modelling damped acoustic waves by a dissipation preserving conformal symplectic method we propose a novel stable and efficient dissipation preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude through introducing the conformal multi symplectic structure the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation the proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law more specifically two subsystems in conjunction with the original damped wave equation are derived one is actually the conservative hamiltonian wave equation and the other is a dissipative linear ordinary differential equation ode system standard symplectic method is devoted to the conservative system whereas the analytical solution is obtained for the ode system an explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the strang splitting technique stability analysis and convergence tests are given thereafter a benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media

3d finite difference modeling of elastic wave propagation in the laplace fourier domain abstractwith the recent interest in the laplace fourier domain full waveform inversion we have developed new heterogeneous 3d fourth and second order staggered grid finite difference schemes for modeling seismic wave propagation in the laplace fourier domain our approach is based on the integro interpolation technique for the velocity stress formulation in the cartesian coordinate system five averaging elastic coefficients and three averaging densities are necessary to describe the heterogeneous medium with harmonic averaging of the bulk and shear moduli and arithmetic averaging of density in the fourth order approximation we improved the accuracy of the scheme using a combination of integral identities for two elementary volumes small and large around spatial gridpoints where the wave variables are defined two solution approaches are provided both of which are solved with iterative krylov methods in the first approach the stress variables are eliminated and a linear system for the veloc

an immersed free surface boundary treatment for seismic wave simulation abstractfinite difference methods are popular for wave simulation within the seismic exploration community thanks to their efficiency however difficulties arise when encountering complex topography due to the regular grid pattern of the finite difference schemes despite alternatives that can handle the free surface with little effort such as the spectral element or discontinuous galerkin s methods incorporating a free surface boundary condition within the finite difference framework is still appealing even at the cost of extra algorithm complexity and stronger requirement of computational resources we present a free surface boundary treatment within the finite difference framework belonging to the family of the immersed boundary methods inherently the presented boundary treatment is separated from the rest of the wave simulation which makes it easy to be integrated in existing finite difference codes specifically we construct an extrapolation operator for each grid point above the free surfa

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

phase error control for fd td methods of second and fourth order accuracy for fd td methods used to solve maxwell s equations we determine the spatial resolution of the discretized domain in terms of the total computation time and the desired phase error it is shown that the spatial step should vary as spl delta x spl sim g e sub spl phi t sub c sup 1 s in order to maintain a prescribed phase error level e sub spl phi throughout the computation time t sub c where s 2 or 4 is the spatial order of accuracy of the scheme and g is a geometric factor significantly we show that the rule of thumb of using 10 20 points per wavelength to determine the spatial cell size for the standard scheme is not optimal our results are verified by numerical simulations in two dimensions with the yee 1966 scheme and a new fourth order accurate fd td scheme

numerical study of the interface errors of finite difference simulations of seismic waves numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion which is only valid for homogeneous media however there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers the error associated with media that have two layers is called an interface error and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite difference schemes fds for homogeneous media we evaluated a detailed numerical study of the interface error for three staggered grid fds that are commonly used in the simulation of seismic wave propagation we determined that a standard staggered grid scheme ssgs also known as the virieux scheme a rotated staggered grid scheme rsgs and a lebedev scheme ls preserve the second order of convergence at horizontal vertical solid solid interfaces when the medium parameters have been properly modified such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density however for a fluid solid interface aligned with the grid line a second order convergence can only be achieved by an ssgs in addition the presence of a fluid solid interface reduces the order of convergence for the ls and the rsgs to a first order of convergence the presence of inclined interfaces makes high order second and more convergence impossible

spatial parallelism of a 3d finite difference velocity stress elastic wave propagation code in a three dimensional isotropic elastic earth the wave equation solution consists of three velocity components and six stresses we discretize the partial derivatives using second order in time and fourth order in space staggered finite difference operators the parallel implementation uses the message passing interface library for platform portability and a spatial decomposition for efficiency most of the communication in the code consists of passing subdomain face information to neighboring processors when the parallel communication is balanced against computation by allocating subdomains of reasonable size we observe excellent scaled speedup allocating subdomains of size 25 times 25 times 25 on each node we achieve efficiencies of 94 on 128 processors of an intel linebreak paragon

a stable free surface boundary condition for two dimensional elastic finite difference wave simulation two of the persistent problems in finite difference solutions of the elastic wave equation are the limited stability range of the free surface boundary condition and the boundary condition s treatment of lateral variations in velocity and density the centered difference approximation presented by alterman and karal 1968 for example remains stable only for greater than 0 30 where and are the shear s and compressional p wave velocities the one sided approximation alterman and rotenberg 1969 and composed approximation ilan et al 1975 have similar restrictions the revised composed approximation of ilan and loewenthal 1976 overcomes this restriction but cannot handle laterally varying media properly

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

attenuating boundary conditions for numerical modeling of acoustic wave propagation four types of boundary conditions dirichlet neumann transmitting and modified transmitting are derived by combining the damped wave equation with corresponding boundary conditions the dirichlet attenuating boundary condition is the easiest to implement for an appropriate choice of attenuation parameter it can achieve a boundary reflection coefficient of a few percent in a one wavelength wide zone the neumann attenuating boundary condition has characteristics similar to the dirichlet attenuating boundary condition but it is numerically more difficult to implement both the transmitting boundary condition and the modified transmitting boundary condition need an absorbing boundary condition at the termination of the attenuating region the modified transmitting boundary condition is the most effective in the suppression of boundary reflections for multidimensional modeling there is no perfect absorbing boundary condition and an approximate absorbing boundary condition is used

sbp sat finite difference discretization of acoustic wave equations on staggered block wise uniform grids abstract we consider the numerical simulation of the acoustic wave equations arising from seismic applications for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency we relax the uniform grid restriction on finite difference methods and allow the grids to be block wise uniform with nonconforming interfaces in doing so variations in the wave speeds of the subterranean media can be accounted for more efficiently staggered grid finite difference operators satisfying the summation by parts sbp property are devised to approximate the spatial derivatives appearing in the acoustic wave equation these operators are applied within each block independently the coupling between blocks is achieved through simultaneous approximation terms sats which impose the interface conditions weakly i e by penalty ratio of the grid spacing of neighboring blocks is allowed to be rational number for which specially designed interpolation formulas are presented these interpolation formulas constitute key pieces of the simultaneous approximation terms the overall discretization is shown to be energy conserving and examined on test cases of both theoretical and practical interests delivering accurate and stable simulation results

two dimensional axisymmetric numerical simulation of supercritical phase conjugation of ultrasound in active solid media in the present study a two dimensional axisymmetric numerical model is developed for supercritical parametric phase conjugation of ultrasound in a solid active element of cylindrical shape and finite length the pseudospectral time domain algorithm pstd is used owing to its efficiency to model large scale problems pstd solves elastic wave equation in time dependent heterogeneous isotropic and axisymmetric anisotropic solids using ffts for high order approximation of the spatial differential operator on staggered grid and a fourth order adams bashforth time integrator in order to truncate the computational domain absorbing boundary conditions are introduced with complex frequency shifted perfectly matched layers this procedure is highly effective at absorbing signals of long time signature the free surface of the active ceramic rod is introduced through the method of images a systematic study of the influence of lateral limitations of the active medium on parametric wave phase conjugation of sound

finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid we describe the application of the rotated staggered grid rsg finite difference technique to the wave equations for anisotropic and viscoelastic media the rsg uses rotated finite difference operators leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position this can be advantageous for modeling wave propagation in anisotropic media or complex media including high contrast discontinuities because no averaging of elastic moduli is needed the rsg can be applied both to displacement stress and to velocity stress finite difference fd schemes whereby the latter are commonly used to model viscoelastic wave propagation with a von neumann style anlysis we estimate the dispersion error of the rsg scheme in general anisotropic media in three different simulation examples all based on previously published problems we demonstrate the application and the accuracy of the proposed numerical approach

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

elastic wave propagation in heterogeneous anisotropic media using the lumped finite element method a numerical technique for wave propagation simulation in 2 d heterogeneous anisotropic structures is presented the scheme is flexible in incorporating arbitrary surface topography inner openings liquid solid boundaries and irregular interfaces and it naturally satisfies the free surface conditions of complex geometrical boundaries the algorithm based on a discretization mesh of triangles and quadrilaterals solves the problem using integral equilibrium around each node instead of satisfying elastodynamic differential equations at each node as in the finite difference method this study is an extension of previous work for the elastic isotropic case besides accounting for anisotropy a simplified quadrilateral grid cell with low computational cost is introduced the transversely isotropic medium with a symmetry axis on the horizontal or vertical plane as typically caused by a system of parallel cracks or fine layers is discussed in detail a 2 d algorithm is presented that can handle the situatio

source independent full waveform inversion of seismic data a set of seismic trace data is collected in an input data set that is first fourier transformed in its entirety into the frequency domain a normalized wavefield is obtained for each trace of the input data set in the frequency domain normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data the normalized wavefield is source independent complex and dimensionless the normalized wavefield is shown to be uniquely defined as the normalized impulse response provided that a certain condition is met for the source this property allows construction of the inversion algorithm disclosed herein without any source or source coupling information the algorithm minimizes the error between data normalized wavefield and the model normalized wavefield the methodology is applicable to any 3 d seismic problem and damping may be easily included in the process

a highly efficient time space domain optimized method with lax wendroff type time discretization for the scalar wave equation abstract in this paper we propose an efficient time space domain optimized optts finite difference scheme to model 2d and 3d scalar wave propagation it adopts piecewise constant interpolation coefficients for several consecutive courant number ranges which avoids the extra time costs caused by loading the coefficients consecutively according to different wave velocities in heterogeneous media the new scheme is capable of suppressing numerical dispersion errors using a large time interval because we introduce a lax wendroff type time discretization which utilizes the wave field information of o m 1 m 2 neighboring grid points in each step with merely o m 1 m 2 computation where m 1 and m 2 are the stencil radii to generate the 2nd order and 4th order spatial derivatives the improved computational efficiency and accuracy of optts are validated by substantial experiments using both theoretical analysis and numerical modeling they show that in heterogeneous models the maximal courant numbers reach 1 0 and 0 85 in 2d and 3d cases respectively without severe temporal error and the scheme is computationally nearly three times faster than its high order counterpart providing a powerful tool for large scale modeling and high resolution imaging

some functions reversing the order of positive operators abstract as a complement to our previous results about the function preserving the operator order we shall show the following reversing version let a and b be positive operators on a hilbert space h satisfying mi b mi 0 let f t be a continuous convex function on m m if g t is a continuous decreasing convex function on m m sp a then for a given 0 a b 0 implies g b i f a where max m t m f m f m f m t m m m g t our main result is to classify complementary inequalities on power means of positive operators as a matter of fact we determine real constants 1 and 1 such that 2 m k s a m k r a 1 m k s a if r s where m k r a j 1 k j a j r 1 r r r 0 is weighted power mean of positive operators a j sp a j m m for some scalars 0 m m and j r such that j 1 k j 1 j 1 k

on the construction and efficiency of staggered numerical differentiators for the wave equation finite difference fd techniques have established themselves as viable tools for the numerical modeling of wave propagation the accuracy and the computational efficiency of numerical modeling can be enhanced by using high order spatial differential operators dablain 1986

computationally efficient three dimensional acoustic finite difference frequency domain seismic modeling in vertical transversely isotropic media with sparse direct solver abstractthe computational burden of frequency domain full waveform inversion fwi of wide aperture fixed spread data is conventionally reduced by limiting the inversion to a few discrete frequencies in this framework frequency domain seismic modeling is performed efficiently for multiple sources by solving the linear system resulting from the discretization of the time harmonic wave equation with the massively parallel sparse direct solver frequency domain seismic modeling based on the sparse direct solver dsfdm requires specific design finite difference stencils of compact support to minimize the computational cost of the lower upper decomposition of the impedance matrix in terms of memory demand and floating point operations a straightforward adaptation of such finite difference stencil originally developed for the isotropic acoustic wave equation is proposed to introduce vertical transverse isotropy vti in the modeling without any extra computational cost the method relies on a fourth ord

elastic least squares reverse time migration abstractwe use elastic least squares reverse time migration lsrtm to invert for the reflectivity images of p and s wave impedances elastic lsrtm solves the linearized elastic wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration numerical tests on synthetic data and field data reveal the advantages of elastic lsrtm over elastic reverse time migration rtm and acoustic lsrtm for our examples the elastic lsrtm images have better resolution and amplitude balancing fewer artifacts and less crosstalk compared with the elastic rtm images the images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic lsrtm images similar to conventional least squares migration elastic lsrtm also requires an accurate estimation of the p and s wave migration velocity models however the problem remains that when there are moderate errors in the velocity model and strong multiples lsrtm

two dimensional method for unconditionally stable elastic wave simulations the stability of the conventional staggered grid finite difference time domain method for elastic wave simulations is limited by the courant condition and material heterogeneity its computational efficiency is significantly hampered when the mesh size is much smaller than a wavelength for geometric modeling accuracy and or with a high impedance contrast an unconditionally stable alternating direction implicit method is proposed to overcome this issue it is based on additive operator splitting that renders tri diagonal matrices for the implicit update of selected field variables theoretical analysis of its stability and grid dispersion error are provided numerical examples further demonstrate its accuracy and efficiency

application of the multiaxial perfectly matched layer m pml to near surface seismic modeling with rayleigh waves perfectly matched layer pml absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling when modeling rayleigh waves with the existence of the free surface the classical pml algorithm becomes unstable when the poisson s ratio of the medium is high numerical errors can accumulate exponentially and terminate the simulation due to computational overflows numerical tests show that the divergence speed of the classical pml has a nonlinear relationship with the poisson s ratio generally the higher the poisson s ratio the faster the classical pml diverges the multiaxial pml m pml attenuates the waves in pmls using different damping profiles that are proportional to each other in orthogonal directions the proportion coefficients of the damping profiles usually vary with the specific model settings if they are set appropriately the m pml algorithm is stable for high poisson s ratio earth models through numerical tests of 40 models with poisson s ratios that varied from 0 10 to 0 49 we found that a constant proportion coefficient of 1 0 for the x and z directional damping profiles is sufficient to stabilize the m pml for all 2d isotropic elastic cases wavefield simulations indicate that the instability of the classical pml is strongly related to the wave phenomena near the free surface when applying the multiaxial technique only in the corners of the pml near the free surface the original m pml technique can be simplified without losing its stability the simplified m pml works efficiently for homogeneous and heterogeneous earth models with high poisson s ratios the analysis in this paper is based on 2d finite difference modeling in the time domain that can easily be extended into the 3d domain with other numerical methods

2d frequency domain elastic full waveform inversion using time domain modeling and a multistep length gradient approach abstractto decouple the parameters in elastic full waveform inversion fwi we evaluated a new multistep length gradient approach to assign individual weights separately for each parameter gradient and search for an optimal step length along the composite gradient direction to perform wavefield extrapolations for the inversion we used parallelized high precision finite element fe modeling in the time domain the inversion was implemented in the frequency domain the data were obtained at every subsurface grid point using the discrete fourier transform at each time domain extrapolation step we also used frequency selection to reduce cycle skipping time windowing to remove the artifacts associated with different source spatial patterns between the test and predicted data and source wavelet estimation at the receivers over the full frequency spectrum by using a fast fourier transform in the inversion the velocity and density reconstructions behaved differently as a low wavenumber tomography for v

efficient and accurate numerical simulation of acoustic wave propagation in a 2d heterogeneous media abstract in this paper a compact fourth order finite difference scheme is derived to solve the 2d acoustic wave equation in heterogenous media the pade approximation is used to obtain fourth order accuracy in both temporal and spatial dimensions and the alternating direction implicit adi technique is used to reduce the computational cost due to the non constant wave velocity the conventional adi method is hard to implement as the algebraic manipulation cannot be used here a novel numerical strategy is proposed in this work so that the compact scheme still maintains fourth order accuracy in time and space the fourth order convergence order was firstly proved by theoretical error analysis then was confirmed by numerical examples it was shown that the proposed method is conditionally stable with a courant friedrichs lewy cfl condition that is comparable to other existing finite difference schemes several numerical examples were solved to demonstrate the efficiency and accuracy of the new algorithm

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

comparison of two algorithms for isotropic elastic p and s vector decomposition the research leading to this paper is supported by the sponsors of the ut dallas geophysical consortium a portion of the computations was done at the texas advanced computing center this paper is contribution no 1270 from the department of geosciences at the university of texas at dallas special thanks go to j brown a shabelansky and two anonymous reviewers for their useful comments and encouragements

decomposed element free galerkin method compared with finite difference method for elastic wave propagation the decomposed element free galerkin defg method is a modified scheme to resolve shortcomings of memory use in element free galerkin efg methods defg solves elastic wave equation problems by alternately updating the stress strain relations and the equations of motion as in the staggered grid finite difference fd method defg requires at most twice the memory space a size comparable to that used by the fd method in addition defg can adopt perfectly matched layer pml absorbing boundary conditions as in the fd case to confirm that defg performs as well as fd a 2d defg under pml boundary conditions was compared with an fd with fourth order spatial accuracy fd4 using an exact analytical solution of ps reflection waves the defg results are as accurate as those obtained by fd4 in a comparison using lamb s problem with eight nodal spaces for the shortest s wavelength defg provides a remarkably accurate rayleigh waveform over a distance of at least 50 wavelengths compared with 10 wavelengths for

dithering of absorbers for efficient finite difference modeling of viscoelastic wave propagation viscoelastic modeling traditionally uses multiple relaxation mechanisms absorbers per wave type in each cell of the numerical grid we show that one absorber per cell suffices when the absorbers of different relaxation frequencies and absorption strengths are dithered in a supercell i e a very small area of several nearby cells the relaxation frequencies used in a supercell can be preselected to cover the bandwidth required in the modeling and distributed locally in a random manner to minimize numerical scattering the absorption strengths however must be found iteratively to yield an average absorption that matches the desired medium over the bandwidth we have tested the dithered absorber method for modeling a variety of 2 d media using a staggered grid finite difference scheme for a uniform medium our result agrees well with the exact analytic solution for inhomogeneous models the dithered absorber solution agrees well with the finite difference solution computed with five absorbers per cel

an optimal absorbing boundary condition for finite difference modeling of acoustic and elastic wave propagation this paper presents an optimal absorbing boundary condition designed to model acoustic and elastic wave propagation in two dimensional and three dimensional media using the finite difference method in this condition extrapolation on the artificial boundaries of a finite difference domain is expressed as a linear combination of wave fields at previous time steps and or interior grids the acoustic and elastic reflection coefficients from the artificial boundaries are derived they are found to be identical to the transfer functions of two cascaded systems the inverse of a causal system and an anticausal system the method makes use of the zeros and poles of reflection coefficients in a complex plane the optimal absorbing boundary condition described in this paper yields on the average reflection coefficients about 10 db smaller than higdon s absorbing boundary condition and about 20 db smaller than reynolds absorbing boundary condition

elastic full waveform inversion for surface topography abstractelastic full waveform inversion efwi attempts to find high resolution model parameters that are able to match observed data exactly by minimizing residuals between the observed and predicted data however irregular surface topography can generate problematic efwi results to resolve this issue an efwi method in the curvilinear system is presented to invert velocities for areas with surface topography this method meshes the regions near the surface topography into body fitted grids and areas off surface regions into rectangular grids we derived the gradient formulas with respect to multiple parameters in the curvilinear system and developed a step search method to calculate the corresponding step lengths wiener filtering is performed on a multiscale decomposition method to invert the velocity from a low to a high wavenumber error analysis using a homogeneous model with a surface topography demonstrates the accuracy and validity of our methods numerical experiments with an elastic hill g

generalization of von neumann analysis for a model of two discrete half spaces the acoustic case evaluating the performance of finite difference algorithms typically uses a technique known as von neumann analysis for a given algorithm application of the technique yields both a dispersion relation valid for the discrete time space grid and a mathematical condition for stability in practice a major shortcoming of conventional von neumann analysis is that it can be applied only to an idealized numerical model that of an infinite homogeneous whole space experience has shown that numerical instabilities often arise in finite difference simulations of wave propagation at interfaces with strong material contrasts these interface instabilities occur even though the conventional von neumann stability criterion may be satisfied at each point of the numerical model to address this issue i generalize von neumann analysis for a model of two half spaces i perform the analysis for the case of acoustic wave propagation using a standard staggered grid finite difference numerical scheme by deriving expres

an unsplit convolutional perfectly matched layer improved at grazing incidence for seismic wave propagation in poroelastic media the perfectly matched layer pml absorbing technique has become popular in numerical modeling in elastic or poroelastic media because of its efficiency in absorbing waves at nongrazing incidence however after numerical discretization at grazing incidence large spurious oscillations are sent back from the pml into the main domain the pml then becomes less efficient when sources are located close to the edge of the truncated physical domain under study for thin slices or for receivers located at a large offset we develop a pml improved at grazing incidence for the poroelastic wave equation based on an unsplit convolutional formulation of the equation as a first order system in velocity and stress we show its efficiency for both nondissipative and dissipative biot porous models based on a fourth order staggered finite difference method used in a thin mesh slice the results obtained are improved significantly compared with those obtained with the classical pml

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

staggered grid spectral element methods for elastic wave simulations in this paper we develop and analyze a new class of spectral element methods for the simulations of elastic wave propagation the major components of the method are the spatial discretization and the choice of interpolation nodes the spatial discretization is based on piecewise polynomial approximation defined on staggered grids the resulting method combines the advantages of both staggered grid based methods and classical non staggered grid based spectral element methods our new method is energy conserving and does not require the use of any numerical flux because of the staggered local continuity of the basis functions our new method also uses radau points as interpolation nodes and the resulting mass matrix is diagonal thus time marching is explicit and is very efficient moreover we give a rigorous proof for the optimal convergence of the method in terms of dispersion we present a numerical study for the numerical dispersion and show that this error is of very high order finally some numerical convergence tests and applications to unbounded domain problems with perfectly matched layer are shown

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

elastic full waveform inversion using the second generation wavelet and an adaptive operator length scheme abstractelastic full waveform inversion fwi can generally yield subsurface model parameters of high spatial resolution however the computational and storage burden is tremendous to mitigate this problem we developed the following three strategies first we derived a time space domain adaptive staggered grid finite difference method the size of the calculation was reduced by using smaller operator lengths for high s wave velocities second we evaluated a multiscale elastic fwi scheme based on the second generation wavelet by which scale decompositions of data and model space were implemented with the scale increasing spatial sampling intervals and time steps became larger therefore the minimum computational cost could be achieved by combining the variable operator length scheme with the multiscale scheme third to economize the memory consumption we extended the efficient boundary storage and checkpointing schemes into elastic fwi only wavefields of several parts in the boundary area betwe

an improved immersed boundary finite difference method for seismic wave propagation modeling with arbitrary surface topography abstractto accurately simulate seismic wave propagation for the purpose of developing modern land data processing tools especially full waveform inversion fwi we have developed an efficient high order finite difference forward modeling algorithm with the capability of handling arbitrarily shaped free surface topography unlike most existing forward modeling algorithms using curvilinear grids to fit irregular surface topography this finite difference algorithm based on an improved immersed boundary method uses a regular cartesian grid system without suffering from staircasing error which is inevitable in a conventional finite difference method in this improved immersed boundary finite difference ibfd algorithm arbitrarily curved surface topography is accounted for by imposing the free surface boundary conditions at exact boundary locations instead of using body conforming grids or refined grids near the boundaries thus greatly reducing the complexity of its preprocessing procedures and the com

