sequential convex subdifferential calculus and sequential lagrange multipliers the aim of this paper is to establish formulas for the subdifferentials of the sum and the composition of convex functions in terms of the subdifferentials of the data functions at nearby points applications to general optimization problems lead to a new notion of sequential lagrange multipliers

generalized bremmer series with rational approximation for the scattering of waves in inhomogeneous media the bremmer series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper sparse matrix representations of these pseudodifferential operators are derived the authors focus on designing sparse matrices keeping the accuracy high at the cost of ignoring any critical scattering angle phenomena such matrix representations follow from rational approximations of the vertical slowness and the transverse laplace operator symbols and of the vertical derivative as they appear in the parabolic equation method sparse matrix representations lead to a fast algorithm an optimization procedure is followed to minimize the errors in the high frequency limit for a given discretization rate the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counterpropagating constituents each of these steps is represented by a sparse matrix equation the resulting algorithm provides an improvement of the parabolic equation method in particular for transient wave phenomena and extends the latter method systematically for backscattered waves

asymptotics for some semilinear hyperbolic equations with non autonomous damping abstract let v and h be hilbert spaces such that v h v with dense and continuous injections consider a linear continuous operator a v v which is assumed to be symmetric monotone and semi coercive given a function f v h and a map w loc 1 1 r r such that lim t t 0 our purpose is to study the asymptotic behavior of the following semilinear hyperbolic equation d 2 u d t 2 t t d u d t t a u t f u t 0 t 0 the nonlinearity f is assumed to be monotone and conservative condition 0 t d t guarantees that some suitable energy function tends toward its minimum the main contribution of this paper is to provide a general result of convergence for the trajectories of e if the quantity t behaves as k t for some 0 1 k 0 and t large enough then u t weakly converges in v toward an equilibrium as t strong convergence in v holds true under compactness or symmetry conditions we also give estimates for the speed of convergence of the energy under some ellipticity like conditions the abstract results are applied to particular semilinear evolution problems at the end of the paper

flux normalized wavefield decomposition and migration of seismic data separation of wavefields into directional components can be accomplished by an eigenvalue decomposition of the accompanying system matrix in conventional pressurenormalized wavefield decomposition the resulting one way wave equations contain an interaction term which depends on the reflectivity function applying directional wavefield decomposition using flux normalized eigenvalue decomposition and disregarding interaction between up and downgoing wavefields these interaction terms were absent by also applying a correction term for transmission loss the result was an improved estimate of the up and downgoing wavefields in the wave equation angle transform a crosscorrelation function in local offset coordinates was fourier transformed to produce an estimate of reflectivity as a function of slowness or angle we normalized this wave equation angle transform with an estimate of the planewave reflection coefficient the flux normalized oneway wave propagation scheme was applied to imaging and to the normalized wave equation angle transform on synthetic and field data this proved the effectiveness of the new methods

application of unsplit convolutional perfectly matched layer for scalar arbitrarily wide angle wave equation abstracta classical split perfectly matched layer pml method has recently been applied to the scalar arbitrarily wide angle wave equation awwe in terms of displacement however the classical split pml obviously increases computational cost and cannot efficiently absorb waves propagating into the absorbing layer at grazing incidence our goal was to improve the computational efficiency of awwe and to enhance the suppression of edge reflections by applying a convolutional pml cpml we reformulated the original awwe as a first order formulation and incorporated the cpml with a general complex frequency shifted stretching operator into the renewed formulation a staggered grid finite difference fd method was adopted to discretize the first order equation system for wavefield depth continuation the first order awwe with the cpml saved memory compared with the original second order awwe with the conventional split pml with the help of numerical examples we verified the correctness of the staggered

target oriented wave equation least squares migration inversion with phase encoded hessian prestack depth migration produces blurred images resulting from limited acquisition apertures complexities in the velocity model and band limited characteristics of seismic waves this distortion can be partially corrected using the model space least squares migration inversion approach where a target oriented wave equation hessian operator is computed explicitly and then inverse filtering is applied iteratively to deblur or invert for the reflectivity however one difficulty is the cost of computing the explicit hessian operator which requires storing a large number of green s functions making it challenging for large scale applications a new method to compute the hessian operator for the wave equation based least squares migration inversion problem modifies the original explicit hessian formula enabling efficient computation of this operator an advantage is that the method eliminates disk storage of green s functions the modifications however also introduce undesired crosstalk artifacts two different phase encoding schemes plane wave phase encoding and random phase encoding suppress the crosstalk when the randomly phase encoded hessian operator is applied to the sigsbee2a synthetic data set an improved subsalt image with more balanced amplitudes is obtained

reducing two way splitting error of ffd method in dual domains the fourier finite difference ffd method is very popular in seismic depth migration but its straightforward 3d extension creates two way splitting error due to ignoring the cross terms of spatial partial derivatives traditional correction schemes either in the spatial domain by the implicit finite difference method or in the wavenumber domain by phase compensation lead to substantially increased computational costs or numerical difficulties for strong velocity contrasts we propose compensating the two way splitting error in dual domains alternately in the spatial and wavenumber domains via fourier transform first we organize the expanded square root operator in terms of two way splitting ffd plus the usually ignored cross terms second we select a group of optimized coefficients to maximize the accuracy of propagation in both inline and crossline directions without yet considering the diagonal directions finally we further optimize the constant coefficient of the compensation part to further improve the overall accuracy of the operator in implementation the compensation terms are similar to the high order corrections of the generalized screen method but their functions are to compensate the two way splitting error rather than the expansion error numerical experiments show that optimized one term compensation can achieve nearly perfect circular impulse responses and the propagation angle with less than 1 error for all azimuths is improved up to 60 from 35 compared with traditional single domain methods our scheme can handle lateral velocity variations even for strong velocity contrasts much more easily with only one additional fourier transform based on the two way splitting ffd method which helps retain the computational efficiency

exact and operator rational approximate solutions of the helmholtz weyl composition equation in underwater acoustics the quadratic profile the application of phase space and path integral methods to both mathematical and computational direct and inverse wave propagation modeling at the level of the scalar one way helmholtz equation is briefly reviewed the role of operator symbols is stressed and their properties briefly discussed the construction of the operator symbol requires the exact or approximate solution of the helmholtz weyl composition equation in the weyl pseudodifferential operator calculus the exact symbols for several quadratic profiles are presented and briefly analyzed the symbols exactly corresponding to a family of operator rational approximations are also presented for the quadratic case these results are used to illustrate several points pertinent to wide angle propagation modeling and the refractive index profile reconstruction problem in underwater acoustics

wide angle fd and ffd migration using complex pade approximations seismic migration by downward continuation using the one way wave equation approximations has two shortcomings imaging steep dip reflectors and handling evanescent waves complex pade approximations allow a better treatment of evanescent modes stabilizing finite difference migration without requiring special treatment for the migration domain boundaries imaging of steep dip reflectors can be improved using several terms in the pade expansion we discuss the implementation and evaluation of wide angle complex pade approximations for finite difference and fourier finite difference migration methods the dispersion relation and the impulsive response of the migration operator provide criteria to select the number of terms and coefficients in the pade expansion this ensures stability for a prescribed maximum propagation direction the implementations are validated on the marmousi model data set and seg eage salt model data

