an immersed free surface boundary treatment for seismic wave simulation abstractfinite difference methods are popular for wave simulation within the seismic exploration community thanks to their efficiency however difficulties arise when encountering complex topography due to the regular grid pattern of the finite difference schemes despite alternatives that can handle the free surface with little effort such as the spectral element or discontinuous galerkin s methods incorporating a free surface boundary condition within the finite difference framework is still appealing even at the cost of extra algorithm complexity and stronger requirement of computational resources we present a free surface boundary treatment within the finite difference framework belonging to the family of the immersed boundary methods inherently the presented boundary treatment is separated from the rest of the wave simulation which makes it easy to be integrated in existing finite difference codes specifically we construct an extrapolation operator for each grid point above the free surfa

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

graph based change point detection we consider the testing and estimation of change points locations where the distribution abruptly changes in a data sequence a new approach based on scan statistics utilizing graphs representing the similarity between observations is proposed the graph based approach is nonparametric and can be applied to any data set as long as an informative similarity measure on the sample space can be defined accurate analytic approximations to the significance of graph based scan statistics for both the single change point and the changed interval alternatives are provided simulations reveal that the new approach has better power than existing approaches when the dimension of the data is moderate to high the new approach is illustrated on two applications the determination of authorship of a classic novel and the detection of change in a network over time

multivariable circle criteria for multiparameter singularly perturbed systems a direct approach to the lur e problem for singularly perturbed systems sps is proposed in contrast to previous results the feedback connection between the linear and nonlinear parts of sps is allowed to depend essentially on both the slow and the fast variables the lur e problem for multiparameter sps is studied by the same framework

hybrid modeling of elastic wave propagation in two dimensional laterally inhomogeneous media we have constructed a hybrid scheme for elastic wave propagation in two dimensional laterally inhomogeneous media the algorithm is based on a combination of finite difference techniques and the boundary integral equation method it involves a dedicated application of each of the two methods to specific domains of the model structure finite difference techniques are applied to calculate a set of boundary values wave field and stress field in the vicinity of the heterogeneous domain the continuation of the near field response is then calculated by means of the boundary integral equation method in a numerical example the hybrid method has been applied to calculate a plane wave response for an elastic lens embedded in a homogeneous environment the example shows that the hybrid scheme enables more efficient modeling with the same accuracy than with pure finite difference calculations

nonreflecting boundary condition in finite element formulation for an elastic wave equation numerical modeling of seismic wavefields using finite difference or finite element methods requires truncation of the model to finite computational domains it is well known that the edges of such truncated models give boundary reflections on the synthetic seismograms an essential step to successful numerical modeling is to eliminate these reflections we present a simple scheme that eliminates such boundary reflections when computing synthetics we also demonstrate the efficiency and robustness of our method on a variety of geological models

frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

a hybrid scheme for absorbing edge reflections in numerical modeling of wave propagation we propose an efficient scheme to absorb reflections from the model boundaries in numerical solutions of wave equations this scheme divides the computational domain into boundary transition and inner areas the wavefields within the inner and boundary areas are computed by the wave equation and the one way wave equation respectively the wavefields within the transition area are determined by a weighted combination of the wavefields computed by the wave equation and the one way wave equation to obtain a smooth variation from the inner area to the boundary via the transition zone the results from our finite difference numerical modeling tests of the 2d acoustic wave equation show that the absorption enforced by this scheme gradually increases with increasing width of the transition area we obtain equally good performance using pseudospectral and finite element modeling with the same scheme our numerical experiments demonstrate that use of 10 grid points for absorbing edge reflections attains nearly perfect absorption

numerical study of the interface errors of finite difference simulations of seismic waves numerical simulations of wave propagation produce different errors and the most well known is numerical dispersion which is only valid for homogeneous media however there is a lack of error studies for heterogeneous media or even for the canonical case of media that have two constant velocity layers the error associated with media that have two layers is called an interface error and it typically converges to zero with a lower order of convergence compared to the theoretical convergence rate of the finite difference schemes fds for homogeneous media we evaluated a detailed numerical study of the interface error for three staggered grid fds that are commonly used in the simulation of seismic wave propagation we determined that a standard staggered grid scheme ssgs also known as the virieux scheme a rotated staggered grid scheme rsgs and a lebedev scheme ls preserve the second order of convergence at horizontal vertical solid solid interfaces when the medium parameters have been properly modified such as by harmonic averaging of finely layered media for the stiffness tensor and arithmetic mean for the density however for a fluid solid interface aligned with the grid line a second order convergence can only be achieved by an ssgs in addition the presence of a fluid solid interface reduces the order of convergence for the ls and the rsgs to a first order of convergence the presence of inclined interfaces makes high order second and more convergence impossible

