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

eigenvalue computations in the context of data sparse approximations of integral operators in this work we consider the numerical solution of a large eigenvalue problem resulting from a finite rank discretization of an integral operator we are interested in computing a few eigenpairs with an iterative method so a matrix representation that allows for fast matrix vector products is required hierarchical matrices are appropriate for this setting and also provide cheap lu decompositions required in the spectral transformation technique we illustrate the use of freely available software tools to address the problem in particular slepc for the eigensolvers and hlib for the construction of h matrices the numerical tests are performed using an astrophysics application results show the benefits of the data sparse representation compared to standard storage schemes in terms of computational cost as well as memory requirements

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

high accuracy wave simulation revised derivation numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation the nearly analytic integration discrete naid method for solving the two dimensional acoustic wave equation has been fully mathematically revised analyzed and tested the naid method is an alternative numerical modeling method for generating synthetic seismograms the acoustic wave equation is first transformed into a system of first order ordinary differential equations odes with respect to time variable t and then directly integrated at a small time interval of tn tn 1 to obtain semi discrete ordinary differential equations the integral kernel is expanded into a truncated taylor series to which the integration operator is explicitly applied the high order temporal derivatives involved in the integral kernel are replaced by high order spatial derivatives which then are approximately calculated as a weighted linear combination of the displacement the particle velocity and their spatial gradients in this article we investigate the theoretical properties of the revised naid method including the discrete error and the stability criteria numerical results for constant and layered velocity models show that comparing to the lax wendroff correction lwc scheme and the staggered grid finite difference method the naid method can effectively suppress the numerical dispersion and source noises caused by the discretization of the acoustic wave equation with too coarse spatial grids or when models have strong velocity contrasts between adjacent layers the proposed naid method has been applied in computing the acoustic wavefields for two heterogeneous models the corner edge model and the marmousi model promising numerical results illustrate that the naid method provides an encouraging tool for large scale and complex wave simulation and inversion problems based on the acoustic equation

equivalent accuracy at a fraction of the cost overcoming temporal dispersion abstractnumerical dispersion in finite difference fd modeling produces coherent artifacts severely constraining the resolution of advanced imaging and inversion techniques conventionally numerical dispersion is reduced by increasing the order of accuracy of the fd operators and we resign ourselves to paying the high computational cost that is incurred assuming no spatial dispersion we have found that fd time dispersion is independent of the medium velocity and the spatial grid for propagation and only depends on the time stepping scheme and the propagation time based on this observation we have devised postpropagation filters to collapse the time dispersion effect of fd modeling our dispersion correction filters are designed by comparing the input waveform with dispersive waveforms obtained by 1d forward modeling these filters are then applied on multidimensional shot records to eliminate the time dispersion by two schemes 1 stationary filtering plus interpolation and 2 nonstationary fil

wavelets generated by using discrete singular convolution kernels this paper explores the connection between wavelet methods and an efficient computational algorithm the discrete singular convolution dsc many new dsc kernels are constructed and they are identified as wavelet scaling functions two approaches are proposed to generate wavelets from dsc kernels two well known examples the canny filter and the mexican hat wavelet are found to be special cases of the present dsc kernel generated wavelets a family of wavelet generators proposed in this paper are found to form an infinite dimensional lie group which has an invariant subgroup of translation and dilation if dsc kernels form an orthogonal system they are found to span a wavelet subspace in a multiresolution analysis

cubic threefolds and abelian varieties of dimension five ii this paper extends joint work with r friedman to show that the closure of the locus of intermediate jacobians of smooth cubic threefolds in the moduli space of principally polarized abelian varieties ppavs of dimension five is an irreducible component of the locus of ppavs whose theta divisor has a point of multiplicity three or more this paper also gives a sharp bound on the multiplicity of a point on the theta divisor of an indecomposable ppav of dimension less than or equal to 5 for dimensions four and five this improves the bound due to j kollar r smith r varley and l ein r lazarsfeld

comparison of methods for extracting adcigs from rtm abstractmethods for extracting angle domain common image gathers adcigs during 2d reverse time migration fall into three main categories direction vector based methods local plane wave decomposition methods and local shift imaging condition methods the direction vector based methods which use either amplitude gradients or phase gradients cannot handle overlapping events because of an assumption of one propagation direction per imaging point per imaging time however the adcigs from the direction vector based methods have the highest angle resolution a new direction vector based method using instantaneous phase gradients in space and time gives the same propagation directions and adcigs as those obtained by the poynting vector or polarization vector based methods where amplitudes are large angles calculated by the phase gradients have larger uncertainties at smaller amplitudes but they do not significantly degrade the adcigs because they contribute only small amplitudes the local plane wave d

