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

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

the solution of the matrix polynomial equation a s x s b s y s c s this note presents an algorithm for finding the solution x s y s of the matrix polynomial equation a s x s b s y s c s since the solution when it exists is not unique we will find the solution for which the rows of x s have a minimal possible degree this procedure will result in a unique solution that is obtained by solving a set of linear equations an example is given to illustrate the procedure

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

modeling of frequency domain elastic wave equation with an average derivative optimal method abstractbased on an average derivative method we developed a new nine point numerical scheme for a frequency domain elastic wave equation compared with the classic nine point scheme this scheme reduces the required number of grid points per wavelength for equal and unequal directional spacings the reduction in the number of grid points increases as the poisson s ratio becomes larger in particular as the poisson s ratio reaches 0 5 the numerical s wave phase velocity of this new scheme becomes zero whereas the classical scheme produces spurious numerical s wave phase velocity numerical examples demonstrate that this new scheme produces more accurate results than the classical scheme at approximately the same computational cost

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

an efficient point algorithm for a linear two stage optimization problem this paper presents an algorithm using sensitivity analysis to solve a linear two stage optimization problem the underlying theory rests on a set of first order optimality conditions that parallel the kuhn tucker conditions associated with a one dimensional parametric linear program the solution to the original problem is uncovered by systematically varying the parameter over the unit interval and solving the corresponding linear program finite convergence is established under nondegenerate assumptions the paper also discusses other solution techniques including branch and bound and vertex enumeration and gives an example highlighting their computational and storage requirements by these measures the algorithm presented here has an overall advantage finally a comparison is drawn between bicriteria and bilevel programming and underscored by way of an example

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

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

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

a note on an efficient point algorithm for a linear two stage optimization problem the efficient point algorithm proposed by j f bard for the computation of the solution of the linear two stage optimization problem does not always converge to the desired solution a counterexample is provided and the reasons for this lack of convergence are discussed

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

application of the multiaxial perfectly matched layer m pml to near surface seismic modeling with rayleigh waves perfectly matched layer pml absorbing boundaries are widely used to suppress spurious edge reflections in seismic modeling when modeling rayleigh waves with the existence of the free surface the classical pml algorithm becomes unstable when the poisson s ratio of the medium is high numerical errors can accumulate exponentially and terminate the simulation due to computational overflows numerical tests show that the divergence speed of the classical pml has a nonlinear relationship with the poisson s ratio generally the higher the poisson s ratio the faster the classical pml diverges the multiaxial pml m pml attenuates the waves in pmls using different damping profiles that are proportional to each other in orthogonal directions the proportion coefficients of the damping profiles usually vary with the specific model settings if they are set appropriately the m pml algorithm is stable for high poisson s ratio earth models through numerical tests of 40 models with poisson s ratios that varied from 0 10 to 0 49 we found that a constant proportion coefficient of 1 0 for the x and z directional damping profiles is sufficient to stabilize the m pml for all 2d isotropic elastic cases wavefield simulations indicate that the instability of the classical pml is strongly related to the wave phenomena near the free surface when applying the multiaxial technique only in the corners of the pml near the free surface the original m pml technique can be simplified without losing its stability the simplified m pml works efficiently for homogeneous and heterogeneous earth models with high poisson s ratios the analysis in this paper is based on 2d finite difference modeling in the time domain that can easily be extended into the 3d domain with other numerical methods

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

numerical investigation of implementation of air earth boundary by acoustic elastic boundary approach the need for incorporating the traction free condition at the air earth boundary for finite difference modeling of seismic wave propagation has been discussed widely a new implementation has been developed for simulating elastic wave propagation in which the free surface condition is replaced by an explicit acoustic elastic boundary detailed comparisons of seismograms with different implementations for the air earth boundary were undertaken using the 2 2 the finite difference operators are second order in time and space and the 2 6 second order in time and sixth order in space standard staggered grid ssg schemes methods used in these comparisons to define the air earth boundary included the stress image method sim the heterogeneous approach the scheme of modifying material properties based on transversely isotropic medium approach the acoustic elastic boundary approach and an analytical approach the method proposed achieves the same or higher accuracy of modeled body waves relative to the sim rayleigh waves calculated using the explicit acoustic elastic boundary approach differ slightly from those calculated using the sim numerical results indicate that when using the 2 2 ssg scheme for sim and our new method a spatial step of 16 points per minimum wavelength is sufficient to achieve 90 accuracy 32 points per minimum wavelength achieves 95 accuracy in modeled rayleigh waves when using the 2 6 ssg scheme for the two methods a spatial step of eight points per minimum wavelength achieves 95 accuracy in modeled rayleigh waves our proposed method is physically reasonable and based on dispersive analysis of simulated seismographs from a layered half space model is highly accurate as a bonus our proposed method is easy to program and slightly faster than the sim