2 d depth migration in transversely isotropic media using explicit operators stable explicit depth extrapolation filters can be used to propagate plane waves corresponding to the qp and qsv quasi p and quasi sv propagation modes for transversely isotropic ti media here i discuss and compare results of two different methods for obtaining the filters for ti media with a vertical axis of symmetry vti the first a modified taylor series method is used to calculate the n coefficients of a finite length filter such that the taylor expansion around vertical propagation matches the spatial fourier transform of the downward continuation operator for vti media second a least squares method is used to calculate the filter coefficients such that the amplitude and phase departures from the ideal response of the downward continuation operator for vti media are minimized over a range of frequencies and propagation angles in both methods the amplitude response of the filter is forced to be less than unity in the evanescent region to achieve stability in general as exemplified in

splitting of operators alternate directions and paraxial approximations for the three dimensional wave equation we design a family of new three dimensional 3 d paraxial equations well adapted to numerical solution by the alternate directions method these equations do not suffer from bad anisotropic effects and the cost for their numerical integration remains cheap in comparison with classical paraxial equations

migration of primaries and multiples using an imaging condition for amplitude normalized separated wavefields abstractmigration of primary and multiple reflections leads to enhanced subsurface illumination and to increased image resolution a joint migration approach using the complete wavefield requires properly imaged primaries and multiples of all orders in recent works primaries and multiples have therefore been imaged separately using the upgoing and downgoing pressure wavefields obtained by decomposing dual sensor streamer data the matches between the corresponding depth images are still not found to be sufficiently accurate so new and more appropriate imaging conditions need to be found we reviewed the classical imaging approach used in one way wave equation migration with the aim of extending it for simultaneous migration of primaries and all orders of multiples based on rayleigh s reciprocity theorem and well known theoretical developments we derived a new imaging condition described in terms of the upgoing pressure wavefield and a filtered version of the downgoing vertical velocity wavefield t

n symplectic algebra of observables in covariant lagrangian field theory n symplectic geometry on the adapted frame bundle l e e of an n m k dimensional fiber bundle e m is used to set up an algebra of observables for covariant lagrangian field theories using the principle bundle l e j1 we lift a lagrangian l j1 r to a lagrangian l l l e r and then use l to define a modified n symplectic potential l on l e the cartan hamilton poincare chp rn valued 1 form if the lifted lagrangian is nonzero then l e d l is an n symplectic manifold to characterize the observables we define a lifted legendre transformation l from l e into le the image ql l l e is a submanifold of le and ql d ql is shown to be an n symplectic manifold we prove the theorem that l l ql and pull back the reduced canonical n symplectic geometry on ql to l e to define the algebras of observables on the n symplectic manifold l e d l to find the reduced n symplectic algebra on ql we set up the equations of n symplectic reduction and apply the general theory to the mo

optimized finite difference operator for broadband seismic wave modeling high resolution image and waveform inversion of smallscale targets requires the handling of high frequency seismic wavefields however conventional finite difference fd methods have strong numerical dispersions in the presence of high frequency components to reduce these numerical dispersions we optimized the constant coefficients of the fd operator by maximizing the wavenumber coverage within a given error limitation we set up three general criteria to enhance the convergence of the algorithm and reduce the optimization effort we selected the error limitation to be 0 0001 this being the smallest in the literature which led to perfect agreement between theoretical analyses and numerical experiments the accuracy of our optimized fd methods can even reach that of much higher order unoptimized fd methods which means great savings of computational efforts and memory demand these advantages become even more apparent with 3d modeling especially for saving memory demand

globally optimized finite difference extrapolator for strongly vti media implicit finite difference fd migration is unconditionally stable and is popular in handling strong velocity variations but its extension to strongly transversely anisotropic media with vertical symmetric axis media is difficult traditional local optimizations generate the optimized coefficients for each pair of thomsen anisotropy parameters independently which can degrade results substantially for large anisotropy variations and lead to a huge table we developed an implicit fd method using the analytic taylor series expansion and used a global optimization scheme to improve its accuracy for wide phase angles we first extended the number of the constant coefficients then we relaxed the coefficient of the time delay extrapolation term by tuning a small factor such that error is less than 0 1 finally we optimized the constant coefficients using a simulated annealing algorithm by constraining that all the error functions on a fine grid of the whole anisotropic region did not exceed 0 5 simultaneously the extended number of the constant coefficients and the relaxed coefficient greatly enhanced the flexibility of matching the dispersion relation and significantly improved the ability of handling strong anisotropy over a much wider range compared with traditional local optimization our scheme does not need any table and table lookup for each order of the fd method only one group of optimized coefficients is enough to handle strong variations in velocity and anisotropy more importantly our global optimization scheme guarantees the accuracy for various possible ranges of anisotropy parameters no matter how strong the anisotropy is for the globally optimized second order fd method the accurate phase angle is up to 58 and the increase is about 18 22 for the globally optimized fourth order fd method the accurate phase angle is up to 77 and the increase is about 22 27

fourier finite difference migration many existing migration schemes cannot simultaneously handle the two most important problems of migration imaging of steep dips and imaging in media with arbitrary velocity variations in all directions for example phase shift k migration is accurate for nearly all dips but it is limited to very simple velocity functions on the other hand finite difference schemes based on one way wave equations consider arbitrary velocity functions but they attenuate steeply dipping events we propose a new hybrid migration method named fourier finite difference migration wherein the downward continuation operator is split into two downward continuation operators one operator is a phase shift operator for a chosen constant background velocity and the other operator is an optimized finite difference operator for the varying component of the velocity function if there is no variation of velocity then only a phase shift operator will be applied automatically on the other hand if there is a strong variatio

