an evd algorithm for para hermitian polynomial matrices an algorithm for computing the eigenvalue decomposition of a para hermitian polynomial matrix is described this amounts to diagonalizing the polynomial matrix by means of a paraunitary similarity transformation the algorithm makes use of elementary paraunitary transformations and constitutes a generalization of the classical jacobi algorithm for conventional hermitian matrix diagonalization a proof of convergence is presented the application to signal processing is highlighted in terms of strong decorrelation and multichannel data compaction some simulated results are presented to demonstrate the capability of the algorithm

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion householder transformations we present a practical and efficient means to compute the singular value decomposition svd of a real or complex quaternion matrix a based on bidiagonalization of a to a real or complex bidiagonal matrix b using quaternionic householder transformations computation of the svd of b using an existing subroutine library such as lapack provides the singular values of a the singular vectors of a are obtained trivially from the product of the householder transformations and the real or complex singular vectors of b we show in the paper that left and right quaternionic householder transformations are different because of the non commutative multiplication of quaternions and we present formulae for computing the householder vector and matrix in each case

regular article a fast spectral algorithm for nonlinear wave equations with linear dispersion spectral algorithms offer very high spatial resolution for a wide range of nonlinear wave equations on periodic domains including well known cases such as the korteweg de vries and nonlinear schrodinger equations for the best computational efficiency one needs also to use high order methods in time while somehow bypassing the usual severe stability restrictions we use linearly implicit multistep methods with the innovation of choosing different methods for different ranges in fourier space high accuracy at low wavenumbers and a stability at high wavenumbers this new approach compares favorably to alternatives such as split step and integrating factor or linearly exact methods

time domain vector potential technique for the meshless radial point interpolation method summary a time domain meshless algorithm based on vector potentials is introduced for the analysis of transient electromagnetic fields the proposed numerical algorithm is a modification of the radial point interpolation method where radial basis functions are used for local interpolation of the vector potentials and their derivatives in the proposed implementation solving the second order vector potential wave equation intrinsically enforces the divergence free property of the electric and magnetic fields furthermore the computational effort associated with the generation of a dual node distribution as required for solving the first order maxwell s equations is avoided the proposed method is validated with several examples of 2d waveguides and filters and the convergence is empirically demonstrated in terms of node density or size of local support domains it is further shown that inhomogeneous node distributions can provide increased convergence rates that is the same accuracy with smaller number of nodes compared with a solution for homogeneous node distribution a comparison of the magnetic vector potential technique with conventional radial point interpolation method is performed highlighting the superiority of the divergence free formulation copyright 2015 john wiley sons ltd

energy stable and high order accurate finite difference methods on staggered grids for wave propagation over distances of many wavelengths high order finite difference methods on staggered grids are widely used due to their excellent dispersion properties however the enforceme

spatial finite difference approximations for wave type equations the simplest finite difference approximations for spatial derivatives are centered explicit and applied to regular equispaced grids well established generalizations include the use of implicit compact approximations and staggered grids we find here that the combination of these two concepts together with high formal order of accuracy is very effective for approximating the first derivatives in space that occur in many wave type pdes

two dimensional axisymmetric numerical simulation of supercritical phase conjugation of ultrasound in active solid media in the present study a two dimensional axisymmetric numerical model is developed for supercritical parametric phase conjugation of ultrasound in a solid active element of cylindrical shape and finite length the pseudospectral time domain algorithm pstd is used owing to its efficiency to model large scale problems pstd solves elastic wave equation in time dependent heterogeneous isotropic and axisymmetric anisotropic solids using ffts for high order approximation of the spatial differential operator on staggered grid and a fourth order adams bashforth time integrator in order to truncate the computational domain absorbing boundary conditions are introduced with complex frequency shifted perfectly matched layers this procedure is highly effective at absorbing signals of long time signature the free surface of the active ceramic rod is introduced through the method of images a systematic study of the influence of lateral limitations of the active medium on parametric wave phase conjugation of sound

