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

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

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

high order finite difference simulations of marine csem surveys using a correspondence principle for wave and diffusion fields the computer time required to solve a typical 3d marine controlled source electromagnetic surveying csem simulation can be reduced by more than one order of magnitude by transforming low frequency maxwell equations in the quasi static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain the dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling second order to eighth order spatial derivative operators are implemented for flexibility fourth order and sixth order methods are the most numerically efficient implementations for this particular scheme an implementation with high order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer the stability condition given for high order staggered derivative operators here should be equally valid for seismic wave simulation the bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain the fields in the fictitious wave domain do not represent observable fields propagation paths and interaction reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain however the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain the propagation paths that contribute most to the diffusive domain fields are airwave shallow water plus typically postcritical events such as refracted and guided waves the transform from the diffusive frequency domain to the fictitious wave domain is an ill posed problem the transform is nonunique this gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency domain fields

a new approach to modeling the electromagnetic response of conductive media we introduce a new and potentially useful method for computing electromagnetic em responses of arbitrary conductivity distributions in the earth the diffusive em field is known to have a unique integral representation in terms of a fictitious wave field that satisfies a wave equation we show that this integral transform can be extended to include vector fields our algorithm takes advantage of this relationship between the wave field and the actual em field specifically numerical computation is carried out for the wave field and the result is transformed back to the em field in the time domain the proposed approach has been successfully demonstrated using two dimensional 2 d models the appropriate te mode diffusion equation in the time domain for the electric field is initially transformed into a scalar wave equation in an imaginary q domain where q is a time like variable the corresponding scalar wave field is computed numerically using an explicit q stepping technique standard finite diffe

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

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

a study of causality structure and dynamics in industrial electricity consumption based on granger network abstract based on industrial electricity consumption we model industrial networks by granger causality method and mst minimum spanning tree and then further stick onto an industrial coupling mechanism from energy consumption perspective first we construct granger causality networks of five provinces in south china of gd gx gz hn and yn based on their industrial electricity consumption data and we demonstrate from a network topology perspective the distribution of weight of links of all industrial electricity consumption granger causality networks approximately follows power law distribution revealing a phenomenon that few industries may bring a tremendous influence on the rest moreover correlation analysis between weight and degree of a node shows that in most granger causality networks both span and strength of influence of a given industry will significantly increase further we analyze the relationship between the thresholds of granger causality significance and density of corresponding networks results show gd and hn could be classified into a group with relatively greater global differentiation in industries and unbalanced industrial development however gx gz and yn are grouped as second cluster with relatively balanced industrial development furthermore using chu liu edmonds mst algorithm we extract graphs of msts or maximal cliques from industrial electricity consumption granger causality networks and research on energy transmission structure feedback loop and bootstrap reliability by analyzing msts we find that only gd gx and yn can be extracted with mst graphs and capture the probable transmission routes of key nodes besides we illustrate all three mst graphs are involved with feedback loops structures with various characteristics gx has complete feed forward section feed back section and feedback loop section yn has only feed forward section and feedback loop section gd has multiple feedback loops section finally we conduct bootstrap reliability tests on msts results show branches of reliability of gd gx and yn are decentralized while those feedback loops sections are only concentrated on highly reliable branches as neither hn nor gz can be extracted with mst graphs we characterize them by maximal cliques it is demonstrated that both industrial systems of hn and gz are significantly grouped into two independent branches and those internal industries inside independent branches are highly correlated however from perspectives of whole south china the causal relationship is only converged on a few of industries for regions without mst graphs like hn and gz the reliability of links is also greater than that of gd gx and yn

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

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

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

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

the pseudospectral method accurate representation of interfaces in elastic wave calculations when finite difference methods are used to solve the elastic wave equation in a discontinuous medium the error has two dominant components dispersive errors lead to artificial wave trains errors from interfaces lead to circular wavefronts emanating from each location where the interface appears jagged to the rectangular grid the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy with this method dispersive errors are essentially eliminated the mappings introduced in this paper also eliminate the other dominant error source test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost although the mapping method is described here in connection with the pseudospectral method it can also be used with high order finite difference approximations

travelling waves on a membrane reflection and transmission at a corner of arbitrary angle ii this is the second part of an investigation into the reflection and transmission of fluid coupled membrane waves at a corner of arbitrary angle in part one of this work abrahams lawrie 1995 an exact solution was obtained for a model problem comprising of a fluid wedge of arbitrary angle 4 bounded by two identical semi infinite plane membranes and forced by a surface wave incident along one face of the wedge the problem was decomposed into a symmetric and an antisymmetric sub problem and closed form expressions for the reflection coefficients r s and r a respectively were derived the solution method incorporates several fundamental advancements on the work of maliuzhinets 1958 and offers a constructive approach by which wedge problems with higher order boundary conditions can be solved easily in this part of the investigation it is demonstrated how for rational wedge angles the formulae of part i can be exploited to yield simple exact or asymptotic expressions for r s r a and therefore the reflection and transmission coefficients for the full problem further a numerical implementation of the analytic solution enables these coefficients to be determined for all wedge angles 0 both for heavy and moderate fluid loading the results confirm the reflection coefficients known previously for a few special wedge angles and highlight several interesting trends in particular it is found that in the heavy fluid loading limit there is a remarkably simple relationship between the phases of r s r a and