rational square root approximations for parabolic equation algorithms in this article stable pade approximations to the function 1 z are derived by choosing a branch cut in the negative half plane the pade coefficients are complex and may be derived analytically to arbitrary order from the corresponding real coefficients associated with the principal branch defined by z 1 i z 0 i z denotes the imaginary part of z the characteristics of the corresponding square root approximation are illustrated for various segments of the complex plane in particular for waveguide problems it is shown that an increasingly accurate representation may be obtained of both the evanescent part of the mode spectrum for the acoustic case and the complex mode spectrum for the elastic case an elastic parabolic equation algorithm is used to illustrate the application of the new pade approximations to a realistic ocean environment including elasticity in the ocean bottom

riemannian wavefield extrapolation riemannian spaces are described by nonorthogonal curvilinear coordinates we generalize one way wavefield extrapolation to semiorthogonal riemannian coordinate systems that include but are not limited to ray coordinate systems we obtain a one way wavefield extrapolation method that can be used for waves propagating in arbitrary directions in contrast to downward continuation which is used for waves propagating mainly in the vertical direction ray coordinate systems can be initiated in many different ways for example from point sources or from plane waves incident at various angles since wavefield propagation happens mostly along the extrapolation direction we can use inexpensive finite difference or mixed domain extrapolators to achieve high angle accuracy the main applications of our method include imaging of steeply dipping or overturning reflections

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

euler poincare reduction for discrete field theories in this note we develop a theory of euler poincare reduction for discrete lagrangian field theories we introduce the concept of euler poincare equations for discrete field theories as well as a natural extension of the moser veselov scheme and show that both are equivalent the resulting discrete field equations are interpreted in terms of discrete differential geometry an application to the theory of discrete harmonic mappings is also briefly discussed

compensation for transmission losses based on one way propagators in the mixed domain most true amplitude migration algorithms based on oneway wave equations involve corrections of geometric spreading and seismic q attenuation however few papers discuss the compensation of transmission losses ctl based on one way wave equations here we present a method to compensate for transmission losses using one way wave propagators for a 2d case the scheme is derived from the lippmann schwinger integral equation the ctl scheme is composed of a transmission term and a phase shift term the transmission term compensates amplitudes while the wave propagates through subsurfaces the transmission term is a function of the vertical wavenumbers of two adjacent heterogeneous screens the phase shift term is a fourier finite difference ffd propagator implemented in a mixed domain via fourier transform the transmission term can be flexibly incorporated into the conventional phaseshift migration algorithm i e ffd at every depth step we analyze the effects of frequency lateral velocity contrast and vertical velocity ratio on the accuracy of the presented formulae numerical examples from a flat model and a fault model with lateral velocity variations are presented to demonstrate the ability of the proposed scheme for compensation of transmission losses

the stabilization of high order multistep schemes for the laguerre one way wave equation solver abstract this paper considers spectral finite difference methods of a high order of accuracy for solving the one way wave equation using the laguerre integral transform with respect to time as the base in order to provide a high spatial accuracy and stability the richardson method can be employed however such an approach requires high computer costs therefore we consider alternative algorithms based on the adams multistep schemes to reach the stability for the one way equation the stabilizing procedures using the spline interpolation were developed this made it possible to efficiently implement a predictor corrector type method thus decreasing computer costs the stability and accuracy of the procedures proposed have been studied based on the implementation of the migration algorithm within a problem of seismic prospecting

k cosymplectic manifolds and lagrangian field theories a geometrical description of classical field theories of first order is given the underlying k cosymplectic structure permits to derive the corresponding field equations

wave equation based seismic illumination analysis we present a wave equation based method for seismic illumination analysis a one way wave equation based generalized screen propagator is used to extrapolate the wavefields from sources and receivers to the subsurface target a local plane wave analysis is used at the target to calculate localized directionalenergyfluxesforbothsourceandreceiver wavefields we construct an illumination matrix using these energy fluxes to quantify the target illumination conditions the target geometry information is used to manipulate the illumination matrix and generate different types of illumination measures the wave equation based approach can properly handle forward multiple scattering phenomena including focusing defocusing diffraction and interference effects itcanbedirectlyappliedtocomplexvelocitymodels velocity model smoothing and fresnel zone smoothing are not required different illumination measurements derived from this method can be applied to target oriented or volumetric illumination analyses this new method is flexible and practical for illumination analysis in complex 2d and 3d velocity models with nontrivial acquisition and target

3 d implicit finite difference migration by multiway splitting we show that 3 d implicit finite difference schemes can be realized by multiway splitting in such a way that the steep dip problem and the problem of numerical anisotropy are overcome the basic idea is as follows we approximate the 3 d square root operator by a sequence of 2 d operators in three four or six directions to solve the azimuth symmetry problem each 2 d square root operator is then approximated by a sequence of implicit 2 d operators to improve steep dip accuracy this sequence contains some unknown coefficients which are calculated by a taylor expansion technique or by an optimization technique in the taylor expansion method the square root and its approximation are expanded into power series by comparing the terms the unknown coefficients are calculated the more 2 d finite difference operators for cascading are taken and the more directions for downward continuation are chosen the more terms from power series can be compared to obtain a higher degree migration operator with better

covariant field theory on frame bundles of fibered manifolds we show that covariant field theory for sections of e m lifts in a natural way to the bundle of vertically adapted linear frames l e our analysis is based on the fact that l e is a principal fiber bundle over the bundle of 1 jets j1 on l e the canonical soldering 1 forms play the role of the contact structure of j1 a lifted lagrangian l l e r is used to construct modified soldering 1 forms which we refer to as the cartan hamilton poincare 1 forms these 1 forms on l e pass to the quotient to define the standard cartan hamilton poincare m form on j1 we derive generalized hamilton jacobi and hamilton equations on l e and show that the hamilton jacobi and canonical equations of caratheodory rund and de donder weyl are obtained as special cases

imaging the total wavefields by reflectivity estimation using amplitude normalized wavefield decomposition field data example abstractto combine imaging information of up and downgoing wavefields using primaries and multiples we have developed an imaging framework based on the subsurface impulse response the latter is expressed at every depth level by a fredholm integral equation defined in terms of the amplitude normalized upgoing pressure and downgoing vertical velocity wavefields the subsurface impulse response can be obtained by inverting the matrix form of the integral equation by performing a multidimensional deconvolution mdd of the upgoing components by the downgoing components this procedure gives the reflectivity matrix by assuming locally reacting media or lateral shift invariance we can describe how the mdd can reduce to simpler deconvolution imaging conditions based on an example acquired with dual sensor towed streamers and simultaneous sources we have developed a real data application of imaging the total up and downgoing wavefields by reflectivity inversion the total upgoing wavefield used in the im

an acoustic wave equation for anisotropic media a wave equation derived using the acoustic medium assumption for p waves in transversely isotropic ti media with a vertical symmetry axis vti media yields a good kinematic approximation to the familiar elastic wave equation for vti media the wavefield solutions obtained using this vti acoustic wave equation are free of shear waves which significantly reduces the computation time compared to the elastic wavefield solutions for exploding reflector type applications from this vti acoustic wave equation the eikonal and transport equations that describe the ray theoretical aspects of wave propagation in a ti medium are derived these equations based on the acoustic assumption shear wave velocity 0 are much simpler than their elastic counterparts yet they yield an accurate description of traveltimes and geometrical amplitudes numerical examples prove the usefulness of this acoustic equation in simulating the kinematic aspects of wave propagation in complex ti models