finite difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid we describe the application of the rotated staggered grid rsg finite difference technique to the wave equations for anisotropic and viscoelastic media the rsg uses rotated finite difference operators leading to a distribution of modeling parameters in an elementary cell where all components of one physical property are located only at one single position this can be advantageous for modeling wave propagation in anisotropic media or complex media including high contrast discontinuities because no averaging of elastic moduli is needed the rsg can be applied both to displacement stress and to velocity stress finite difference fd schemes whereby the latter are commonly used to model viscoelastic wave propagation with a von neumann style anlysis we estimate the dispersion error of the rsg scheme in general anisotropic media in three different simulation examples all based on previously published problems we demonstrate the application and the accuracy of the proposed numerical approach

hybrid time domain solvers for the maxwell equations in 2d numerical simulation is an important tool in understanding the electromagnetic field and how it interacts with the environment different topics for time domain finite difference fdtd and finite element fetd methods for maxwell s equations are treated in this thesis subcell models are of vital importance for the efficient modeling of small objects that are not resolved by the grid a novel model for thin sheets using shell elements is proposed this approach has the advantage of taking into account discontinuities in the normal component of the electric field unlike previous models based on impedance boundary conditions ibcs several results are presented to illustrate the capabilities of the shell element approach waveguides are of fundamental importance in many microwave applications for example in antenna feeds the key issues of excitation and truncation of waveguides are addressed a complex frequency shifted form of the uniaxial perfectly matched layer upml absorbing boundary condition abc in fetd is developed prism elements are used to promote automatic grid generation and enhance the performance results are presented where reflection errors below 70db are obtained for different types of waveguides including inhomogeneous cases excitation and analysis via the scattering parameters are achieved using waveguide modes computed by a general frequency domain mode solver for the vector helmholtz equation huygens surfaces are used in both fdtd and fetd for excitation in waveguide ports inverse problems have received an increased interest due to the availability of powerful computers an important application is non destructive evaluation of material a time domain minimization approach is presented where exact gradients are computed using the adjoint problem the approach is applied to a general form of maxwell s equations including dispersive media and upml successful reconstruction examples are presented both using synthetic and experimental measurement data parameter reduction of complex geometries using simplified models is an interesting topic that leads to an inverse problem gradients for subcell parameters are derived and a successful reconstruction example is presented for a combined dielectric sheet and slot geometry

elastic wave propagation in heterogeneous anisotropic media using the lumped finite element method a numerical technique for wave propagation simulation in 2 d heterogeneous anisotropic structures is presented the scheme is flexible in incorporating arbitrary surface topography inner openings liquid solid boundaries and irregular interfaces and it naturally satisfies the free surface conditions of complex geometrical boundaries the algorithm based on a discretization mesh of triangles and quadrilaterals solves the problem using integral equilibrium around each node instead of satisfying elastodynamic differential equations at each node as in the finite difference method this study is an extension of previous work for the elastic isotropic case besides accounting for anisotropy a simplified quadrilateral grid cell with low computational cost is introduced the transversely isotropic medium with a symmetry axis on the horizontal or vertical plane as typically caused by a system of parallel cracks or fine layers is discussed in detail a 2 d algorithm is presented that can handle the situatio

high order finite differences and the pseudospectral method on staggered grids finite difference approximations for the first derivative valid halfway between equidistant gridpoints are in general much more accurate than the corresponding approximations which are valid at gridpoints the pseudospectral fourier method can be viewed as the limit of finite difference approximations when the order of accuracy tends to infinity a fundamentally different and more accurate pseudospectral method is obtained from the halfway approximations this study derives these and related methods and discusses their accuracies

three dimensional elastic modeling by the fourier method earlier work on three dimensional forward modeling is extended to elastic waves using the equations of con servation of momentum and the stress strain relations for an isotropic elastic medium undergoing infinitesimal deformation in addition to arbitrary compressional or p wave velocity and density variation in lateral and vertical directions elastic modeling permits shear or s wave velocity variation as well the elastic wave equation is solved using a generalization of the method for the acoustic case computation of each time step begins by computing six strain components by per forming nine spatial partial differentiation operations on the three displacement components from the pre vious time step the six strains and two lame constants are linearly combined to yield six stress components

