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

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

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

types in the n adic tree and minimal analytic gaps abstract we develop some technical tools for the study of the record structure of the n adic tree and we give as an application the list of the nine minimal analytic gaps

on block triangular preconditioned iteration methods for solving the helmholtz equation abstract to further improve the efficiency of solving the helmholtz equation in heterogeneous media with large wavenumber the krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two by two linear system derived from the discrete helmholtz equation we further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods the superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods we also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber roughly furthermore the computational wave fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm

optimization over the efficient set abstract in this paper we study the problem of optimizing a linear function over the set of efficient solutions for a vector maximization problem this problem arises whenever a linear function is available which acts as a criterion for measuring the importance of or for discriminating among the efficient alternatives that are available using the concepts of an efficient direction and of a direction of recession we develop necessary and sufficient conditions for this problem to be unbounded we also present necessary and sufficient conditions for efficient and for arbitrary solutions of the underlying vector maximization problem to be optimal solutions for this problem some of these conditions are geometric in nature others algebraic the algebraic conditions suggest potential computational procedures for finding an optimal solution to the problem finally we give conditions under which the set of optimal solutions possesses certain special properties

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

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

assessment of rod behaviour theories used in spectral finite element modelling abstract in this work different theories of rods have been discussed and compared the investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves these are various single two mode and three mode theories including the elementary classical love and mindlin herrmann approaches as well as new two three and four mode theories proposed by the authors dispersion curves associated with each theory obtained by the use of hamilton s principle have been presented and discussed in the paper the investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application great attention has been paid on properties limitations as well as difficulties associated with the use of the theories the results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures

boundary element and finite element coupling for aeroacoustics simulations we consider the scattering of acoustic perturbations in the presence of a flow we suppose that the space can be split into a zone where the flow is uniform and a zone where the flow is potential in the first zone we apply a prandtl glauert transformation to recover the helmholtz equation the well known setting of boundary element method for the helmholtz equation is available in the second zone the flow quantities are space dependent we have to consider a local resolution namely the finite element method herein we carry out the coupling of these two methods and present various applications and validation test cases the source term is given through the decomposition of an incident acoustic field on a section of the computational domain s boundary validations against analytic another numerical method and measurements on different test cases are presented

3d finite difference synthetic acoustic log in cylindrical coordinates a finite difference method of numerical simulation of sonic logging has been developed and implemented the very general statement is considered the surrounding medium is allowed to be 3d heterogeneous and a source can be located at any point inside or outside the well to provide a maximally precise description of the sharpest interface of the problem the interface between the well and surrounding formations we use cylindrical coordinates with the axis directed along the well in order to avoid an excessive azimuth inflation of grid cells with an increase of radius we perform periodical refinement of the grid step in the azimuth direction in order to truncate the area of computations perfectly matched layer pml for cylindrical coordinates is developed and implemented its main advantage in comparison with other approaches is an extremely low level of artificial reflections and the absence of necessity to perform splitting of variables in the azimuth direction based on this numerical method a software oriented for the use of parallel computations is developed and implemented under message passing interface library results of numerical experiments for a well with completion embedded within layered elastic background with a vertical planar crack are presented and discussed

3d rotated and standard staggered finite difference solutions to biot s poroelastic wave equations stability condition and dispersion analysis a fourth order in space and second order in time 3d staggered sg and rotated staggered grid rsg method for the solution of biot s equation are presented the numerical dispersion and stability conditions are derived using a von neumann analysis the exact stability condition is calculated from the roots of a 12th order polynomial and therefore no nontrivial expression exists to overcome this a 1d stability condition is usually generalized to three dimensions it is shown that in certain cases the 1d approximate stability condition is violated by a 3d sg method the rsg method obeys the approximate 1d stability condition for the material properties and spatiotemporal scales in the examples shown both methods have been verified against an analytical solution for an infinite homogeneous porous medium with a misfit error of less than 0 5 a free surface has been implement ed to test the accuracy of this boundary condition it also serves as a test of the methods to include high material contrasts th

multidirectional vector based elastic reverse time migration and angle domain common image gathers with approximate wavefield decomposition of p and s waves abstractelastic reverse time migration e rtm has limitations when the migration velocities contain strong contrasts first the traditional scheme of p s wave mode separation is based on helmholtz s equations which ignore the conversion between p and s waves at the current separation time thus it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition second the vector based imaging condition needs to use the reflection image normal and it also cannot give the correct polarity of the pp image in all possible conditions third the angle domain common image gathers adcigs calculated using the poynting vectors pvs do not consider the wave interferences that happen at each reflector therefore smooth models are often used for e rtm we relax this condition by proposing an improved data flow that involves three new contributions the first contribution is an improved system of p s wave mode

