wave field separation in two dimensional anisotropic media until recently the term elastic usually implied two dimensional 2 d and isotropic in this limited context the divergence and curl operators have found wide use as wave separation operators for example mora 1987 used them in his inversion method to allow separate correlation of p and s arrivals although the separation is buried in the math and not obvious clayton 1981 used them explicitly in several modeling and inversion methods devaney and oristaglio 1986 used closely related operators to separate p and s arrivals in elastic vsp data

2d and 3d elastic wavefield vector decomposition in the wavenumber domain for vti media a pragmatic decomposition of a vector wavefield into p and s waves is based on the helmholtz theory and the christoffel equation it is applicable to vti media when the plane wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber except for the kiss singularities on the vti symmetry axis unlike divergence and curl which separate the wavefield into a scalar and a vector field the decomposed p and s wavefields are both vector fields with correct amplitude phase and physical units if the vector components of decomposed wavefields are added they reconstruct those of the original input wavefield wavefield propagation in any portions of a vti medium that have the same polarization distribution i e the same eigenvector in the wavenumber domain have the same decomposition operators and can be recon structed with a single 3d fourier transform for each operator e g one for p waves and one for s waves this applies to isotropic wavefields a

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

particle method solution of two dimensional convection diffusion equations abstract we present a new method for solving two dimensional convection dominated convection diffusion equations containing spatially and temporally localized source terms the method uses grid free transport particles which carry point values of the concentration gradient move by convection and undergo random walk to simulate diffusion no numerical diffusion is introduced equations for the evolution of the gradient values are solved the concentration is recovered from the particle data by solving poisson equations the method is applied to problems in rotational and elongational flow fields numerical results demonstrating convergence of the method are presented

fast algorithms for elastic wave mode separation and vector decomposition using low rank approximation for anisotropic media abstractwave mode separation and vector decomposition are significantly more expensive than wavefield extrapolation and are the computational bottleneck for elastic reverse time migration in heterogeneous anisotropic media we have expressed elastic wave mode separation and vector decomposition for anisotropic media as space wavenumber domain operations in the form of fourier integral operators and developed fast algorithms for their implementation using their low rank approximations synthetic data generated from 2d and 3d models demonstrated that these methods are accurate and efficient

accuracy of the random vortex method for a problem with non smooth initial conditions abstract the accuracy of the random vortex method in approximating the solution of the navier stokes equation is investigated using the model problem of a circular vortex as suggested by milinazzo and saffman 13 the method consists of partitioning the vorticity into vortex blobs these blobs are moved via two actions first a blob is deterministically moved under the action of the velocity field associated with the other blobs then to simulate viscosity a random component is added to the position of the blob for this model problem the nonlinear terms of the navier stokes equation vanish thus the major error inherent in the deterministic component of the method vanishes consequently for this model problem concentration is on the interaction of the deterministic and random components of the method results show that the accuracy of the method depends heavily on the initial distribution and strength of the computational elements i e the vortex blobs with the right choice of initial conditions it is found that e1 t l t a t a t is o r 1 2 n 1 2 where l t and a t are respectively the exact and computed angular moment of vorticity distribution at time t for n vortex blobs at a reynolds number r

partially reflected diffusion the radiation reactive or robin boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries the underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary however the relation between the reaction constant in the robin boundary condition and the reflection probability is not well defined in this paper we define the partially reflected process as a limit of the markovian jump process generated by the euler scheme for the underlying ito dynamics with partial boundary reflection trajectories that cross the boundary are terminated with probability p sqrt delta t and otherwise are reflected in a normal or oblique direction we use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the fokker planck equation in a half space wi