the pseudospectral method accurate representation of interfaces in elastic wave calculations when finite difference methods are used to solve the elastic wave equation in a discontinuous medium the error has two dominant components dispersive errors lead to artificial wave trains errors from interfaces lead to circular wavefronts emanating from each location where the interface appears jagged to the rectangular grid the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy with this method dispersive errors are essentially eliminated the mappings introduced in this paper also eliminate the other dominant error source test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost although the mapping method is described here in connection with the pseudospectral method it can also be used with high order finite difference approximations

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

an l 1 refined projection approximate solution of the radiation transfer equation in stellar atmospheres this paper deals with the numerical approximation of the solution of a weakly singular integral equation of the second kind which appears in astrophysics the reference space is the complex banach space of lebesgue integrable functions on a bounded interval whose amplitude represents the optical thickness of the atmosphere the kernel of the integral operator is defined through the first exponential integral function and depends on the albedo of the media the numerical approximation is based on a sequence of piecewise constant projections along the common annihilator of the corresponding local means in order to produce high precision solutions without solving large scale linear systems we develop an iterative refinement technique of a low order approximation for this scheme parallelization of matrix computations is suitable

a frequency space 2 d scalar wave extrapolator using extended 25 point finite difference operator finite difference frequency domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s the generation of synthetic seismograms with the time domain finite difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse time migration this became possible with the rapid development of high performance computers however the space frequency x finite difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory therefore finite difference time domain modeling is much more popular among exploration geophysicists a limitation of the space frequency domain is that the recently developed nine point scheme still requires that g the number of grid points per wavelength be 5 this value is greater than for most other numerical modeling techniques for example the pseudospectral scheme

singularities of the prym theta divisor for the jacobian of a curve the riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve this paper proves an analogous theorem for prym varieties applications of this theorem to cubic threefolds and prym varieties of dimension five are also considered

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

regularization and datuming of seismic data by weighted damped least squares the possibility of improving regularization datuming of seismic data is investigated by treating wavefield extrapolation as an inversion problem weighted damped least squares is then used to produce the regularized datumed wavefield regularization datuming is extremely costly because of computing the hessian so an efficient approximation is introduced approximation is achieved by computing a limited number of diagonals in the operators involved real and synthetic data examples demonstrate the utility of this approach for synthetic data regularization datuming is demonstrated for large extrapolation distances using a highly irregular recording array without approximation regularization datuming returns a regularized wavefield with reduced operator artifacts when compared to a nonregularizing method such as generalized phase shift plus interpolation pspi approximate regularization datuming returns a regularized wavefield for approximately two orders of magnitude less in cost but it is dip limi

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

accurate depth migration by a generalized phase shift method a new depth migration method derived in the space frequency domain is based on a generalized phase shift method for the downward continuation of surface data for a laterally variable velocity structure the fourier spatial components are no longer eigenvectors of the wave equation and therefore a rigorous application of the phase shift method would seem to require finding the eigenvectors by a matrix diagonalization at every depth step however a recently derived expansion technique enables phase shift accuracy to be obtained without resorting to a costly matrix diagonalization the new technique is applied to the migration of zero offset time sections as with the laterally uniform velocity case the evanescent components of the solution need to be isolated and eliminated in this case by the application of a spatially variant high cut filter tests performed on the new method show that it is more accurate and efficient than standard integration techniques such as the runge kutta method or the taylor

symplectic scheme and the poynting vector in reverse time migration abstractwe developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method rem this solution can be used for seismic modeling and reverse time migration rtm in seismic modeling and rtm spatial derivatives are usually calculated by finite differences fds or by the fourier method and the time evolution is normally obtained by a second order fd approach if the spatial derivatives are computed by higher order fd schemes then the time step needs to be small enough to avoid numerical dispersion therefore increasing the computational time however by using rem with the fourier method for the spatial derivatives we can apply the proposed method to propagate the wavefield for larger time steps moreover if the appropriate number of expansion terms is chosen this method is unconditionally stable and propagates seismic waves free of numerical dispersion the use of a symplectic numerical scheme provides the solution of the wave equation an

