,and then show that the result can be generalized to a class of other balance laws.We perform extensive one and two dimensional simulations to verify the properties of these schemes such as the exact preservation of the balance laws for certain steady state solutions,the non-oscillatory property for general solutions with discontinuities,and the genuine high order accuracy in smooth regions.