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

scalar reverse time depth migration of prestack elastic seismic data reflected p to p and p to s converted seismic waves in a two component elastic common source gather generated with a p wave source in a two dimensional model can be imaged by two independent scalar reverse time depth migrations the inputs to migration are pure p and s waves that are extracted by divergence and curl calculations during shallow extrapolation of the elastic data recorded at the earth s surface for both p to p and p to s converted reflected waves the imaging time at each point is the p wave traveltime from the source to that point the extracted p wave is reverse time extrapolated and imaged with a p velocity model using a finite difference solution of the scalar wave equation the extracted s wave is reverse time extrapolated and imaged similarly but with an s velocity model converted s wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources synthetic examples show that the algorithm gives satisfactory resul

vortex simulation of laminar recirculating flow abstract the accuracy of the random vortex method a numerical scheme that combines the representation of the vorticity field by a number of lagrangian vortex elements of finite cores with a stochastic simulation of diffusion using random walk is investigated the converged solutions for a 2 dimensional entry flow in a channel and a recirculating flow behind a rearward facing step both in the laminar range of reynolds numbers are compared with analytical results and experimental data respectively it is shown that the computed results converge as the strength of computational vortex elements is decreased and that the method is accurate to within the experimental errors the convergence properties are used to devise a scheme for optimizing the choice of the numerical parameters

spectral function methods for nonlinear diffusion equations two spectral function methods are developed for linear and nonlinear diffusion equations in one dimension where the nonlinearity is in the inhomogeneous term and occurs as a power of the solution in the single spectral function method polynomial spectral functions in the spatial variable are introduced the spectral resolution of the diffusion equation in the hilbert space spanned by these functions yields a system of ordinary differential equations which is then integrated in discrete steps of the time variable the double spectral method introduces polynomial spectral functions in both space and time variables and thereby eliminates the need for time integration through application of an iterative algorithm both methods are compared against analytical solutions for the linear cases and against the numerical solutions for the nonlinear cases the second spectral function method was found to be more efficient than the first by a factor of 6 in the case of nonlinear problems

seismic modeling in viscoelastic media anelasticity is usually caused by a large number of physical mechanisms which can be modeled by different microstructural theories a general way to take all these mechanisms into account is to use a phenomenologic model such a model which is consistent with the properties of anelastic media can be represented mechanically by a combination of springs and dash pots a suitable system can be constructed by the parallel connection of several standard linear elements and is referred to as the general standard linear solid rheology two relaxation functions that describe the dilatational and shear dissipation mechanisms of the medium are needed this model properly describes the short and long term behaviors of materials with memory and is the basis for describing viscoelastic wave propagation this work presents two dimensional 2 d and three dimensional 3 d forward modeling in linear viscoelastic media the theory implements boltzmann s superposition principle based on a spectrum of relaxation mechanisms

boundary conditions for stochastic solutions of the convection diffusion equation stochastic methods offer an attractively simple solution to complex transport controlled problems and have a wide range of physical chemical and biological applications stochastic methods do not suffer from the numerical diffusion that plagues grid based methods but they typically lose accuracy in the vicinity of interfacial boundaries in this work we introduce some ideas and algorithms that can be used to implement boundary conditions in stochastic simulations of the convection diffusion equation with accuracies comparable to the bulk phase the algorithms have been tested in two dimensional channel flows over a range of peclet numbers and compared with independent finite difference calculations

phase correction in separating p and s waves in elastic data two dimensional elastic data containing reflected p waves and converted s wave generated by a p source may be separated using dilatation and rotation calculation sun 1999 the algorithm is a combination of elastic full wavefield extrapolation sun and mcmechan 1986 chang and mcmechan 1987 1994 and wave type separation using dilatation divergence and rotation curl calculations dellinger and etgen 1990 it includes 1 downward extrapolating the multicomponent elastic data in an elastic velocity model using the elastic wave equation 2 calculating the dilatation to represent pure p waves and calculating the rotation to represent pure s waves at some depth and 3 upward extrapolating the dilatation in a p velocity model and upward extrapolating the rotation in an s velocity model using the acoustic wave equation for each

