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

a mass redistributed finite element method mr fem for acoustic problems using triangular mesh the accuracy of numerical results using standard finite element method fem in acoustic problems will deteriorate with increasing frequency due to the dispersion error such dispersion error depends on the balance between the stiffness and mass of discretization equation systems this paper reports an improved finite element method fem for solving acoustic problems by re distributing the mass in the mass matrix to tune the balance aiming to minimize the dispersion errors this is done by shifting the integration point locations when computing the entries of the mass matrix while ensuring the mass conservation the new method is verified through the detailed numerical error analysis and a strategy is also proposed for the best mass redistribution in terms of minimizing dispersion error the relative dispersion error of present mass redistributed finite element method mr fem is found to be much smaller than the fem solution in both theoretical prediction and numerical examination the present mr fem works well by using the linear triangular elements that can be generated automatically which enables automation in computation and saving computational cost in mesh generation numerical examples demonstrate the advantages of mr fem in comparison with the standard fem using the same triangular meshes and quadrilateral meshes

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

extended nonlinear schrodinger equation with higher order odd and even terms and its rogue wave solutions a a and n a acknowledge the support of the australian research council discovery project dp110102068 and also thank the volkswagen foundation for financial support

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

moving breathers and breather to soliton conversions for the hirota equation we find that the hirota equation admits breather to soliton conversion at special values of the solution eigenvalues this occurs for the first order as well as higher orders of breather solutions an analytic expression for the condition of the transformation is given and several examples of transformations are presented the values of these special eigenvalues depend on two free parameters that are present in the hirota equation we also find that higher order breathers generally have complicated quasi periodic oscillations along the direction of propagation various breather solutions are considered including the particular case of second order breathers of the nonlinear schrodinger equation

harmonic analysis on the free product of two cyclic groups abstract let g z r z x a b a r b s e where r s 2 a natural length is defined for the elements of g and 1 denotes the characteristic function of the set of elements of length 1 we study the convolution c algebra a 1 generated by 1 we obtain explicitly the associated plancherel measure orthogonal polynomials and spherical functions and show that a 1 is not a maximal abelian subalgebra of the reduced group c algebra of g closely related to work by figa talamanca picardello pytlik and others on free groups this study is complicated by the fact that a 1 does not consist of radial functions

free group representations from vector valued multiplicative functions ii let gamma be a non commutative free group on finitely many generators in a previous work two of the authors have constructed the class of multiplicative representations of gamma and proved them irreducible as representation of gamma ltimes lambda c omega in this paper we analyze multiplicative representations as representations of gamma and we prove a criterium for irreducibility based on the growth of their matrix coefficients

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

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

comparison of source independent methods of elastic waveform inversion in this paper we investigate several source independent methods of nonlinear full waveform inversion of multicomponent elastic wave data this includes iterative estimation of source signature ies standard trace normalization stn and average trace normalization atn inversion methods all are based on the finite element method in the frequency domain one synthetic elastic crosshole model is used to compare the recovered images with all these methods as well as the known source signature kss inversion method the numerical experiments show that the ies method is superior to both stn and atn methods in two component elastic wave inversion in the frequency domain when the source signature is unknown the stn and atn methods have limitations associated with near zero amplitudes or polarity reversals in traces from one of the components which destroy the energy balance in the normalized traces and cause a loss of frequency information but the atn method is somewhat superior to the stn method in suppressing random noise and improving stability as the developed formulas and the numerical experiments show we suggest the ies method as a practical procedure for multicomponent seismic inversion

applying finite element analysis to the memory variable formulation of wave propagation in anelastic media finite element methods are applied to solution of seismic wave motion in linear viscoelastic media using the memory variable formalism the displacements are represented as a superposition of a set of basis functions it is shown that if memory variables are represented using the spatial derivatives of those basis functions rather than the basis functions themselves the equations to be solved are simpler and require less computer memory using this formulation results for sh waves in one and two dimensions are calculated using a simple explicit finite element in space finite difference in time scheme these results agree with those found with a method of lines solution results in a homogeneous medium also agree with the frequency domain solutions of kjartansson s constant q method

a bayesian approach to estimate uncertainty for full waveform inversion using a priori information from depth migration abstractfull waveform inversion fwi enables us to obtain high resolution subsurface images however estimating model uncertainties associated with this technique is still a challenging problem we have used a bayesian inference framework to estimate model uncertainties associated with fwi the uncertainties were assessed based on an a posteriori covariance operator evaluated at the maximum a posteriori model for the prior distribution we have used a spatially nonstationary covariance operator based on a plane wave construction with local dips measured from migrated images preconditioned frequency domain fwi was used to estimate the maximum a posteriori model efficient manipulation of the posterior covariance was based on a low rank approximation of the data misfit hessian preconditioned by the prior covariance operator the strong decay of the singular values indicated that data were mostly informative about a low dimensional subspace of model parameters to reduce computational cost of the random