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

a stable free surface boundary condition for two dimensional elastic finite difference wave simulation two of the persistent problems in finite difference solutions of the elastic wave equation are the limited stability range of the free surface boundary condition and the boundary condition s treatment of lateral variations in velocity and density the centered difference approximation presented by alterman and karal 1968 for example remains stable only for greater than 0 30 where and are the shear s and compressional p wave velocities the one sided approximation alterman and rotenberg 1969 and composed approximation ilan et al 1975 have similar restrictions the revised composed approximation of ilan and loewenthal 1976 overcomes this restriction but cannot handle laterally varying media properly

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

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

interface conditions for acoustic and elastic wave propagation the divergence theorem is used to handle the physics required at interfaces for acoustic and elastic wave propagation in heterogeneous media the physics required at regular and irregular interfaces is incorporated into numerical schemes by integrating across the interface the technique which can be used with many numerical schemes is applied to finite differences a derivation of the acoustic wave equation which is readily handled by this integration scheme is outlined since this form of the equation is equivalent to the scalar sh wave equation the scheme can be applied to this equation also each component of the elastic p sv equation is presented in divergence form to apply this integration scheme naturally incorporating the continuity of the normal and tangential stresses required at regular and irregular interfaces

asymptotic distribution free change point detection for multivariate and non euclidean data we consider the testing and estimation of change points locations where the distribution abruptly changes in a sequence of multivariate or non euclidean observations we study a nonparametric framework that utilizes similarity information among observations which can be applied to various data types as long as an informative similarity measure on the sample space can be defined the existing approach along this line has low power and or biased estimates for change points under some common scenarios we address these problems by considering new tests based on similarity information simulation studies show that the new approaches exhibit substantial improvements in detecting and estimating change points in addition under some mild conditions the new test statistics are asymptotically distribution free under the null hypothesis of no change analytic p value approximations to the significance of the new test statistics for the single change point alternative and changed interval alternative are derived making the new approaches easy off the shelf tools for large datasets the new approaches are illustrated in an analysis of new york taxi data

a staggered grid convolutional differentiator for elastic wave modelling the computation of derivatives in governing partial differential equations is one of the most investigated subjects in the numerical simulation of physical wave propagation an analytical staggered grid convolutional differentiator cd for first order velocity stress elastic wave equations is derived in this paper by inverse fourier transformation of the band limited spectrum of a first derivative operator a taper window function is used to truncate the infinite staggered grid cd stencil the truncated cd operator is almost as accurate as the analytical solution and as efficient as the finite difference fd method the selection of window functions will influence the accuracy of the cd operator in wave simulation we search for the optimal gaussian windows for different order cds by minimizing the spectral error of the derivative and comparing the windows with the normal hanning window function for tapering the cd operators it is found that the optimal gaussian window appears to be similar to the hanning window function for tapering the same cd operator we investigate the accuracy of the windowed cd operator and the staggered grid fd method with different orders compared to the conventional staggered grid fd method a short staggered grid cd operator achieves an accuracy equivalent to that of a long fd operator with lower computational costs for example an 8th order staggered grid cd operator can achieve the same accuracy of a 16th order staggered grid fd algorithm but with half of the computational resources and time required numerical examples from a homogeneous model and a crustal waveguide model are used to illustrate the superiority of the cd operators over the conventional staggered grid fd operators for the simulation of wave propagations

a wraparound elimination technique for the pseudospectral wave synthesis using an antiperiodic extension of the wavefield the pseudospectral method e g kosloff and baysal 1982 is an attractive alternative to numerical modeling schemes such as finite difference or finite element because it requires several orders of magnitude less computer memory and computation time in the pseudospectral method the field variables are expanded in terms of fourier interpolation polynomials which impose periodic boundary conditions on the equation of motion numerical differentiation is then implemented via the discrete fourier transform the periodicity condition causes the periodically extended wavefield on either side of the computational domain to propagate in from the sides interfering with the actual solution this phenomenon is called wraparound in many current applications e g cerjan et al 1985 an absorbing boundary condition has been used to damp the wraparound phases through a gradual reduction of the wavefield amplitude in the vicinity of the grid boundary such a method works effectively when the appropriate absorpt

