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

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

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

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

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

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

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

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

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

limit distributions of u statistics resampled by symmetric stable laws if y i and v i are independent random sequences such thaty i are i i d random variables belonging to the normal domain of attraction of a symmetric stable law 0 2 andv i are i i d random variables then the limit distributions of u statistics n 1 alpha sum limits 1 leqq i t ldots i d leqq n y i 1 ldots y i d f v i 1 ldots v i d coincide with the probability laws of multiple stochastic integralsx d f f t 1 t d dx t d with respect to a symmetric stable processx t

digitization scheme that assures faithful reconstruction of plane figures in the present extensive work we propose a digitization scheme for a broad class of plane figures it includes arbitrary polygons possibly non convex and with holes and plane sets whose boundary consists of smooth curves or straight segments our approach is based on an appropriate scaling of the original continuous real object so that the obtained magnified object and its appropriately constructed digitization feature analogous geometric properties as a bi product of the presented theory we prove the strong np hardness of the problem of obtaining an optimal i e with a minimal number of facets polyhedral reconstruction in which the facets are trapezoids or triangles this result implies the strong np hardness of the general polyhedral reconstruction problem which was a long standing open problem as another major result we show that the proposed digitization scheme allows faithful reconstruction of plane figures from the considered general class the reconstructed set features the basic properties of the original object such as location of curve and straight segments and inflection points of its boundary

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

on a p stable multiple integral ii we construct a multiple strictly p stable stochastic integral as a dunford type integral with respect to a l q valued vector measure 1qp 2

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

on the multiple stable integral we prove that the n fold stochastic integral mathop smallint limits left 0 1 right n f left t 1 t n right dz left t 1 right dz left t n right with respect to a p stable levy process z t t 0 1 p 1 2 exists as a continuous linear mapping from the zygmund space l p log l n 1 of functions f 0 1 n to l q for any q p this strengthens the author s earlier result 15 the proof is based on an interpolation theorem due to bennett and rudnick 2

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

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

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

efficient and accurate modeling of ocean bottom seismometer data using reciprocity abstractseismic experiments in which the number of sources is considerably larger than the number of receivers occur regularly an important example is the collection of crustal scale seismic data using ocean bottom seismometers and marine sources we describe a method to accurately and efficiently compute synthetic seismograms for such experiments by using finite differences and reciprocity we show numerically how to decompose an explosive source into its equivalent body force components using the staggered grid finite difference technique with a fourth order approximation for the spatial derivative and a second order approximation for the temporal derivative this decomposition results in a source configuration where the equivalent body forces are defined in 12 points around the point where the ex plosive source is applied we then use the derived equivalent body forces for the explosive source and seismic reciprocity theorems to convert the common receiver gather to a common shot gather the method is

stochastic integrals a combinatorial approach a combinatorial definition of multiple stochastic integrals is given in the setting of random measures it is shown that some properties of such stochastic integrals formerly known to hold in special cases are instances of combinatorial identities on the lattice of partitions of a set the notion of stochastic sequences of binomial type is introduced as a generalization of special polynomial sequences occuring in stochastic integration such as hermite poisson charlier and kravchuk polynomials it is shown that identities for such polynomial sets have a common origin

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

guarding a set of line segments in the plane we consider the following problem given a finite set of straight line segments in the plane find a set of points of minimum size so that every segment contains at least one point in the set this problem can be interpreted as looking for a minimum number of locations of policemen guards cameras or other sensors that can observe a network of streets corridors tunnels tubes etc we show that the problem is strongly np complete even for a set of segments with a cubic graph structure but in p for tree structures

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

random multilinear forms on etudie la convergence des formes multilineaires f n n k x n 1 x n k dans des variables aleatoires independantes symetriques

a new insight into the reciprocity principle reciprocity is usually applied to wavefields associated with concentrated point forces and point receivers but reciprocity has a much wider application potential in many cases however it is not used at its full potential because 1 a variety of source and receiver types are not considered or 2 its implementation is not well understood we obtain reciprocity relations for inhomogeneous anisotropic viscoelastic solids and for distributed sources and receivers and test these relations with a full wave numerical modeling algorithm the theory and the numerical experiments show that in addition to the usual relations involving directional forces 1 the diagonal components of the strain tensor are reciprocal for dipole sources single couple without moment 2 the off diagonal components of the stress tensor are reciprocal for double couples with moments 3 the dilatation due to a directional force is reciprocal to the particle velocity due to a dilatational source and 4 some combinations of the off diagonal strains are reciprocal for single couples with moments

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

contraction and decoupling inequalities for multilinear forms and u statistics we prove decoupling inequalities for random polynomials in independent random variables with coefficients in vector space we use various means of comparison including rearrangement invariant norms e g orlicz and lorentz norms tail distributions tightness hypercontractivity and so forth

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

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

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

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

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

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

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

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

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

double stochastic integrals random quadratic forms and random series in orlicz spaces let x t t geq 0 be a process with independent symmetric and stationary increments and let xi i be i i d symmetric real random variables we provide a characterization of functions f s t s t geq 0 such that the double integral int int f s t dx s dx t exists a characterization of infinite matrices alpha ij such that the double series sum alpha ij xi i xi j converges a s and a characterization of orlicz space l psi valued sequences a i for which the series sum a i xi i converges a s in l psi the above three problems are closely related

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

partitioning a polygonal region into trapezoids the problem of partitioning a polygonal region into a minimum number of trapezoids with two horizontal sides is discussed a triangle with a horizontal side is considered to be a trapezoid with two horizontal sides one of which is degenerate first a method of achieving a minimum partition is presented the number m of the trapezoids in the minimum partition of a polygonal region p is shown to be m n w h d 1 where n w and h are the number of vertices windows holes and horizontal edges of p respectively and d is the cardinality of a maximum independent set of the straight lines in the plane graph associated with p next this problem is shown to be polynomially equivalent to the problem of finding a maximum independent set of a straight lines in the plane graph and consequently it is shown to be np complete however for a polygonal region without windows an o n 2 time algorithm for partitioning it into a minimum number of trapezoids is presented finally an o n log n time approximation algorithm with the performance bound 3 is presented

comparison of moments for tangent sequences of random variables it is shown that for all tangent sequences dn and en of nonnegative or conditionally symmetric random variables and for every function satisfying the growth condition 2x x the following inequality holds e phi left mathop sup limits n left sum limits k 1 n d k right right leqq ce phi left mathop sup limits n left sum limits k 1 n e k right right this generalizes results of j zinn and proves a conjecture of s kwapien and w a woyczynski

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

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

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

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

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

multiple integration with respect to poisson and levy processes necessary and sufficient conditions are given for the existence of a multiple stochastic integral of the form fdx1 dxd where x1 xd are components of a positive or symmetric pure jump type levy process in d conditions are also given for a sequence of integrals of this type to converge in probability to zero or infinity or to be tight all arguments proceed via reduction to the special case of poisson integrals

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

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