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

k symplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k symplectic geometry we discuss the relation between the lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied in particular systems with symmetry and poisson sigma models

algebraic constructions in the category of lie algebroids the aim of this paper is to give the first general and abstract treatment of the algebraic properties of lie algebroids the concept of lie algebroid was introduced in 1967 by pradines ls as the basic infinitesimal invariant of a differentiable groupoid the construction of the lie algebroid of a differentiable groupoid is a direct generalization of the construction of the lie algebra of a lie group and 183 described a full lie theory for differentiable groupoids and lie algebroids encompassing many phenomena in the foundations of differential geometry a more detailed account and further references are given in 13 however at this stage the algebraic properties of lie algebroids were not pursued in 13 iii section 2 iv section 11 one of us gave a fairly detailed account of the abstract algebra of lie algebroids over a fixed base lie algebroids are vector bundles with several additional structures and a category of lie algebroids on a given base and morphisms which preserve that base has properties similar to those of the category of lie algebras using this algebra it was demonstrated in 13 that the lie theory of a category of locally trivial differentiable lie groupoids over a fixed base and the corresponding category of transitive lie algebroids over that base is coextensive with the standard theory of connections in principal bundles we now propose to show that the lie functor from the category of all differentiable groupoids over arbitrary bases and arbitrary smooth morphisms to the category of all lie algebroids preserves the basic algebraic constructions known to be possible in the category of differentiable groupoids the main work is to give abstract formulations of the corresponding lie algebroid concepts the groupoid results with which we

least squares wave equation migration for avp ava inversion we present an acoustic migration inversion algorithm that uses extended double square root wave equation migration and modeling operators to minimize a constrained least squares data misfit function least squares migration we employ an imaging principle that allows for the extraction of ray parameter domain common image gathers cigs from the propagated seismic wavefield the cigs exhibit amplitude variations as a function of half offset ray parameter avp closely related to the amplitude variation with reflection angle ava our least squares wave equation migration inversion is constrained by a smoothing regularization along the ray parameter this approach is based on the idea that rapid amplitude changes or discontinuities along the ray parameter axis result from noise incomplete wavefield sampling and numerical operator artifacts these discontinuities should therefore be penalized in the inversion the performance of the proposed algorithm is examined with two synthetic examples in the first

factorization and path integration of the helmholtz equation numerical algorithms the propagator for the reduced scalar helmholtz equation plays a significant role in both analytical and computational studies of acoustic direct wave propagation path functional integrals are taken to provide the principal representation of the propagator and are computed directly the path integral is the primary tool in extending the classical fourier methods so appropriate for wave propagation in homogeneous media to inhomogeneous media for transversely inhomogeneous environments the n dimensional helmholtz equation can be exactly factored into separate forward and backward one way wave equations a parabolic based one way phase space path integral construction provides the generalization of the tappert hardin split step fft algorithm to the full one way factored helmholtz wave equation these extended marching algorithms can readily accommodate density profiles and range updating and further in conjunction with imbedding methods provide the basis for incorporating backscatter effects in a complementary manner for general range dependent environments elliptic based two way path integral constructions lead to an approximate representation of the propagator feynman garrod and a natural statistical monte carlo means of evaluation taken together the path integrals provide the basis for a global analysis in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process the one way marching algorithms are applied to ocean acoustic environments seismological environments and extreme model environments designed to establish their range of validity and manner of breakdown

arbitrarily wide angle wave equations for complex media by combining various ideas related to one way wave equations owwes half space stiffness relation special finite element discretization and complex coordinate stretching a systematic procedure is developed for deriving a series of highly accurate space domain versions of owwes the resulting procedure is applicable to complex media where the governing equation full wave equation is a second order differential system making the procedure applicable for general heterogeneous anisotropic porous viscoelastic media owing to their high accuracy in representing waves propagating in an arbitrarily wide range of angles the resulting equations are named arbitrarily wide angle wave equations awwes in order to illustrate the proposed procedure awwes are derived for one way propagation in acoustic as well as elastic media while acoustic awwes can be considered as modified versions of well known space domain owwes based on rational approximations of the square root operator the elastic awwes are significantly different from the existing elastic owwes unlike the existing elastic owwes elastic awwes are displacement based and are applicable to general anisotropic media furthermore awwes are simple in their form and appear amenable to easy numerical implementation

a stable one way wave propagator for vti media for the purpose of one way wave equation imaging a pseudoscreen propagator is developed for transversely isotropic media with vertical axes of symmetry the phase shift for propagation through a depth slice is decomposed into three terms a gazdag phase shift for propagation in a laterally homogeneous reference medium a correction for lateral variability of vertical propagation and a remaining wide angle term for oblique directions of propagation based on rational function approximation for this remaining wide angle term a fourier finite difference ffd approach with four way splitting is applied fourth order pade approximation is unsatisfactory in anelliptic media for large propagation angles with respect to the vertical direction therefore a method of coefficient optimization is developed in conjunction with a method of choosing an adequate homogeneous reference medium in a depth slice by symmetrizing the finite difference operators and because of the choice of the optimized coefficients the propagator is stable in the sense that the least squares norm of the wavefield measured for a frequency depth slice does not grow with increasing depth of propagation a small amount of artificial damping is applied to suppress artifacts that appear at the critical angle defined by the velocities in the reference medium and the actual medium synthetic examples confirm that good kinematic accuracy is achieved for a wide range of propagation angles typically up to 60

classical field theory on lie algebroids variational aspects the variational formalism for classical field theories is extended to the setting of lie algebroids given a lagrangian function we study the problem of finding critical points of the action functional when we restrict the fields to being morphisms of lie algebroids in addition to the standard case our formalism includes as particular examples the case of systems with symmetry covariant euler poincare and lagrange poincare cases sigma models or chern simons theories

