A stable staggered-grid finite-difference scheme for acoustic modeling beyond conventional stability limit

Staggered-grid finite-difference(SGFD)schemes have been widely used in acoustic wave modeling for geophysical problems.Many improved methods are proposed to enhance the accuracy of numerical modeling.However,these methods are inevitably limited by the maximum Courant-Friedrichs-Lewy(CFL)numbers,making them unstable when modeling with large time sampling intervals or small grid spacings.To solve this problem,we extend a stable SGFD scheme by controlling SGFD dispersion relations and maximizing the maximum CFL numbers.First,to improve modeling stability,we minimize the error between the FD dispersion relation and the exact relation in the given wave-number region,and make the FD dispersion approach a given function outside the given wave-number area,thus breaking the conventional limits of the maximum CFL number.Second,to obtain high modeling accuracy,we use the SGFD scheme based on the Remez algorithm to compute the FD coefficients.In addition,the hybrid absorbing boundary condition is adopted to suppress boundary reflections and we find a suitable weighting coefficient for the proposed scheme.Theoretical derivation and numerical modeling demonstrate that the proposed scheme can maintain high accuracy in the modeling process and the value of the maximum CFL number of the proposed scheme can exceed that of the conventional SGFD scheme when adopting a small maximum effective wavenumber,indicating that the proposed scheme improves stability during the modeling.

An adaptive finite-difference method for seismic traveltime modeling based on 3D eikonal equation

3D eikonal equation is a partial differential equation for the calculation of first-arrival traveltimes and has been widely applied in many scopes such as ray tracing,source localization,reflection migration,seismic monitoring and tomographic imaging.In recent years,many advanced methods have been developed to solve the 3D eikonal equation in heterogeneous media.However,there are still challenges for the stable and accurate calculation of first-arrival traveltimes in 3D strongly inhomogeneous media.In this paper,we propose an adaptive finite-difference(AFD)method to numerically solve the 3D eikonal equation.The novel method makes full use of the advantages of different local operators characterizing different seismic wave types to calculate factors and traveltimes,and then the most accurate factor and traveltime are adaptively selected for the convergent updating based on the Fermat principle.Combined with global fast sweeping describing seismic waves propagating along eight directions in 3D media,our novel method can achieve the robust calculation of first-arrival traveltimes with high precision at grid points either near source point or far away from source point even in a velocity model with large and sharp contrasts.Several numerical examples show the good performance of the AFD method,which will be beneficial to many scientific applications.

Viscoacoustic prestack reverse time migration based onthe optimal time-space domain high-order finite-difference method

Prestack reverse time migration （RTM） is an accurate imaging method ofsubsurface media. The viscoacoustic prestack RTM is of practical significance because itconsiders the viscosity of the subsurface media. One of the steps of RTM is solving thewave equation and extrapolating the wave field forward and backward; therefore, solvingaccurately and efficiently the wave equation affects the imaging results and the efficiencyof RTM. In this study, we use the optimal time-space domain dispersion high-order finite-difference （FD） method to solve the viscoacoustic wave equation. Dispersion analysis andnumerical simulations show that the optimal time-space domain FD method is more accurateand suppresses the numerical dispersion. We use hybrid absorbing boundary conditions tohandle the boundary reflection. We also use source-normalized cross-correlation imagingconditions for migration and apply Laplace filtering to remove the low-frequency noise.Numerical modeling suggests that the viscoacoustic wave equation RTM has higher imagingresolution than the acoustic wave equation RTM when the viscosity of the subsurface isconsidered. In addition, for the wave field extrapolation, we use the adaptive variable-lengthFD operator to calculate the spatial derivatives and improve the computational efficiencywithout compromising the accuracy of the numerical solution.

Three-dimensional acoustic wave equation modeling based on the optimal finite-difference scheme

Generally, FD coefficients can be obtained by using Taylor series expansion （TE） or optimization methods to minimize the dispersion error. However, the TE-based FD method only achieves high modeling precision over a limited range of wavenumbers, and produces large numerical dispersion beyond this range. The optimal FD scheme based on least squares （LS） can guarantee high precision over a larger range of wavenumbers and obtain the best optimization solution at small computational cost. We extend the LS-based optimal FD scheme from two-dimensional （2D） forward modeling to three-dimensional （3D） and develop a 3D acoustic optimal FD method with high efficiency, wide range of high accuracy and adaptability to parallel computing. Dispersion analysis and forward modeling demonstrate that the developed FD method suppresses numerical dispersion. Finally, we use the developed FD method to source wavefield extrapolation and receiver wavefield extrapolation in 3D RTM. To decrease the computation time and storage requirements, the 3D RTM is implemented by combining the efficient boundary storage with checkpointing strategies on GPU. 3D RTM imaging results suggest that the 3D optimal FD method has higher precision than conventional methods.