the pseudospectral method comparisons with finite differences for the elastic wave equation the pseudospectral or fourier method has been used recently by several investigators for forward seismic modeling the method is introduced here in two different ways as a limit of finite differences of increasing orders and by trigonometric interpolation an argument based on spectral analysis of a model equation shows that the pseudospectral method for the accuracies and integration times typical of forward elastic seismic modeling may require in each space dimension as little as a quarter the number of grid points compared to a fourth order finite difference scheme and one sixteenth the number of points as a second order finite difference scheme for the total number of points in two dimensions these factors become 1 16 and 1 256 respectively in three dimensions they become 1 64 and 1 4 096 respectively in a series of test calculations on the two dimensional elastic wave equation only minor degradations are found in cases with variable coefficients and discontinuous interfaces

mimetic seismic wave modeling including topography on deformed staggered grids abstractfinite difference methods for modeling seismic waves are known to be inaccurate when including a realistic topography due to the large dispersion errors that appear in the modelled surface waves and the scattering introduced by the staircase approximation to the topography as a consequence alternatives to finite difference methods have been proposed to circumvent these issues we present a new numerical scheme for 3d elastic wave propagation in the presence of strong topography this finite difference scheme is based upon a staggered grid of the lebedev type or fully staggered grid fsg it uses a grid deformation strategy to make a regular cartesian grid conform to a topographic surface in addition the scheme uses a mimetic approach to accurately solve the free surface condition and hence allows for a less restrictive grid spacing criterion in the computations the scheme can use high order operators for the spatial derivatives and obtain low dispersion results with as few as six points pe

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

acoustic diffraction from the junction of two flat plates a general solution is developed for the acoustic and structural scattered response from the junction of two flat plates under unilateral fluid loading the plates are modelled by the classical theory of flexure and the solution is found using the wiener hopf technique for the dual integral equations for the unknown pressure on the plates explicit formulae are obtained for the pressure transform when the plates are in welded or clamped contact and corresponding explicit expressions are given for the various diffraction coefficients associated with the fluid structure interaction the magnitudes of the reflection and transmission coefficients of structural waves display very simple analytic forms at low frequency

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

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

application of the rational function approximation to wiener hopf calculations for fluid loaded plates this paper discusses the wiener hopf analysis for the solution of fluid loaded plate problems when the planar acoustic impedance is represented by a rational function approximation with such a formulation the factorization of the integrand kernel can immediately be written once the system poles are determined the approach is applied to several representative problems and the approximate results are shown to be in agreement with those of exact solutions

lead lag cross sectional structure and detection of correlated anticorrelated regime shifts application to the volatilities of inflation and economic growth rates abstract we have recently introduced the thermal optimal path top method to investigate the real time lead lag structure between two time series the top method consists in searching for a robust noise averaged optimal path of the distance matrix along which the two time series have the greatest similarity here we generalize the top method by introducing a more general definition of distance which takes into account possible regime shifts between positive and negative correlations this generalization to track possible changes of correlation signs is able to identify possible transitions from one convention or consensus to another numerical simulations on synthetic time series verify that the new top method performs as expected even in the presence of substantial noise we then apply it to investigate changes of convention in the dependence structure between the historical volatilities of the usa inflation rate and economic growth rate several measures show that the new top method significantly outperforms standard cross correlation methods

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

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

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

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

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

maximizing the cfl number of stable time space domain explicit finite difference modeling abstract the time space domain explicit finite difference fd method numerically solves wave equation by applying fd to approximate its spatial and temporal derivatives this method always faces the stability problem i theoretically analyze stability condition of this method for the acoustic wave equation and find that the maximum courant friedrichs lewy cfl number of stable modeling depends on the maximum value of the spatial fd dispersion relation the conventional method commonly determines spatial fd coefficients by approximating the spatial fd dispersion to the exact relation within the given wavenumber area however outside this area the dispersion relation and hence the maximum cfl number is uncontrollable to make it controllable i propose to simultaneously approximate the spatial fd dispersion relation to a given function outside this area further i develop an algorithm to maximize the maximum cfl number stability analysis and modeling examples demonstrate that the proposed method has better stability than the conventional method especially when the maximum effective wavenumber frequency is small for the 2d case the conventional method performs stable simulation using a cfl number of no more than 0 707 but the new method can perform stable simulation using a large cfl number for example 1 6 4 8 and 9 0 and non long spatial fd stencils

