migration velocity analysis using traveltime wavefield tomography abstractwe introduce an integrated wave equation technique for migration velocity analysis mva that consists of three steps 1 forming the extended data 2 approximating the correct transmitted wavefield and 3 using wavefield tomography to update the velocity model in the first step the crosscorrelation imaging condition is relaxed to produce other nonzero lag common image gathers cig that combined form a common image cube cic slicing the cic at different crosscorrelation lags forms a series of cigs flattened events will occur in the cigs at a lag other than the zero lag when an incorrect velocity model is used in the migration in the second step for each event on the cig we pick the focusing depth and crosscorrelation lag at which it is flattest we then model a green s function by seeding a source at the focusing depth using one way wave equation modeling then shift the modeled wavefield with the focusing crosscorrelation lag this process is repeated for the other primary events

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

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

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

the jacobian determinant revisited this paper is devoted to the study of the jacobian determinant of a map g from a smooth bounded open subset of rn into rn n 2 more generally could be a smooth bounded open subset of an n dimensional manifold starting with the seminal work of c b morrey 27 y reshetnyak 32 and j ball 1 it has been known that one can define the distributional jacobian determinant det g under fairly weak assumption on g in particular it is defined for all maps g w 1 n2 n 1 and also for all

on the total variation of the jacobian abstract a characterization of the total variation tv u of the jacobian determinant det du is obtained for some classes of functions u r 2 r 2 outside the traditional regularity space w 1 2 r 2 in particular explicit formulas are deduced for functions that are locally lipschitz continuous away from a given one point singularity x 0 i e u w 1 p r 2 w 1 x 0 r 2 for some p 1

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

weak continuity and lower semicontinuity results for determinants weak continuity properties of minors and lower semicontinuity properties of functionals with polyconvex integrands are addressed in this paper in particular it is shown that if un is bounded in open image in new window and if u bv n are such that un u in l1 n and

on the optimality of certain sobolev exponents for the weak continuity of determinants keywords optimality of sobolev exponents weak continuity of determinant radon measure reference caa article 1992 005doi 10 1016 0022 1236 92 90071 pview record in web of science record created on 2008 11 25 modified on 2017 05 12

harmonic maps with defects two problems concerning maps with point singularities from a domain c 3 tos2 are solved the first is to determine the minimum energy of when the location and topological degree of the singularities are prescribed in the second problem is the unit ball and g is given on we show that the only cases in whichg x x minimizes the energy isg const org x rx withr a rotation extensions of these problems are also solved e g points are replaced by holes 3 s2 is replaced by n sn 1 or by n pn 1 the latter being appropriate for the theory of liquid crystals

w1 p quasiconvexity and variational problems for multiple integrals abstract variational problems for the multiple integral i u g u x dx where r m and u r n are studied a new condition on g called w1 p quasiconvexity is introduced which generalizes in a natural way the quasiconvexity condition of c b morrey it being shown in particular to be necessary for sequential weak lower semicontinuity of i in w 1 p r n and for the existence of minimizers for certain related integrals counterexamples are given concerning the weak continuity properties of jacobians in w 1 p r n p n m an existence theorem for nonlinear elastostatics is proved under optimal growth hypotheses

improving explicit seismic depth migration with a stabilizing wiener filter and spatial resampling we present a new approach to the design and implementation of explicit wavefield extrapolation for seismic depth migration in the space frequency domain instability of the wavefield extrapolation operator is addressed by splitting the operator into two parts one to control phase accuracy and a second to improve stability the first partial operator is simply a windowed version of the exact operator for a half step the second partial operator is designed using the wiener filter method as a band limited least squares inverse of the first the final wavefield extrapolation operator for a full step is formed as a convolution of the first partial operator with the complex conjugate of the second this resulting wavefield extrapolation operator can be designed to have any desired length and is generally more stable and more accurate than a simple windowed operator of similar length additional stability is gained by reducing the amount of evanescent filtering and by spatially downsampling the lower tempor

efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion linearized inversion of surface seismic data for a model of the earth s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth s subsurface this sensitivity or jacobian matrix is usually quite expensive to estimate for all but the simplest model parameterizations we exploit the numerical structure of the finite element method modern sparse matrix technology and source receiver reciprocity to develop an algorithm that explicitly calculates the jacobian matrix at only the cost of a forward model solution furthermore we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the hessian matrix as predicted by the classical gauss newton method our method is applicable to the full suite of wave scattering problems amenable to finite element forward modeling we demonstrate our method through some simple 2 d synthetic examples