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

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

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

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

the poisson transform and representations of a free group abstract let g be a free group with r generators 1 r g which plays the same role of the laplace beltrami operator on semisimple lie groups are characterized furthermore an analytic family of representations z of g on functions on the boundary is considered defined by z x p z x x 1 where p x is the poisson kernel relative to the action of g on it is proved that for 0 s re z z is uniformly bounded on an appropriate hilbert space h s finally the uniform boundedness of other special representations of g obtained by considering the free group either as a subgroup of the group of all isometries of a tree or as a subgroup of gl 2 q p is proved

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

simple intuitive relations concerning the effect of boundaries on the stability of the fdtd method simple intuitive relations are derived to study the effect of some boundaries on the courant friedrich levy number cfln in the finite difference time domain method a novel von neumann stability analysis is proposed to study the effect of perfect electric conductor perfect magnetic conductor and perfect electromagnetic conductor boundaries on the cfln also gershgorin s theorem is applied to determine the cfln at the interface between two dielectrics studies show that the aforesaid boundaries do not restrict the cfln of the computational space

a recursive green s function technique for acoustic scattering from heterogeneous objects a fast efficient algorithm for computing acoustic fields scattered by inhomogeneous objects in an otherwise homogeneous space is presented the algorithm called the recursive green s function method rgfm constructs the domain green s function by recursively combining known green s functions from smaller subdomains the fields on the scatterer surface are then computed using a boundary integral formulation proper implementation of the rgfm results in a storage requirement of o n and computational costs of o n3 2 and o n2 for two and three dimensional problems respectively where n is the total number of discrete points in the inhomogeneous region results are compared with those obtained from exact solutions to show the accuracy of the method

a new 3 d numerical approach to the solution of pdes with optimal accuracy on irregular domains and cartesian meshes abstract a new 3 d numerical approach for the time dependent wave and heat equations as well as for the time independent laplace equation on irregular domains with the dirichlet boundary conditions has been developed trivial cartesian meshes and simple 27 point uniform and nonuniform stencil equations are used for 3 d irregular domains the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique very small distances 0 1 h 1 0 9 h where h is the grid size between the grid points of a cartesian mesh and the boundary do not worsen the accuracy of the new technique at similar 27 point stencils the accuracy of the new approach is two orders higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the fifth order tetrahedral finite elements with much wider stencils the wave and heat equations can be uniformly treated with the new approach the order of the time derivative in these equations does not affect the coefficients of the stencil equations of the semi discrete systems the new approach can be directly applied to other partial differential equations

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

a recipe for stability of finite difference wave equation computations finite difference solutions to the wave equation are pervasive in the modeling of seismic wave propagation kelly and marfurt 1990 and in seismic imaging bording and lines 1997 that is they are useful for the forward problem modeling and the inverse problem migration in computational solutions to the wave equation it is necessary to be aware of conditions for numerical stability in this short note we examine a convenient recipe for insuring stability in our finite difference solutions to the wave equation the stability analysis for finite difference solutions of partial differential equations is handled using a method originally developed by von neumann and described by press et al 1986 p 827 830

dispersion minimizing quadrature rules for c1 quadratic isogeometric analysis abstract we develop quadrature rules for the isogeometric analysis of wave propagation and structural vibrations that minimize the discrete dispersion error of the approximation the rules are optimal in the sense that they only require two quadrature points per element to minimize the dispersion error barto et al 2017 1 and they are equivalent to the optimized blending rules we recently described our approach further simplifies the numerical integration instead of blending two three point standard quadrature rules we construct directly a single two point quadrature rule that reduces the dispersion error to the same order for uniform meshes with periodic boundary conditions also we present a 2 5 point rule for both uniform and non uniform meshes with arbitrary boundary conditions consequently we reduce the computational cost by using the proposed quadrature rules various numerical examples demonstrate the performance of these quadrature rules

probability measures on groups and some related ideals in group algebras abstract let g be a locally compact group random walks on g some factorization problems in l 1 g and the significance for these of the amenability of g are studied these topics are linked in this paper via ideals in l 1 g of the form l 1 g e where is a probability measure on g cohen s factorization theorem and some ideas from ergodic theory play an important part