modeling of pseudoacoustic p waves in orthorhombic media with a low rank approximation abstractwavefield extrapolation in pseudoacoustic orthorhombic anisotropic media suffers from wave mode coupling and stability limitations in the parameter range we use the dispersion relation for scalar wave propagation in pseudoacoustic orthorhombic media to model acoustic wavefields the wavenumber domain application of the laplacian operator allows us to propagate the p waves exclusively without imposing any conditions on the parameter range of stability it also allows us to avoid dispersion artifacts commonly associated with evaluating the laplacian operator in space domain using practical finite difference stencils to handle the corresponding space wavenumber mixed domain operator we apply the low rank approximation approach considering the number of parameters necessary to describe orthorhombic anisotropy the low rank approach yields space wavenumber decomposition of the extrapolator operator that is dependent on space location regardless of the parameters a feature necessary for orthorhomb

implicit finite difference simulations of seismic wave propagation we propose a new finite difference modeling method implicit both in space and in time for the scalar wave equation we use a three level implicit splitting time integration method for the temporal derivative and implicit finite difference operators of arbitrary order for the spatial derivatives both the implicit splitting time integration method and the implicit spatial finite difference operators require solving systems of linear equations we show that it is possible to merge these two sets of linear systems one from implicit temporal discretizations and the other from implicit spatial discretizations to reduce the amount of computations to develop a highly efficient and accurate seismic modeling algorithm we give the complete derivations of the implicit splitting time integration method and the implicit spatial finite difference operators and present the resulting discretized formulas for the scalar wave equation we conduct a thorough numerical analysis on grid dispersions of this new implicit

stability and convergence of staggered runge kutta schemes for semilinear wave equations a staggered runge kutta staggered rk scheme is a runge kutta type scheme using a time staggered grid as proposed by ghrist et al in 2000 6 afterwards verwer in two papers investigated the efficiency of a scheme proposed by ghrist et al 6 for linear wave equations we study stability and convergence properties of this scheme for semilinear wave equations in particular we prove convergence of a fully discrete scheme obtained by applying the staggered rk scheme to the mol approximation of the equation

a staggered fourth order accurate explicit finite difference scheme for the time domain maxwell s equations abstract we consider a model explicit fourth order staggered finite difference method for the hyperbolic maxwell s equations appropriate fourth order accurate extrapolation and one sided difference operators are derived in order to complete the scheme near metal boundaries and dielectric interfaces an eigenvalue analysis of the overall scheme provides a necessary but not sufficient stability condition and indicates long time stability numerical results verify both the stability analysis and the scheme s fourth order convergence rate over complex domains that include dielectric interfaces and perfectly conducting surfaces for a fixed error level we find the fourth order scheme is computationally cheaper in comparison to the yee scheme by more than an order of magnitude some open problems encountered in the application of such high order schemes are also discussed

an algorithm for calculating the qr and singular value decompositions of polynomial matrices in this paper a new algorithm for calculating the qr decomposition qrd of a polynomial matrix is introduced this algorithm amounts to transforming a polynomial matrix to upper triangular form by application of a series of paraunitary matrices such as elementary delay and rotation matrices it is shown that this algorithm can also be used to formulate the singular value decomposition svd of a polynomial matrix which essentially amounts to diagonalizing a polynomial matrix again by application of a series of paraunitary matrices example matrices are used to demonstrate both types of decomposition mathematical proofs of convergence of both decompositions are also outlined finally a possible application of such decompositions in multichannel signal processing is discussed