convergence and comparison analysis of some numerical schwarz methods domain decomposition methods are a natural means for solving partial differential equations on multi processors the spatial domain of the equation is expressed as a collection of overlapping subdomains and the solution of an associated equation is solved on each of these subdomains the global solution is then obtained by piecing together the subsolutions in some manner for elliptic equations the global solution is obtained by iterating on the subdomains in a fashion that resembles the classical schwarz alternating method in this paper we examine the convergence behavior of different subdomain iteration procedures as well as different subdomain approximations for elliptic equations it is shown that certain iterative procedures are equivalent to block gauss siedel and jacobi methods using different subdomain approximations an inner outer iterative procedure is defined m matrix analysis yields a comparison of different inner outer iterations

elastic least squares one way wave equation migration abstractleast squares migration seeks a reflectivity model that fits the observed data it is used to compensate for acquisition noise poor sampling of sources and receivers on the surface as well as poor illumination of the subsurface to date least squares migration has been mainly restricted to the imaging of acoustic wavefields we have developed an extension of one way wave equation least squares migration for elastic wavefields in isotropic media least squares migration is an iterative method that requires a forward and an adjoint operator in elastic least squares one way wave equation migration the forward operator generates data components from multiparameter images by recursive wavefield decomposition extrapolation and recomposition conversely the adjoint operator generates multiparameter images from data components by recursively applying the adjoint of the wavefield recomposition extrapolation and wavefield decomposition operators we use an extended imaging condition and regularize

migration velocity analysis using the common image cube cic migration velocity analysis mva is commonly performed in the image domain in conjunction with ray based tomography to update the velocity model this approach can be challenging in the presence of large velocity errors as it may require many mva iterations before converging to a model that can focus the events in the image domain we introduced a downward continuation based domain for carrying out mva that is more flexible than conventional domains this approach consists of two steps 1 forming the common image cube cic and 2 modeling the green s functions in the first step the cross correlation imaging condition is relaxed to produce more than the zero lag common image gather cig slicing these data at different lags forms a series of cigs whereas a conventional cig can be obtained by slicing the cube at the zero lag when the velocity model used for the migration differs from the true velocity model properly flattened events may occur in cigs other than the zero lag in the second step f

p wave slowness surface approximation for tilted orthorhombic media abstractwe have developed an analytic and approximate formula for vertical slowness components of down and upgoing plane p waves in 3d tilted orthorhombic media a perturbation method and shanks transform were used to derive the approximation for slowness surface of p waves in tilted orthorhombic media we have also quantitatively described the validity range of the radial horizontal slowness components for the proposed formula the validity range was affected by the strength of the anellipticity of an orthorhombic medium the stronger the anellipticity the smaller the validity range numerical examples determined that the proposed formula is accurate for tilted orthorhombic media with weak to strong anellipticity we have also evaluated in detail the application of the proposed formula on calculating the p wave intercept time in the p domain for horizontally layered tilted orthorhombic models our formula is useful for ray tracing phase shift migration and p domain intercept time approximation f

extended local born fourier migration method a migration approach based on a local application of the born approximation within each extrapolation interval contains a singularity that can make direct application unstable previous authors have suggested adding an imaginary part to the vertical wavenumber to eliminate the singularity however their method requires that the reference slowness must be the maximum slowness of a given layer consequently the slowness perturbations are larger than those when the average slowness is selected as a reference slowness therefore its applicability is limited we develop an extended local born fourier migration method that circumvents the singularity problem of the local born solution and makes it possible to choose the average slowness as a reference slowness it is computationally efficient because of the use of a fast fourier transform algorithm it can handle wider angles or steeper interfaces and scattering effects of heterogeneities more accurately than the split step fourier ssf method which acco

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

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

designing stable extrapolators for explicit depth extrapolation of 2d and 3d wavefields using projections onto convex sets we have developed a robust algorithm for designing explicit depth extrapolation operators using the projections onto convex sets pocs method the operators are optimal in the sense that they satisfy all required extrapolation design characteristics in addition we propose a simple modification of the pocs algorithm modified pocs or mpocs that further enhances the stability of extrapolated wavefields and reduces the number of iterations required to design such operators to approximately 2 of that required for the basic pocs design algorithm various synthetic tests show that 25 coefficient 1d extrapolation operators which have 13 unique coefficients can accommodate dip angles up to 70 we migrated the seg eage salt model data with the operators and compare our results with images obtained via extrapolators based on modified taylor series and with wavefield extrapolation techniques such as phase shift plus interpolation pspi and split step fourier the mpocs algorithm provides practically stable