iterative finite difference solution analysis of acoustic wave equation in the laplace fourier domain we have investigated numerical characteristics of iterative solutions to the acoustic wave equation in the laplacefourier lf domain we transformed the time domain acoustic wave equation into the lf domain the transformed equation was discretized with finite differences and was solved with iterative methods finite difference modeling experiments demonstrate that iterative methods require an infinitesimal stopping tolerance to accurately compute the pressure field especially at long offsets to understand the requirement for such infinitesimal tolerance values we analyzed the evolution of intermediate solution vectors residual vectors and search direction vectors during the iteration the analysis showed that the requirement arises from the fact that in the solution space the amplitude of the pressure field varies more than sixty orders of magnitude on the common log scale accordingly we propose a rule of thumb for choosing a proper stopping tolerance value we also examined numerical dispersion errors in terms of the grid sampling resolutions per skin depth and wavelength we found that despite the similarity of the form of the acoustic wave and electromagnetic diffusion equations the former is different from the latter due to the fact that in the lf domain the skin depth of the acoustic wave equation is decoupled from its wavelength this aspect requires that in the lf domain its grid size be determined by considering the minimum grid sampling resolutions based not only the wavelength but also the skin depth

accuracy of finite difference and finite element modeling of the scalar and elastic wave equations numerical solutions of the scalar and elastic wave equations have greatly aided geophysicists in both forward modeling and migration of seismic wave fields in complicated geologic media and they promise to be invaluable in solving the full inverse problem this paper quantitatively compares finite difference and finite element solutions of the scalar and elastic hyperbolic wave equations for the most popular implicit and explicit time domain and frequency domain techniques in addition to versatility and ease of implementation it is imperative that one choose the most cost effective solution technique for a fixed degree of accuracy to be of value a solution technique must be able to minimize 1 numerical attenuation or amplification 2 polarization errors 3 numerical anisotropy 4 errors in phase and group velocities 5 extraneous numerical parasitic modes 6 numerical diffraction and scattering and 7 errors and transmission coefficients this paper shows that in homogeneous media the

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

spherical functions and harmonic analysis on free groups let f r 1 be a free group with r generators in this paper we study a principal series and a complementary series of irreducible unitary representations of f which are defined through the action of f on its poisson boundary relative to a simple random walk we show that the regular representation of f can be written as a direct integral of the representations of the principal series and that the resulting harmonic analysis on the free group bears a close resemblance with the harmonic analysis of sl 2 ir

dispersion minimized mass for isogeometric analysis abstract we introduce the dispersion minimized mass for isogeometric analysis to approximate the structural vibration which we model as a second order differential eigenvalue problem the dispersion minimized mass reduces the eigenvalue error significantly from the optimum order of 2 p to the superconvergence order of 2 p 2 for the p th order isogeometric elements with maximum continuity which in return leads to a more accurate method we first establish the dispersion error where the leading error term is explicitly written in terms of the stiffness and mass entries for arbitrary polynomial order isogeometric elements we derive the dispersion minimized mass in one dimension by solving a p dimensional local matrix problem for the p th order approximation and then extend it to multiple dimensions on tensor product grids we show that the dispersion minimized mass can also be obtained by approximating the mass matrix using optimally blended quadratures we generalize the lower order quadrature blending results to arbitrary polynomial order isogeometric approximations as well as to arbitrary quadrature rules various numerical examples validate the eigenvalue and eigenfunction error estimates

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

waveform inversion using a logarithmic wavefield although waveform inversion has been studied extensively since its beginning 20 years ago applications to seismic field data have been limited and most of those applications have been for global seismology or engineering seismology scale problems not for exploration scale data as an alternative to classical waveform inversion we propose the use of a new objective function constructed by taking the logarithm of wavefields allowing consideration of three types of objective function namely amplitude only phase only or both in our waveform inversion we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function we compute the steepest descent directions by using a matrix formalism derived from a frequency domain finite element finite difference modeling technique our numerical algorithms are similar to those of reverse time migration and waveform inversion based on the adjoint state of the wave equation in order to demonstrate the practical applicability of our algorithm we use a synthetic data set from the marmousi model and seismic data collected from the korean continental shelf for noisefree synthetic data the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion when random noise is added the inverted velocity model is also close to the true marmousi model but when frequencies below 5 hz are removed from the data the velocity structure is not as good as those for the noise free and noisy data for field data we compare the time domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short period features of the subsurface although we use wrapped phases in our examples we still obtain reasonable results we expect that if we were to use correctly unwrapped phases in the inversion algorithm we would obtain better results