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

time step n tupling for wave equations abstractwe have constructed novel temporal discretizations for wave equations we first select an explicit time integrator that is of second order leading to classic time marching schemes in which the next value of the wavefield at the discrete time t t is computed from current values known at time t and the previous time t t then we determine how the time step can be doubled tripled or generally n tupled producing a new time stepping method in which the next value of the wavefield at the discrete time t n t is computed from current values known at time t and the previous time t n t in between time values of the wavefield are eliminated using the fourier method to calculate space derivatives the new time integrators allow larger stable time steps than traditional time integrators however like the lax wendroff procedure they require more computational effort per time step because the new schemes are developed from the classic second order time stepping scheme they will have the same proper

frequency dispersive materials for 3 d hybrid solvers in time domain the recursive convolution method to treat linear dispersive materials in the finite difference time domain fdtd is here generalized to an explicit finite volume solver and an implicit finite element solver both solvers are interfaced to fdtd resulting in two hybrid solvers the stability of the solvers is analyzed and the accuracy is demonstrated in several scattering cases where a plane wave illuminates a sphere with complex permittivity excellent agreement with the analytical mie series solution is obtained for materials of debye and lorentz type as well as for a material consisting of two lorentz poles

stability and numerical dispersion analysis of a fourth order accurate fdtd method in order to obtain high order accuracy a fourth order accurate finite difference time domain fdtd method is presented by kyu pyung hwang unlike conventional fdtd methods a staggered backward differentiation scheme instead of the leapfrog scheme is used to approximate the temporal partial differential operator however the high order of its characteristic equation derived by the von neumann method makes the analysis of its numerical dispersion and stability very difficult in automatic control theory there are two effective methods for the stability analysis i e the routh hurwitz test and the jury test the combination of the von neumann method with each of the two can strictly derive the stability condition which only makes use of the coefficients of its characteristic equation without numerically solving it the method of analysis in this paper is also applicable in the stability and numerical analysis of other high order accurate fdtd methods

lowrank seismic wave extrapolation on a staggered grid we evaluated a new spectral method and a new finite difference fd method for seismic wave extrapolation in time using staggered temporal and spatial grids we derived a wave extrapolation operator using a lowrank decomposition for a first order system of wave equations and designed the corresponding fd scheme the proposed methods extend previously proposed lowrank and lowrank fd wave extrapolation methods from the cases of constant density to those of variable density dispersion analysis demonstrated that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion the method of manufactured solutions coupled with mesh refinement was used to verify each method and to compare numerical errors tests on 2d synthetic examples demonstrated that the proposed method is highly accurate and stable the proposed methods can be used for seismic modeling or reverse time migration

correlograms for non stationary autoregressions graphical descriptions of temporal dependence two methods are available for computing these statistics one based on autocorrelations and the other on scaled autocovariances for a stationary time series the resulting plots are nearly identical when it comes to time series exhibiting non stationary features these methods can lead to very different results this has two consequences incorrect inferences can be drawn when confusing these concepts better discrimination between stationary and non stationarity is achieved when using autocorrelations instead of or along with the autocovariances which are commonly used in statistical software

classroom note calculation of weights in finite difference formulas the classical techniques for determining weights in finite difference formulas were either computationally slow or very limited in their scope e g specialized recursions for centered and staggered approximations for adams bashforth adams moulton and bdf formulas for odes etc two recent algorithms overcome these problems for equispaced grids such weights can be found very conveniently with a two line algorithm when using a symbolic language such as mathematica reducing to one line in the case of explicit approximations for arbitrarily spaced grids we describe a computationally very inexpensive numerical algorithm

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

a spectral scheme for wave propagation simulation in 3 d elastic anisotropic media this work presents a new scheme for wave propagation simulation in three dimensional 3 d elastic anisotropic media the algorithm is based on the rapid expansion method rem as a time integration algorithm and the fourier pseudospectral method for computation of the spatial derivatives the rem expands the evolution operator of the second order wave equation in terms of chebychev polynomials constituting an optimal series expansion with exponential convergence the modeling allows arbitrary elastic coefficients and density in lateral and vertical directions numerical methods which are based on finite difference techniques in time and space are not efficient when applied to realistic 3 d models since they require considerable computer memory and time to obtain accurate results on the other hand the fourier method permits a significant reduction of the working space and the rem algorithm gives machine accuracy with half the computational effort as the usual second order temporal differencing sch