asymptotic expansions for game theoretic riccati equations and stabilization with disturbance attenuation for singularly perturbed systems abstract if the systems to be controlled are singularly perturbed then the riccati equations which appear in h problems are difficult to solve due to the presence of the small parameter the present paper describes and validates asymptotic expansions showing how they may be used in h control furthermore we construct a composite controller based on the solution of slow and fast problems which guarantees a desired achievable performance level for the full order plant as e approaches zero

asymptotic properties of input output operators norm associated with singularly perturbed systems with multiplicative white noise in this paper we study the problem of the asymptotic property of the norm of input output operators related to a class of singularly perturbed stochastic linear systems the system is under perturbation of multiplicative white noise by using reduction order and boundary layer techniques it is shown that the norm of the operator of the perturbed system is less than a given number gamma when the small perturbation varepsilon tends to zero if both the related norms of the reduced subsystem and the boundary layer subsystem are less than gamma furthermore a stabilizing robust controller is designed which is independent of perturbation varepsilon

a new graph based two sample test for multivariate and object data abstracttwo sample tests for multivariate data and especially for non euclidean data are not well explored this article presents a novel test statistic based on a similarity graph constructed on the pooled observations from the two samples it can be applied to multivariate data and non euclidean data as long as a dissimilarity measure on the sample space can be defined which can usually be provided by domain experts existing tests based on a similarity graph lack power either for location or for scale alternatives the new test uses a common pattern that was overlooked previously and works for both types of alternatives the test exhibits substantial power gains in simulation studies its asymptotic permutation null distribution is derived and shown to work well under finite samples facilitating its application to large datasets the new test is illustrated on two applications the assessment of covariate balance in a matched observational study and the comparison of network data under different co

finite difference predictions of p sv wave propagation inside submerged solids i liquid solid interface conditions a finite difference scheme has been developed to analyze internal strains in submerged elastic solids of irregular geometry subjected to ultrasonic wave sources that simulate a clinical lithotripter in part i of this paper the finite difference formulation that accounts for arbitrary liquid solid interfaces is presented and sample numerical results are discussed two different methods for discretizing the liquid solid interface conditions are developed the first treats the interface conditions explicitly the second integrates the heterogeneous wave equations across the interface using the divergence theorem both schemes account for varying grid sizes and give similar results for a test problem consisting of a radially diverging source incident on the rectangular solid the time sequence obtained numerically for strain at the center of a rectangular solid matches well with the experimental results s m gracewski et al j acoust soc am 94 652 661 1993 in terms of the arrival times and the relative amplitudes of the peaks in addition strain contours are plotted to visualize the propagation of p longitudinal and s shear vertical waves inside a circular solid the reflection from the concave back surface of the circular solid has a focusing effect with the subsequent formation of focal zones known as caustics where peak strains occur in part ii of this paper the finite difference scheme is used to study the effects of geometry changes on the internal stresses and caustics predicted in model stones subjected to lithotripter pulses

robust stabilization of two time scale systems with respect to the normalized coprime factorization the present paper has two main purposes the first is to provide an explicit state space solution of the robust stabilization problem with respect to the nlcf for two time scale systems in this construction only the solutions of the control and of the filtering riccati equations associated with the slow and with the fast dynamics of the nominal plant are used these solutions do not depend upon the singular perturbation and therefore all computations are well conditioned the second purpose of the paper is to investigate the case when an k independent robust controller can be used for the robust stabilization of a two time scale nominal system two constructions of such controllers are developed under some additional assumptions numerical examples illustrate the theoretical developments

nonreflecting boundary conditions in elastodynamics for finite element methods based upon off surface boundary integral equations in this work an off surface boundary integral osbi method is presented as a mesh termination scheme for solving large or infinite domain problems of elastodynamics the boundary integral equation is discretized using finite element shape functions and the neumann boundary condition term is solved for in terms of the dirichlet boundary condition term this expression is then substituted into associated finite element formulation for the interior problem comparisons are made using the new osbi technique the dtn method of givoli and keller and several other local nonreflecting boundary conditions the proposed boundary condition is shown to be accurate is well suited for use with finite element methods and is competitive with the dtn method