estimating the critical time step in explicit dynamics using the lanczos method the goal of our paper is to demonstrate the cost effective use of the lanczos method for estimating the critical time step in an explicit transient dynamics code the lanczos method can provide a significantly larger estimate for the critical time step than an element based method the typical scheme however the lanczos method represents a more expensive method for calculating a critical time step than element based methods our paper shows how the additional cost of the lanczos method can be amortized over a number of time steps and lead to an overall decrease in run time for an explicit transient dynamics code we present an adaptive hybrid scheme that synthesizes the lanczos based and element based estimates and allows us to run near the critical time step estimate provided by the lanczos method

perturbational and nonperturbational inversion of rayleigh wave velocities the inversion of rayleigh wave dispersion curves is a classic geophysical inverse problem we have developed a set of matlab codes that performs forward modeling and inversion of rayleigh wave phase or group velocity measurements we describe two different methods of inversion a perturbational method based on finite elements and a nonperturbational method based on the recently developed dix type relation for rayleigh waves in practice the nonperturbational method can be used to provide a good starting model that can be iteratively improved with the perturbational method although the perturbational method is well known we solve the forward problem using an eigenvalue eigenvector solver instead of the conventional approach of root finding features of the codes include the ability to handle any mix of phase or group velocity measurements combinations of modes of any order the presence of a surface water layer computation of partial derivatives due to changes in material properties and layer boundaries and the implementation of an automatic grid of layers that is optimally suited for the depth sensitivity of rayleigh waves

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

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

rogue waves and rational solutions of the hirota equation the hirota equation is a modified nonlinear schrodinger equation nlse that takes into account higher order dispersion and time delay corrections to the cubic nonlinearity in describing wave propagation in the ocean and optical fibers it can be viewed as an approximation which is more accurate than the nlse we have modified the darboux transformation technique to show how to construct the hierarchy of rational solutions of the hirota equation we present explicit forms for the two lower order solutions each one is a regular nonsingular rational solution with a single maximum that can describe a rogue wave in this model numerical simulations reveal the appearance of these solutions in a chaotic field generated from a perturbed continuous wave solution

radial functions on free groups and a decomposition of the regular representation into irreducible components this opens a possibility of using methods of commutative harmonic analysis to study certain convolution algebras of radial functions on g the fist who considered the subalgebra of radial functions in the regular c algebra of g was j m cohen 1 he proved that it is isomorphic to c 2 2 the algebra of continuous functions on the interval 2 2 where 2k 1 and k is the number of free generators in g in the paper we study other such algebras one of our results is theorem 2 1 that the algebra r g of radial functions in 1 g has the ellipse

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

3d frequency domain finite difference viscoelastic wave modeling using weighted average 27 point operators with optimal coefficients abstractexperimental data suggest that the viscoelastic behavior of rocks is more easily and accurately described in the frequency domain than in the time domain supporting the idea of simulating seismic wave propagation in the frequency domain we evaluated weighted averaged 27 point finite difference operators for 3d viscoelastic wave modeling in the frequency domain within the proposed framework we developed general equations for normalized phase velocities that can be used with arbitrary finite difference operators three sets of weighting coefficients for second order central finite difference operators that minimize the numerical dispersion for up to five grid points per wavelength were found using a damped least squares ls criterion as well as a global optimization scheme based on l1 and l2 norm criteria the three sets produced very similar dispersion curves and improvement provided by global optimization appeared marginal in this respect we also evaluated a discrete form for the heterogen

an analytic family of uniformly bounded representations of free groups harmonic analysis on a free group f has attracted considerable attention in the last ten years or so there seem to be two reason for that one is the discovery of deep analogy of certain aspects of harmonic analysis on a free group and harmonic analysis on sl 2 r cf e g fundamental works of p cartier 4 a figa talamanca and m a picardello 8 the other being the interest in the c algebra generated by the regular representation of f cf a connes 5 j cuntz 6 u haagerup 10 m pimsner and d voiculescu 14

