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

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

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

accelerating hessian free gauss newton full waveform inversion via l bfgs preconditioned conjugate gradient algorithm abstractfull waveform inversion fwi has emerged as a powerful strategy for estimating subsurface model parameters by iteratively minimizing the difference between synthetic data and observed data the hessian free hf optimization method represents an attractive alternative to newton type and gradient based optimization methods at each iteration the hf approach obtains the search direction by approximately solving the newton linear system using a matrix free conjugate gradient cg algorithm the main drawback with hf optimization is that the cg algorithm requires many iterations in our research we develop and compare different preconditioning schemes for the cg algorithm to accelerate the hf gauss newton gn method traditionally preconditioners are designed as diagonal hessian approximations we additionally use a new pseudo diagonal gn hessian as a preconditioner making use of the reciprocal property of green s function furthermore we have developed an l bfgs inverse hessian preconditioning

simultaneous multifrequency inversion of full waveform seismic data we present a simultaneous multifrequency inversion approach for seismic data interpretation this algorithm inverts all frequency data components simultaneously a data weighting scheme balances the contributions from different frequency data components so the inversion process does not become dominated by high frequency data components which produce a velocity image with many artifacts a gauss newton minimization approach achieves a high convergence rate and an accurate reconstructed velocity image by introducing a modified adjoint formulation we can calculate the jacobian matrix efficiently allowing the material properties in the perfectly matched layers pmls to be updated automatically during the inversion process this feature ensures the correct behavior of the inversion and implies that the algorithm is appropriate for realistic applications where a priori information of the background medium is unavailable two different regularization schemes an l2 norm and a weighted l2 norm function a

a dispersion minimizing compact finite difference scheme for the 2d helmholtz equation in this paper we present a dispersion minimizing compact finite difference scheme for solving the 2d helmholtz equation which is a fourth order scheme the error between the numerical wavenumber and the exact wavenumber is analyzed and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

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

dispersion reducing methods for edge discretizations of the electric vector wave equation we present a novel strategy for minimizing the numerical dispersion error in edge discretizations of the time domain electric vector wave equation on square meshes based on the mimetic finite difference mfd method we compare this strategy called m adaptation to two other discretizations also based on square meshes one is the lowest order nedelec edge element discretization the other is a modified quadrature approach gy adaptation proposed by guddati and yue for the acoustic wave equation in two dimensions all three discrete methods use the same edge based degrees of freedom while the temporal discretization is performed using the standard explicit leapfrog scheme to obtain efficient and explicit time stepping methods the three schemes are further mass lumped we perform a dispersion and stability analysis for the presented schemes and compare all three methods in terms of their stability regions and phase error our results indicate that the method produced by gy adaptation and the nedelec method are both second order accurate for numerical dispersion but differ in the order of their numerical anisotropy fourth order versus second order respectively the result of m adaptation is a discretization that is fourth order accurate for numerical dispersion as well as numerical anisotropy numerical simulations are provided that illustrate the theoretical results

analysis of the numerical dispersion of waves in saturated poroelastic media abstract this work presents the analysis of the numerical dispersion properties of a finite element method used to approximate the solution of the equations of motion for a fluid saturated porous elastic solid a biot medium in the two dimensional case and the low frequency regime the finite element method employed comprises a nonconforming rectangular element for the approximation of each component of the displacement vector in the solid phase and the raviart thomas nedelec mixed finite element space of zero order for the fluid phase the study is carried out by constructing and analyzing analytic and numerical dimensionless dispersion relations and by evaluating derived quantities such as dimensionless phase and group velocities and dimensionless attenuation for the three type of waves predicted by biot s equations of motion as a measure of the numerical distortion it is observed that the finite element procedure introduces both numerical dispersion and anisotropy in all three waves in a similar fashion however only the slow wave displayed a significant frequency dependent behavior it was also observed that the loss of accuracy is more important for the dimensionless attenuation than for the dimensionless group or phase velocities the analysis presented yields lower bounds for the number of points per wavelength needed to reach a desired accuracy in the dimensionless phase and group velocities and attenuation coefficients in biot media