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

explicit finite element perfectly matched layer for transient three dimensional elastic waves the use of a perfectly matched layer pml model is an efficient approach toward the bounded domain modelling of wave propagation on unbounded domains this paper formulates a three dimensional pml for elastic waves by building upon previous work by the author and implements it in a displacement based finite element setting the novel contribution of this paper over the previous work is in making this finite element implementation suitable for explicit time integration thus making it practicable for use in large scale three dimensional dynamic analyses an efficient method of calculating the strain terms in the pml is developed in order to take advantage of the lack of the overhead of solving equations at each time step the pml formulation is studied and validated first for a semi infinite bar and then for the classical soil structure interaction problems of a square flexible footing on a i half space ii layer on a half space and iii layer on a rigid base numerical results for these problems demonstrate that the pml models produce highly accurate results with small bounded domains and at low computational cost and that these models are long time stable with critical time step sizes similar to those of corresponding fully elastic models copyright 2008 john wiley sons ltd

perfectly matched layers for transient elastodynamics of unbounded domains one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outward from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies in a recent work computer methods in applied mechanics and engineering 2003 192 1337 1375 the authors presented inter alia time harmonic governing equations of pmls for anti plane and for plane strain motion of visco elastic media this paper presents a corresponding time domain displacement based governing equations of these pmls and b displacement based finite element implementations of these equations suitable for direct transient analysis the finite element implementation of the anti plane pml is found to be symmetric whereas that of the plane strain pml is not numerical results are presented for the anti plane motion of a semi infinite layer on a rigid base and for the classical soil structure interaction problems of a rigid strip footing on i a half plane ii a layer on a half plane and iii a layer on a rigid base these results demonstrate the high accuracy achievable by pml models even with small bounded domains

technical communique on linear robust h controllers for a class of nonlinear singular perturbed systems it is shown in this paper that any state feedback output feedback robust h controller of the linearized singularly perturbed system yields a local solution of the nonlinear h control problem for the original nonlinear singularly perturbed system

h sup spl infin optimal control for singularly perturbed systems ii imperfect state measurements for pt i see ibid vol 29 no 2 p 401 423 1993 in this paper we study the h sup spl infin optimal control of singularly perturbed linear systems under general imperfect measurements for both finite and infinite horizon formulations using a differential game theoretic approach we first show that as the singular perturbation parameter say spl epsiv 0 approaches zero the optimal disturbance attenuation level for the full order system under a quadratic performance index converges to a value that is bounded above by and in some cases equal to the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate slow and fast quadratic cost functions with the bound being computable independently of e and knowing only the slow and fast dynamics of the system we then construct a controller based on the slow subsystem only and obtain conditions under which it delivers a desired performance level even though the fast dynamics are completely neglected the ultimate performance level achieved by this slow controller can be uniformly improved upon however by a composite controller that uses some feedback from the output of the fast subsystem we construct one such controller via a two step sequential procedure which uses static feedback from the fast output and dynamic feedback from an appropriate slow output each one obtained by solving appropriate spl epsiv independent lower dimensional h sup spl infin optimal control problems under some informational constraints we provide a detailed analysis of the performance achieved by this lower dimensional spl epsiv independent composite controller when applied to the full order system and illustrate the theory with some numerical results on some prototype systems

brief robust h minimum entropy static output feedback control of singularly perturbed systems the problem of designing static output feedback e independent controllers for linear time invariant singularly perturbed systems is considered the controller is required to satisfy a prescribed h norm bound and to minimize the closed loop entropy at s for all small enough e the optimal controller gain is designed on the basis of generalized riccati and lyapunov equations with symmetric block 2 2 that are coupled via a projection this gain is either purely fast purely slow or a composite one depending on the structure of the output coupling matrix a well posed problem with a finite value of entropy for e 0 is obtained by assuming that the entropy of the fast subproblem is zero or by scaling the matrices of the system in the first case the optimal controller is the one that minimizes the entropy of the corresponding descriptor system

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

improved frequency domain elastic wave modeling using weighted averaging difference operators we develop a new finite difference scheme that reduces the number of grid points per wavelength required in frequency domain elastic modeling our approach computes weighted averages of the spatial second order derivative and the mass acceleration terms using a 25 point computational stencil by determining the weighting coefficients to minimize numerical dispersion and numerical anisotropy we reduce the number of grid points to 3 3 per shear wavelength with a resulting error in velocities smaller than 1 our choice of grid points reduces the computer memory needed to store the complex impedance matrix to 4 of that for a conventional second order scheme and to 54 of that for a combined second order scheme the 25 point weighted averaging scheme of this paper makes it possible to accurately simulate realistic models numerical examples show that this technique can achieve the same accurate solutions with fewer grid points than those from previous frequency domain second order schemes our technique can be extended directly to 3 d elastic modeling the computational efficiency will be even greater than that realized for 2 d models