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 robust solution of the generalized polynomial bezout identity in this paper algorithms for the computation of all matrices of the generalized polynomial bezout identity are proposed the algorithms are based on the computation of minimal polynomial basis for the right null spaces of certain polynomial matrices for this reason an algorithm for the computation of minimal polynomial bases is also proposed since this algorithm relies solely on singular value decompositions of certain real matrices formed with the coefficients of the polynomial matrix whose minimal polynomial bases one is interested in finding it can be said to be robust

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

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

an improved vacuum formulation for 2d finite difference modeling of rayleigh waves including surface topography and internal discontinuities rayleigh waves are generated along the free surface and their propagation can be strongly influenced by surface topography modeling of rayleigh waves in the near surface in the presence of topography is fundamental to the study of surface waves in environmental and engineering geophysics for simulation of rayleigh waves the traction free boundary condition needs to be satisfied on the free surface avacuum formulation naturally incorporates surface topographyinfinite difference fd modeling by treating the surface grid nodes as the internal grid nodes however theconventionalvacuumformulationdoesnotcompletely fulfill the free surface boundary condition and becomes unstableformodelingusinghigh orderfdoperators wedeveloped a stable vacuum formulation that fully satisfies the free surface boundary condition by choosing an appropriate combination of the staggered grid form and a parameter averaging scheme the elastic parameters on the topographic free surface are updated with exactly the same treatment as internal grid nodes the improved vacuum formulation can accurately and stably simulate rayleigh waves along the topographic surface for homogeneous and heterogeneous elastic models with high poisson s ratios 0 4 this method requires fewer grid points per wavelength than the stress image based methods internal discontinuities in a model can be handled without modification of the algorithm onlyminorchangesarerequiredtoimplementtheimproved vacuum formulation in existing 2d fd modeling codes

an average derivative optimal scheme for modeling of the frequency domain 3d elastic wave equation because of its high computational cost we needed to develop an efficient numerical scheme for the frequency domain 3d elastic wave equation in addition the numerical scheme should be applicable to media with a liquid solid interface to address these two issues we have developed a new average derivative optimal 27 point scheme with arbitrary directional grid intervals and a corresponding numerical dispersion analysis for the frequency domain 3d elastic wave equation the novelty of this scheme is that its optimal coefficients depend on the ratio of the directional grid intervals and poisson s ratio in this way this scheme can be applied to media with a liquid solid interface and a computational grid with arbitrary directional grid intervals for media with a variable poisson s ratio we have developed an effective and stable interpolation method for optimization coefficients compared with the classic 19 point scheme this new scheme reduces the required number of grid points per wavelength for equal and unequal directional grid intervals the reduction of the number of grid points increases as the poisson s ratio becomes larger in particular the numerical s wave phase velocity of this new scheme becomes zero whereas the classic 19 point scheme produces a spurious numerical s wave phase velocity as poisson s ratio reaches 0 5 we have performed numerical examples to develop the theoretical analysis

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

a simple approach to solving the diophantine equation based on the state space concepts a simple approach to finding all polynomial matrix solutions of the diophantine equation is proposed the procedure presented is very simple in comparison to earlier ones unlike earlier ones it is not necessary to solve any equation only two constant matrices which could be selected at random are required all solutions are expressed in an explicit formula form

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

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

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

on the polynomial matrix equation ax yb c in this note an algorithm for finding a unique solution x y of the bilateral polynominal matrix equation ax yb c is presented the solution is obtained by solving a set of linear equations relations between the bilateral polynomial equation and the unilateral one are also discussed

on solving diophantine equations by real matrix manipulation this note presents simple algorithms for obtaining the solutions of the diophantine equation our methods can produce classes of all solutions with lower degree than a specified number the previous algorithms involve some troublesome computations e g the calculation of both the controllability indexes and the observability indexes or the solution of a pole assignment problem etc our contribution is that our algorithm requires only basic matrix operations such as addition subtraction multiplication and inversion of given real matrices in addition by solving simple linear equations the class of all minimum degree solutions can be given therefore the computational efforts are reduced compared with previous algorithms

a bilevel bottleneck programming problem this paper studies a bilevel programming problem with linear constraints and in which the objective functions at both levels are concave bottleneck functions which are to be minimized the problem is a non convex programming problem it is shown that an optimal solution to the problem is attainable at an extreme point of the underlying region the outer level objective function values are ranked in increasing order until a value is reached one of the solutions corresponding to which is feasible for the problem this solution is then the required global optimal solution

solving a bilevel linear program when the inner decision maker controls few variables abstract this paper presents an algorithm that always finds an optimum of a bilevel linear program blp problem if one exists moreover if the inner decision maker does not control too many variables the algorithm is efficient in particular if the number of variables controlled by the inner decision maker can be considered to be constant then the algorithm is a polynomial time algorithm the algorithm is also well suited to parallel implementation

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