Finite-difference numerical modeling with even-order accuracy in two-phase anisotropic media

To improve the accuracy of the conventional finite-difference method, finitedifference numerical modeling methods of any even-order accuracy are recommended. We introduce any even-order accuracy difference schemes of any-order derivatives derived from Taylor series expansion. Then, a finite-difference numerical modeling method with any evenorder accuracy is utilized to simulate seismic wave propagation in two-phase anisotropic media. Results indicate that modeling accuracy improves with the increase of difference accuracy order number. It is essential to find the optimal order number, grid size, and time step to balance modeling precision and computational complexity. Four kinds of waves, static mode in the source point, SV wave cusps, reflection and transmission waves are observed in two-phase anisotropic media through modeling.

Finite-difference modeling of surface waves in poroelastic media and stress mirror conditions

During seismic wave propagation on a free surface, a strong material contrast boundary develops in response to interference by P- and S- waves to create a surface-wave phenomenon. To accurately determine the effects of this interface on surface-wave propagation, the boundary conditions must be accurately modeled. In this paper, we present a numerical approach based on the dynamic poroelasticity for a space–time-domain staggered-grid finite-difference simulation in porous media that contain a free-surface boundary. We propose a generalized stess mirror formulation of the free-surface boundary for solids and fluids in porous media for the grid mesh on which lays the free-surface plane. Its analog is that used for elastic media, which is suitable for precise and stable Rayleigh-type surface-wave modeling. The results of our analysis of first kind of Rayleigh （R1） waves obtained by this model demonstrate that the discretization of the mesh in a similar way to that for elastic media can realize stable numerical solutions with acceptable precision. We present numerical examples demonstrating the efficiency and accuracy of our proposed method.

2-D elastic wave modeling with frequency-space 25-point finite-difference operators

Numerical simulation in the frequency-space domain has inherent advantages, such as： it is possible to simulate wave propagation from multiple sources simultaneously; there are no cumulative errors; only the interesting frequencies can be selected; and it is more suitable for wave propagation in viscoelastic media. The only obstacle to using the method is the requirement of huge computer storage. We extend the compressed format for storing the coefficient matrix. It can reduce the required computer storage dramatically. We get the optimal coefficients by least-squares method to suppress the numerical dispersion and adopt the perfectly matched layer （PML） boundary conditions to eliminate the artificial boundary reflections. Using larger grid intervals decreases computer storage requirements and provides high computational efficiency. Numerical experiments demonstrate that these means are economic and effective, providing a good basis for elastic wave imaging and inversion.

3D finite-difference modeling algorithm and anomaly features of ZTEM

The Z-Axis tiPPer eiectromagnetic （ZTEM） technique is based on a frequency-domain airbome electromagnetic system that measures the natural magnetic field. A survey area was divided into several blocks by using the Maxwell＇s equations, and the magnetic components at the center of each edge of the grid cell are evaluated by applying the staggered-grid finite-difference method. The tipper and its divergence are derived to complete the 3D ZTEM forward modeling algorithm. A synthetic model is then used to compare the responses with those of 2D finite-element forward modeling to verify the accuracy of the algorithm. ZTEM offers high horizontal resolution to both simple and complex distributions of conductivity. This work is the theoretical foundation for the interpretation of ZTEM data and the study of 3D ZTEM inversion.

Optimized staggered-grid finite-difference operators using window functions

The staggered-grid finite-difference （SGFD） method has been widely used in seismic forward modeling. The precision of the forward modeling results directly affects the results of the subsequent seismic inversion and migration. Numerical dispersion is one of the problems in this method. The window function method can reduce dispersion by replacing the finite-difference operators with window operators, obtained by truncating the spatial convolution series of the pseudospectral method. Although the window operators have high precision in the low-wavenumber domain, their precision decreases rapidly in the high-wavenumber domain. We develop a least squares optimization method to enhance the precision of operators obtained by the window function method. We transform the SGFD problem into a least squares problem and find the best solution iteratively. The window operator is chosen as the initial value and the optimized domain is set by the error threshold. The conjugate gradient method is also adopted to increase the stability of the solution. Approximation error analysis and numerical simulation results suggest that the proposed method increases the precision of the window function operators and decreases the numerical dispersion.