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

comparison of taylor finite difference and window finite difference and their application in fdtd the finite difference time domain fdtd method is an important tool in numerical electromagnetic simulation there are many ways to construct a finite difference approximation such as the taylor series expansion theorem the filtering theory etc this paper aims to provide the comparison between the taylor finite difference tfd scheme based on the taylor series expansion theorem and the window finite difference wfd scheme based on the filtering theory their properties have been examined in detail separately in addition the formula of the generalized finite difference gfd scheme is presented which can include both the tfd scheme and the wfd scheme furthermore their application in the numerical solution of maxwell s equations is presented the formulas for the stability criterion and the numerical dispersion relation are derived and analyzed in order to evaluate their performance more accurately a new definition of error is presented upon it the effect of several factors including the grid resolution the courant number and the aspect ratio of the cell on the performance of the numerical dispersion is examined

music algorithm for vector sensors array using biquaternions in this paper we use a biquaternion formalism to model vector sensor signals carrying polarization information this allows a concise and elegant way of handling signals with eight dimensional 8 d vector valued samples using this model we derive a biquaternionic version of the well known array processing music algorithm and we show its superiority to classically used long vector approach new results on biquaternion valued matrix spectral analysis are presented of particular interest for the biquaternion music bq music algorithm is the decomposition of the spectral matrix of the data into orthogonal subspaces we propose an effective algorithm to compute such an orthogonal decomposition of the observation space via the eigenvalue decomposition evd of a hermitian biquaternionic matrix by means of a newly defined quantity the quaternion adjoint matrix the bq music estimator is derived and simulation results illustrate its performances compared with two other approaches in polarized antenna processing lv music and psa music the proposed algorithm is shown to be superior in several aspects to the existing approaches compared with lv music the bq music algorithm is more robust to modelization errors and coherent noise while it can detect less sources in comparaison with psa music our approach exhibits more accurate estimation of direction of arrival doa for a small number of sources while keeping the polarization information accessible

3 d wave simulation in anelastic media using the kelvin voigt constitutive equation we design a numerical algorithm for wave simulation in anelastic media in the presence of free surface which can be used to model seismic waves at the earth s surface and ultrasonic waves in heterogeneous materials the stress strain relation is based on the kelvin voigt mechanical model which has the advantage of not requiring additional field variables the model requires two anelastic parameters and twice the spatial derivatives of the lossless case the high frequency components of the wave field are more attenuated than the low frequency components with the attenuation factors being approximately proportional to the square of the frequency the modeling simulates 3 d waves by using the fourier and chebyshev methods to compute the spatial derivatives along the horizontal and vertical directions respectively we stretch the mesh in the vertical direction to increase the minimum grid spacing and reduce the computational cost instabilities of the chebyshev differential operator due to the implementation of the free surface boundary conditions are solved with a characteristic approach where the characteristic variables are evaluated at the source central frequency the results of the modeling are verified by comparisons to the analytical solutions for lamb s problem and propagation in unbounded homogeneous media examples illustrating the propagation of rayleigh and love waves are presented

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

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

simulation of anisotropic wave propagation based upon a spectral element method we introduce a numerical approach for modeling elastic wave propagation in 2 d and 3 d fully anisotropic media based upon a spectral element method the technique solves a weak formulation of the wave equation which is discretized using a high order polynomial representation on a finite element mesh for isotropic media the spectral element method is known for its high degree of accuracy its ability to handle complex model geometries and its low computational cost we show that the method can be extended to fully anisotropic media the mass matrix obtained is diagonal by construction which leads to a very efficient fully explicit solver we demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2 d transversely isotropic test case and by comparing its predictions against those based upon a finite difference method for a 2 d heterogeneous anisotropic medium we show its generality and its flexibility by modeling wave propagation in a 3 d transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid

numerical integration of damped maxwell equations we study the numerical time integration of maxwell s equations from electromagnetism following the method of lines approach we start from a general semidiscrete maxwell system for which a number of time integration methods are considered these methods have in common an explicit treatment of the curl terms central in our investigation is the question how to efficiently raise the temporal convergence order beyond the standard order of two in particular in the presence of an explicitly or implicitly treated damping term which models conduction