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

an efficient symplectic reverse time migration method using a local nearly analytic discrete operator in acoustic transversely isotropic media with a vertical symmetry axis abstractreverse time migration rtm has become an extremely important imaging method due to its ability of handling complicated velocity models in recent years there has been an increasing need to perform rtm with anisotropic velocity models such as transversely isotropic media with a vertical symmetry axis vti finite difference fd is one of the most widely used numerical methods in rtm however it often suffers from serious numerical dispersion resulting in significant loss of imaging resolution over the years a nearly analytic discrete operator has been developed to approximate the high order spatial partial differential operators which is effective in suppressing numerical dispersion using this operator we developed a nearly analytic symplectic nas method to numerically solve acoustic vti wave equations which preserves the symplectic structure we have investigated its properties including the stability condition numerical dispersion relation and computational efficiency also we a

tukey classification of some ideals on ω and the lattices of weakly compact sets in banach spaces abstract we study the lattice structure of the family of weakly compact subsets of the unit ball b x of a separable banach space x equipped with the inclusion relation this structure is denoted by k b x and also with the parametrized family of almost inclusion relations k l e b x where e 0 this structure is denoted by ak b x tukey equivalence between partially ordered sets and a suitable extension to deal with ak b x are used assuming the axiom of analytic determinacy we prove that separable banach spaces fall into four categories namely k b x is equivalent either to a singleton or to or to the family k q of compact subsets of the rational numbers or to the family c of all finite subsets of the continuum also under the axiom of analytic determinacy a similar classification of ak b x is obtained for separable banach spaces not containing l 1 we prove in zfc that k b x ak b x are equivalent to either 0 k q or c the lattice structure of the family of all weakly null subsequences of an unconditional basis is also studied

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

spectral element method for elastic and acoustic waves in frequency domain numerical techniques in time domain are widespread in seismic and acoustic modeling in some applications however frequency domain techniques can be advantageous over the time domain approach when narrow band results are desired especially if multiple sources can be handled more conveniently in the frequency domain moreover the medium attenuation effects can be more accurately and conveniently modeled in the frequency domain in this paper we present a spectral element method sem in frequency domain to simulate elastic and acoustic waves in anisotropic heterogeneous and lossy media the sem is based upon the finite element framework and has exponential convergence because of the use of gll basis functions the anisotropic perfectly matched layer is employed to truncate the boundary for unbounded problems compared with the conventional finite element method the number of unknowns in the sem is significantly reduced and higher order accuracy is obtained due to its spectral accuracy to account for the acoustic solid interaction the domain decomposition method ddm based upon the discontinuous galerkin spectral element method is proposed numerical experiments show the proposed method can be an efficient alternative for accurate calculation of elastic and acoustic waves in frequency domain

accurate and efficient seismic modeling in random media the optimum method for seismic modeling in random media must 1 be highly accurate to be sensitive to subtle effects of wave propagation 2 allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths this is necessary to obtain statistically meaningful overall attributes of wavefields high order staggered grid finite difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling investigations for random media reveal that both methods lead to nearly identical wavefields the small differences can be attributed mainly to differences in the numerical dispersion this result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous the differentiator can be longer than the medium scale leng

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

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

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

3 d finite difference modeling of elastic waves in borehole environments full wavefield sonic logs may be simulated for arbitrarily complicated three dimensional 3 d borehole environments using 3 d elastic finite differences to ensure reliability through large contrasts in poisson s ratio across mud casing lithology contacts a staggered grid finite difference formulation is used cylindrical symmetry is not assumed so the responses of features such as asymmetrical washouts and dipping structure are easily obtained when features are asymmetrical seismic responses vary significantly at various points around the hole circumference at any depth even for simple hole geometries observed responses are complicated because of coupling between waves inside and outside the hole observations are also sensitive to the source receiver separation output formats include fixed time snapshots of displacement divergence and curl and seismograms for the center and edge of a borehole this allows detailed arrival identification and interpretation to our knowledge this is the most