anisotropic complex pade hybrid finite difference depth migration standard real valued finite difference fd and fourier finite difference ffd migrations cannot handle evanescent waves correctly which can lead to numerical instabilities in the presence of strong velocity variations a possible solution to these problems is the complex pade approximation which avoids problems with evanescent waves by rotating the branch cut of the complex square root we have applied this approximation to the acoustic wave equation for vertical transversely isotropic media to derive more stable fd and hybrid fd ffd migrations for such media our analysis of the dispersion relation of the new method indicates that it should provide more stable migration results with fewer artifacts and higher accuracy at steep dips our studies lead to the conclusion that the rotation angle of the branch cut that should yield the most stable image is 60 for fd migration as confirmed by numerical impulse responses and work with synthetic data

optimized chebyshev fourier migration a wide angle dual domain method for media with strong velocity contrasts a wide angle propagator is essential when imaging complex media with strong lateral velocity contrasts in one way waveequation migration we have developed a dual domain one way propagator using truncated chebyshev polynomials and a globally optimized scheme our method increases the accuracy of the expanded square root operator by adding two high order terms to the traditional split step fourier propagator first we approximate the square root operator using taylor expansion around the reference background velocity then we apply the first kindchebyshevpolynomialstoeconomizetheresultsofthe taylor expansion finally we optimize the constant coefficients using the globally optimized scheme which are fixed and feasible for arbitrary velocity models theoretical analysis and numericalexperimentshavedemonstratedthatthemethodhasvery highaccuracyandexceedstheunoptimizedfourierfinite difference propagator for the entire range of practical velocity contrasts the accurate propagation angle of the method is always about 60 under the relative error of 1 for complex media with weak moderate and even strong lateral velocity contrasts the method allows us to handle wide angle propagations and strong lateral velocity contrast simultaneously by using fourier transform alone only four 2d fourier transforms are required for each step of 3d wavefield extrapolation and the computing cost is similar to that of the fourier finite difference method compared with the finite difference method our method has no twowaysplittingerrori e azimuthal anisotropyerrorfor3dcases and almost no numerical dispersion for coarse grids in addition it has strong potential to be accelerated when an enhanced fast fouriertransformalgorithmemerges

reduced classical field theories k cosymplectic formalism on lie algebroids in this paper we introduce a geometric description of lagrangian and hamiltonian classical field theories on lie algebroids in the framework of k cosymplectic geometry we discuss the relation between lagrangian and hamiltonian descriptions through a convenient notion of legendre transformation the theory is a natural generalization of the standard one in addition other interesting examples are studied mainly on reduction of classical field theories

lagrangian submanifolds and dynamics on lie algebroids in some previous papers a geometric description of lagrangian mechanics on lie algebroids has been developed in this topical review we give a hamiltonian description of mechanics on lie algebroids in addition we introduce the notion of a lagrangian submanifold of a symplectic lie algebroid and we prove that the lagrangian hamiltonian dynamics on lie algebroids may be described in terms of lagrangian submanifolds of symplectic lie algebroids the lagrangian hamiltonian formalism on lie algebroids permits us to deal with lagrangian hamiltonian functions not defined necessarily on tangent cotangent bundles thus we may apply our results to the projection of lagrangian hamiltonian functions which are invariant under the action of a symmetry lie group as a consequence we obtain that lagrange poincare hamilton poincare equations are the euler lagrange hamilton equations associated with the corresponding atiyah algebroid moreover we prove that lagrange poincare hamilton poincare equations are the local equations defining certain lagrangian submanifolds of symplectic atiyah algebroids

migration with arbitrarily wide angle wave equations we develop a new scalar migration technique that is highly accurate for imaging steep dips in heterogeneous media this method is based on arbitrarily wide angle wave equations awwes that are highly accurate space domain one way wave equations and have a form similar to the 15 equation the accuracy of the proposed method is increased by introducing auxiliary variables as well as adjusting the parameters of the approximation poststack migration is carried out by downward continuation using the awwe for which we have developed a stable explicit double marching scheme up to 80 accuracy is achieved by second order awwe migration with only 2 3 times the computational effort of the 15 equation and requiring almost the same storage we illustrate the performance of awwe migration using impulse response graphs a single dipping reflector and a slice of the seg eage salt model

scalar generalized screen algorithms in transversely isotropic media with a vertical symmetry axis the scalar generalized screen method in isotropic media is extended here to transversely isotropic media with a vertical symmetry axis vti although wave propagation in a transversely isotropic medium is essentially elastic we use an equivalent acoustic system of equations for the qp waves which we prove to be accurate for both the dispersion relation and the polarization angle in the case of mild anisotropy the enhanced accuracy of the generalized screen method as compared to the split step fourier methods allows the extension to vti media the generalized screen expansion of the one way propagator follows closely the method used in the isotropic case the medium is defined in terms of a background and a perturbation the generalized screen expansion of the vertical slowness is based upon an expansion of the medium parameters simultaneously into magnitude and smoothness of variation we cast the theory into numerical algorithms and assess the accuracy of the generalized screen method in a partic

reduction in principal bundles covariant lagrange poincare equations a general reduction theory for field theoretical lagrangians on principal fiber bundles is presented the reduced variational problem as well as the associated equations are formulated the link between the solutions of the original problem and that of the reduced problem is discussed and an obstruction to the reconstruction of the solutions is isolated the important case of semidirect products is discussed in detail and several concrete examples are presented

the hartley transform in seismic imaging phase shift migration techniques that attempt to account for lateral velocity variations make substantial use of the fast fourier transform fft generally the hermitian symmetry of the complex valued fourier transform causes computational redundancies in terms of the number of operations and memory requirements in practice a combination of the fft with the well known real to complex fourier transform is often used to avoid such complications as an alternative means to the fourier transform we introduce the inherently real valued non symmetric hartley transform into phase shift migration techniques by this we automatically avoid the hermitian symmetry resulting in an optimized algorithm that is comparable in efficiency to algorithms based on the real to complex fft we derive the phase shift operator in the hartley domain for migration in two and three dimensions and formulate phase shift plus interpolation split step migration and split step double square root prestack migration in terms of the

optimum split step fourier 3d depth migration developments and practical aspects we present an efficient scheme for depth extrapolation of wide angle 3d wavefields in laterally heterogeneous media the scheme improves the so called optimum split step fourier method by introducing a frequency independent cascaded operator with spatially varying coefficients the developments improve the approximation of the optimum split step fourier cascaded operator to the exact phase shift operator of a varying velocity in the presence of strong lateral velocity variations and they naturally lead to frequency dependent varying step depth extrapolations that reduce computational cost significantly the resulting scheme can be implemented alternatively in spatial and wavenumber domains using fast fourier transforms ffts the accuracy of the first order approximate algorithm is similar to that of the second order optimum split step fourier method in modeling wide angle propagation through strong laterally varying media similar to the optimum split step fourier method the scheme is superior to methods such as the generalized screen and fourier finite difference we demonstrate the scheme s accuracy by comparing it with 3d two way finite difference modeling comparisons with the 3d prestack kirchhoff depth migration of a real 3d data set demonstrate the practical application of the proposed method

stable wide angle fourier finite difference downward extrapolation of 3 d wavefields i present an unconditionally stable implicit finite difference operator that corrects the constant velocity phase shift operator for lateral velocity variations the method is based on the fourier finite difference ffd method contrary to previous results my correction operator is stable even when the medium velocity has sharp discontinuities and the reference velocity is higher than the medium velocity the stability of the new correction enables the definition of a new downward continuation method based on the interpolation of two wavefields the first wavefield is obtained by applying the ffd correction starting from a reference velocity lower than the medium velocity the second wavefield is obtained by applying the ffd correction starting from a reference velocity higher than the medium velocity the proposed fourier finite difference plus interpolation ffdpi method combines the advantages of the ffd technique with the advantages of interpolation a simple and economical procedure for defining

directional illumination analysis using the local exponential frame we have developed an efficient method of directional illuminationanalysisinthelocalangledomainusinglocalexponential frame beamlets the space domain wavefields with different shot receiver geometries are decomposed into the local angle domain by using the local exponential beamlets which form a tight frame with the redundancy ratio two and are implemented by a linear combination of local cosine and localsinetransforms becauseofthefastalgorithmsofthelocal cosine sine transforms this method is much more efficient than the previously used decomposition methods in directional illumination analysis such as the local slant stackingmethodandthegabor daubechiesframemethod theresults of directional illumination di maps and the acquisitiondipresponsesadrforthe2dseg eagesaltmodel and the 45 shot 3d seg eage model demonstrated the validity and feasibility of our method compared with the illumination results using local slant stacking decomposition thenewmethodproducesilluminationmapsofsimilarquality but it does so a few times faster furthermore because of its high computational efficiency and saving in memory usage the new method makes the 3d directional illumination analysisreadilyapplicableintheindustry

subdifferential calculus using ϵ subdifferentials in applications of convex analysis it is important to be able to calculate the subdifferentials of various combinations of proper and lower semicontinuous convex functions such as the sum of two such functions or their inf convolution epi sum as well as the pre composition of a convex function with an affine map or the marginal function obtained from a convex function and a linear map the classical formulas for such calculations all require additional hypotheses same of which may be difficult to check or are not always satisfied in a given variational problem in this paper we present formulas for the subdifferentials of such combinations without assuming any additional hypotheses by utilizing subdifferentials this wider applicability comes at the price of somewhat more complicated formulas

derivation and application of extended parabolic wave theories ii path integral representations the n dimensional reduced scalar helmholtz equation for a transversely inhomogeneous medium is naturally related to parabolic propagation models through 1 the n dimensional extended parabolic weyl pseudodifferential equation and 2 an imbedding in an n 1 dimensional parabolic schrodinger equation the first relationship provides the basis for the parabolic based hamiltonian phase space path integral representation of the half space propagator the second relationship provides the basis for the elliptic based path integral representations associated with feynman and fradkin feynman and garrod and feynman and dewitt morette exact and approximate path integral constructions are derived for the homogeneous and transversely inhomogeneous cases corresponding to both narrow and wide angle extended parabolic wave theories the path integrals allow for a global perspective of the transition from elliptic to parabolic wave theory in addition to providing a unifying framework for dynamical approximations resolution of the square root operator and the concept of an underlying stochastic process

fourier finite difference wave propagation we introduce a novel technique for seismic wave extrapolation in time the technique involves cascading a fourier transform operator and a finite difference operator to form a chain operator fourier finite differences ffd we derive the ffd operator from a pseudoanalytical solution of the acoustic wave equation two dimensional synthetic examples demonstrate that the ffd operator can have high accuracy and stability in complex velocity media applying the ffd method to the anisotropic case overcomes some disadvantages of other methods such as the coupling of qp waves and qsv waves the ffd method can be applied to enhance accuracy and stability of seismic imaging by reverse time migration

the polysymplectic hamiltonian formalism in field theory and calculus of variations i the local case on presente un formalisme hamiltonien geometrique invariant pour des problemes variationnels a integrale multiple et des theories de champ le formalisme est base sur la notion de forme polysymplectique qui est une generalisation a valeur vectorielle des formes symplectiques on etudie des equations hamiltoniennes des transformations canoniques des systemes de lagrange des symetries des applications en pulsion de theorie du champ et une classification des systemes de theorie du champ g homogenes sur une generalisation des orbites coadjointes

prestack migration by split step dsr the double square root dsr prestack migration equation though defined for depth variable velocity can be used to image media with strong velocity variations using a phase shift plus interpolation pspi or split step correction the split step method is based on applying a phase shift correction to the extrapolated wavefield a correction that attempts to compensate for the lateral velocity variations i show how to extend dsr prestack migration to lateral velocity media and exemplify the method by applying the new algorithm to the marmousi data set the split step dsr migration is very fast and offers excellent imaging results

generalization of the bremmer coupling series an operator formalism is developed to expand the acoustic wave field in a multi dimensionally smoothly varying medium generated by a source localized in space and time into a sum of constituents each of which can be interpreted as a wave that has traveled up and down with respect to a direction of preference a definite number of times this expansion is a generalization of the bremmer coupling series the condition of smoothness of the medium relates to the width of the signature of the source in the configuration both the existence and the convergence in the weak sense of the expansion are discussed the operator calculus involved leads to a natural generalization of the concept of slowness surface to multi dimensionally smoothly varying media the operator associated with the corresponding generalized vertical slowness induces the full one way wave operator in the type of media under consideration in addition a wavefield decomposition operator as well as an interaction operator that couples the decomposed constituents are derived

the gunther s formalism in classical field theory momentum map and reduction the polysymplectic formalism in local field theory developed by gunther j diff geom 25 23 1987 is revised a new approach and new results on momentum maps and reduction are given

k symplectic structures the basic properties of k symplectic structures are introduced and developed in analogy with the well known symplectic differential geometry examples of such structures related to k symplectic lie algebras are given