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

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

self sorting mixed radix fast fourier transforms abstract some examples are given of the uses of the fft algorithm in numerical weather prediction the algorithm is derived by means of matrix factorization in such a way that the many possible variants can be obtained in a unified manner particular attention is paid to self sorting variants of the algorithm but skeleton fortran routines are included for all the canonical forms the fft works by decomposing the discrete fourier transform into a sequence of short transforms algorithms for these short transforms are presented and used to derive operation counts for the whole algorithm the implementation of the fft on vector computers is described and in the final section it is demonstrated how savings can be achieved in the case of two dimensional transforms

efficient 3d frequency response modeling with spectral accuracy by the rapid expansion method frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the fourier transform the former approach requires solving systems of linear equations which becomes progressively difficult to tackle for larger scale models and for higher frequency components on the contrary the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses commonly used explicit time integration methods correspond to the truncated taylor series approximations that can cause significant errors for large time steps the rapid expansion method rem uses the chebyshev expansion and offers an optimal solution to the second order in time wave equations when applying the fourier transform to the time domain wavefield solution computed by the rem we can derive a frequency response modeling formula that has the same form as the original time domain rem equation but with different summation coefficients in particular the summation coefficients for the frequency response modeling formula corresponds to the fourier transform of those for the time domain modeling equation as a result we can directly compute frequency responses from the chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods when combined with the pseudospectral method in space this new frequency response modeling method can produce spectrally accurate results with high efficiency

anisotropic wave propagation through finite difference grids an algorithm is presented to solve the elastic wave equation by replacing the partial differentials with finite differences it enables wave propagation to be simulated in three dimensions through generally anisotropic and heterogeneous models the space derivatives are calculated using discrete convolution sums while the time derivatives are replaced by a truncated taylor expansion a centered finite difference scheme in cartesian coordinates is used for the space derivatives leading to staggered grids the use of finite difference approximations to the partial derivatives results in a frequency dependent error in the group and phase velocities of waves for anisotropic media the use of staggered grids implies that some of the elements of the stress and strain tensors must be interpolated to calculate the hook sum this interpolation induces an additional error in the wave properties the overall error depends on the precision of the derivative and interpolation operators the anisotropic symmetry system its orientation and the degree of anisotropy the dispersion relation for the homogeneous case was derived for the proposed scheme since we use a general description of convolution sums to describe the finite difference operators the numerical wave properties can be calculated for any space operator and an arbitrary homogeneous elastic model in particular phase and group velocities of the three wave types can be determined in any direction we demonstrate that waves can be modeled accurately even through models with strong anisotropy when the operators are properly designed

the qrd and svd of matrices over a real algebra abstract recent work in the field of signal processing has shown that the singular value decomposition of a matrix with entries in certain real algebras can be a powerful tool in this article we show how to generalise the qr decomposition and svd to a wide class of real algebras including all finite dimensional semi simple algebras twisted group algebras and clifford algebras two approaches are described for computing the qrd svd one jacobi method with a generalised givens rotation and one based on the artin wedderburn theorem

application of normalized pseudo laplacian to elastic wave modeling on staggered grids we extended the pseudo laplacian to staggered grids based on the concept of normalized pseudo laplacian and applied it to constructing the pseudoanalytical formulations for the variable density acoustic wave equation and the elastic wave equation acoustic wavefields only contain p waves and therefore only p wave pseudo laplacians are required for acoustic wave propagation in comparison two sets of staggered grid pseudo laplacians are needed in the elastic case in order to properly compensate for time stepping errors for both p waves and s waves we gave a thorough derivation of the pseudoanalytical method for the elastic wave equation based on normalized pseudo laplacians implemented on staggered grids and presented the resulting complete discretized formulas we proved that the staggered grid pseudo laplacian reduces to the pseudo laplacian for the scalar wave equation on standard grids when using zero compensation velocities for normalized pseudo laplacians the pseudoanalytical formulas simply re