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

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

an algorithm for optimizing over the weakly efficient set abstract in this paper a bisecting search algorithm is developed for solving the problem p of optimizing a linear function over the set of weakly efficient solutions of a multiple objective linear program we show that problem p can arise in a variety of practical situations the algorithm for solving problem p is guaranteed to find an optimal or approximately optimal solution for the problem in a finite number of steps using a fortran computer code called conmin as an aid we have solved ten test problems using our proposed algorithm this preliminary computational experience seems to indicate that the algorithm is quite practical for relatively small problems

elastic wavefield separation based on the helmholtz decomposition abstractdivergence and curl operators used for the decomposition of p and s wave modes in elastic reverse time migration rtm change the amplitudes units and phases of extrapolated wavefields i separate the p and s waves in elastic media based on the helmholtz decomposition the decomposed wavefields based on this approach have the same amplitudes units and phases as the extrapolated wavefields to avoid expensive multidimensional integrals in the helmholtz decomposition i introduce a fast poisson solver to efficiently solve the vector poisson s equation this fast algorithm allows us to reduce computational complexity from o n2 to o n log n where n is the total number of grid points because the decomposed p and s waves are vector fields i use vector imaging conditions to construct pp ps ss and sp images several 2d numerical examples demonstrate that this approach allows us to accurately and efficiently decompose p and s waves in elastic media in addition elastic rtm images based

a density version of the halpern lauchli theorem abstract we prove a density version of the halpern lauchli theorem this settles in the affirmative a conjecture of r laver specifically let us say that a tree t is homogeneous if t has a unique root and there exists an integer b 2 such that every t t has exactly b immediate successors we show that for every d 1 and every tuple t 1 t d of homogeneous trees if d is a subset of the level product of t 1 t d satisfying lim sup n d t 1 n t d n t 1 n t d n 0 then there exist strong subtrees s 1 s d of t 1 t d having a common level set such that the level product of s 1 s d is a subset of d

absorbing boundary condition for the elastic wave equation velocity stress formulation this paper describes an absorbing boundary condition for finite difference modeling of elastic wave propagation in two and three dimensions the boundary condition is particularly effective for obliquely incident waves typically quite troublesome for absorbing boundaries analytical predictions of the boundary reflection coefficients of a few percent or less for angles of incidence up to 89 are verified in example finite difference applications the algorithm is appropriate for use in a velocity stress finite difference vs fd formulation it is computationally simpler than a similar absorbing boundary given previously for the standard displacement formulation a second algorithm is presented which may be advantageous when the boundary of interest is exposed to strong evanescent waves both algorithms require that the adjacent elastic medium be locally homogeneous

ramsey families of subtrees of the dyadic tree we show that for every rooted finitely branching pruned tree t of height w there exists a family f which consists of order isomorphic to t subtrees of the dyadic tree c 0 1 n with the following properties i the family f is a g subset of 2 c ii every perfect subtree of c contains a member of f iii if k is an analytic subset of f then for every perfect subtree s of c there exists a perfect subtree s of s such that the set a f a c s either is contained in or is disjoint from k

a dual form of ramsey s theorem abstract let k where is the set of all natural numbers ramsey s theorem deals with colorings of the k element subsets of our dual form deals with colorings of the k element partitions of let k respectively be the set of all partitions of having exactly k respectively infinitely many blocks given x let x k be the set of all y k such that y is coarser than x dual ramsey theorem if k c 0 c t 1 where each c i is borel then there exists x such that x k c i for some i l dual galvin prikry theorem same as before with k replaced by we also obtain dual forms of theorems of ellentuck and mathias our results also provide an infinitary generalization of the graham rothschild parameter set theorem trans amer math soc 159 1971 257 292 and a new proof of the halpern lauchli theorem trans amer math soc 124 1966 360 367

an improved optimal nearly analytical discretized method for 2d scalar wave equation in heterogeneous media based on the modified nearly analytical discrete operator abstractconventional finite difference methods will encounter strong numerical dispersion on coarse grids or when higher frequencies are used the nearly analytic discrete method nadm and its improved version developed recently can suppress numerical dispersion efficiently however the nadm is imperfect especially in long time seismic wave propagation simulation to overcome this limitation by minimizing the energy error function we have developed a modified optimal nearly analytical discretized mnad method the stencil of mnad for the 2d wave equation is a novel diamond stencil mnad has two major advantages the first one is that mnad can suppress numerical dispersion effectively on coarse grids which significantly improved computational efficiency and reduced memory demands the second advantage is that the energy error of mnad is much smaller after long time simulation the numerical dispersion analysis shows that the maximum phase velocity error was 5 92 even if only two sampling points w