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

h optimal control for singularly perturbed systems part i perfect state measurements we study the h optimal control of singularly perturbed linear systems under perfect state measurements for both finite and infinite horizons using a differential game theoretic approach we show that as the singular perturbation parameter approaches zero the optimal disturbance attenuation level for the full order system under a quadratic performance index converges to a value that is bounded above by the maximum of the optimal disturbance attenuation levels for the slow and fast subsystems under appropriate slow and fast quadratic cost functions furthermore we construct a composite controller based on the solution of the slow and fast games which guarantees a desired achievable performance level for the full order plant as approaches zero a slow controller however is not generally robust in this sense but still under some conditions which are delineated in the paper the fast dynamics can be totally ignored the paper also studies optimality when the controller includes a feedforward term in the disturbance and presents some numerical examples to illustrate the theoretical results

absorbing boundary conditions for elastic waves absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains the boundary conditions developed here for elastic waves are generalizations of ones developed earlier for acoustic waves these conditions are based on compositions of simple first order differential operators the formulas can be applied without modification to problems in both two and three dimensions the boundary conditions are stable for all values of the ratio of p wave velocity to s wave velocity and they are effective near a free surface and in a horizontally stratified medium the boundary conditions are approximated with simple finite difference equations that use values of the solution only along grid lines perpendicular to the boundary this property facilitates implementation especially near a free surface and at other corners of the computational domain

finite difference modeling of the monopole acoustic logs in a horizontally stratified porous formation monopole acoustic logs in a homogeneous fluid saturated porous formation can be simulated by the real axis integration rai method to analytically solve biot s equations 1956a j acoust soc am 28 168 178 1956b j acoust soc am 28 179 191 1962 j appl phys 33 1482 1498 which govern the wave propagation in poro elastic media such analytical solution generally is impossible for horizontally stratified formations which are common in reality in this paper a velocity stress finite difference time domain fdtd algorithm is proposed to solve the problem this algorithm considers both the low frequency viscous force and the high frequency inertial force in poro elastic media extending its application to a wider frequency range compared to existing algorithms which are only valid in the low frequency limit the perfectly matched layer pml is applied as an absorbing boundary condition to truncate the computational region a pml technique without splitting the fields is extended to the poro elastic wave problem the fdtd algorithm is validated by comparisons against the rai method in a variety of formations with different velocities and permeabilities the acoustic logs in a horizontally stratified porous formation are simulated with the proposed fdtd algorithm

generalised nonlinear l 2 l filtering of discrete time markov jump descriptor systems this paper is devoted to the l2 l filter design problem for nonlinear discrete time markov jump descriptor systems subject to partially unknown transition probabilities the partially unknown transition probabilities are modelled via the polytopic uncertainties the objective is to propose a generalised nonlinear full order filter design method such that the resulting filtering error system is regular casual and stochastically stable and a prescribed l2 l attenuation level is satisfied for the autonomous discrete time descriptor system subject to lipschitz nonlinear condition by introducing some slack matrix variables a mode dependent stability criterion is established it cannot only ensure the regularity casuality and stochastic stability of system but also guarantee the considered system has a unique solution based on this obtained criterion a sufficient condition in terms of linear matrix inequalities lmis is derived such that the resulting filtering error system is regular casual s

a rectangular grid lattice spring model for modeling elastic waves in poisson s solidsrectangular lsm to model elastic waves lattice spring model lsm combined with the velocity verlet algorithm is a newly developed scheme for modeling elastic waves propagation in solid media unlike conventional wave equations lsm is established on the basis of micro mechanics of the subsurface media which enjoys better dynamic characteristics of elastic systems we present a rectangular grid lsm scheme for elastic waves simulation in poisson solids and the direction dependent elastic constants are deduced to keep the isotropy of the discrete grid the stability condition and numerical dispersion properties of lsm are discussed and compared with other numerical methods two and three dimensional numerical simulations are carried out using the rectangular grid lsm as well as the second and fourth order accuracy staggered finite difference method fdm wavefields obtained by lsm are fairly similar with those by analytical solution ana and fdm which demonstrates the correctness of the proposed scheme and its capability of modeling elast

on a state space approach in robust control for singularly perturbed systems a direct state space approach for an estimate of the graph metric of singularly perturbed systems sps is proposed based on a new version of the tikhonov theorem on uniform convergence of the slow variables in a stable sps over the l unit ball of inputs in an estimate of the first order to the small parameter for a 2 graph metric between sps and its state decoupled form is obtained for both time varying and time invariant cases moreover a similar estimate of a graph metric between sps and its slow system when the fast system satisfies some additional h conditions is given furthermore the robust state feedback control for sps is also considered in an analogous manner an advantage there is that infinitely many composite feedback controls are obtained directly bypassing the traditional arguments used in sps theory such as approximation of solutions of riccati equations and the implicit function theorem