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

regularization and datuming of seismic data by weighted damped least squares the possibility of improving regularization datuming of seismic data is investigated by treating wavefield extrapolation as an inversion problem weighted damped least squares is then used to produce the regularized datumed wavefield regularization datuming is extremely costly because of computing the hessian so an efficient approximation is introduced approximation is achieved by computing a limited number of diagonals in the operators involved real and synthetic data examples demonstrate the utility of this approach for synthetic data regularization datuming is demonstrated for large extrapolation distances using a highly irregular recording array without approximation regularization datuming returns a regularized wavefield with reduced operator artifacts when compared to a nonregularizing method such as generalized phase shift plus interpolation pspi approximate regularization datuming returns a regularized wavefield for approximately two orders of magnitude less in cost but it is dip limi

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

integrability for the jacobian of orientation preserving mappings assume f is a map from into rn where is a bounded domain in rn such that d ln logl s with 0 s 1 i e d n log 1 d s then j l log l 1 s k for any compact subset k where j det d when s 0 we recover a well known result of muller j reine angew math 412 1990 20 34 while the case s 1 was obtained by iwaniec and sbordone arch rational mech anal to appear

q compensated least squares reverse time migration using low rank one step wave extrapolation abstractattenuation of seismic waves needs to be taken into account to improve the accuracy of seismic imaging in viscoacoustic media reverse time migration rtm can be performed with q compensation which is also known as q rtm least squares rtm lsrtm has also been shown to be able to compensate for attenuation through linearized inversion however seismic attenuation may significantly slow down the convergence rate of the least squares iterative inversion process without proper preconditioning we have found that incorporating attenuation compensation into lsrtm can improve the speed of convergence in attenuating media obtaining high quality images within the first few iterations based on the low rank one step seismic modeling operator in viscoacoustic media we have derived its adjoint operator using nonstationary filtering theory the proposed forward and adjoint operators can be efficiently applied to propagate viscoacoustic waves and to implement attenuation compensation recognizing that

cal h 1 estimates of jacobians by subdeterminants let f omega rightarrow bbb r n be a mapping in the sobolev space w 1 n 1 loc omega bbb r n n geq 2 we assume that the cofactors of the differential matrix df x belong to l frac n n 1 omega then among other things we prove that the jacobian determinant detdf lies in the hardy space cal h 1 omega

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

theory of nonstationary linear filtering in the fourier domain with application to time variant filtering a general linear theory describes the extension of the convolutional method to nonstationary processes this theory can apply any linear nonstationary filter with arbitrary time and frequency variation in the time fourier or mixed domains the filter application equations and the expressions to move the filter between domains are all ordinary fourier transforms or generalized convolutional integrals nonstationary transforms such as the wavelet transform are not required there are many possible applications of this theory including the one way propagation of waves through complex media time migration normal moveout removal time variant filtering and forward and inverse q filtering two complementary nonstationary filters are developed by generalizing the stationary convolution integral the first called nonstationary convolution corresponds to the linear superposition of scaled impulse responses of a nonstationary filter the second called nonstationary combination does not correspond to such a superposition but is shown to be a linear process capable of achieving arbitrarily abrupt temporal variations in the output frequency spectrum both extensions have stationary convolution as a limiting form and in the discrete case can be formulated as matrix operations fourier transformation shows that both filter types are nonstationary filter integrals in the fourier domain as well this result is a generalization of the convolution theorem for stationary signals because as the filter becomes stationary in one domain the integral in the other domain collapses to a scalar multiplication for discrete signals stationary filters are a matrix multiplication of the input signal spectrum by a diagonal spectral matrix while nonstationary filters require off diagonal terms for quasi stationary filters a computational advantage is obtained by computing only the significant terms near the diagonal unlike stationary theory a mixed domain of time and frequency is also possible in this context the nonstationary filter is applied simultaneously with the transform from time to frequency or the reverse nonstationary convolution becomes a generalized forward fourier integral and nonstationary combination is a generalized inverse fourier integral