modeling elastic fields across irregular boundaries geologists often see the earth as homogeneous blocks separated by smoothly curving boundaries in contrast computer modeling algorithms based on finite difference schemes require elastic constants to be specified on the vertices of a regular rectangular grid how can we convert a continuous geological model into a form suitable for a finite difference grid one common way is to lay the finite difference grid down on the continuous geological model and use whatever elastic constants happen to lie beneath each of the grid points

time varying lead lag structure between the crude oil spot and futures markets abstract the linkages between crude oil spot and futures markets are widely studied in previous literatures most of these conventional methods are static linear and parametric in order to overcome this difficulty we employ the nonparametric and nonlinear symmetric thermal optimal path tops method to determine the time dependent lead lag relationship between two time series based on the daily weekly monthly spot and futures returns for maturities of one two three and four months of the west texas intermediate wti crude oil benchmark we apply the tops method together with self consistent test to investigate the time varying lead lag dependences between spot and futures markets from 1987 to 2017 we find that for total three data sets with different frequencies the tops paths indicate an alternate lead lag structure instead of a dominance between spot and futures markets in most of the time periods for the daily data the lead lag relationship between the two markets exists within one day except on some specific days moreover without considering the units of time on average the lead lag steps between the two markets are diminishing with the increasing frequency roughly speaking these periods with strong lead lag signals coincide with major influential changes in the oil and stock markets and the geopolitics which can reflect real economic financial and geopolitical regime changes all in all the results show that the lead lag relationship between the spot and futures oil markets exists only temporarily

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

approximations for the two dimensional green s function of a fluid loaded plate simple analytical formulas to predict the acoustic component of the two dimensional 2 d green s function of a fluid loaded plate are presented the formulas do not require any side computations such as numerical zero finding or numerical integrations the simplest result obtained provides a good estimate of the green s function for a very broad range of frequencies being degraded only near the coincidence frequency a somewhat more complicated formula almost uniformly valid in frequency and yielding reasonable accuracy even at coincidence is also presented numerical results comparing the asymptotic approximations to the exact green s function are presented and show significant improvement over a basic steepest descents based approximation for large values of x very high accuracy can be achieved with relative errors of less than 0 1 for other less stringent applications the approximations are accurate to within 10 at fairly small distances from the source requiring only kfx 1 and k0x 0 5

two dimensional simulation of rayleigh waves with staggered sine cosine transforms and variable grid spacing simulation of rayleigh waves requires high accuracy and an adequate spatial sampling at the surface discrete cosine and sine transforms are used to compute spatial derivatives along the direction perpendicular to the surface of the earth unlike the standard fourier method these transforms allow nonperiodicboundaryconditionstobesatisfied inparticular the stress free conditions at the surface because simulation of surface waves requires more points per minimum wavelengthatthesurfacethansimulationofbodywaves theequispaced grid is not efficient to overcome this problem a grid compression is performed at the surface to obtain a denser spatial sampling grid size is minimal at the surface and increases with depth until reaching at a relatively shallow depth the grid points per wavelength required by the body waves the stress free boundary conditions are naturally handled by expanding the appropriate stress components in terms of the discrete sine transform the wave equation is solved in the particle velocity and stress formulation using a runge kutta time integration and the convolutional pml cpmlmethodtopreventreflectionsfromthemeshboundaries the simulations are very accurate for shallow sources andreceiversandlargeoffsets

on the internal interfaces in finite difference schemes abstractimplementing sharp internal interfaces in finite difference schemes with high spatial accuracy is challenging the propagation of fields in a locally homogeneous part of a model can be performed with spectral accuracy the implementations of interfaces are generally considered accurate to at best second order this situation can be improved by a proper band limitation of the simulation grid interfaces can be located anywhere on the grid however the fine detail information regarding the interface location must be imprinted correctly in the coarse simulation grid this can be done by starting out with a representation of a sharp material jump in the wavenumber domain and limiting the highest wavenumber to the maximum wavenumber allowed for the simulation grid the resulting wavenumber representation is then transformed to the space domain an alternative procedure is to create a fine grid model that is low pass filtered to remove wavenumbers above the maximum wavenumber allowed for the coarse s

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

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

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

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

acoustic radiation from an insonified elastic plate with a line discontinuity this paper deals with the development of analytic models for the prediction of the acoustic radiated field from an infinite elastic plate with a single line force and line moment impedance discontinuity due to an incident plane acoustic wave the solution is in the form of a fourier integral with a kernel having ten poles the integral is evaluated by three methods the first uses the steepest descent path sdp method leading to a solution that decays as 1 k0r 1 2 the second method used conformal transformations and the modified saddle point msp method where all ten poles of the integrand are factored out this second method yields a solution that has complementary error functions and an asymptotic series in k0r the third method employs a transformation of the integrand to effect an efficient and fastly convergent numerical integration algorithm in general the msp asymptotic series solution and the numerical integration yielded numerically identical results however while the sdp solution pre