a comparison of different propagation schemes for the time dependent schro dinger equation abstract a comparison of three widely used time propagation algorithms for the time dependent schrodinger equation is described a typical evolution problem is chosen to demonstrate the efficiency and accuracy of the various methods on a numerical grid using a pseudo spectral fft spatial representation for scattering and bound state evolution the methods used second order differencing split operator propagation chebyshev polynomial expansion are discussed in terms of their applicability to various classes of dynamic problems a new method is introduced which is based upon a low order lanczos technique this method appears to offer an accurate and flexible alternative to the existing techniques overall the chebyshev method is recommended for time independent potentials and the lanczos method for time dependent potentials

a two way nonreflecting wave equation in seismic modeling and in migration it is often desirable to use a wave equation with varying velocity but constant density which does not produce interlayer reverberations the conventional approach has been to use a one way wave equation which allows energy to propagate in one dominant direction only typically this direction being either upward or downward claerbout 1972 we introduce a two way wave equation which gives highly reduced reflection coefficients for transmission across material boundaries for homogeneous regions of space however this wave equation becomes identical to the full acoustic wave equation possible applications of this wave equation for forward modeling and for migration are illustrated with simple models

construction of data sparse mathcal h 2 matrices by hierarchical compression hierarchical matrices mathcal h matrices provide an elegant approach to handling large densely populated matrices the matrix is split into a hierarchy of blocks and each block is approximated by a low rank matrix in factorized form it has been demonstrated that this representation can be used to treat integral and partial differential equations solve matrix equations from the field of control theory and evaluate matrix functions efficiently mathcal h 2 matrices use a refined representation that employs a multilevel structure in order to reduce the storage requirements of hierarchical matrices it has been shown that mathcal h 2 matrices can significantly reduce storage requirements for large problems in particular when combined with modern error control schemes until now all algorithms for constructing an efficient approximation of a general matrix by an mathcal h 2 matrix required a representation of the entire original matrix to be kept in storage therefore the storage requirements of mathcal h 2 matrix algorithms could be far larger than those of the final approximation this paper presents a new approach that allows us to construct an mathcal h 2 matrix without storing the entire original matrix the central idea is to approximate submatrices and combine them by an efficient new algorithm to form approximations of larger matrices until the entire matrix has been treated using this new approach many mathcal h matrix algorithms can be refitted easily to compute results in the more efficient representation possible applications include efficient matrix arithmetics for the construction of preconditioners the approximation of matrix functions or solutions of matrix equations or efficient compression schemes based on the popular cross approximation algorithms

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

an exact adjoint operation pair in time extrapolation and its application in least squares reverse time migration to reduce the migration artifacts arising from incomplete data or inaccurate operators instead of migrating data with the adjoint of the forward modeling operator a least squares migration often is considered least squares migration requires a forward modeling operator and its adjoint in a derivation of the mathematically correct adjoint operator to a given forward time extrapolation modeling operator the exact adjoint of the derived operator is obtained by formulating an explicit matrix equation for the forward operation and transposing it the programs that implement the exact adjoint operator pair are verified by the dot product test the derived exact adjoint operator turns out to differ from the conventional reverse time migration rtm operator an implementation of wavefield extrapolation backward in time examples with synthetic data show that migration using the exact adjoint operator gives similar results for a conventional rtm operator and that least squares rtm is quite successful in reducing most migration artifacts the least squares solution using the exact adjoint pair produces a model that fits the data better than one using a conventional rtm operator pair

modeling of piezoelectric transducers with combined pseudospectral and finite difference methods a new hybrid finite difference fd and pseudospectral ps method adapted to the modeling of piezoelectric transducers pzts is presented the time dependent equations of propagation are solved using the ps method while the electric field induced in the piezoelectric material is determined through a fd representation the purpose of this combination is to keep the advantages of both methods in one model the adaptability of fd representation to model piezoelectric elements with various geometries and materials and the low number of nodes per wavelength required by the ps method this approach is implemented to obtain an accurate algorithm to simulate the propagation of acoustic waves over large distances directly coupled to the calculation of the electric field created inside the piezoelectric material which is difficult with classical algorithms these operations are computed using variables located on spatially and temporally staggered grids which attenuate gibbs phenomenon and increase the algorithm s accuracy the two dimensional modeling of a pzt plate excited by a 50 mhz sinusoidal electrical signal is performed the results are successfully compared to those obtained using the finite element fe algorithm of atila software with configurations spatially and temporally adapted to the fe requirements the cost efficiency of the fd ps time domain method is quantified and verified

new full wave phase shift approach to solve the helmholtz acoustic wave equation for modeling we have developed a method of solving the helmholtz equation based on a new way to generalize the one way wave equation impose correct boundary conditions and eliminate exponentially growing evanescent waves the full two way nature of the helmholtz equation is included but the equation is converted into a pseudo one way form in the framework of a generalized phase shift structure consisting of two coupled first order partial differential equations for wave propagation with depth a new algorithm based on the particular structure of the coupling between p z and 2p z2 is introduced to treat this problem by an explicit approach more precisely in a depth marching strategy the wave operator is decomposed into the sum of two matrices the first one is a propagator in a reference velocity medium whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity the initial conditions are generated by solving the lippmann schwinger integral e

an accurate and efficient scheme for wave propagation in linear viscoelastic media the problem of wave propagation in a linear viscoelastic medium can be described mathematically as an exponential evolution operator of the form em t acting on a vector representing the initial conditions where m is a spatial operator matrix and t is the time variable techniques like finite difference for instance are based on a taylor expansion of this evolution operator we propose an optimal polynomial approximation of em t based on the powerful method of interpolation in the complex plane in a domain which includes the eigenvalues of the matrix m the new time integration technique is implemented to solve the isotropic viscoacoustic equation of motion the algorithm is tested for the problem of wave propagation in a homogeneous medium and compared with second order temporal differencing and the spectral chebychev method the algorithm solves efficiently and with machine accuracy the problem of seismic wave propagation with a dissipation mechanism in the sonic band a characteristic of sedimen

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

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

recursive integral time extrapolation methods for waves a comparative review abstractwe compared several families of algorithms for recursive integral time extrapolation rite algorithms for waves in isotropic and anisotropic media these methods allow simulating accurate wave extrapolation with little numerical dispersion even when using larger time steps than are usually possible for conventional finite difference methods these various rite algorithms all share the use of mixed space wavenumber domain operators derived from fourier integral solutions of single mode wave equations we evaluated a taxonomy for rite methods based on how they approximated the influence of medium heterogeneity one family of methods uses mixed domain series expansions to provide accurate approximations to heterogeneous extrapolators even for large time steps we compared several methods for deriving coefficients for such series approximations another family of methods uses interpolation between different homogeneous extrapolations to approximate heterogeneous time extrapolation such methods can b

an effective imaging condition for reverse time migration using wavefield decomposition reverse time migration rtm exhibits great superiority over other imaging algorithms in handling steeply dipping structures and complicated velocity models however low frequency high amplitude noises commonly seen in a typical rtm image have been one of the major concerns because they can seriously contaminate the signals in the image if they are not handled properly we propose a new imaging condition to effectively and efficiently eliminate these specific noises from the image the method works by first decomposing the source and receiver wavefields to their one way propagation components followed by applying a correlation based imaging condition to the appropriate combinations of the decomposed wavefields we first give the physical explanation of the principle of such noises in the conventional rtm image then we provide the detailed mathematical theory for the new imaging condition finally we propose an efficient scheme for its numerical implementation it replaces the computationally intensiv

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

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

stable depth extrapolation of seismic wavefields by a neumann series several migration methods fail to work when applied to complex geological structures with strong lateral heterogeneity the generalized migration in the frequency wavenumber f k domain based on a convolution with a slowness inverse of velocity dependent operator is capable of downward continuation of wavefield in media with strong vertical and lateral variations of velocity unfortunately this method as presented in the literature is potentially unstable we propose a new stable extrapolator based on the solution of the integral fredholm equation which describes a one way wave equation in the form of a neumann series the resulting algorithm of depth migration is implemented in both the frequency wavenumber f k and frequency space f x domains and takes into account arbitrary lateral gradients of velocity using a low frequency filter in x f domain that is the sum of the power series the computation time of depth migration by a neumann series is slightly longer than for split step fourier