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

comparison of two algorithms for isotropic elastic p and s vector decomposition the research leading to this paper is supported by the sponsors of the ut dallas geophysical consortium a portion of the computations was done at the texas advanced computing center this paper is contribution no 1270 from the department of geosciences at the university of texas at dallas special thanks go to j brown a shabelansky and two anonymous reviewers for their useful comments and encouragements

amplitude balancing in separating p and s waves in 2d and 3d elastic seismic data the reflected p and s waves in elastic displacement component data recorded at the earth s surface are separated by reverse time downward extrapolation of the data in an elastic computational model followed by calculations to give divergence dilatation and curl rotation at a selected reference depth the surface data are then reconstructed by separate forward time upward scalar extrapolations from the reference depth of the magnitude of the divergence and curl wavefields and extraction of the separated p and s waves respectively at the top of the models a p wave amplitude will change by a factor that is inversely proportional to the p velocity when it is transformed from displacement to divergence and an s wave amplitude will change by a factor that is inversely proportional to the s velocity when it is transformed from displacement to curl consequently the ratio of the p to the s wave amplitude the p s amplitude ratio in the form of divergence and curl postseparation is different

hydrodynamics in two dimensions and vortex theory we consider the navier stokes equation for a viscous and incompressible fluid inr2 we show that such an equation may be interpreted as a mean field equation vlasov like limit for a system of particles called vortices interacting via a logarithmic potential on which in addition a stochastic perturbation is acting more precisely we prove that the solutions of the navier stokes equation may be approximated in a suitable way by finite dimensional diffusion processes with the diffusion constant related to the viscosity as a particular case when the diffusion constant is zero the finite dimensional theory reduces to the usual deterministic vortex theory and the limiting equation reduces to the euler equation

a hierarchical elastic full waveform inversion scheme based on wavefield separation and the multistep length approach abstractelastic full waveform inversion fwi updates model parameters by minimizing the residuals of the p and s wavefields resulting in more local minima and serious nonlinearity in addition the coupling of different parameters degrades the inversion results to address these problems we have developed a hierarchical elastic fwi scheme based on wavefield separation and a multistep length gradient approach first we have derived the gradients expressed by different wave modes analyzed the crosstalk between various parameters and evaluated the sensitivity of separated p wave separated s wave and p and s wave misfit functions then a practical four stage inversion workflow was developed in the first stage conventional fwi is used to achieve rough estimates of the p and s wave velocities in the second stage we only invert the p wave velocity applying the separated p wavefields when strong s wave energy is involved or we merely update the s wave velocity by matching the separated s wavefiel

p and s decomposition in anisotropic media with localized low rank approximations abstractwe have developed a p and s wave decomposition algorithm based on windowed fourier transforms and a localized low rank approximation with improved scalability and efficiency for anisotropic wavefields the model and wavefield are divided into rectangular blocks that do not have to be geologically constrained low rank approximations and p and s decomposition are performed separately in each block an overlap add method reduces artifacts at block boundaries caused by fourier transforms at wavefield truncations limited communication is required between blocks localization allows a lower rank to be used than global low rank approximations while maintaining the same quality of decomposition the algorithm is scalable making p and s decomposition possible in complicated 3d models tests with 2d and 3d synthetic data indicate good p and s decomposition results

semi lagrangian methods for parabolic problems in divergence form semi lagrangian methods have traditionally been developed in the framework of hyperbolic equations but several extensions of the semi lagrangian approach to diffusion and advection diffusion problems have been proposed recently these extensions are mostly based on probabilistic arguments and share the common feature of treating second order operators in trace form which makes them unsuitable for mass conservative models like the classical formulations of turbulent diffusion employed in computational fluid dynamics we propose here some basic ideas for treating second order operators in divergence form a general framework for constructing consistent schemes in one space dimension is presented and a specific case of nonconservative discretization is discussed in detail and analyzed finally an extension to possibly nonlinear problems in an arbitrary number of dimensions is proposed although the resulting discretization approach is only of first order in time numerical results in a number of test

convergence of a random walk method for the burgers equation we show that the solution of the burgers equation can be approximated in ital l sup 1 bold r the within ital o ital m sup minus 1 4 ln ital m sup 2 by a random walk method generated by ital o ital m particles the nonlinear advection term of the equation is approximated by advecting the particles in a velocity field induced by the particles the diffusive term is approximated by adding an appropriate random perturbation to the particle positions it is also shown that the corresponding viscuous splitting algorithm approximates the solution of the burgers equation in ital l sup 1 bold r to within ital o ital k when ital k is the size of the time step this work provides the first proof of convergence in a strong sense for a random walk method in which the related advection equation allows for the formation of shocks

graph decomposition is np complete a complete proof of holyer s conjecture an h decomposition of a graph g v e is a partition of e into subgraphs isomorphic to h given a fixed graph h the h decomposition problem is to determine whether an input graph g admits an h decomposition in 1980 holyer conjectured that h decomposition is np complete whenever h is connected and has three edges or more some partial results have been obtained since then a complete proof of holyer s conjecture is the content of this paper the characterization problem of all graphs h for which h decomposition is np complete is hence reduced to graphs where every connected component contains at most two edges

temporal high order staggered grid finite difference schemes for elastic wave propagation abstractthe traditional high order finite difference fd methods approximate the spatial derivatives to arbitrary even order accuracy whereas the time discretization is still of second order accuracy temporal high order fd methods can improve the accuracy in time greatly however the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation we have developed two temporal high order staggered grid fd sfd schemes for modeling elastic wave propagation a new stencil containing the points on the axis and a few off axial points is introduced to approximate the spatial derivatives we derive the dispersion relations of the elastic wave equation based on the new stencil and we estimate fd coefficients by the taylor series expansion te the te based scheme can achieve 2m th order spatial and 2n th order temporal accuracy n 5 we further optimize the coefficients of fd operators using a combination of te and least squares ls the fd coefficients at the

incorporation of attenuation into time domain computations of seismic wave fields the only numerically tractable way yet found to incorporate attenuation into numerical time domain computations of seismic wave fields is to approximate the viscoelastic modulus by a low order rational function of frequency the coefficients of this function can be determined by the pade approximation our test computations show however that this approximation generally is of poor quality therefore we suggest a new approach which is based on the rheological model of the generalized maxwell body which has a modulus of the desired rational form we choose the relaxation frequencies logarithmically equidistant in the frequency band of interest and determine the weight factors by simple numerical curve fitting to an arbitrary q law this approach is superior to the method above both in accuracy and in computational efficiency for most practical applications approximations of orders 2 or 3 are sufficient the computing time and memory requirements for a finite difference calculation are then approximately twice those of a purely elastic calculation as a first application of the method we compute sh channel waves in discontinuous coal seams with q 50 within the coal the results show that the high frequency airy phase is strongly attenuated this indicates that care has to be taken in comparing the results of purely elastic model calculations of the propagation of seam waves with experimental data

grid free simulation of diffusion using random walk methods abstract the simulation of diffusion of a continuum field by the random walk displacement of a set of particles is investigated in detail computational particles are used to transport elements of the gradients of the diffusive concentration one dimensional and quasi one dimensional cases are treated for a generalized diffusion variable different types of boundary conditions of the diffusion equation are considered as well as the extensions to a system of coupled diffusion equations the reaction diffusion equations and the convection diffusion set of equations it is shown that by using concentration gradients in the random walk process statistical errors are reduced and each realization of the numerical solution is a representation of the exact solution moreover transport of higher order derivatives can be utilized to improve the smoothness the algorithm is grid free and the computational elements move to follow the gradients thus it is self adaptive and uniform resolution is attained for all times the method is particularly suitable for the simulation of diffusion in systems which involve more than one transport mechanism when the diffusivity is small and when lagrangian elements are used to model the other mechanisms

the numerical analysis of random particle methods applied to vlasov poisson fokker planck kinetic equations we devise and study a random particle method for approximating vlasov poisson fokker planck systems such a proposed scheme takes into account the fact that the trajectories of a particle undergoing brownian motion due to collisions with the medium or background particles can be obtained as the solutions of stochastic differential equations i e the langevin equations these equations are the precise analogs of the deterministic hamiltonian system in the collisionless model the particle approximation in particular simulates the action of viscosity by the use of independent wiener processes brownian motions the error analysis generalizes that of ganguly and victory siam j numer anal 26 1989 pp 249 288 by incorporating into the consistency estimates the sampling errors generated by the random motion of the particles these particular errors are the dominant component of the consistency errors and are estimated by applying bennett s inequality for estimating tail probabilities for standard

generalized random walk algorithm for the numerical modeling of complex diffusion processes a generalized form of the random walk algorithm to simulate diffusion processes is introduced unlike the usual approach at a given time all the particles from a grid node are simultaneously scattered using the bernoulli repartition this procedure saves memory and computing time and no restrictions are imposed for the maximum number of particles to be used in simulations we prove that for simple diffusion the method generalizes the finite difference scheme and gives the same precision for large enough number of particles as an example simulations of diffusion in random velocity field are performed and the main features of the stochastic mathematical model are numerically tested 2003 elsevier science b v all rights reserved