an adaptable 17 point scheme for high accuracy frequency domain acoustic wave modeling in 2d constant density media abstractfrequency domain modeling is a crucial part of full waveform inversion because of the huge computation problem it represents several techniques have been applied to reduce the number of grid points per wavelength among them a rotating coordinate system has been widely used however scholars using rotating coordinate systems did not obtain solutions under rectangular grid conditions which are extremely common in real data based on the average derivative method we have developed a new finite difference scheme that used a 17 point operator to approximate spatial derivatives and the mass acceleration term the new method can be applied to square and rectangular grids after optimizing the number of grid points per shortest wavelength was reduced to less than three with phase velocity errors no larger than 1 to suppress the reflection from the boundary we applied perfectly matched layer boundary conditions numerical tests further proved the validity of our adaptable 17 point scheme

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

a frequency space 2 d scalar wave extrapolator using extended 25 point finite difference operator finite difference frequency domain modeling for the generation of synthetic seismograms and crosshole tomography has been an active field of research since the 1980s the generation of synthetic seismograms with the time domain finite difference technique has achieved considerable success for waveform crosshole tomography and for wider applications in seismic reverse time migration this became possible with the rapid development of high performance computers however the space frequency x finite difference modeling technique is still beyond the capability of the modern supercomputer in terms of both cost and computer memory therefore finite difference time domain modeling is much more popular among exploration geophysicists a limitation of the space frequency domain is that the recently developed nine point scheme still requires that g the number of grid points per wavelength be 5 this value is greater than for most other numerical modeling techniques for example the pseudospectral scheme

dispersion analysis of an average derivative optimal scheme for laplace domain scalar wave equation laplace domain modeling is an important foundation of laplace domain full waveform inversion however dispersion analysis for laplace domain numerical schemes has not been completely established this hampers the construction and optimization of laplace domain modeling schemes by defining a pseudowavelength as a scaled skin depth i establish a method for laplace domain numerical dispersion analysis that is parallel to its frequency domain counterpart this method is then applied to an averagederivative nine point scheme for laplace domain scalar wave equation within the relative error of 1 the laplace domain average derivative optimal scheme requires four grid points per smallest pseudowavelength whereas the classic five point scheme requires 13 grid points per smallest pseudowavelength for general directional sampling intervals the average derivative optimal scheme is more accurate than the classic five point scheme for the same sampling intervals by using much smaller sampling intervals the classic fivepoint scheme can approach the accuracy of the averagederivative optimal scheme but the corresponding cost is much higher in terms of storage requirement and computational time

traveltime and amplitude calculations using the damped wave solution because of its computational efficiency prestack kirchhoff depth migration remains the method of choice for all but the most complicated geological depth structures further improvement in computational speed and amplitude estimation will allow us to use such technology more routinely and generate better images to this end we developed a new accurate and economical algorithm to calculate first arrival traveltimes and amplitudes for an arbitrarily complex earth model our method is based on numerical solutions of the wave equation obtained by using well established finite difference or finite element modeling algorithms in the laplace domain where a damping term is naturally incorporated in the wave equation we show that solving the strongly damped wave equation is equivalent to solving the eikonal and transport equations simultaneously at a fixed reference frequency which properly accounts for caustics and other problems encountered in ray theory using our algorithm we can easily calculate first arrival traveltimes for given models we present numerical examples for 2 d acoustic models having irregular topography and complex geological structure using a finite element modeling code

a multigrid based preconditioned krylov subspace method for the helmholtz equation with pml abstract in this paper we generalize the complex shifted laplacian preconditioner to the complex shifted laplacian pml preconditioner for the helmholtz equation with perfectly matched layer helmholtz pml equation the helmholtz pml equation is discretized by an optimal 9 point difference scheme and the preconditioned linear system is solved by the krylov subspace method especially by the biconjugate gradient stabilized method bi cgstab the spectral analysis of the linear system is given and a new matrix based interpolation operator is proposed for the multigrid method which is used to approximately invert the preconditioner the numerical experiments are presented to illustrate the efficiency of the preconditioned bi cgstab method with the multigrid based on the new interpolation operator also numerical results are given for comparing the performance of the new interpolation operator with that of classic bilinear interpolation operator and the one suggested in erlangga et al 2006 10