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

dispersion reducing finite elements for transient acoustics this paper focuses on increasing the accuracy of low order four node quadrilateral finite elements for the transient analysis of wave propagation modified integration rules originally proposed for time harmonic problems provide the basis for the proposed technique the modified integration rules shift the integration points to locations away from the conventional gauss or gauss lobatto integration points with the goal of reducing the discretization errors specifically the dispersion error presented here is an extension of the idea to time dependent analysis using implicit as well as explicit time stepping schemes the locations of the stiffness integration points remain unchanged from those in time harmonic case on the other hand the locations of the integration points for the mass matrix depend on the time stepping scheme and the step size furthermore the central difference method needs to be modified from its conventional form to facilitate fully explicit computation the superior performance of the proposed algorithms is illustrated with the help of several numerical examples

optimal blended spectral element operators for acoustic wave modeling spectral element methods based on high order polynomials are among the most commonly used techniques for computing accurate simulations of wave propagation phenomena in complex media however to retain computational efficiency very high order polynomials cannot be used and errors such as numerical dispersion and numerical anisotropy cannot be totally avoided in the present work we devise an approach for reducing such errors by considering modified discrete wave operators we analyze consistent and lumped operators together with blended operators weighted averages of consistent and lumped operators furthermore using the operator blending approach and a novel dispersion analysis method we develop optimal spectral element operators that have increased numerical accuracy without resorting to very high order operators the new operators are faster and computationally more efficient than consistent operators our approach is based on the tensor product decomposition of the element matrices into 1d f

state transition induced by higher order effects and background frequency the state transition between the peregrine rogue wave and w shaped traveling wave induced by higher order effects and background frequency is studied we find that this intriguing transition described by an exact explicit rational solution is consistent with the modulation instability mi analysis that involves a mi region and a stability region in a low perturbation frequency region in particular the link between the mi growth rate and the transition characteristic analytically demonstrates that the localization characteristic of transition is positively associated with the reciprocal of the zero frequency growth rate furthermore we investigate the case for nonlinear interplay of multilocalized waves it is interesting that the interaction of second order waves in the stability region features a line structure rather than an elastic interaction between two w shaped traveling waves

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

second order nonlinear schrodinger equation breather solutions in the degenerate and rogue wave limits we present an explicit analytic form for the two breather solution of the nonlinear schr odinger equation with imaginary eigenvalues it describes various nonlinear combinations of akhmediev breathers and kuznetsov ma solitons the degenerate case when the two eigenvalues coincide is quite involved the standard inverse scattering technique does not generally provide an answer to this scenario we show here that the solution can still be found as a special limit of the general second order expression and appears as a mixture of polynomials with trigonometric and hyperbolic functions a further restriction of this particular case where the two eigenvalues are equal to i produces the second order rogue wave with two free parameters considered as differential shifts the illustrations reveal a precarious dependence of wave profile on the degenerate eigenvalues and differential shifts thus we establish a hierarchy of second order solutions revealing the interrelated nature of the general case the rogue wave and the degenerate breathers

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 high order numerical approach with cartesian meshes for modeling of wave propagation and heat transfer on irregular domains with inhomogeneous materials abstract recently we have developed a new numerical approach for pdes with constant coefficients on irregular domains and cartesian meshes in this paper we extend it to a much more general case of pdes with variable coefficients that have a lot of applications e g the modeling of functionally graded materials the inhomogeneous materials obtained by 3 d printing and many others here we consider the 2 d wave and heat equations for isotropic and anisotropic inhomogeneous materials the idea of the extension to the case of pdes with variable coefficients is based on the representation of the stencil coefficients as functions of the mesh size this leads to the increase in the size of the local system of algebraic equations solved for each grid point of the new approach however this does not change the size of the global system of semidiscrete equations and practically does not increase the computational costs of the proposed technique similar to our previous technique the new 2 d approach with compact 9 point stencils uses trivial cartesian meshes for complex irregular domains and provides the fourth order of accuracy for the wave and heat equations with variable coefficients the calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of the new technique at similar 9 point stencils the accuracy of the new approach is much higher than that for the linear finite elements the numerical results for irregular domains show that at the same number of degrees of freedom the new approach is even much more accurate than the high order up to the third order finite elements with much wider stencils the wave and heat equations are uniformly treated with the new approach

full wave directional illumination analysis in the frequency domain directional illumination analysis based on one way wave equations has been studied extensively however its inherent limitations e g one way propagation wide angle error and amplitude inaccuracy can severely hinder its applications for accurate survey design and true reflection imaging corrections in complex media we have analyzed the illumination in the frequency domain using full two way wave propagators considering the extensive computation and huge storage required for time domain methods and the fact that the illumination is frequency dependent this full wave analysis can provide frequency dependent full angle true amplitude illumination not only for the downgoing waves but also for the upgoing waves including turning waves and reflected waves two methods were considered to decompose the full wavefield into the local angle domain a direct full dimensional decomposition and more efficient split step decomposition composed of three lower dimensional decompositions the results of illumination analysis demonstrated the advantages of this method the two decomposition methods produced similar results