idr s a family of simple and fast algorithms for solving large nonsymmetric systems of linear equations we present idr s a new family of efficient short recurrence methods for large nonsymmetric systems of linear equations the new methods are based on the induced dimension reduction idr method proposed by sonneveld in 1980 idr s generates residuals that are forced to be in a sequence of nested subspaces although idr s behaves like an iterative method in exact arithmetic it computes the true solution using at most n n s matrix vector products with n the problem size and s the codimension of a fixed subspace we describe the algorithm and the underlying theory and present numerical experiments to illustrate the theoretical properties of the method and its performance for systems arising from different applications our experiments show that idr s is competitive with or superior to most bi cg based methods and outperforms bi cgstab when s 1

finite difference schemes for elastic waves based on the integration approach the authors present a second order explicit finite difference scheme for elastic waves in 2 d nonhomogeneous media these schemes are based on integrating the equations of motion and the stress free surface conditions across the discontinuities before discretizing them on a grid as an alternative for the free surface treatment a scheme using zero density above the surface is suggested this scheme is first order and is shown to be a natural consequence of the integrated equations of motion and is called a vacuum formalism these schemes remove instabilities encountered in earlier integration schemes the consistency study reveals a close link between the vacuum formalism and the integrated discretized stress free condition giving priority to the vacuum formalism when a material discontinuity reaches the free surface the two presented free surface treatments coincide in the sense of the limit grid size 0 for lateral homogeneity at or near the free surface

application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media we present and analyze a perfectly matched absorbing layer model for the velocity stress formulation of elastodynamics the principal idea of this method consists of introducing an absorbing layer in which we decompose each component of the unknown into two auxiliary components a component orthogonal to the boundary and a component parallel to it a system of equations governing these new unknowns then is constructed a damping term finally is introduced for the component orthogonal to the boundary this layer model has the property of generating no reflection at the interface between the free medium and the artificial absorbing medium in practice both the boundary condition introduced at the outer boundary of the layer and the dispersion resulting from the numerical scheme produce a small reflection which can be controlled even with very thin layers as we will show with several experiments this model gives very satisfactory results namely the reflection coefficient even in the case of heterogeneous anisotropic media is about 1 for a layer thickness of five space discretization steps

perfectly matched layers for time harmonic elastodynamics of unbounded domains theory and finite element implementation abstract one approach to the numerical solution of a wave equation on an unbounded domain uses a bounded domain surrounded by an absorbing boundary or layer that absorbs waves propagating outwards from the bounded domain a perfectly matched layer pml is an unphysical absorbing layer model for linear wave equations that absorbs almost perfectly outgoing waves of all non tangential angles of incidence and of all non zero frequencies this paper develops the pml concept for time harmonic elastodynamics in cartesian coordinates utilising insights obtained with electromagnetics pmls and presents a novel displacement based symmetric finite element implementation of the pml for time harmonic plane strain or three dimensional motion the pml concept is illustrated through the example of a one dimensional rod on elastic foundation and through the anti plane motion of a two dimensional continuum the concept is explored in detail through analytical and numerical results from a pml model of the semi infinite rod on elastic foundation and through numerical results for the anti plane motion of a semi infinite layer on a rigid base numerical results are presented for the classical soil structure interaction problems of a rigid strip footing on a i half plane ii layer on a half plane and iii layer on a rigid base the analytical and numerical results obtained for these canonical problems demonstrate the high accuracy achievable by pml models even with small bounded domains

stable difference approximations for the elastic wave equation in second order formulation we consider the three dimensional elastic wave equation for an isotropic heterogeneous material subject to a stress free boundary condition building on our recently developed theory for difference methods for second order hyperbolic systems h o kreiss n a petersson j ystrom siam j numer anal 40 2002 pp 1940 1967 we develop an explicit second order accurate technique which is stable for all ratios of longitudinal over transverse phase velocities the spatial discretization is self adjoint and the stability is obtained through an energy estimate seismic events are often modeled using singular source terms and we devise a technique to place sources independently of the grid while retaining second order accuracy away from the source several numerical examples are given

brief paper h control for singularly perturbed systems this paper derives a set of e independent sufficient and necessary conditions for the h control problem for singularly perturbed systems in a different approach from previous ones a sub optimal controller is explicitly constructed we observe that the controller is also singularly perturbed with its fast slow part being an h sub optimal controller for the fast slow sub system of the original system respectively e independent h controller design is also considered the main contribution of this paper is that our results apply to both standard and non standard singularly perturbed systems