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

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

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

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

numerical analysis of a narrow angle one way elastic wave equation and extension to curvilinear coordinates in this paper we review the finite difference implementation of a narrow angle one way vector wave equation for elastic 3d media extrapolation is performed in the frequency domain where the second order accurate lateral spatial difference operators are sufficiently accurate for narrow angle propagation we perform a numerical analysis of the finite difference scheme to highlight the stability and dispersion characteristics the von neumann stability criterion indicates that extracting a reference phase during the extrapolation step noticeably improves the forward marching scheme and dispersion analysis shows that numerical grid anisotropy is minimal for the propagation path lengths source pulse spectral content and angular range of forward propagation of interest although the algorithm is reasonable its computational efficiency is limited by the second order accurate extrapolation step therefore the extrapolation scheme can be improved we extend the cartesian narrow angle formulation to curvilinear coordinates where the computational grid tracks the true wavefront in a reference medium and the wavefield derivative normal to the reference wavefront is evaluated locally using the cartesian propagator an example of curvilinear extrapolation for a simple model consisting of a high velocity sphere within a homogeneous background velocity structure shows that the narrow angle propagator is capable of modeling frequency dependent geometric spreading and diffraction effects in curvilinear coordinates

image normalization of wiener hopf operators and boundary transmission value problems for a junction of two half planes abstract the present paper deals with an application of the image normalization technique for certain classes of wiener hopf operators whos associated to ill posed boundary transmission value problems we briefly describe the method of normalization and then apply it to boundary transmission value problems issued from diffraction problems for a junction of two half planes which are relevant in mathematical physics applications for each boundary transmission value problem we analyze the conditions under which the associated operator and the equivalent who are not normally solvable and define the corresponding image normalized operators

acoustic diffraction by a fluid loaded membrane corner the complete solution is derived for an acoustic plane wave incident on a membrane which is stretched over a line constraint forming a wedge shaped fluid region of interior angle 2 the exact two dimensional solution is developed using malyuzhinets functions and integral transforms explicit formulae are presented for the acoustic and structural diffraction coefficients for 2 and numerical results are given for the frequency and angular dependence of the membrane wave excitation coefficients for identical membranes on either side of the vertex

modeling of the acoustic wave equation with transform methods numerical methods are described for the simulation of wave phenomena with application to the modeling of seismic data two separate topics are studied the first deals with the solution of the acoustic wave equation the second topic treats wave phenomena whose direction of propagation is restricted within 90 degrees from a given axis in the numerical methods developed here the wave field is advanced in time by using standard time differencing schemes on the other hand expressions including space derivative terms are computed by fourier transform methods this approach to computing derivatives minimizes truncation errors another benefit of transform methods becomes evident when attempting to restrict propagation to upward moving waves e g to avoid multiple reflections constraints imposed on the direction of the wave propagation are accomplished most precisely in the wavenumber domain the error analysis of the algorithms shows that truncation errors are due mainly to time discretization such er

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

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

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

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

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

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

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

time compact difference methods for wave propagation in discontinuous media in b gustafsson and e mossberg siam j sci comput 26 2004 pp 259 271 a fourth order one step method was constructed for the solution of wave propagation problems the method is based on the first order system form of the pde and uses a staggered grid both in space and time the method was also applied with good results to a problem with discontinuous coefficients without using any special procedure across the discontinuity in this paper we will analyze a model problem from acoustics and demonstrate the theoretical foundation for this behavior furthermore we shall present more detailed numerical experiments which confirm the theoretical results

computation of the homogeneous and forced solutions of a finite length line driven submerged plate a formulation is developed to predict the vibration response of a finite length submerged plate due to a line drive the formulation starts by describing the fluid in terms of elliptic cylinder coordinates which allows the fluid loading term to be expressed in terms of mathieu functions by moving the fluid loading term to the right hand side of the equation it is considered to be a force the operator that remains on the left hand side is the same as that of the in vacuo plate a fourth order constant coefficient ordinary differential equation therefore the problem appears to be an inhomogeneous ordinary differential equation the solution that results has the same form as that of the in vacuo plate the sum of a forced solution and four homogeneous solutions each of which is multiplied by an arbitrary constant these constants are then chosen to satisfy the structural boundary conditions on the two ends of the plate results for the finite plate are compared to the infinite plate in both the wave number and spatial domains the theoretical predictions of the plate velocity response are also compared to results from finite element analysis and show reasonable agreement over a large frequency range

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed