an optimal 9 point finite difference frequency space 2 d scalar wave extrapolator in this study a new finite difference technique is designed to reduce the number of grid points needed in frequency space domain modeling the new algorithm uses optimal nine point operators for the approximation of the laplacian and the mass acceleration terms the coefficients can be found by using the steepest descent method so that the best normalized phase curves can be obtained this method reduces the number of grid points per wavelength to 4 or less with consequent reductions of computer memory and cpu time that are factors of tens less than those involved in the conventional second order approximation formula when a band type solver is used on a scalar machine

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

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

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

inhomogeneous differential realization boson fermion realization of lie superalgebras and their indecomposable representations an inhomogeneous differential realization ihdr of lie superalgebras which plays an important role in the search for quasi exactly solvable problems qesp of quantum mechanics is studied the corresponding inhomogeneous boson fermion realization ihbfr of lie superalgebras is obtained by making use of the ihbfr a new kind of indecomposable and irreducible representations of lie superalgebras is studied on the universal enveloping algebra of heisenberg weyl superalgebra and on its subspaces and quotient spaces the finite dimensional representations of lie superalgebras are naturally obtained on the subspaces of fock space as examples the ihdr of lie superalgebra spl m 1 is given and the indecomposable and irreducible representations of lie superalgebra su 2 1 are studied in detail

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

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

three dimensional elastic modeling by the fourier method earlier work on three dimensional forward modeling is extended to elastic waves using the equations of con servation of momentum and the stress strain relations for an isotropic elastic medium undergoing infinitesimal deformation in addition to arbitrary compressional or p wave velocity and density variation in lateral and vertical directions elastic modeling permits shear or s wave velocity variation as well the elastic wave equation is solved using a generalization of the method for the acoustic case computation of each time step begins by computing six strain components by per forming nine spatial partial differentiation operations on the three displacement components from the pre vious time step the six strains and two lame constants are linearly combined to yield six stress components

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

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

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

staggered mesh for the anisotropic and viscoelastic wave equation computation of the spatial derivatives with nonlocal differential operators such as the fourier pseudospectral method may cause strong numerical artifacts in the form of noncausal ringing this situation happens when regular grids are used the problem is attacked by using a staggered pseudospectral technique with a different scheme for each rheological relation the nature and causes of acausal ringing in regular grid methods and the reasons why staggered grid methods eliminate this problem are explained in papers by fornberg 1990 and ozdenvar and mcmechan 1996 thus the objective here is not to propose a new method but to develop the algorithm for the viscoelastic and transversely isotropic vti wave equation for which the technique can be implemented without interpolation the algorithm is illustrated for one physical situation that requires very high accuracy such as a fluid solid interface where very large contrasts in material properties occur the staggered grid solution is noise free i

a spectral scheme for wave propagation simulation in 3 d elastic anisotropic media this work presents a new scheme for wave propagation simulation in three dimensional 3 d elastic anisotropic media the algorithm is based on the rapid expansion method rem as a time integration algorithm and the fourier pseudospectral method for computation of the spatial derivatives the rem expands the evolution operator of the second order wave equation in terms of chebychev polynomials constituting an optimal series expansion with exponential convergence the modeling allows arbitrary elastic coefficients and density in lateral and vertical directions numerical methods which are based on finite difference techniques in time and space are not efficient when applied to realistic 3 d models since they require considerable computer memory and time to obtain accurate results on the other hand the fourier method permits a significant reduction of the working space and the rem algorithm gives machine accuracy with half the computational effort as the usual second order temporal differencing sch

multivalued generalized nonlinear contractive maps and fixed points abstract we introduce some notions of generalized nonlinear contractive maps and prove some fixed point results for such maps consequently several known fixed point results are either improved or generalized including the corresponding recent fixed point results of ciric l b ciric multivalued nonlinear contraction mappings nonlinear anal 71 2009 2716 2723 klim and wardowski d klim d wardowski fixed point theorems for set valued contractions in complete metric spaces j math anal appl 334 2007 132 139 feng and liu y feng s liu fixed point theorems for multivalued contractive mappings and multivalued caristi type mappings j math anal appl 317 2006 103 112 and mizoguchi and takahashi n mizoguchi w takahashi fixed point theorems for multivalued mappings on complete metric spaces j math anal appl 141 1989 177 188

analysis of higher order finite difference schemes in 3 d reverse time migration the design and usefulness of practical algorithms for 3 d reverse time depth migration is examined while demonstrating data applications of the algorithm we evaluate quantitatively the accuracy of the finite difference operator from second order to eighth order for the scalar wave equation by comparing numerical and analytical solutions the results clearly show the advantage of using higher order finite difference schemes especially from second order to fourth order for space derivatives hence a finite difference method with the accuracy of fourth order in space and second order in time is applied to 3 d full scalar wave equations in reverse time migration considerable savings in cpu and memory are obtained by using larger horizontal grid spacings than the vertical spacings therefore we derive dispersion and stability conditions for such unequal grid spacing the 3 d reverse time migration of data from the hibernia field shows an image improvement over 2 d migrations despite the fact that much of the structure was considered to be approximately 2 d stable nondispersive finite difference methods of higher order can allow for tractable and efficient 3 d reverse time migration solutions

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

modeling of the acoustic wave equation with transform methods numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data two separate topics are studied the first deals with the solution of the acoustic wave equation the second topic treats wave phenomena whose direction of propagation is restricted within 90 degrees from a given axis in the numerical methods developed here the wave field is advanced in time by using standard time differencing schemes on the other hand expressions including space derivative terms are computed by fourier transform methods this approach to computing derivatives minimizes truncation errors another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves e g to avoid multiple reflections constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain the error analysis of the algorithms shows that truncation errors are due mainly to time discretization such er

finite difference modeling in transversely isotropic media we developed a modeling algorithm for transversely isotropic media that uses finite difference operators in a staggered grid staggered grid schemes are more stable than the conventional finite difference methods because the differences are actually based on half the grid spacing this modeling algorithm uses the full elastic wave equation that makes possible the modeling of all kinds of waves propagating in transversely isotropic media the spatial derivatives are represented by fourth order finite difference operators while the time derivative is represented by a secondorder finite difference operator the algorithm has no limitation on the acquisition geometry or on the heterogeneity of the media the program is currently formulated to work in a 2 d transversely isotropic medium but can readily be extended to 3 d snapshots can be obtained at any time with no additional computational cost a four layer model is used to show the usefulness of the method horizontal and vertical component seismograms a

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

seismic modeling by optimizing regularized staggered grid finite difference operators using a time space domain dispersion relationship preserving method the staggered grid finite difference fd method is widely used in numerical simulation of the wave equation with stability conditions grid dispersion often exists because of the discretization of the time and the spatial derivatives in thewave equation therefore suppressing grid dispersion is a key problem for the staggered grid fd schemes to reduce the grid dispersion the traditional method uses high order staggered grid schemes in the space domain however the wave is propagated in the time and space domain simultaneously therefore some researchers proposed to derive staggered grid fd schemes based on the time space domain dispersion relationship however such methods were restricted to low frequencies and special angles of propagation we have developed a regularizing technique to tackle the ill conditioned property of the symmetric linear system and to stably provide approximate solutions of the fd coefficients for acoustic wave equations dispersion analysis and seismic numerical simulations determined that the proposed method satisfies the dispersion relationship over a much wider range of frequencies and angles of propagation and can ensure fd coefficients being solved via a well posed linear system and hence improve the forward modeling precision

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

finite difference time domain dispersion reduction schemes the finite difference time domain fdtd although recognized as a flexible robust and simple to implement method for solving complex electromagnetic problems is subject to numerical dispersion errors in addition to the traditional ways for reducing dispersion i e increasing sampling rate and using higher order degrees of accuracy a number of schemes have been proposed recently in this work an unified methodology for deriving new difference schemes is presented it is based on certain modifications of the characteristic equation that accompanies any given discretized version of the wave equation the method is duly compared with existing schemes and verified numerically