accurate viscoelastic modeling by frequency domain finite differences using rotated operators the viscoelastic wave equation is an integro differential equation that requires special methods when using time domain numerical finite difference methods in the frequency domain the integral terms are easily represented by complex valued elastic media properties there are further significant advantages to using the frequency domain if the forward or the inverse problem requires modeling or inverting a large number of prestack source gathers numerical modeling is expensive for seismic data because of the large number of wavelenghths typically separating sources from receivers which results in a need for a large number of grid points a major obstacle to using frequency domain methods is the consequent storage requirements to reduce these we maximize the accuracy and simultaneously minimize the spatial extent of the numerical operators we achieve this by extending earlier published methods introduced for the viscoacoustic case to the viscoelastic case this requires the formulation of two new nume

on a wavelet based method for the numerical simulation of wave propagation a wavelet based method for the numerical simulation of acoustic and elastic wave propagation is developed using a displacement velocity formulation and treating spatial derivatives with linear operators the wave equations are rewritten as a system of equations whose evolution in time is controlled by first order derivatives the linear operators for spatial derivatives are implemented in wavelet bases using an operator projection technique with nonstandard forms of wavelet transform using a semigroup approach the discretized solution in time can be represented in an explicit recursive form based on taylor expansion of exponential functions of operator matrices the boundary conditions are implemented by augmenting the system of equations with equivalent force terms at the boundaries the wavelet based method is applied to the acoustic wave equation with rigid boundary conditions at both ends in 1 d domain and to the elastic wave equation with a traction free boundary conditions at a free surface in 2 d spatial media the method can be applied directly to media with plane surfaces and surface topography can be included with the aid of distortion of the grid describing the properties of the medium the numerical results are compared with analytic solutions based on the cagniard technique and show high accuracy the wavelet based approach is also demonstrated for complex media including highly varying topography or stochastic heterogeneity with rapid variations in physical parameters these examples indicate the value of the approach as an accurate and stable tool for the simulation of wave propagation in general complex media

absorbing boundary conditions and surface waves one of the major problems in numerically simulating waves traveling in the earth is that an artificial boundary must be introduced to produce unique solutions to eliminate the spurious reflections introduced by this artificial boundary we use a damping expression based on analogies to shock absorbers this method can reduce the amplitude of the reflected wave to any pre specified value and is successful for waves at any angle of incidence the method can eliminate unwanted reflections from the surface reflections at the corners of the model and waves reflected off an interface that strike the artificial boundary many of the boundary conditions currently used in the numerical solution of waves are approximations to perfectly absorbing boundary conditions and depend upon the angle of incidence of the incoming wave at the artificial boundary stability problems often occur with these boundary conditions the method we use at the artificial boundary allows use of stable dirichlet or von neumann condition

implicit system approach to the robust stability for a class of singularly perturbed linear systems abstract asymptotic stability and the complex stability radius of a class of singularly perturbed systems of linear differential algebraic equations daes are studied the asymptotic behavior of the stability radius for a singularly perturbed implicit system is characterized as the parameter in the leading term tends to zero the main results are obtained in direct and short ways which involve some basic results in linear algebra and classical analysis only our results can be extended to other singular perturbation problems for daes of more general form

absorbing boundary conditions for acoustic and elastic waves in stratified media abstract absorbing boundary conditions are needed for computing numerical models of wave motions in unbounded spatial domains prior progress on this problem for acoustic and elastic waves has generally been concerned with waves propagating through uniform media the present paper is concerned with waves in stratified media which are of interest for example in geophysical problems suppose that the medium consists of homogeneous layers separated by parallel horizontal interfaces and suppose that absorbing boundary conditions are needed along a vertical computational boundary the boundary conditions that are described in this paper are based on a quantity known as the ray parameter according to snell s law this parameter remains the same when a plane wave propagates through a stratified medium and undergoes reflection refraction and in the case of elastic waves conversion once can therefore use the same absorbing boundary conditions in all layers for acoustic waves the absorption properties are the same in all layers for elastic waves the absorption properties vary somewhat from one layer to another however one still obtains good absorption in all layers even in the presence of strong contrasts between layers the boundary conditions are also effective in absorbing rayleigh waves which propagate along free surfaces of elastic media the boundary formulas developed here can be applied without modification to problems in both two and three dimensions

the singular minimum entropy h control problem abstract in this paper we search for controllers which minimize an entropy function of the closed loop transfer matrix under the constraint of internal stability and under the constraint that the closed loop transfer matrix has h norm less than some a priori given bound we find an explicit expression for the infimum moreover we give a characterization when the infimum is attained contrary to the regular case for the singular minimum entropy h control problem the infimum is not always attained