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

modeling of pure acoustic wave in tilted transversely isotropic media using optimized pseudo differential operators abstract conventional modeling and imaging for tilted transversely isotropic tti media may suffer from numerical instabilities and shear artifacts due to the coupling of the p and sv wave modes in the coupled equations on the contrary the decoupled equations for tti media provide a more stable solution due to the separated p and s wave modes because the decoupled equations involve complicated pseudo differential operators in space it is more convenient to apply the pseudo spectral method to these equations however the second order time stepping scheme of the pseudo spectral method may suffer from time stepping errors and instabilities for a large time step we have developed an optimized pseudo differential operator opo that incorporates not only the spectral evaluation of the pseudo differential operator but also a temporal correction term that would effectively compensate the time stepping errors associated with the time wavenumber domain extrapolation of the wave equation the opo was const

3 d depth migration via mcclellan transformations three dimensional seismic wavefields may be extrapolated in depth one frequency at a time by two dimensional convolution with a circularly symmetric frequency and velocity dependent filter this depth extrapolation performed for each frequency independently lies at the heart of 3 d finite difference depth migration the computational efficiency of 3 d depth migration depends directly on the efficiency of this depth extrapolation mcclellan transformations provide an efficient method for both designing and implementing two dimensional digital filters that have a particular form of symmetry such as the circularly symmetric depth extrapolation filters used in 3 d depth migration given the coefficients of one dimensional frequency and velocity dependent filters used to accomplish 2 d depth migration mcclellan transformations lead to a simple and efficient algorithm for 3 d depth migration 3 d depth migration via mcclellan transformations is simple because the coefficients of two dimensional depth

theory and modeling of constant q p and s waves using fractional time derivatives i have developed and solved the constant q model for the attenuation of p and s waves in the time domain using a new modeling algorithm based on fractional derivatives the model requires time derivatives of order m 2 applied to the strain components where m 0 1 and 1 tan 1 1 q with q the p wave or s wave quality factor the derivatives are computed with the grunwald letnikov and central difference fractional approximations which are extensions of the standard finite difference operators for derivatives of integer order the modeling uses the fourier method to compute the spatial derivatives and therefore can handle complex geometries and general material property variability i verified the results by comparison with the 2d analytical solution obtained for wave propagation in homogeneous pierre shale moreover the modeling algorithm was used to compute synthetic seismograms in heterogeneous media corresponding to a crosswell seismic experiment

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

suppressing the unwanted reflections of the full wave equation in many instances in exploration geophysics we are interested in the so called one way wave equation this equation allows the wave fields to propagate in the positive depth direction but not in the reverse z direction some modeling and migration methods such as the f k method stolt 1978 and the phase shift method gazdag 1978 produce in a natural way the one way wave equation the main advantage of the one way wave equation is that it does not give rise to multiples or interlayer reverberations and enables the observation of primary events only

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

a first order k space model for elastic wave propagation in heterogeneous media a pseudospectral model of linear elastic wave propagation is described based on the first order stress velocity equations of elastodynamics k space adjustments to the spectral gradient calculations are derived from the dyadic green s function solution to the second order elastic wave equation and used to a ensure the solution is exact for homogeneous wave propagation for timesteps of arbitrarily large size and b also allows larger time steps without loss of accuracy in heterogeneous media the formulation in k space allows the wavefield to be split easily into compressional and shear parts a perfectly matched layer pml absorbing boundary condition was developed to effectively impose a radiation condition on the wavefield the staggered grid which is essential for accurate simulations is described along with other practical details of the implementation the model is verified through comparison with exact solutions for canonical examples and further examples are given to show the efficiency of the method for practical problems the efficiency of the model is by virtue of the reduced point per wavelength requirement the use of the fast fourier transform fft to calculate the gradients in k space and larger time steps made possible by the k space adjustments

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

determinantal characterization of canonical curves and combinatorial theta identities we characterize genus g canonical curves by the vanishing of combinatorial products of g 1 determinants of brill noether matrices this also implies the characterization of canonical curves in terms of g 2 g 3 2 theta identities a remarkable mechanism based on a basis of h 0 k c expressed in terms of szego kernels reduces such identities to a simple rank condition for matrices whose entries are logarithmic derivatives of theta functions such a basis together with the fay trisecant identity also leads to the solution of the question of expressing the determinant of brill noether matrices in terms of theta functions without using the problematic klein fay section