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

elastic imaging with exact wavefield extrapolation for application to ocean bottom 4c seismic data abstracta central component of imaging methods is receiver side wavefield backpropagation or extrapolation in which the wavefield from a physical source scattered at any point in the subsurface is estimated from data recorded by receivers located near or at the earth s surface elastic reverse time migration usually accomplishes wavefield extrapolation by simultaneous reversed time injection of the particle displacements or velocities recorded at each receiver location into a wavefield modeling code here we formulate an exact integral expression based on reciprocity theory that uses a combination of velocity stress recordings and quadrupole dipole backpropagating sources rather than the commonly used approximate formula involving only particle velocity data and dipole backpropagating sources the latter approximation results in two types of nonphysical waves in the scattered wavefield estimate first each arrival contained in the data is injected upward and downward rather than unidirectionally as

time space domain mesh free finite difference based on least squares for 2d acoustic wave modeling abstractfinite difference fd methods approximate derivatives through a weighted summation of function values from neighboring nodes traditionally these neighboring nodes are assumed to be distributed regularly such as in square or rectangular lattices to improve geometric flexibility one option is to develop fd in a mesh free discretization in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra etc these mesh free fds have had successful applications especially in computational geoscience however they are all space domain fd schemes in which fd coefficients are derived by approximating spatial derivatives individually in the space domain for acoustic wave modeling it has been proven that space domain fd methods normally have higher dispersion error than time space domain fd methods which determine fd coefficients by approximating the time space domain disp

two dimensional nonlinear inversion of seismic waveforms numerical results the nonlinear problem of inversion of seismic waveforms can be set up using least squares methods the inverse problem is then reduced to the problem of minimizing a lp nonquadratic function in a space of many 104to106 variables using gradient methods leads to iterative algorithms each iteration implying a forward propagation generated by the actual sources a backward propagation generated by the data residuals acting as if they were sources and a correlation at each point of the space of the two fields thus obtained which gives the updated model the quality of the results of any inverse method depends heavily on the realism of the forward modeling finite difference schemes are a good choice relative to realism because although they are time consuming they give excellent results numerical tests performed with multioffset synthetic data from a two dimensional model prove the feasibility of the approach if only surface recorded reflections are used the high spatial frequency content of the

a disjoint unions theorem for trees abstract we prove an infinitary disjoint unions theorem for level products of trees to implement the proof we develop a hales jewett type result for words indexed by a level product of trees

an improved rotated staggered grid finite difference method with fourth order temporal accuracy for elastic wave modeling in anisotropic media abstract the rotated staggered grid rsg finite difference method is a powerful tool for elastic wave modeling in 2d anisotropic media where the symmetry axes of anisotropy are not aligned with the coordinate axes we develop an improved rsg scheme with fourth order temporal accuracy to reduce the numerical dispersion associated with prolonged wave propagation or a large temporal step size the high order temporal accuracy is achieved by including high order temporal derivatives which can be converted to high order spatial derivatives to reduce computational cost dispersion analysis and numerical tests show that our method exhibits very low temporal dispersion even with a large temporal step size for elastic wave modeling in complex anisotropic media using the same temporal step size our method is more accurate than the conventional rsg scheme our improved rsg scheme is therefore suitable for prolonged modeling of elastic wave propagation in 2d anisotropic media

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

2d isotropic elastic gaussian beam migration for common shot multicomponent records gaussian beam migration is flexible and adaptable for imaging geologically complicated areas it avoids some limitations of traditional ray based migration approaches such as amplitude singularity in caustic zones and inability to image multiple arrivals previous studies on gaussian beam migration mainly focused on acoustic media we develop a 2d isotropic elastic gaussian beam migration scheme for common shot multicomponent records the proposed method extrapolates p and s mode wavefields simultaneously using the kirchhoff helmholtz integral solution of the isotropic elastodynamic equation we separate the extrapolated wavefields into compressional and shear modes using the helmholtz decomposition and then implement a modified dot product imaging condition this approach enables us to produce clear pp images and avoid polarity reversal issues for ps images in addition based on the theory of wavefield approximation in the effective vicinity of central rays we derive a formula to compute the propaga