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

wide angle one way wave equations a one way wave equation also known as a paraxial or parabolic wave equation is a differential equation that permits wave propagation in certain directions only such equations are used regularly in underwater acoustics in geophysics and as energy absorbing numerical boundary conditions the design of a one way wave equation is connected with the approximation of 1 s2 1 2 on 1 1 by a rational function which has usually been carried out by pade approximation this article presents coefficients for l2 l and other alternative classes of approximants that have better wide angle behavior for theoretical results establishing the well posedness of these wide angle equations see the work of trefethen and halpern well posedness of one way wave equations and absorbing boundary conditions math comput 47 421 435 1986

schouten nijenhuis brackets the poisson and graded poisson schouten nijenhuis algebras of symmetric and antisymmetric contravariant tensor fields respectively on an n dimensional manifold m are shown to be n symplectic this is accomplished by showing that both brackets may be defined in a unified way using the n symplectic structure on the bundle of linear frames lm of m new results in n symplectic geometry are presented and then used to give globally defined representations of the hamiltonian operators defined by the schouten nijenhuis brackets

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

preconditioned acoustic least squares two way wave equation migration with exact adjoint operator abstractwe have investigated the problem of designing the forward operator and its exact adjoint for two way wave equation least squares migration we study the problem in the time domain and pay particular attention to the individual operators that are required by the algorithm we derive our algorithm using the language of linear algebra and establish a simple path to design forward and adjoint operators that pass the dot product test we also found that the exact adjoint operator is not equal to the classic reverse time migration algorithm for instance one must pay particular attention to boundary conditions to compute the exact adjoint that accurately passes the dot product test forward and adjoint operators are adopted to solve the so called least squares reverse time migration problem via the method of conjugate gradients we also examine a preconditioning strategy to invert extended images

optimized implicit finite difference and fourier finite difference migration for vti media propagation velocity of seismic waves in heterogeneous vti media depends not only on spatial location but also on their propagation direction which leads to a much more complex dispersion relation than in isotropic media as a result designing implicit finite difference fd schemes for wavefield extrapolation in anisotropic media through analytic taylor series expansion is more difficult implicit fd and fourier finite difference ffd schemes are developed for vertical transversely isotropic vti media based on function fitting the dispersion relation of vti media is approximated with a rational function and its coefficients are estimated by weighted least squares optimization because these coefficients are functions of thomsen anisotropy parameters e and and vary laterally in heterogeneous vti media they are calculated before wavefield extrapolation and stored in a table implicit fd and ffd schemes for vti media are almost the same as for isotropic media except that coefficients are looke

optimization of the parameters in complex pade fourier finite difference migration abstractcomplex pade fourier finite difference migration is a stable one way wave equation technique that allows for better treatment of evanescent modes than its real counterpart in this way producing fewer artifacts as for real fourier finite difference ffd migration its parameters can be optimized to improve the imaging of steeply dipping reflectors the dip limitation of the ffd operator depends on the variation of the velocity field we have developed a wide angle approximation for the one way continuation operator by means of optimization of the pade coefficients and the most important velocity dependent parameter we have evaluated the achieved quality of the approximate dispersion relation in dependence on the chosen function of the ratio between the model and reference velocities under consideration of the number of terms in the pade approximation and the branch cut rotation angle the optimized parameters are chosen based on the migration results and the computational cost we found that by

stable depth extrapolation of seismic wavefields by a neumann series several migration methods fail to work when applied to complex geological structures with strong lateral heterogeneity the generalized migration in the frequency wavenumber f k domain based on a convolution with a slowness inverse of velocity dependent operator is capable of downward continuation of wavefield in media with strong vertical and lateral variations of velocity unfortunately this method as presented in the literature is potentially unstable we propose a new stable extrapolator based on the solution of the integral fredholm equation which describes a one way wave equation in the form of a neumann series the resulting algorithm of depth migration is implemented in both the frequency wavenumber f k and frequency space f x domains and takes into account arbitrary lateral gradients of velocity using a low frequency filter in x f domain that is the sum of the power series the computation time of depth migration by a neumann series is slightly longer than for split step fourier

subsurface reflectivity estimation from imaging of primaries and multiples using amplitude normalized separated wavefields abstractseveral studies have shown the benefits of including multiple reflections together with primaries in the structural imaging of subsurface reflectors however to characterize the reflector properties there is a need to compensate for propagation effects due to multiple scattering and to properly combine the information from primaries and all orders of multiples from this perspective and based on the wave equation and rayleigh s reciprocity theorem recent works have suggested computing the subsurface image from the green s function reflection response or reflectivity by inverting a fredholm integral equation in the frequency space domain by following claerbout s imaging principle and assuming locally reacting media the integral equation may be reduced to a trace by trace deconvolution imaging condition for a complex overburden and considering that the structure of the subsurface is angle dependent this trace by trace deconvolution does not properly solve the fredholm integral equation we

one way wave equation migration in log polar coordinates we obtained acoustic wave and wavefield extrapolation equations in log polar coordinates lpcs and tried to enhance the imaging to achieve this goal it was necessary to decrease the angle between the wavefield extrapolation axis and wave propagation direction in the one way wave equation migration wem if we were unable to carry it out more reflection wave energy would be lost in the migration process it was concluded that the wavefield extrapolation operator in lpcs at low frequencies has a large wavelike region and at high frequencies it can mute the evanescent energy in these coordinate systems an extrapolation operator can readily lend itself to high order finitedifference schemes therefore even with the use of inexpensive operators wem in lpcs can clearly image varied horizontal and vertical events in complex geologic structures using wide angle and turning waves in these coordinates we did not encounter any problems with reflections from opposing dips dispersion played important roles not only as a filter operator but also as a gain function prestack and poststack migration results were obtained with extrapolation methods in different coordinate systems and it was concluded that migration in lpcs can image steeply dipping events in a much better way when compared with other methods

optimized explicit finite difference schemes for spatial derivatives using maximum norm we propose an optimized scheme using the maximum norm and the simulated annealing the maximum norm offers the largest set of possible solutions for solvers to search the explicit finite difference operator is greatly improved by our optimized scheme we use a tight error limitation to make accuracy improvement to be high and solid our optimized scheme allows greater saving of computational cost and memory demand conventional explicit finite difference methods have difficulties in handling high frequency components due to strong numerical dispersions one can reduce the numerical dispersions by optimizing the constant coefficients of the finite difference operator different from traditional optimized schemes that use the 2 norm and the least squares we propose to construct the objective functions using the maximum norm and solve the objective functions using the simulated annealing algorithm both theoretical analyses and numerical experiments show that our optimized scheme is superior to traditional optimized schemes with regard to the following three aspects first it provides us with much more flexibility when designing the objective functions thus we can use various possible forms and contents to make the objective functions more reasonable second it allows for tighter error limitation which is shown to be necessary to avoid rapid error accumulations for simulations on large scale models with long travel times finally it is powerful to obtain the optimized coefficients that are much closer to the theoretical limits which means greater savings in computational efforts and memory demand

gravity from lie algebroid morphisms inspired by the poisson sigma model and its relation to 2d gravity we consider models governing morphisms from t to any lie algebroid e where is regarded as a d dimensional spacetime manifold we address the question of minimal conditions to be placed on a bilinear expression in the 1 form fields sij x aiaj so as to permit an interpretation as a metric on this becomes a simple compatibility condition of the e tensor s with the chosen lie algebroid structure on e for the standard lie algebroid e tm the additional structure is identified with a riemannian foliation of m in the poisson case e t m with a sub riemannian structure which is poisson invariant with respect to its annihilator bundle for integrable image of s this means that the induced riemannian leaves should be invariant with respect to all hamiltonian vector fields of functions which are locally constant on this foliation this provides a huge class of new gravity models in d dimensions embedding known 2d and 3d models as particular examples