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

the distributional hyper jacobian determinants in fractional sobolev spaces abstract in this paper we give a positive answer to the question about hyper jacobian determinants and associated minors raised by baer jerison inspired by recent works of brezis nguyen and baer jerison on jacobian and hessian determinants we establish the weak continuity and fundamental representation for the distributional mth jacobian minors of degree r in the fractional sobolev space w m m r r r n moreover the result is optimal in the framework of fractional sobolev spaces i e all mth jacobian minors of degree r are well defined in w s p r n if and only if w s p r n w m m r m r n

derivation of forward and adjoint operators for least squares shot profile split step migration the forward and adjoint operators for shot profile least squares migration are derived the forward operator is demigration and the adjoint operator is migration the demigration operator is derived from the born approximation the process begins with a green s function that allows for a laterally varying migration velocity model using the split step approximation next the earth is divided into horizontal layers and within each layer the migration velocity model is made to be constant with respect to depth for a given layer 1 the source side wavefield is propagated down to its top using the background wavefield this gives a background wavefield incident at the layer s upper boundary 2 the layer s contribution to the scattered wavefield is computed using the born approximation to the scattered wavefield and the background wavefield 3 next its scattered wavefield is propagated back up to the measurement surface using again the background wavefield the measured wavefield is approximated by the sum of scattered wavefields from each layer in the derivation of the measured wavefield the shot profile migration geometry is used for each shot the resulting wavefield modeling operator takes the form of a fredholm integral equation of the first kind and this is used to write down its adjoint the shot profile migration operator this forward adjoint pair is used for shot profile least squares migration shot profile least squares migration is illustrated with two synthetic examples the first uses data collected over a four layer acoustic model and the second uses data from the sigsbee 2a model

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

improving the efficiency of elastic wave mode separation for heterogeneous tilted transverse isotropic media wave mode separation for ti transversely isotropic models can be carried out by nonstationary filtering the elastic wavefields with localized filters these filters are constructed based on the polarization vectors obtained by solving the christoffel equation using local medium parameters this procedure although accurate is computationally expensive especially in 3d we develop an efficient method for wave mode separation which exploits the same general idea of projecting wavefields onto polarization vectors the method consists of two steps 1 separate wave modes in the wavenumber domain at a number of reference models to obtain the same number of partially separated wavefields then transform all the wavefields to the space domain 2 interpolate the wavefields obtained in step 1 in the space domain using the spatially variable model parameters the new method resembles the phaseshift plus interpolation pspi technique which interpolates the wavefields that are reconstructed at several reference velocities synthetic examples indicate that the separation followed by interpolation is effective for models with complex geology the new technique has the benefits of speed and accuracy

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

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

parallel interval newton like schwarz methods for almost linear parabolic problems we introduce parallel interval newton schwarz like methods for nonlinear systems of equations arising from discretizations of almost linear parabolic problems by applying interval techniques we get global convergence properties and verified enclosures parallelism is introduced by domain decomposition numerical results from a sgi altix 3700 are included

krawczyk like algorithms for the solution of systems of nonlinear equations in this paper we present interval arithmetic methods for the solution of systems of nonlinear equations these methods are based on the algorithm of r krawczyk 5 and a modification introduced in 3 starting with an interval vector containing a solution of the given system these methods converge to that solution under relatively weak conditions i e it can be said that the convergence is almost global

new full wave phase shift approach to solve the helmholtz acoustic wave equation for modeling we have developed a method of solving the helmholtz equation based on a new way to generalize the one way wave equation impose correct boundary conditions and eliminate exponentially growing evanescent waves the full two way nature of the helmholtz equation is included but the equation is converted into a pseudo one way form in the framework of a generalized phase shift structure consisting of two coupled first order partial differential equations for wave propagation with depth a new algorithm based on the particular structure of the coupling between p z and 2p z2 is introduced to treat this problem by an explicit approach more precisely in a depth marching strategy the wave operator is decomposed into the sum of two matrices the first one is a propagator in a reference velocity medium whereas the second one is a perturbation term which takes into account the vertical and lateral variation of the velocity the initial conditions are generated by solving the lippmann schwinger integral e