a nearly analytic discrete method for solving the acoustic wave equations in the frequency domain abstractwe have developed a nearly analytic discrete nad method to discretize frequency domain acoustic wave equations with an absorbing boundary condition we evaluate in detail the discrete process of wave equations to derive a linear system the sparse structure and eigenproperties of its coefficient matrix also called the impedance matrix were analyzed to reveal the intrinsic difficulty in solving the linear system efficiently to accelerate the forward modeling process in the frequency domain we introduce a class of inexact rotated block triangular preconditioners incorporated with krylov subspace methods to solve this linear system and test their numerical behaviors by comparing with other two commonly used preconditioners to this end we perform wavefield simulation by the nad method and another two conventional numerical schemes in various media numerical dispersion analysis and waveform comparison are also implemented for these numerical schemes our results show the superiority of our pro

an efficient helmholtz solver for acoustic transversely isotropic media abstractthe acoustic approximation even for anisotropic media is widely used in current industry imaging and inversion algorithms mainly because p waves constitute most of the energy recorded in seismic exploration the resulting acoustic formulas tend to be simpler resulting in more efficient implementations and they depend on fewer medium parameters however conventional solutions of the acoustic wave equation with higher order derivatives suffer from s wave artifacts thus we separate the quasi p wave propagation in anisotropic media into the elliptic anisotropic operator free of the artifacts and the nonelliptic anisotropic components which form a pseudodifferential operator we then develop a separable approximation of the dispersion relation of nonelliptic anisotropic components specifically for transversely isotropic media finally we iteratively solve the simpler lower order elliptical wave equation for a modified source function that includes the nonelliptical terms represented in the

a numerical scheme for solving a class of logarithmic integral equations arisen from two dimensional helmholtz equations using local thin plate splines abstract this paper presents a numerical method for solving logarithmic fredholm integral equations which occur as a reformulation of two dimensional helmholtz equations over the unit circle with the robin boundary conditions the method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions the local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain to compute logarithm like singular integrals appeared in the method we use a particular nonuniform gauss legendre quadrature rule since the scheme does not require any mesh generations on the domain it can be identified as a meshless method the error estimate of the proposed method is presented numerical results are included to show the validity and efficiency of the new technique these results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines moreover it seems that the algorithm of the presented approach is attractive and easy to implement on computers

the estimates of approximation by using a new type of weighted modulus of continuity in this paper we introduce a new type modulus of continuity for function f belonging to a particular weighted subspace of c 0 and show that it has some properties of ordinary modulus of continuity we obtain some estimates of approximation of functions with respect to a suitable weighted norm via the new type moduli of continuity finally we give some examples

computationally efficient three dimensional acoustic finite difference frequency domain seismic modeling in vertical transversely isotropic media with sparse direct solver abstractthe computational burden of frequency domain full waveform inversion fwi of wide aperture fixed spread data is conventionally reduced by limiting the inversion to a few discrete frequencies in this framework frequency domain seismic modeling is performed efficiently for multiple sources by solving the linear system resulting from the discretization of the time harmonic wave equation with the massively parallel sparse direct solver frequency domain seismic modeling based on the sparse direct solver dsfdm requires specific design finite difference stencils of compact support to minimize the computational cost of the lower upper decomposition of the impedance matrix in terms of memory demand and floating point operations a straightforward adaptation of such finite difference stencil originally developed for the isotropic acoustic wave equation is proposed to introduce vertical transverse isotropy vti in the modeling without any extra computational cost the method relies on a fourth ord

efficient calculation of the steepest descent direction for source independent seismic waveform inversion an amplitude approach in seismic waveform inversion if we have no information on source signature we need to invert seismic data and source signature either simultaneously or successively in order to avoid the iterative update of the source signature in waveform inversion based on classical local optimization techniques we propose two source independent objective functions using amplitude spectra of fourier transformed wavefields one is constructed by normalizing the amplitude spectra of observed data and modeled data with respect to the respective reference amplitudes the other is achieved by cross multiplying the amplitude spectra of observed data and modeled data with the respective reference amplitudes in the computation of the steepest descent direction we circumvent explicitly computing the jacobian by employing a matrix formalism of the wave equation in the frequency domain through numerical examples for the marmousi model we demonstrate that our inversion algorithms can reproduce the subsurface velocity structure without estimating source signature

a fourth order optimal finite difference scheme for the helmholtz equation with pml abstract in this paper 17 point and 25 point finite difference fd schemes for the helmholtz equation with perfectly matched layer pml in the two dimensional domain are presented it is shown that the 17 point fd scheme is inconsistent in the presence of pml however the 25 point fd scheme is pointwise consistent an error analysis for the numerical approximation of the exact wavenumber is also presented we present the global and refined 25 point finite difference schemes based on minimizing the numerical dispersion numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

a helmholtz iterative solver for 3d seismic imaging problems a preconditioned iterative solver for the 3d frequency domain wave equation applied to seismic problems is evaluated the preconditioner corresponds to an approximate inverse of a heavily damped wave equation deduced from the undamped wave equation the approximate inverse is computed with one multigrid cycle numerical results show that the method is robust and that the number of iterations increases roughly linearly with frequency when the grid spacing is adapted to keep a constant number of discretization points per wavelength to evaluate the relevance of this iterative solver the number of floating point operations required for two imaging problems are roughly evaluated this rough estimate indicates that the time domain migration approach is more than one order of magnitude faster the full wave form tomography based on a least squares formulation and a scale separation approach has the same complexity in both domains

a dispersion minimizing finite difference scheme and preconditioned solver for the 3d helmholtz equation in this paper a new 27 point finite difference method is presented for solving the 3d helmholtz equation with perfectly matched layer pml which is a second order scheme and pointwise consistent with the equation an error analysis is made between the numerical wavenumber and the exact wavenumber and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters a full coarsening multigrid based preconditioned bi cgstab method is developed for solving the linear system stemming from the helmholtz equation with pml by the finite difference scheme the shifted laplacian is extended to precondition the 3d helmholtz equation and a spectral analysis is given the discrete preconditioned system is solved by the bi cgstab method with a multigrid method used to invert the preconditioner approximately full coarsening multigrid is employed and a new matrix based prolongation operator is constructed accordingly numerical results are presented to demonstrate the efficiency of both the new 27 point finite difference scheme with refined parameters and the preconditioned bi cgstab method with the 3d full coarsening multigrid

an optimal 25 point finite difference scheme for the helmholtz equation with pml in this paper we present an optimal 25 point finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in two dimensional domain based on minimizing the numerical dispersion we propose the refined choice strategy for choosing optimal parameters of the 25 point finite difference scheme numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion

an improved optimal nearly analytical discretized method for 2d scalar wave equation in heterogeneous media based on the modified nearly analytical discrete operator abstractconventional finite difference methods will encounter strong numerical dispersion on coarse grids or when higher frequencies are used the nearly analytic discrete method nadm and its improved version developed recently can suppress numerical dispersion efficiently however the nadm is imperfect especially in long time seismic wave propagation simulation to overcome this limitation by minimizing the energy error function we have developed a modified optimal nearly analytical discretized mnad method the stencil of mnad for the 2d wave equation is a novel diamond stencil mnad has two major advantages the first one is that mnad can suppress numerical dispersion effectively on coarse grids which significantly improved computational efficiency and reduced memory demands the second advantage is that the energy error of mnad is much smaller after long time simulation the numerical dispersion analysis shows that the maximum phase velocity error was 5 92 even if only two sampling points w