Numerical simulation of seismic wavefields in TTI media using the rotated staggered-grid compact finite-difference scheme

Numerical simulation in transverse isotropic media with tilted symmetry axis（TTI） using the standard staggered-grid finite-difference scheme（SSG）results in errors caused by averaging or interpolation. In order to eliminate the errors, a method of rotated staggered-grid finite-difference scheme（RSG） is proposed. However, the RSG brings serious numerical dispersion. The compact staggered-grid finite-difference scheme（CSG） is an implicit difference scheme, which use fewer grid points to suppress dispersion more effectively than the SSG. This paper combines the CSG with the RSG to derive a rotated staggered-grid compact finite-difference scheme（RSGC）. The numerical experiments indicate that the RSGC has weaker numerical dispersion and better accuracy than the RSG.

Finite-difference calculation of traveltimes based on rectangular grid

To the most of velocity fields, the traveltimes of the first break that seismic waves propagate along rays can be computed on a 2-D or 3-D numerical grid by finite-difference extrapolation. Under ensuring accuracy, to improve calculating efficiency and adaptability, the calculation method of first-arrival traveltime of finite-difference is de- rived based on any rectangular grid and a local plane wavefront approximation. In addition, head waves and scat- tering waves are properly treated and shadow and caustic zones cannot be encountered, which appear in traditional ray-tracing. The testes of two simple models and the complex Marmousi model show that the method has higher accuracy and adaptability to complex structure with strong vertical and lateral velocity variation, and Kirchhoff prestack depth migration based on this method can basically achieve the position imaging effects of wave equation prestack depth migration in major structures and targets. Because of not taking account of the later arrivals energy, the effect of its amplitude preservation is worse than that by wave equation method, but its computing efficiency is higher than that by total Green′s function method and wave equation method.

Acoustic finite-difference modeling beyond conventional Courant-Friedrichs-Lewy stability limit:Approach based on variable-length temporal and spatial operators

Conventional finite-difference(FD)methods cannot model acoustic wave propagation beyond Courant-Friedrichs-Lewy(CFL)numbers 0.707 and 0.577 for two-dimensional(2D)and three-dimensional(3D)equal spacing cases,respectively,thereby limiting time step selection.Based on the definition of temporal and spatial FD operators,we propose a variable-length temporal and spatial operator strategy to model wave propagation beyond those CFL numbers while preserving accuracy.First,to simulate wave propagation beyond the conventional CFL stability limit,the lengths of the temporal operators are modified to exceed the lengths of the spatial operators for high-velocity zones.Second,to preserve the modeling accuracy,the velocity-dependent lengths of the temporal and spatial operators are adaptively varied.The maximum CFL numbers for the proposed method can reach 1.25 and 1.0 in high velocity contrast 2D and 3D simulation examples,respectively.We demonstrate the effectiveness of our method by modeling wave propagation in simple and complex media.

Seismic wavefield modeling based on time-domain symplectic and Fourier finite-difference method

Seismic wavefield modeling is important for improving seismic data processing and interpretation. Calculations of wavefield propagation are sometimes not stable when forward modeling of seismic wave uses large time steps for long times. Based on the Hamiltonian expression of the acoustic wave equation, we propose a structure-preserving method for seismic wavefield modeling by applying the symplectic finite-difference method on time grids and the Fourier finite-difference method on space grids to solve the acoustic wave equation. The proposed method is called the symplectic Fourier finite-difference （symplectic FFD） method, and offers high computational accuracy and improves the computational stability. Using acoustic approximation, we extend the method to anisotropic media. We discuss the calculations in the symplectic FFD method for seismic wavefield modeling of isotropic and anisotropic media, and use the BP salt model and BP TTI model to test the proposed method. The numerical examples suggest that the proposed method can be used in seismic modeling of strongly variable velocities, offering high computational accuracy and low numerical dispersion. The symplectic FFD method overcomes the residual qSV wave of seismic modeling in anisotropic media and maintains the stability of the wavefield propagation for large time steps.

Optimization of the seismic processing phase-shift plus finite-difference migration operator based on a hybrid genetic and simulated annealing algorithm

Although the phase-shift seismic processing method has characteristics of high accuracy, good stability, high efficiency, and high-dip imaging, it is not able to adapt to strong lateral velocity variation. To overcome this defect, a finite-difference method in the frequency-space domain is introduced in the migration process, because it can adapt to strong lateral velocity variation and the coefficient is optimized by a hybrid genetic and simulated annealing algorithm. The two measures improve the precision of the approximation dispersion equation. Thus, the imaging effect is improved for areas of high-dip structure and strong lateral velocity variation. The migration imaging of a 2-D SEG/EAGE salt dome model proves that a better imaging effect in these areas is achieved by optimized phase-shift migration operator plus a finite-difference method based on a hybrid genetic and simulated annealing algorithm. The method proposed in this paper is better than conventional methods in imaging of areas of high-dip angle and strong lateral velocity variation.