3d wavefield extrapolation with optimum split step fourier method a one way propagator is proposed for more accurately modeling wide angle wavefields in the presence of severe lateral variations of the velocity the method adds a higher order correction to improve the split step fourier method by directly designing a cascaded operator that matches the exact phase shift operator of a varying velocity using an optimization scheme the coefficients in the cascaded operator are determined according to the local velocity distribution and the prescribed angular range of wavefield propagation the proposed algorithm is implemented alternately in spatial and wavenumber domains using fast fourier transforms as in the split step fourier and generalized screen methods this algorithm can achieve higher accuracy than the generalized screen method for wide angle wavefields although the same numerical scheme is used with comparable computational cost no extra error arises for the proposed algorithm when used for 3d wave propagation in contrast to methods that introduce an implic

true amplitude one way wave equation migration in the mixed domain one way wave equation migration is a powerful imaging tool for locating accurately reflectors in complex geologic structures however the classical formulation of one way wave equations does not provide accurate amplitudes for the reflectors when dynamic information is required after migration such as studies for amplitude variation with angle or when the correct amplitudes of the reflectors in the zero offset images are needed some modifications to the one way wave equations are required the new equations which are called true amplitude one way wave equations provide amplitudes that are equivalent to those provided by the leading order of the ray theoretical approximation through the modification of the transverse laplacian operator with dependence of lateral velocity variations the introduction of a new term associated with the amplitudes and the modification of the source representation in a smoothly varying vertical medium the extrapolation of the wavefields with the true amplitude one way

globally optimal finite difference schemes based on least squares abstractspatial finite difference fd coefficients are usually determined by the taylor series expansion te or optimization methods the former can provide high accuracy on a smaller wavenumber or frequency zone and the latter can give moderate accuracy on a larger zone present optimization methods applied to calculate fd coefficients are generally gradient like or global optimization like algorithms and thus iterations are involved they are more computationally expensive and sometimes the global solution may not be found i examined second order spatial derivatives and computed the optimized spatial fd coefficients over the given wavenumber range using the least squares ls method the results indicated that the fd accuracy increased with increasing operator length and decreasing wavenumber range therefore for the given error and operator length globally optimal spatial fd coefficients can be easily obtained some optimal fd coefficients were given i developed schemes to obtain optimized ls

explicit time dependent schrodinger propagators the authors compute explicitly the time dependent schrodinger and heat propagator for the potentials lambda 2m m 1 cosh2 lambda x a delta function potential several cases of periodic delta function potentials and a delta function potential in a square well they also discuss the semiclassical approximation for the delta function potential

finite difference modeling with adaptive variable length spatial operators most finite difference simulation algorithms use fixed length spatial operators to compute spatial derivatives the choice of length is dictated by computing cost stability and dispersion criteria that are satisfied globally we propose finite difference schemes with adaptive variable length spatial operators to decrease computing costs significantly without reducing accuracy these schemes adopt long operators in regions of low velocity and short operators in regions of high velocity two methods automatically determine variable operator lengths dispersion analysis along with 1d and 2d modeling demonstrates the validity and efficiency of our schemes in addition a hybrid absorbing boundary condition helps reduce unwanted reflections from model boundaries our scheme is more efficient than those based on variable grid methods for modeling migration and inversion of models with complex velocity structures because the latter require local grid refinement which usually increases memory requirements

derivation and application of extended parabolic wave theories i the factorized helmholtz equation the reduced scalar helmholtz equation for a transversely inhomogeneous half space supplemented with an outgoing radiation condition and an appropriate boundary condition on the initial value plane defines a direct acoustic propagation model this elliptic formulation admits a factorization and is subsequently equivalent to a first order weyl pseudodifferential equation which is recognized as an extended parabolic propagation model perturbation treatments of the appropriate weyl composition equation result in a systematic development of approximate wave theories which extend the narrow angle weak inhomogeneity and weak gradient ordinary parabolic schrodinger approximation the analysis further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic studies the wave theories foreshadow computational algorithms the inclusion of range dependent effects and the extension to 1 the vector formulation appropriate for elas

design of one way wavefield extrapolation operators using smooth functions in wlsq optimization many depth migration methods use one way frequency space depth extrapolation methods these methods are generally considered to be expensive so it is important to find the most efficient way of implementing them this usually means making spatial convolution operators that are as short as possible applying the extrapolation operators in a recursive way using small depth steps also demands that the operators do not amplify the wavefield at every depth step weighted least squares is an appropriate method to use for designing extrapolation operators that are accurate and efficient and that remain stable in a recursive algorithm the extrapolated wavefields calculated with these operators are comparable with the extrapolation results obtained with other known operator design techniques as the remez exchange method and nonlinear optimization in this paper the weighted least squares technique is refined by using different model functions by smoothing the phase and amplitude transition at the evanescent c

global optimization of generalized nonhyperbolic moveout approximation for long offset normal moveout the approximation of normal moveout is essential for estimating the anisotropy parameters of anisotropic media the generalized nonhyperbolic moveout approximation gma brings a considerable improvement in accuracy compared to known analytical approximations however this is still prone to relatively large errors in the presence of relatively long offsets and large anisotropic parameters which would degrade the inversion accuracy due to error accumulation in velocity analysis we optimize the constant coefficients for all possible groupings of the anellipticity parameter and the offset to depth ratio within practical ranges theoretical analyses and numerical experiments show that the traditional optimization scheme using 2 norm solved by the least squares method could not provide an error constrained result in addition a direct optimization without constant coefficient extension would not lead to a satisfactory accuracy improvement we construct the objective function using the maximum norm and so

hamiltonian systems on k cosymplectic manifolds the hamiltonian framework on symplectic and cosymplectic manifolds is extended in order to consider classical field theories to do this the notion of k cosymplectic manifold is introduced and a suitable hamiltonian formalism is developed so that the field equations for scalar and vector hamiltonian functions are derived