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

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

a parallel multigrid based preconditioner for the 3d heterogeneous high frequency helmholtz equation we investigate the parallel performance of an iterative solver for 3d heterogeneous helmholtz problems related to applications in seismic wave propagation for large 3d problems the computation is no longer feasible on a single processor and the memory requirements increase rapidly therefore parallelization of the solver is needed we employ a complex shifted laplace preconditioner combined with the bi cgstab iterative method and use a multigrid method to approximate the inverse of the resulting preconditioning operator a 3d multigrid method with 2d semi coarsening is employed we show numerical results for large problems arising in geophysical applications

an implicit finite difference operator for the helmholtz equation abstractwe have developed an implicit finite difference operator for the laplacian and applied it to solving the helmholtz equation for computing the seismic responses in the frequency domain this implicit operator can greatly improve the accuracy of the simulation results without adding significant extra computational cost compared with the corresponding conventional explicit finite difference scheme we achieved this by taking advantage of the inherently implicit nature of the helmholtz equation and merging together the two linear systems one from the implicit finite difference discretization of the laplacian and the other from the discretization of the helmholtz equation itself the end result of this simple yet important merging manipulation is a single linear system similar to the one resulting from the conventional explicit finite difference discretizations without involving any differentiation matrix inversions we analyzed grid dispersions of the discrete helmholtz equation to show the accura

waveform inversion using a logarithmic wavefield although waveform inversion has been studied extensively since its beginning 20 years ago applications to seismic field data have been limited and most of those applications have been for global seismology or engineering seismology scale problems not for exploration scale data as an alternative to classical waveform inversion we propose the use of a new objective function constructed by taking the logarithm of wavefields allowing consideration of three types of objective function namely amplitude only phase only or both in our waveform inversion we estimate the source signature as well as the velocity structure by including functions of amplitudes and phases of the source signature in the objective function we compute the steepest descent directions by using a matrix formalism derived from a frequency domain finite element finite difference modeling technique our numerical algorithms are similar to those of reverse time migration and waveform inversion based on the adjoint state of the wave equation in order to demonstrate the practical applicability of our algorithm we use a synthetic data set from the marmousi model and seismic data collected from the korean continental shelf for noisefree synthetic data the velocity structure produced by our inversion algorithm is closer to the true velocity structure than that obtained with conventional waveform inversion when random noise is added the inverted velocity model is also close to the true marmousi model but when frequencies below 5 hz are removed from the data the velocity structure is not as good as those for the noise free and noisy data for field data we compare the time domain synthetic seismograms generated for the velocity model inverted by our algorithm with real seismograms and find that the results show that our inversion algorithm reveals short period features of the subsurface although we use wrapped phases in our examples we still obtain reasonable results we expect that if we were to use correctly unwrapped phases in the inversion algorithm we would obtain better results

a dispersion minimizing scheme for the 3 d helmholtz equation based on ray theory we develop a new dispersion minimizing compact finite difference scheme for the helmholtz equation in 2 and 3 dimensions the scheme is based on a newly developed ray theory for difference equations a discrete helmholtz operator and a discrete operator to be applied to the source and the wavefields are constructed their coefficients are piecewise polynomial functions of hk chosen such that phase and amplitude errors are minimal the phase errors of the scheme are very small approximately as small as those of the 2 d quasi stabilized fem method and substantially smaller than those of alternatives in 3 d assuming the same number of gridpoints per wavelength is used in numerical experiments accurate solutions are obtained in constant and smoothly varying media using meshes with only five to six points per wavelength and wave propagation over hundreds of wavelengths when used as a coarse level discretization in a multigrid method the scheme can even be used with down to three points per wavelength tests on 3 d examples with up to 108 degrees of freedom show that with a recently developed hybrid solver the use of coarser meshes can lead to corresponding savings in computation time resulting in good simulation times compared to the literature