A Finite-Difference Approach to the Time-Dependent Mild-Slope Equation

A finite-difference approach is used to develop a time-dependent mild-slope equation incorporating the effects of bottom dissipation and nonlinearity. The Enler predietor-corrector method and the three-point finite-difference method with varying spatial steps are adopted to discretize the time derivatives and the two-dimensional horizontal ones, respectively, thus leading both the time and spatial derivatives to the second-order accuracy. The boundary conditions for the present model are treated on the basis of the general conditions for open and fixed boundaries with an arbitrary reflection coefficient and phase shift. Both the linear and nonlinear versions of the numerical model are applied to the wave propagation and transformation over an elliptic shoal on a sloping beach, respectively, and the linear version is applied to the simulation of wave propagation in a fully open rectangular harbor. From comparison of numerical results with theoretical or experimental ones, it is found that they are in reasonable agreement.

Uniform stable conformal convolutional perfectly matched layer for enlarged cell technique conformal finite-difference time-domain method

Based on conformal construction of physical model in a three-dimensional Cartesian grid,an integral-based conformal convolutional perfectly matched layer（CPML） is given for solving the truncation problem of the open port when the enlarged cell technique conformal finite-difference time-domain（ECT-CFDTD） method is used to simulate the wave propagation inside a perfect electric conductor（PEC） waveguide.The algorithm has the same numerical stability as the ECT-CFDTD method.For the long-time propagation problems of an evanescent wave in a waveguide,several numerical simulations are performed to analyze the reflection error by sweeping the constitutive parameters of the integral-based conformal CPML.Our numerical results show that the integral-based conformal CPML can be used to efficiently truncate the open port of the waveguide.

Parallel computation of unified finite-difference time-domain for underwater sound scattering

In this work, we treat scattering objects, water, surface and bottom in a truly unified manner in a parallel finitedifference time-domain （FDTD） scheme, which is suitable for distributed parallel computing in a message passing interface （MPI） programming environment. The algorithm is implemented on a cluster-based high performance computer system. Parallel computation is performed with different division methods in 2D and 3D situations. Based on analysis of main factors affecting the speedup rate and parallel efficiency, data communication is reduced by selecting a suitable scheme of task division. A desirable scheme is recommended, giving a higher speedup rate and better efficiency. The results indicate that the unified parallel FDTD algorithm provides a solution to the numerical computation of acoustic scattering.

Investigation of three-pulse photon echo in thick crystal using finite-difference time-domain method

This paper investigates the phenomenon of three-pulse photon echo in thick rare-earth ions doped crystal whose thickness is far larger than 0.002 cm which is adopted in previous works.The influence of thickness on the three-pulse photon echo＇s amplitude and efficiency is analyzed with the Maxwell-Bloch equations solved by finite-difference timedomain method.We demonstrate that the amplitude of three-pulse echo will increase with the increasing of thickness and the optimum thickness to generate three-pulse photon echo is 0.3 cm for Tm^（3＋）：YAG when the attenuation of the input pulse is taken into account.Meanwhile,we find the expression 0.09 exp（α＇L）,which is previously employed to describe the relationship between echo＇s efficiency and thickness,should be modified as 1.3 · 0.09 exp（2.4 ·α＇L） with the propagation of echo considered.

A wavelet finite-difference method for numerical simulation of wave propagation in fluid-saturated porous media

In this paper, we consider numerical simulation of wave propagation in fluidsaturated porous media. A wavelet finite-difference method is proposed to solve the 2-D elastic wave equation. The algorithm combines flexibility and computational efficiency of wavelet multi-resolution method with easy implementation of the finite-difference method. The orthogonal wavelet basis provides a natural framework, which adapt spatial grids to local wavefield properties. Numerical results show usefulness of the approach as an accurate and stable tool for simulation of wave propagation in fluid-saturated porous media.

Accuracy of the staggered-grid finite-difference method of the acoustic wave equation for marine seismic reflection modeling

Seismic wave modeling is a cornerstone of geophysical data acquisition, processing, and interpretation, for which finite-difference methods are often applied. In this paper, we extend the velocity- pressure formulation of the acoustic wave equation to marine seismic modeling using the staggered-grid finite-difference method. The scheme is developed using a fourth-order spatial and a second-order temporal operator. Then, we define a stability coefficient (SC) and calculate its maximum value under the stability condition. Based on the dispersion relationship, we conduct a detailed dispersion analysis for submarine sediments in terms of the phase and group velocity over a range of angles, stability coefficients, and orders. We also compare the numerical solution with the exact solution for a P-wave line source in a homogeneous submarine model. Additionally, the numerical results determined by a Marmousi2 model with a rugged seafloor indicate that this method is sufficient for modeling complex submarine structures.