pseudo analytical finite difference elastic wave extrapolation based on the k space method abstractcost effective elastic wave modeling is the key to practical elastic reverse time migration and full waveform inversion implementations we have developed an efficient elastic pseudo analytical finite difference pafd scheme for elastic wave extrapolation the elastic pafd scheme is based on a modified pseudo spectral method k space method in which a pseudo analytical operator is used to ensure the high accuracy of elastic wave extrapolation however the k space method is motivated for a pure wave mode and thus its application in coupled first order elastic wave equations may cause the elastic pseudo analytical operators to suffer from crosstalk between the p and s wavefields the approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal this is done first by performing a predictor corrector strategy in first order elastic wave equations to eliminate those errors during wave extrapolation considering the massive computational cost in

migration of seismic data by phase shift plus interpolation under the horizontally layered velocity assumption migration is defined by a set of independent ordinary differential equations in the wavenumber frequency domain the wave components are extrapolated downward by rotating their phases this paper shows that one can generalize the concepts of the phase shift method to media having lateral velocity variations the wave extrapolation procedure consists of two steps in the first step the wave field is extrapolated by the phase shift method using l laterally uniform velocity fields the intermediate result is l reference wave fields in the second step the actual wave field is computed by interpolation from the reference wave fields the phase shift plus interpolation pspi method is unconditionally stable and lends itself conveniently to migration of three dimensional data the performance of the methods is demonstrated on synthetic examples the pspi migration results are then compared with those obtained from a finite difference method

quaternion music for vector sensor array processing this paper considers the problem of direction of arrival doa and polarization parameters estimation in the case of multiple polarized sources impinging on a vector sensor array the quaternion model is used and a data covariance model is proposed using quaternion formalism a comparison between long vector orthogonality and quaternion vector orthogonality is also performed and its implications for signal subspace estimation are discussed consequently a music like algorithm is presented allowing estimation of wave s doas and polarization parameters the algorithm is tested in numerical simulations and performance analysis is conducted when compared with other music like algorithms for vector sensor array the newly proposed algorithm results in a reduction by half of memory requirements for representation of data covariance model and reduces the computational effort for equivalent performance this paper also illustrates a compact and elegant way of dealing with multicomponent complex valued data

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

two results concerning the stability of staggered multistep methods staggered spatial node layouts are widely used in the context of solving wave equations when this idea of staggering is extended to time integrators for wave equations one finds methods that offer both better accuracy and less restrictive stability conditions for an equivalent computational cost here we prove two results concerning staggered multistep methods we first prove that staggered backwards differentiation methods are unstable for all orders p ge 5 we then extend dahlquist s first stability barrier to explicit staggered multistep methods

determination of finite difference weights using scaled binomial windows the finite difference method evaluates a derivative through a weighted summation of function values from neighboring grid nodes conventional finite difference weights can be calculated either from taylor series expansions or by lagrange interpolation polynomials the finite difference method can be interpreted as a truncated convolutional counterpart of the pseudospectral method in the space domain for this reason we also can derive finite difference operators by truncating the convolution series of the pseudospectral method various truncation windows can be employed for this purpose and they result in finite difference operators with different dispersion properties we found that there exists two families of scaled binomial windows that can be used to derive conventional finite difference operators analytically with a minor change these scaled binomial windows can also be used to derive optimized finite difference operators with enhanced dispersion properties

some elementary distribution theory for an autoregression fitted to a random walk a permutation argument is used to derive some properties of regression coefficients in an autoregressive time series model with a polynomial trend and a unit root