which data residual norm for robust elastic frequency domain full waveform inversion elastic full waveform inversion is an ill posed data fitting procedure that is sensitive to noise inaccuracies of the starting model definitionofmultiparameterclasses andinaccuratemodeling of wavefield amplitudes we have investigated the performance of different minimization functionals as the least squares norm 2 the least absolute values norm 1 and combinations of both the huber and so called hybrid criteria with reference to twonoisyoffshorevalhallmodelandonshoreoverthrustmodel synthetic data sets the four minimization functionals were implemented in 2d elastic frequency domain full waveform inversion fwi where efficient multiscale strategies were designed by successive inversions of a few increasing frequencies for the offshore and onshore case studies the 1 norm provided the most reliable models for p and s wave velocities vp and vs even when strongly decimated data sets that correspond to fewfrequencieswereusedintheinversionandwhenoutlierspolluted the data the 2 norm can provide reliable results in the presence of uniform white noise for vp and vs if the data redundancyisincreasedbyrefiningthefrequencysamplingintervalin the inversion at the expense of computational efficiency the 1 norm and the huber and hybrid criteria unlike the 2 norm allowforsuccessfulimagingofthevsmodelfromnoisydataina soft seabed environment where the p to s waves have a small footprint in the data however the huber and hybrid criteria are sensitive to a threshold criterion that controls the transition between the criteria and that requires tedious trial and error investigations for reliable estimation the 1 norm provides a robust alternativetothe2 normforinvertingdecimateddatasetsinthe frameworkofefficientfrequency domainfwi

a general optimal method for a 2d frequency domain finite difference solution of scalar wave equation abstractwe have developed a general optimal method for 2d frequency domain finite difference simulation of the scalar wave equation for a given finite difference stencil this method can generate the dispersion equation and optimize the expansion coefficients many commonly used frequency domain finite difference schemes e g grids with different numbers of points rotated grids and grid spaces with different aspect ratios can be derived as special cases under this framework the possibility of expanding this method to 3d does exist based on the 2d scalar wave equation the optimized coefficients of 25 point 9 point 17 point and 15 point schemes have been worked out the dispersion analysis indicates that our 25 point scheme has much higher accuracy than the average derivative method 25 point scheme the number of grid points per the smallest wavelength is reduced from 2 78 to 2 13 for a maximum phase velocity errors of 1 the synthetic seismograms and the wavefield snapshots calculated using ou

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

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

a multigrid method for the helmholtz equation with optimized coarse grid corrections we study the convergence of multigrid schemes for the helmholtz equation focusing in particular on the choice of the coarse scale operators let g rm c denote the number of points per wavelength at the coarse level if the coarse scale solutions are to approximate the true solutions then the oscillatory nature of the solutions implies the requirement g rm c 2 however in examples the requirement is more like g rm c gtrsim 10 in a trade off involving also the amount of damping present and the number of multigrid iterations we conjecture that this is caused by the difference in phase speeds between the coarse and fine scale operators standard 5 point finite differences in two dimensions are our first example a new coarse scale 9 point operator is constructed to match the fine scale phase speeds we then compare phase speeds and multigrid performance of standard schemes with a scheme using the new operator the required g rm c is reduced from about 10 to about 3 5 with less dam

3d frequency domain finite difference viscoelastic wave modeling using weighted average 27 point operators with optimal coefficients abstractexperimental data suggest that the viscoelastic behavior of rocks is more easily and accurately described in the frequency domain than in the time domain supporting the idea of simulating seismic wave propagation in the frequency domain we evaluated weighted averaged 27 point finite difference operators for 3d viscoelastic wave modeling in the frequency domain within the proposed framework we developed general equations for normalized phase velocities that can be used with arbitrary finite difference operators three sets of weighting coefficients for second order central finite difference operators that minimize the numerical dispersion for up to five grid points per wavelength were found using a damped least squares ls criterion as well as a global optimization scheme based on l1 and l2 norm criteria the three sets produced very similar dispersion curves and improvement provided by global optimization appeared marginal in this respect we also evaluated a discrete form for the heterogen

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

optimal fourth order staggered grid finite difference scheme for 3d frequency domain viscoelastic wave modeling we investigate an optimal fourth order staggered grid finite difference scheme for 3d frequency domain viscoelastic wave modeling an anti lumped mass strategy is incorporated to minimize the numerical dispersion the optimal finite difference coefficients and the mass weighting coefficients are obtained by minimizing the misfit between the normalized phase velocities and the unity an iterative damped least squares method the levenberg marquardt algorithm is utilized for the optimization dispersion analysis shows that the optimal fourth order scheme presents less grid dispersion and anisotropy than the conventional fourth order scheme with respect to different poisson s ratios moreover only 3 7 grid points per minimum shear wavelength are required to keep the error of the group velocities below 1 the memory cost is then greatly reduced due to a coarser sampling a parallel iterative method named carp cg is used to solve the large ill conditioned linear system for the frequency domain modeling validations are conducted with respect to both the analytic viscoacoustic and viscoelastic solutions compared with the conventional fourth order scheme the optimal scheme generates wavefields having smaller error under the same discretization setups profiles of the wavefields are presented to confirm better agreement between the optimal results and the analytic solutions

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

on solvable lie groups of negative ricci curvature we study solvable lie groups which admit a left invariant metric of strictly negative ricci curvature we obtain necessary and sufficient conditions of the existence of such a metric for lie groups the nilradical of whose lie algebra is either abelian or heisenberg or standard filiform and discuss some open questions

a dispersion minimizing subgridding finite difference scheme for the helmholtz equation with pml in this paper we present a dispersion minimizing subgridding finite difference scheme for solving the helmholtz equation with perfectly matched layer pml in the two dimensional domain which is a second order scheme and pointwise consistent with the equation subgrids are used to discretize the computational domain and a refined choice strategy based on minimizing the numerical dispersion is proposed for choosing weight parameters for transitional nodes numerical experiments are given to illustrate that the newly proposed schemes can produce highly accurate seismic modeling results with enhanced efficiency compared to uniform grids

a stabilized radial basis finite difference rbf fd method with hybrid kernels abstract recent developments have made it possible to overcome grid based limitations of finite difference fd methods by adopting the kernel based meshless framework using radial basis functions rbfs such an approach provides a meshless implementation and is referred to as the radial basis generated finite difference rbf fd method in this paper we propose a stabilized rbf fd approach with a hybrid kernel generated through a hybridization of the gaussian and cubic rbf this hybrid kernel was found to improve the condition of the system matrix consequently the linear system can be solved with direct solvers which leads to a significant reduction in the computational cost as compared to standard rbf fd methods coupled with present stable algorithms unlike other rbf fd approaches the eigenvalue spectra of differentiation matrices were found to be stable irrespective of irregularity and the size of the stencils as an application we solve the frequency domain acoustic wave equation in a 2d half space in order to suppress spurious reflections from truncated computational boundaries absorbing boundary conditions have been effectively implemented

sensitivity kernels for seismic fresnel volume tomography fresnel volume tomography fvt offers higher resolution and better accuracy than conventional seismic raypath tomography a key problem in fvt is the sensitivity kernel we propose amplitude and traveltime sensitivity kernels expressed directly with green s functions for transmitted waves for 2d 3d homogeneous heterogeneous media the green s functions are calculated with a finite difference operator of the full wave equation in the frequency space domain in the special case of homogeneous media we analyze the properties of the sensitivity kernels extensively and gain new insight into these properties according to the constructive interference of waves the spatial distribution ranges of the monochromatic sensitivity kernels in fvt differ from each other greatly and are 1 8 2 8 3 8 and 4 8 periods of seismic waves respectively for 2d amplitude 3d amplitude 2d traveltime and 3d traveltime conditions we also have a new understanding of the relationship between raypath tomography and fvt within the first fresnel volume of the dominant frequency the band limited sensitivity kernels of fvt in homogeneous media or smoothly heterogeneous media are very close to those of the dominant frequency thus it is practical to replace the band limited sensitivity kernel with a few selected frequencies or even the single dominant frequency to save computation when performing band limited fvt the numerical experiment proves that fvt using our sensitivity kernels can achieve more accurate results than traditional raypath tomography