window controlled cmp crosscorrelation analysis for surface waves in laterally heterogeneous media cmp crosscorrelation cmpcc analysis of surface waves enhances lateral resolution of surface wave analyses we found the technique of window controlled cmpcc analysis which applies two kinds of spatial windows to further improve the lateral resolution of cmpcc analysis first a spatial weighting function given by the number of crosscorrelation pairs is applied to cmpcc gathers because the number of crosscorrelation pairs is concentrated near the cmp the lateral resolution in extracting dispersion curves on cmps can be improved second crosscorrelation pairs with longer receiver spacing are excluded to further improve lateral resolution although removing crosscorrelation pairs generally decreases the accuracy of phase velocity estimations the required accuracy to estimate phase velocities is maintained by considering the wavenumber resolution defined for given receiver configurations when applied to a synthetic data set simulating a laterally heterogeneous structure window controlled cmpcc analysis improved the retrieval of the lateral variation in local dispersion curves beneath each cmp we also applied the method to field seismic data across a major fault the window controlled cmpcc analysis improved lateral variations of the inverted s wave velocity structure without degrading the accuracy of s wave velocity estimations we discovered that window controlled cmpcc analysis is effective in improving lateral resolution of dispersion curve estimations with respect to the original cmpcc analysis

love wave waveform inversion in time domain for shallow shear wave velocity abstracthigh frequency surface wave techniques are widely used to estimate s wave velocity of near surface materials surface wave methods based on inversions of dispersion curves are only suitable to laterally homogeneous or smoothly laterally varying heterogeneous earth models due to the layered model assumption during calculation of dispersion curves waveform inversion directly fits the waveform of observed data and it can be applied to any kinds of earth models we have used the love wave waveform inversion in the time domain to estimate near surface s wave velocity we used the finite difference method as the forward modeling method the source effect was removed by the deconvolution technique which made our method independent of the source wavelet we defined the difference between the deconvolved observed and calculated waveform as the misfit function we divided the model into different sizes of blocks depending on the resolution of the love waves and we updated the s wave velocity of each blo

wave propagation in heterogeneous media effects of fine scale heterogeneity we study the multiple scattering effects caused by fine scale heterogeneity for this purpose it is unreasonable to rely on a linearization of the dependency of the wavefield in the parameters that describe the medium therefore the only tools that correctly model wave propagation are based on the numerical solution of the two way wave equation by finite differences or finite elements a fine grid provides a straightforward approach to account for fine scale heterogeneity in this situation there is no need for high order schemes variational methods allow us to exhibit a numerical scheme that accounts for heterogeneity and that is by construction stable provided a stability condition is fulfilled this condition is a sufficient stability condition contrary to the classical necessary stability conditions in addition general mathematical results prove the finite difference solution is close to the solution of the wave equation when the grid is fine enough the multiple scattering effects caused by fine scale heterogeneity are very important in particular we observe that imaging the so computed synthetic data by standard migration techniques that assume a linearization of the above mentioned dependency shows a strongly noise corrupted image this illustrates the importance of preprocessing data to remove the effect of multiple scattering we try to improve the signal to noise ratio by removing multiples related to the free surface although significant noise reduction is achieved even more sophisticated preprocessing is required to obtain a clear image of the subsurface

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

simulation of acoustic wavefields in heterogeneous media a robust method for automatic suppression of numerical dispersion numerical dispersion limits the application of numerical simulation methods for solving the acoustic wave equation in largescalecomputation thenearlyanalyticdiscretemethodnadm and its improved version for suppressing numerical dispersion weredevelopedrecently thisnewmethodisarefinementoftwo previousmethodsandfurtherincreasestheabilityofsuppressing numerical dispersion for modeling acoustic wave propagation in 2d heterogeneous media which uses acoustic wave displacement particle velocity and their gradients to restructure the acoustic wavefield via the truncated taylor expansion and the high order interpolation approximate method for the method proposed weinvestigateitsimplementationandcompareitwith the higher order lax wendroff correction lwc scheme the original nearly analytic discrete method nadm and its improved version with regard to numerical dispersion computational costs and computer storage requirements the numerical dispersionrelationsprovidedbytherefinedalgorithmfor1dand 2d cases are analyzed as well as the numerical results obtained bythismethodagainsttheexactsolutionforthe2dacousticcase numerical results show that the refined method gives no visible numericaldispersionforverylargespatialgridincrements itcan simulatehigh frequencywavepropagationforagivengridinterval and automatically suppress the numerical dispersion when the acoustic wave equation is discretized when too few samples per wavelength are used or when models have large velocity contrasts unlike the high order lwc methods our present methodcansavesubstantialcomputationalcostsandmemoryrequirements because very large grid increments can be used the refined method can be used for the simulation of large scale wavefields