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

seismic depth imaging with the gabor transform wavefield extrapolation in depth a vital component of wave equation depth migration is accomplished by repeatedly applying a mathematical operator that propagates the wavefield across a single depth step thus creating a depth marching scheme the phase shift method of wavefield extrapolation is fast and stable however it can be cumbersome to adapt to lateral velocity variations we address the extension of phase shift extrapolation to lateral velocity variations by using a spatial gabor transform instead of the normal fourier transform the gabor transform also known as the windowed fourier transform is applied to the lateral spatial coordinates as a windowed discrete fourier transform where the entire set of windows is required to sum to unity within each window a split step fourier phase shift is applied the most novel element of our algorithm is an adaptive partitioning scheme that relates window width to lateral velocity gradient such that the estimated spatial positioning error is bounded b

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

wavefield extrapolation by nonstationary phase shift the phase shift method of wavefield extrapolation applies a phase shift in the fourier domain to deduce a scalar wavefield at one depth level given its value at another the phase shift operator varies with frequency and wavenumber and assumes constant velocity across the extrapolation step we use nonstationary filter theory to generalize this method to nonstationary phase shift nsps which allows the phase shift to vary laterally depending upon the local propagation velocity for comparison we derive an analytic form for the popular phase shift plus interpolation pspi method in the limit of an exhaustive set of reference velocities nsps and this limiting form of pspi can be written as generalized fourier integrals which reduce to ordinary phase shift in the constant velocity limit in the x domain these processes are the transpose of each other however only nsps has the physical interpretation of forming the scaled linear superposition of laterally variable impulse responses i e huygen

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

extended local rytov fourier migration method we develop a novel depth migration method termed the extended local rytov fourier elrf migration method it is based on the scalar wave equation and a local application of the rytov approximation within each extrapolation interval wavefields are fourier transformed back and forth between the frequency space and frequency wavenumber domains during wavefield extrapolation the lateral slowness variations are taken into account in the frequency space domain the method is efficient due to the use of a fast fourier transform algorithm under the small angle approximation the elrf method leads to the split step fourier ssf method that is unconditionally stable the elrf method and the extended local born fourier elbf method that we previously developed can handle wider propagation angles than the ssf method and account for the phase and amplitude changes due to the lateral variations of slowness whereas the ssf method only accounts for the phase changes the stability of the elrf method is controlled

on the distributional jacobian of maps from sn into sn in fractional sobolev and holder spaces h brezis and l nirenberg proved that if gk c 0 s n s n and g2 c 0 s n s n n 1 are such that gk g in bmo s n then deggk degg on the other hand if g 2 c 1 s n s n then kronecker s formula asserts that degg 1 jsnj r sn det rg d consequently r sn det rgk d converges to r sn det rg d provided gk g in bmo s n in the same spirit we consider the quantity j g r sn det rg d for all 2 c 1 s n r and study the convergence of j gk in particular we prove that j gk converges to j g for any 2 c 1 s n r if gk converges to g in c 0 s n for some n 1 n surprisingly this result is optimal when n 1 in the case n 1 we prove that if gk g almost everywhere and lim supk 1jgk gjbmo is suciently small then j gk j g for any 2 c 1 s 1 r we also establish bounds for j g which are motivated by the works of j bourgain h brezis and h m nguyen and h m nguyen we pay special attention to the case n 1

linear complementarity problem a novel approach to design finite impulse response wavefield extrapolation filters abstractin this paper the problem of complex valued finite impulse response fir wavefield extrapolation filter design was considered as a linear complementarity problem lcp lcp is not an optimization technique because there is no objective function to optimize however quadratic programming one of the applications of lcp can be used to find an optimal solution for the 1d fir wavefield extrapolation filter quadratic programs are an extremely important source of applications of lcp in fact several algorithms for quadratic programs are based on lcp we found that fir wavefield extrapolation filter design problem can be written as a quadratic program and then finally to an equivalent lcp there are two families of algorithms available to solve for lcp 1 direct pivoting based algorithms and 2 indirect iterative algorithms in this study the lcp has been solved using direct and indirect algorithms to show the effectiveness of the proposed method the seg eage salt velocity model data ha