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

vector based elastic reverse time migration based on scalar imaging condition abstractthe scalar images pp ps sp and ss of elastic reverse time migration ertm can be generated by applying an imaging condition as crosscorrelation of pure wave modes in conventional ertm helmholtz decomposition is commonly applied in wavefield separation which leads to a polarity reversal problem in converted wave images because of the opposite polarity distributions of the s wavefields polarity reversal of the converted wave image will cause destructive interference when stacking over multiple shots besides in the 3d case the curl calculation generates a vector s wave which makes it impossible to produce scalar ps sp and ss images with the crosscorrelation imaging condition we evaluate a vector based ertm vb ertm method to address these problems in vb ertm an amplitude preserved wavefield separation method based on decoupled elastic wave equation is exploited to obtain the pure wave modes the output separated wavefields are both vectorial to obtain the scalar images the scala

particle method and its convergence for scalar conservation laws a particle method is presented for solving the scalar conservation laws the stability of the method is analyzed and its convergence to the unique entropy solution is proved the main feature of this method is that particles with same sign of mass are allowed to interpenetrate but those with different sign of mass cancel each other when they meet in their movement this feature makes the method highly adaptive another main feature of this method is that its convergence to the entropy solution in the l 1 norm is proved even if it is not written in conservation form and the l 1 convergent rate for the riemann solutions a shock wave or a rarefaction wave is shown to be the order of o varepsilon 2 delta t or o varepsilon 2 respectively where varepsilon and delta t are the size of the particle mass and the timestep the total number of particles is o varepsilon 1 some numerical experiments are presented to illustrate the performance of this method

combination of the meshless finite difference approach with the monte carlo random walk technique for solution of elliptic problems abstract this paper proposes a stochastic approach for the fast and effective numerical analysis of the second order elliptic differential equations it is based upon the well known monte carlo mc method with a random walk rw technique carried out on the grid of points this method allows for accurate estimation of the solution of the differential equation at selected point s of the domain and or its boundary it extends the standard formulation of the monte carlo random walk mc rw approach by means of its appropriate combination with the meshless version of the finite difference method in this manner the proposed approach may deal with elliptic equations in more general non homogeneous form as well as boundary conditions of both essential and natural types moreover arbitrarily irregular clouds of nodes may be used with no a priori imposed nodes structure therefore the meshless mc rw approach may be applied to the significantly wider class of problems with more complex geometry this concept was examined on variety of 2d boundary value problems selected numerical results are presented and discussed a simple matlab code is included as well

viscoelastic finite difference modeling real earth media disperse and attenuate propagating mechanical waves this anelastic behavior can be described well by a viscoelastic model we have developed a finite difference simulator to model wave propagation in viscoelastic media the finite difference method was chosen in favor of other methods for several reasons finite difference codes are more portable than for example pseudospectral codes moreover finite difference schemes provide a convenient environment in which to define complicated boundaries a staggered scheme of second order accuracy in time and fourth order accuracy in space appears to be optimally efficient because of intrinsic dispersion no fixed grid points per wavelength rule can be given instead we present tables which enable a choice of grid parameters for a given level of accuracy since the scheme models energy absorption natural and efficient absorbing boundaries may be implemented merely by changing the parameters near the grid boundary the viscoelastic scheme is

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

an efficient scheme for convection dominated transport partial differential equations describing transport processes involving a significant effect of the flow velocity may be solved efficiently and easily using a simple algorithm the algorithm is based on the propagator s or green s functions for the equations of transport theory the numerical method employed is always at least as fast as finite differencing and it reduces to a finite difference method in the short time step limit but is especially efficient in cases where flow dominates over diffusion and is consequently widely applicable in kinetic theory and fluid dynamics using this method the ion distribution function and the potential in a plasma sheath were calculated in the presence of charge exchange collisions for a wide range of neutral densities