numerical analysis of a narrow angle one way elastic wave equation and extension to curvilinear coordinates in this paper we review the finite difference implementation of a narrow angle one way vector wave equation for elastic 3d media extrapolation is performed in the frequency domain where the second order accurate lateral spatial difference operators are sufficiently accurate for narrow angle propagation we perform a numerical analysis of the finite difference scheme to highlight the stability and dispersion characteristics the von neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths source pulse spectral content and angular range of forward propagation of interest although the algorithm is reasonable its computational efficiency is limited by the second order accurate extrapolation step therefore the extrapolation scheme can be improved we extend the cartesian narrow angle formulation to curvilinear coordinates where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the cartesian propagator an example of curvilinear extrapolation for a simple model consisting of a high velocity sphere within a homogeneous background velocity structure shows that the narrow angle propagator is capable of modeling frequency dependent geometric spreading and diffraction effects in curvilinear coordinates

elimination of numerical dispersion in finite difference modeling and migration by flux corrected transport finite difference acoustic wave modeling and reverse time depth migration based on the full wave equation are general approaches that can take into account arbitrary variations in velocity and density and can handle turning waves as well however conventional finite difference methods for solving the acoustic or elastic wave equation suffer from numerical dispersion when too few samples per wavelength are used the flux corrected transport fct algorithm adapted from hydrodynamics reduces the numerical dispersion in finite difference wavefield continuation the flux correction procedure endeavors to incorporate diffusion into the wavefield continuation process only where needed to suppress the numerical dispersion incorporating the flux correction procedure in conventional finite difference modeling or reverse time migration can provide finite difference solutions with no numerical dispersion even for impulsive sources the fct correction which can be applied to finite difference approximations of any order in space and time is an efficient alternative to use for finite difference approximations of increasing order through demonstrations of modeling and migration on both synthetic and field data we show the benefits of the fct algorithm as well as its inability to fully recover resolution lost when the spatial sampling becomes too coarse

unstructured grid method for stress wave propagation in elastic media a new numerical method named unstructured grid method is presented to calculate two dimensional stress wave propagation problems in elastic media this method is based on the dynamic equilibrium equations of the investigated lumps that are formed among the auxiliary unstructured grids the approach to the solution is obtained by calculating the node displacements and the central point stresses of the spatial grids alternately the stability criterion of the method is given and tested numerically analysis of dispersion for numerical phase velocity is studied the rule is provided to determine the spatial step size of discretization mesh so as to ensure the numerical accuracy of the unstructured grid method numerical results are compared with the available exact solutions and with the solutions of central difference method for demonstrating the good accuracy of the new method the unstructured grid method possesses higher calculating speed than the central difference method finally the dynamic behavior of a square plate with a hole under an impact loading is researched

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

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

time space domain dispersion relation based finite difference method with arbitrary even order accuracy for the 2d acoustic wave equation high order finite difference fd methods have been widely used for numerical solution of acoustic wave equations it has been reported that the modeling accuracy is of 2nd order when the conventional 2m th order space domain fd and the 2nd order time domain fd stencils are directly used to solve the acoustic wave equation under the same discretization the present version of time space domain dispersion relation based fd method can improve the accuracy from 2nd order to 2m th order along eight directions for the 2d acoustic wave equation to increase the accuracy further we propose a new fd stencil for 2d acoustic wave equation modeling this new time space domain dispersion relation based fd stencil can reach the same arbitrary even order accuracy along all directions and is more accurate and more stable than the conventional one for the same m dispersion analysis and modeling examples demonstrate its advantages

necessary conditions for suboptimization over the weakly efficient set associated to generalized invex multiobjective programming abstract the problem of optimizing a real valued function over the weakly efficient set associated to a multiobjective program is examined two types of necessary conditions for suboptimizing the solution of the general problem are presented which are somewhat similar to theorem 4 1 of s bolintineanu j optim theory appl 78 1993 579 598 but under much weaker assumptions

stable and accurate wave propagation in discontinuous media a time stable discretization is derived for the second order wave equation with discontinuous coefficients the discontinuity corresponds to inhomogeneity in the underlying medium and is treated by splitting the domain each homogeneous sub domain is discretized using narrow diagonal summation by parts operators and then patched to its neighbors by using a penalty method leading to fully explicit time integration this discretization yields a time stable and efficient scheme the analysis is verified by numerical simulations in one dimension using high order finite difference discretizations and in three dimensions using an unstructured finite volume discretization