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

lifting degree and distributional jacobian revisited let g i 0 1 s1 if g vmo we may write g ei for some vmo this is unique modulo 2 see 13 and the earlier work 14 there is no control of bmo in terms of g bmo since we always have g bmo 2 and bmo can be arbitrarily large recall however that when g bmo is sufficiently small there is a linear estimate bmo c g bmo see 13 theorem 4 14 and remark 0 2 below we are going to establish that a norm slightly stronger than g bmo does control bmo consider for 1 p 0 s 1 the fractional sobolev space ws p i equipped with its standard seminorm

efficient simultaneous wave equation statics and trace regularization by series approximation to address the problems of irregular trace spacing and statics correction simultaneous regularization and wave equation statics we statics are implemented by least squares inversion in general inversion is found to be intractable in three dimensions so series approximation is made to reduce significantly the number of required integrals the resulting operator is suitable for direct inversion or for use with gradient methods real and synthetic data are used to determine the viability of the inversion for synthetic data even for severe velocity variation and topography inversion converges to an acceptable solution and aliasing is reduced significantly similarly for real data inversion is found to return an antialiased regularized result with we statics applied

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

optimal function spaces for continuity of the hessian determinant as a distribution abstract we establish optimal continuity results for the action of the hessian determinant on spaces of besov type into the space of distributions on r n in particular inspired by recent work of brezis and nguyen on the distributional jacobian determinant we show that the action is continuous on the besov space of fractional order b 2 2 n n and that all continuity results in this scale of besov spaces are consequences of this result a key ingredient in the argument is the characterization of b 2 2 n n as the space of traces of functions in the sobolev space w 2 n r n 2 on the subspace r n of codimension 2 the most delicate and elaborate part of the analysis is the construction of a counterexample to continuity in b 2 2 n p with p n

block row recursive least squares migration abstractrecursive estimates of large systems of equations in the context of least squares fitting is a common practice in different fields of study for example recursive adaptive filtering is extensively used in signal processing and control applications the necessity of solving least squares problems recursively stems from the need for fast real time signal processing strategies the computational cost of least squares solvers can also limit the applicability of this technique in geophysical problems we have considered a recursive least squares solution for least squares wave equation migration with sliding windows involving several fixed rank downdating and updating computations this technique can be applied for dynamic and stationary processes if we use enough data in each windowed setup the spectrum of the preconditioned system is clustered around one and the method will converge superlinearly with probability one numerical experiments were performed to test the effectiveness of the technique

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

the design of stable sparse wavefield extrapolators using projections onto convex sets abstractwe present the results of poststack explicit depth migration of the well known 2d seg eage salt model zero offset seismic data using sparse wavefield extrapolators the extrapolators are designed to be sparse by forcing some of the very small complex valued coefficients magnitude values to be zero the proposed extrapolators design method combines the previously reported modified projections onto convex sets mpocs for designing explicit depth frequency space f x wavefield extrapolation operators with hard thresholding of the small extrapolators coefficients magnitude the real and imaginary parts of the mpocs operators with small magnitudes are replaced by zeros during the mpocs algorithm iterations the migrated result of the seg eage salt model data using such sparse designed operators shows comparable migrated results using the nonsparse version of the mpocs extrapolation operators as well as the image obtained using the well known phase shift plus interpolation pspi migration techn

uniform asymptotic expansion of the generalized bremmer series the bremmer coupling series solution of the wave equation in generally inhomogeneous media requires the introduction of pseudodifferential operators in this paper in two dimensions uniform asymptotic expansions of the schwartz kernels of these operators are derived also we derive a uniform asymptotic expansion of the one way propagator appearing in the series we focus on designing closed form representations valid in the high frequency limit taking into account critical scattering angle phenomena our expansion is not limited by propagation angle in principle the uniform asymptotic expansion of a kernel follows by matching its asymptotic behaviors away from and near its diagonal the bremmer series solver consists of three steps directional decomposition into up and downgoing waves one way propagation and interaction of the counter propagating constituents each of these steps is represented here by a kernel for which a uniform asymptotic expansion is found the associated algorithm provid

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

prestack depth migration by symmetric nonstationary phase shift a new depth migration method suitable for heterogeneous media is presented the well known phase shift plus interpolation pspi method and the recently introduced nonstationary phase shift nsps method are combined into a single symmetric operator with improved accuracy and stability and with similar computational effort for prestack depth migration the symmetric operator is used in a recursive wavefield extrapolation to compute incident and reflected wavefields at any desired depth and the ratio of the incident and reflected wavefields at a particular depth is used to estimate seismic reflectivity when the velocity model is made piecewise constant laterally the symmetric extrapolation operator can be computed efficiently using ordinary phase shift extrapolation for a series of reference velocities and appropriate spatial windowing migration of the marmousi synthetic data set by symmetric nonstationary phase shift snps provides an image that compares favorably with an image of the zero offset reflectivity derived from the marmousi velocity model