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

artificial boundary conditions for 2d problems in geophysics abstract the numerical solution of a class of two dimensional problems in geophysics is considered the earth is modeled as a semi infinite acoustic or elastic medium in order to make the computational domain finite a semi circular artificial boundary ob is introduced thus truncating the original semi infinite domain an exact nonlocal relation is then derived on ob this relation is used as a boundary condition on ob which is combined with the finite element scheme employed in the computational domain the approach is applied to the helmholtz equation modeling time harmonic acoustic waves and to problems in elastostatics numerical examples validate the method and demonstrate its performance

3d finite difference frequency domain modeling of visco acoustic wave propagation using a massively parallel direct solver a feasibility study we present a finite difference frequency domain method for 3d visco acoustic wave propagation modeling in the frequency domain the underlying numerical problem is the resolution of a large sparse system of linear equations whose right hand side term is the source this system is solved with a massively parallel direct solver we first present an optimal 3d finite difference stencil for frequency domain modeling the method is based on a parsimonious staggered grid method differential operators are discretized with second order accurate staggered grid stencils on different rotated coordinate systems to mitigate numerical anisotropy an antilumped mass strategy is implemented to minimize numerical dispersion the stencil incorporates 27 grid points and spans two grid intervals dispersion analysis shows that four grid points per wavelength provide accurate simulations in the 3d domain to assess the feasibility of the method for frequency domain full waveform inversion we computed simulations in the 3d seg eage overthrust model for frequencies 5 7 and 10 hz results confirm the huge memory requirement of the factorization several hundred figabytes but also the cpu efficiency of the resolution phase few seconds per shot heuristic scalability analysis suggests that the memory complexity of the factorization is o 35n 4 for a n 3 grid our method may provide a suitable tool to perform frequency domain full waveform inversion using a large distributed memory platform further investigation is still necessary to assess more quantitatively the respective merits and drawbacks of time and frequency domain modeling of wave propagation to perform 3d full waveform inversion

fast 3d frequency domain full waveform inversion with a parallel block low rank multifrontal direct solver application to obc data from the north sea wide azimuth long offset obc obn surveys provide a suitable framework to perform computationally efficient frequency domain full waveform inversion fwi with a few discrete frequencies frequency domain seismic modeling is performed efficiently with moderate computational resources for a large number of sources with a sparse multifrontal direct solver gauss elimination techniques for sparse matrices approximate solutions of the time harmonic wave equation are computed using a block low rank blr approximation leading to a significant reduction in the operation count and in the volume of communication during the lu factorization as well as offering a great potential for reduction in the memory demand moreover the sparsity of the seismic source vectors is exploited to speed up the forward elimination step during the computation of the monochromatic wavefields the relevance and the computational efficiency of the frequency domain fwi performed in the visco acoustic vti approximation is shown with a real 3d obc case study from the north sea the fwi subsurface models show a dramatic resolution improvement relative to the initial model built by reflection traveltime tomography the amplitude errors introduced in the modeled wavefields by the blr approximation for different low rank thresholds have a negligible footprint in the fwi results with respect to a standard multifrontal sparse direct factorization and without compromise on the accuracy of the imaging the blr approximation can bring a reduction of the lu factor size by a factor up to three this reduction is not yet exploited to reduce the effective memory usage ongoing work the flop reduction can be larger than a factor of 10 and can bring a factor of time reduction of around three moreover this reduction factor tends to increase with frequency namely with the matrix size frequency domain visco acoustic vti fwi can be viewed as an efficient tool to build an initial model for elastic fwi of 4 c obc data

analysis of a cartesian pml approximation to acoustic scattering problems in r2 abstract in this paper we consider a cartesian pml approximation to solutions of acoustic scattering problems on an unbounded domain in r 2 the perfectly matched layer pml technique in a curvilinear coordinate system has been researched for acoustic scattering applications both in theory and computation our goal will be to extend the results of spherical cylindrical pml to pml in cartesian coordinates that is the well posedness of cartesian pml approximation on both the unbounded and truncated domains the exponential convergence of approximate solutions as a function of domain size is also shown we note that once the stability and convergence of the continuous truncated problem has been achieved the analysis of the resulting finite element approximations is then classical finally the results of numerical computations illustrating the theory and efficiency of the cartesian pml approach will be given