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

a symplectic method for structure preserving modelling of damped acoustic waves in this paper a symplectic method for structure preserving modelling of the damped acoustic wave equation is introduced the equation is traditionally solved using non symplectic schemes however these schemes corrupt some intrinsic properties of the equation such as the conservation of both precision and the damping property in long term calculations in the method presented an explicit second order symplectic scheme is used for the time discretization whereas physical space is discretized by the discrete singular convolution differentiator the performance of the proposed scheme has been tested and verified using numerical simulations of the attenuating scalar seismic wave equation scalar seismic wave field modelling experiments on a heterogeneous medium with both damping and high parameter contrasts demonstrate the superior performance of the approach presented for suppression of numerical dispersion long term computational experiments display the remarkable capability of the approach presented for long time simulations of damped acoustic wave equations promising numerical results suggest that the approach is suitable for high precision and long time numerical simulations of wave equations with damping terms as it has a structure preserving property for the damping term

a comparative study of finite element and spectral element methods in seismic wavefield modeling the computational accuracy and efficiency of finite element method and spectral element method sem are investigated thoroughly in time domain elastic wavefield modeling the diagonal mass matrices of the fem and sem free from matrix inversion are compared comprehensively by making full use of the mass lumped technique and quadrature rules we investigate the fem and sem based respectively on quadrilateral with the polynomials of degrees one and two and on triangular grids with the polynomials of degrees one and three generally the numerical solutions based on quadrilateral grids have a higher precision than those computed on triangular grids when the same order of polynomials is used the fem has a comparable accuracy to the sem with the same number of interpolant points in view of the triangular and quadrilateral sems the former suffers from larger computational costs and relatively lower accuracy compared with the latter furthermore the convergence study proves that the triangular sem produces consistently larger errors than the quadrilateral sem for any order and element sizes however the triangular sem can adapt to arbitrary complex geometries effectively in terms of efficiency the fem has an efficiency comparable with the sem on condition that the order of interpolation polynomials is identical in addition a perfectly matched layer pml boundary condition in variational form is deduced by introducing four intermediate variables in frequency domain the pml avoids convolution calculation and obtains an exact solution through inverse fourier transform in time domain the numerical examples verify the validity and effectiveness of the pml

multiscale modeling of acoustic wave propagation in 2d media abstractconventional finite difference methods produce accurate solutions to the acoustic and elastic wave equation for many applications but they face significant challenges when material properties vary significantly over distances less than the grid size this challenge is likely to occur in reservoir characterization studies because important reservoir heterogeneity can be present on scales of several meters to ten meters here we describe a new multiscale finite element method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine and coarse scale grids the wave equation is solved on a coarse grid but it uses basis functions that are generated from the fine grid and allow the representation of the fine scale variation of the wavefield on the coarser grid time stepping also takes place on the coarse grid providing further speed gains another important property of the method is that the basis functions are only computed once and time savin

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion

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

unconditional families in banach spaces it is shown that for every separable banach space x with non separable dual the space x contains an unconditional family of size x the proof is based on ramsey theory for trees and finite products of perfect sets of reals among its consequences it is proved that every dual banach space has a separable quotient

reverse time migration in acoustic vti media using a high order stereo operator abstractwith the capability of handling complicated velocity models reverse time migration rtm has become a powerful imaging method improving imaging accuracy and computational efficiency are two significant but challenging tasks in the applications of rtm despite being the most popular numerical technique applied in rtm finite difference fd methods often suffer from undesirable numerical dispersion leading to a noticeable loss of imaging resolution a new and effective fd operator called the high order stereo operator has been developed to approximate the partial differential operators in the wave equation from which a numerical scheme called the three step stereo method tsm has been developed and has shown effectiveness in suppressing numerical dispersion numerical results show that compared with the conventional numerical methods such as the lax wendroff correction lwc scheme and the staggered grid sg fd method this new method significantly reduces numerical dispersion and computati

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