negative ricci curvature on some non solvable lie groups ii we construct many examples of lie groups admitting a left invariant metric of negative ricci curvature we study lie algebras which are semidirect products mathfrak l mathfrak a oplus mathfrak u imes mathfrak n and we obtain examples where mathfrak u is any semisimple compact real lie algebra mathfrak a is one dimensional and mathfrak n is a representation of mathfrak u which satisfies some conditions in particular when mathfrak u mathfrak s mathfrak u m mathfrak s mathfrak o m or mathfrak s mathfrak p m and mathfrak n is a representation of mathfrak u in some space of homogeneous polynomials we show that these conditions are indeed satisfied in the case mathfrak u mathfrak s mathfrak u 2 we get a more general construction where mathfrak n can be any nilpotent lie algebra where mathfrak s mathfrak u 2 acts by derivations we also prove a general result in the case when mathfrak u is a semisimple lie algebra of non compact type

a dispersion minimizing finite difference scheme for the helmholtz equation based on point weighting abstract in this paper we develop a new dispersion minimizing finite difference scheme for the helmholtz equation with perfectly matched layer pml in two dimensional domain which is a second order 9 point scheme to discretize the second derivative operator we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme based on minimizing the numerical dispersion the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy the new scheme is simple rotation free and pointwise consistent with the equation which is different from the classical rotated 9 point difference scheme obtained by combining the cartesian coordinate system and the rotated system moreover it is a robust scheme even if the step sizes of different directions are not equal convergence analysis and dispersion analysis are given several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme

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

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

an optimal compact sixth order finite difference scheme for the helmholtz equation abstract in this paper we present an optimal compact finite difference scheme for solving the 2d helmholtz equation a convergence analysis is given to show that the scheme is sixth order in accuracy based on minimizing the numerical dispersion a refined optimization rule for choosing the scheme s weight parameters is proposed numerical results are presented to demonstrate the efficiency and accuracy of the compact finite difference scheme with refined parameters

fast huygens sweeping methods for multiarrival green s functions of helmholtz equations in the high frequency regime abstractmultiarrival green s functions are essential in seismic modeling migration and inversion huygens kirchhoff hk integrals provide a bridge to integrate locally valid first arrival green s functions into a globally valid multiarrival green s function we have designed robust and accurate finite difference methods to compute first arrival traveltimes and amplitudes so that first arrival green s functions can be constructed rapidly we adapted a fast butterfly algorithm to evaluate discretized hk integrals the resulting fast huygens sweeping method has the following unique features 1 it precomputes a set of local traveltime and amplitude tables 2 it automatically takes care of caustics 3 it constructs green s functions of the helmholtz equation for arbitrary frequencies and for many point sources and 4 for a fixed number of points per wavelength it constructs each green s function in nearly optimal complexity o n log n in terms of the total number of mesh points n where the prefa

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

an average derivative optimal scheme for frequency domain scalar wave equation forward modeling is an important foundation of full waveform inversion the rotated optimal nine point scheme is an efficient algorithm for frequency domain 2d scalar wave equation simulation but this scheme fails when directional sampling intervals are different to overcome the restriction on directional sampling intervals of the rotated optimal nine point scheme i introduce a new finite difference algorithm based on an average derivative technique this new algorithm uses a nine point operator to approximate spatial derivatives and mass acceleration term the coefficients can be determined by minimizing phase velocity dispersion errors the resulting nine point optimal scheme applies to equal and unequal directional sampling intervals and can be regarded a generalization of the rotated optimal nine point scheme compared to the classical five point scheme the number of grid points per smallest wavelength is reduced from 13 to less than four by this new nine point optimal scheme for equal and unequal directional sampling intervals three numerical examples are presented to demonstrate the theoretical analysis the average derivative algorithm is also extended to a 2d viscous scalar wave equation and a 3d scalar wave equation