stable and high order accurate difference methods for the elastic wave equation in discontinuous media in this paper we develop a stable and systematic procedure for numerical treatment of elastic waves in discontinuous and layered media we consider both planar and curved interfaces where media parameters are allowed to be discontinuous the key feature is the highly accurate and provably stable treatment of interfaces where media discontinuities arise we discretize in space using high order accurate finite difference schemes that satisfy the summation by parts rule conditions at layer interfaces are imposed weakly using penalties by deriving lower bounds of the penalty strength and constructing discrete energy estimates we prove time stability we present numerical experiments in two space dimensions to illustrate the usefulness of the proposed method for simulations involving typical interface phenomena in elastic materials the numerical experiments verify high order accuracy and time stability

finite difference modeling of electromagnetic wave propagation in dispersive and attenuating media realistic modeling of electromagnetic wave propagation in the radar frequency band requires a full solution of maxwell s equations as well as an adequate description of the material properties we present a finite difference time domain fdtd solution of maxwell s equations that allows accounting for the frequency dependence of the dielectric permittivity and electrical conductivity typical of many near surface materials this algorithm is second order accurate in time and fourth order accurate in space conditionally stable and computationally only marginally more expensive than its standard equivalent without frequency dependent material properties empirical rules on spatial wavefield sampling are derived through systematic investigations of the influence of various parameter combinations on the numerical dispersion curves since this algorithm intrinsically models energy absorption efficient absorbing boundaries are implemented by surrounding the computational domain by a thin 2 dominant waveleng

finite difference modeling of biot s poroelastic equations across all frequencies an explicit time stepping finite difference scheme is presented for solving biot s equations of poroelasticity across the entire band of frequencies in the general case for which viscous boundary layers in the pores must be accounted for the time domain version of darcy s law contains a convolution integral it is shown how to efficiently and directly perform the convolution so that the darcy velocity can be properly updated at each time step at frequencies that are low enough compared to the onset of viscous boundary layers no memory terms are required at higher frequencies the number of memory terms required is the same as the number of time points it takes to sample accurately the wavelet being used in practice we never use more than 20 memory terms and often considerably fewer allowing for the convolution makes the scheme even more stable even larger time steps might be used than it is when the convolution is entirely neglected the accuracy of the scheme is confirmed by comparing numerical examples to exact analytic results

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

on a wavelet based method for the numerical simulation of wave propagation a wavelet based method for the numerical simulation of acoustic and elastic wave propagation is developed using a displacement velocity formulation and treating spatial derivatives with linear operators the wave equations are rewritten as a system of equations whose evolution in time is controlled by first order derivatives the linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform using a semigroup approach the discretized solution in time can be represented in an explicit recursive form based on taylor expansion of exponential functions of operator matrices the boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries the wavelet based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1 d domain and to the elastic wave equation with a traction free boundary conditions at a free surface in 2 d spatial media the method can be applied directly to media with plane surfaces and surface topography can be included with the aid of distortion of the grid describing the properties of the medium the numerical results are compared with analytic solutions based on the cagniard technique and show high accuracy the wavelet based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters these examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media

a two scale solution algorithm for the elastic wave equation operator based upscaling is a two scale algorithm that speeds up the solution of the wave equation by producing a coarse grid solution which incorporates much of the local fine scale solution information we present the first implementation of operator upscaling for the elastic wave equation by using the velocity displacement formulation of the three dimensional elastic wave equation basis functions that are linear in all three directions and applying mass lumping the subgrid solve first stage of the two step algorithm reduces to solving explicit difference equations at the second stage of the algorithm we upscale both velocity and displacement by using local subgrid information to formulate the coarse grid problem the coarse grid system matrix is independent of time sparse and banded this paper explores both serial and parallel implementations of the method the main simplifying assumption of the method special zero boundary conditions imposed on coarse blocks in the first stage of the algorithm leads to an easily parallelizable algorithm because very little communication is required between processors in fact for this upscaling implementation calculation of the load vector for the coarse solve dominates the cost of a time step we show that for a homogeneous medium convergence is second order in space and time so long as both the coarse and fine grids are simultaneously refined a series of heterogeneous medium numerical experiments demonstrate that the upscaled solution captures the fine scale fluctuations in the input parameters accurately most notable for use in a seismic inversion algorithm the upscaling algorithm accurately locates the depth of reflectors interface changes

a compact high order alternating direction implicit method for three dimensional acoustic wave equation with variable coefficient abstract efficient and accurate numerical simulation of seismic wave propagation is important in various geophysical applications such as seismic full waveform inversion fwi problem however due to the large size of the physical domain and requirement on low numerical dispersion many existing numerical methods are inefficient for numerical modelling of seismic wave propagation in an heterogeneous media despite the great efforts that have been devoted during the past decades it still remains a challenging task in the development of efficient and accurate finite difference method for multi dimensional acoustic wave equation with variable velocity in this paper we proposed a pade approximation based finite difference scheme for solving the acoustic wave equation in three dimensional heterogeneous media the new method is obtained by combining the pade approximation and a novel algebraic manipulation the efficiency of the new algorithm is further improved through the alternative directional implicit adi method the stability of the new algorithm has been theoretically proved by energy method the new method is conditionally stable with a better courant friedrichs lewy condition cfl condition which has been verified numerically extensive numerical examples have been solved which demonstrated that the new method is accurate efficient and stable

estimation of elastic constants for hti media using gauss newton and full newton multiparameter full waveform inversion abstractin seismic full waveform inversion fwi subsurface parameters are estimated by iteratively minimizing the difference between the modeled and the observed data we have considered the problem of estimating the elastic constants of a fractured medium using multiparameter fwi and modeling naturally fractured reservoirs as equivalent anisotropic media multiparameter fwi although promising remains exposed to a range of challenges one being the parameter crosstalk problem resulting from the overlap of frechet derivative wavefields parameter crosstalk is strongly influenced by the form of the scattering pattern for each parameter we have derived 3d radiation patterns associated with scattering from a range of elastic constants in general anisotropic media then we developed scattering patterns specific to a horizontal transverse isotropic hti medium to draw conclusions about parameter crosstalk in fwi bare gradients exhibit crosstalk as well as artifacts caused by doubly scattered energy in

a method for correcting acoustic finite difference amplitudes for elastic effects we present a new method for correcting the amplitudes of arrivals in an acoustic finite difference simulation for elastic effects in this method we selectively compute an estimate of the error incurred when the acoustic wave equation is used to approximate the behavior of the elastic wave equation this error estimate is used to generate an effective source field in a second acoustic simulation the result of this second simulation is then applied as a correction to the original acoustic simulation the overall cost is approximately twice that of an acoustic simulation but substantially less than the cost of an elastic simulation because both simulations are acoustic no s waves are generated so dispersed converted waves are avoided we tested the characteristics of the method on a simple synthetic model designed to simulate propagation through a strong acoustic impedance contrast representative of sedimentary geology it corrected amplitudes to high accuracy for reflected arrivals over a wide range of incidence angles we also evaluated results from simulations on more complex models that demonstrated that the method was applicable in realistic sedimentary models containing a wide range of seismic contrasts however its accuracy was reduced for wide angle reflections from very high impedance contrasts such as a shallow top salt interface we examined the influence of modeling at coarse grid resolutions in which converted s waves in the equivalent elastic simulation are dispersed these results provide some validation for the accuracy of the method when applied using finite difference grids designed for acoustic modeling the method appears to offer a cost effective means of modeling elastic amplitudes for p wave arrivals in a useful range of velocity models it has several potential applications in imaging and inversion

finite volume evolution galerkin method for hyperbolic conservation laws with spatially varying flux functions we present a generalization of the finite volume evolution galerkin scheme m lukacova medvid ova j saibertov a g warnecke finite volume evolution galerkin methods for nonlinear hyperbolic systems j comp phys 2002 183 533 562 m lukacova medvid ova k w morton g warnecke finite volume evolution galerkin fveg methods for hyperbolic problems siam j sci comput 2004 26 1 30 for hyperbolic systems with spatially varying flux functions our goal is to develop a genuinely multi dimensional numerical scheme for wave propagation problems in a heterogeneous media we illustrate our methodology for acoustic waves in a heterogeneous medium but the results can be generalized to more complex systems the finite volume evolution galerkin fveg method is a predictor corrector method combining the finite volume corrector step with the evolutionary predictor step in order to evolve fluxes along the cell interfaces we use multi dimensional approximate evolution operator the latter is constructed using the theory of bicharacteristics under the assumption of spatially dependent wave speeds to approximate heterogeneous medium a staggered grid approach is used several numerical experiments for wave propagation with continuous as well as discontinuous wave speeds confirm the robustness and reliability of the new fveg scheme

optimal staggered grid finite difference schemes based on the minimax approximation method with the remez algorithm abstractfinite difference fd schemes especially staggered grid fd sfd schemes have been widely implemented for wave extrapolation in numerical modeling whereas the conventional approach to compute the sfd coefficients is based on the taylor series expansion te method which leads to unignorable great errors at large wavenumbers in the solution of wave equations we have developed new optimal explicit sfd esfd and implicit sfd isfd schemes based on the minimax approximation ma method with a remez algorithm to enhance the numerical modeling accuracy starting from the wavenumber dispersion relations we derived the optimal esfd and isfd coefficients by using the ma method to construct the objective functions and solve the objective functions with the remez algorithm we adopt the ma based esfd and isfd coefficients to solve the spatial derivatives of the elastic wave equations and perform numerical modeling numerical analyses indicated that the ma based esfd and isfd schemes can overcome th

an efficient multiscale method for time domain waveform tomography this efficient multiscale method for time domain waveform tomography incorporates filters that are more efficient than hamming window filters a strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain a staggered grid explicit finite difference method with fourth order accuracy in space and second order accuracy in time is used for forward modeling and the adjoint calculation the adjoint method is utilized in inverting for an efficient computation of the gradient directions in the multiscale approach multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography the method is applied successfully to 1d and 2d heterogeneous models it can accurately recover low and high wavenumber components of the velocity models the inversion result for the 2d model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional single scale method

finite difference modeling of wave propagation in a fluid solid configuration finite difference fd techniques are widely used to model wave propagation through complex structures two main sources of error can be identified 1 from numerical dispersion and numerical anisotropy and 2 by modeling the response of internal grid boundaries conventional discretization criteria to reduce the effects of numerical dispersion and numerical anisotropy have long been established 5 8 gridpoints per wavelength for a fourth order accurate fd scheme we analyze the second source of errors comparing different staggered grid fd solutions to the cagniard de hoop solution in models with fluid solid contacts our results confirm that it is sufficient to rely on conventional discretization criteria if the fluid solid interface is aligned with the grid if accurate modeling of the scholte wave is required then a new imaging method we propose should be used to allow for conventional sampling of the wavefield to minimize numerical dispersion however for an interface not aligned with the grid irregular interfaces a spatial sampling of at least 15 gridpoints per minimum wavelength is required to obtain acceptable results particularly in seismic seabed applications where scholte waves may need to be modeled more accurately

an optimal nearly analytic discrete weighted runge kutta discontinuous galerkin hybrid method for acoustic wavefield modeling abstractthe newly developed optimal nearly analytic discrete onad and the weighted runge kutta discontinuous galerkin wrkdg methods can effectively suppress the numerical dispersion caused by discretizing wave equations but it is difficult for onad to implement on flexible meshes whereas the wrkdg has high computational cost for wavefield simulations we have developed a new hybrid algorithm by combining the onad method with the wrkdg method in this hybrid algorithm the computational domain was split into several subdomains in which the subdomain for the onad method used regular cartesian grids whereas the subdomain for the wrkdg method used triangular grids the hybrid method was at least third order spatially accurate we have applied the proposed method to simulate the scalar wavefields for different models including a homogeneous model a rough topography model a fracture model and a cave model the numerical results found that the hybrid method can deal with complicated geometrical struc

computational homogenization of elasticity on a staggered grid summary in this article we propose to discretize the problem of linear elastic homogenization by finite differences on a staggered grid and introduce fast and robust solvers our method shares some properties with the fft based homogenization technique of moulinec and suquet which has received widespread attention recently because of its robustness and computational speed these similarities include the use of fft and the resulting performing solvers the staggered grid discretization however offers three crucial improvements firstly solutions obtained by our method are completely devoid of the spurious oscillations characterizing solutions obtained by moulinec suquet s discretization secondly the iteration numbers of our solvers are bounded independently of the grid size and the contrast in particular our solvers converge for three dimensional porous structures which cannot be handled by moulinec suquet s method thirdly the finite difference discretization allows for algorithmic variants with lower memory consumption more precisely it is possible to reduce the memory consumption of the moulinec suquet algorithms by 50 we underline the effectiveness and the applicability of our methods by several numerical experiments of industrial scale copyright 2015 john wiley sons ltd

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

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

uniform dispersion reduction schemes for the one dimensional wave equation in isotropic media finite difference fd methods for the wave equation in general suffer from numerical dispersion although fd methods based on accuracy give good dispersion at low frequencies waves tend to disperse for higher wavenumber in this work we will give a unified methodology to derive dispersion reduction fd schemes for the one dimensional wave equation and this new method can reduce dispersion error uniformly for all wavenumbers up to the nyquist stability criteria are given and stability analysis is done for each generated scheme

a symplectic procedure for two dimensional coupled elastic wave equations using radial basis functions interpolation abstract this study proposes a meshless symplectic algorithm for two dimensional coupled elastic waves which satisfy the equations of linear isotropic elasticity on a rectangular domain with appropriate initial and boundary conditions this method is based on a combination of radial basis functions rbfs collocation and geometric integrators the method involves two steps first the spatial discretization by using rbfs interpolation transforms the seismic waves into a finite dimensional hamiltonian system then evolves the semi discretized system by using symplectic integrators furthermore the conservations of energy and symplectic structure are investigated the truncation error and global error of the method are estimated numerical experiments show that the scheme is efficient easy to implement with the scattered knots and possesses a good behavior in the long time integration