an energy conserving parabolic equation for elastic media the accuracy of the parabolic equation pe method is improved for range dependent elastic media the energy conserving elastic pe is a generalization of the energy conserving acoustic pe a range dependent waveguide is approximated by a sequence of range independent regions a linear approximation is derived for the nonlinear condition corresponding to conservation of compressional energy flux at the vertical interfaces between range independent regions the rotated elastic pe is applied to generate reference solutions for demonstrating accuracy the energy conserving elastic pe is applied to solve problems involving the propagation of an interface wave along a sloping ocean bottom and a problem involving mode cutoff and coupling into shear wave beams in the ocean bottom the latter problem is analogous to a cutoff problem that was first solved with the acoustic pe f b jensen and w a kuperman j acoust soc am 67 1564 1566 1980

invariant solutions of certain nonlinear evolution type equations with small parameters the fisher equation which arises in the study of reaction diffusion waves in biology does not display a high level of symmetry properties consequently only travelling wave solutions are obtainable using the method of invariants this has a direct bearing on studying perturbed forms of the equation which may arise from considering e g damping or dissipative factors we show here how one can get around this limitation by appending some unknown function to the perturbation and obtain interesting practical results using invariants the ideas have significant consequences for equations which do not admit large class of symmetry properties the method used in this analysis is then extended to other classes of evolution type equations that involve perturbations for e g the kdv type equations

corner treatments for high order local absorbing boundary conditions in high frequency acoustic scattering this paper deals with the design and validation of accurate local absorbing boundary conditions set on convex polygonal computational domains for the finite element solution of high frequency acoustic scattering problems while high order absorbing boundary conditions habcs are accurate for smooth fictitious boundaries the precision of the solution drops in the presence of corners if no specific treatment is applied we present and analyze two strategies to preserve the accuracy of pade type habcs at corners first by using compatibility relations derived for right angle corners and second by regularizing the boundary at the corner we show that the former strategy is well adapted to right corners and efficient for nearly right corners while the later is better for very obtuse corners numerical results are proposed to analyze and compare the approaches for two and three dimensional problems

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

a separable kernel decomposition method for approximating the dsr continuation operator we develop a separable kernel decomposition method for approximating the double square root dsr continuation operator in one way migrations in this paper this new approach is a further development of separable approximations of the single square root ssr operator the separable approximation of the dsr operator generally involves solving a complicated nonlinear system of integral equations instead of solving this nonlinear system directly our new method consists of repeatedly applying the separable kernel technique developed for the two variable ssr operator to the multivariable dsr operator numerical experiments demonstrate the efficiency of the proposed method we illustrate the fast convergence of the obtained separable approximation we also demonstrate the capability of this novel approximation for imaging an area with geologic complexities through synthetic data

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

a two way parabolic equation for elastic media the parabolic equation pe method is extended to handle wave propagation and scattering in range dependent elastic waveguides which are approximated by a sequence of range independent regions the two way elastic pe involves an efficient pe based scattering method for computing reflected and transmitted fields at the vertical interfaces separating range independent regions the one way elastic pe is used to propagate the outgoing and incoming fields through the range independent regions with two way range marching in addition to providing the backscattered field the two way pe provides a correction to the outgoing field the self starter is generalized to handle a source in a solid layer

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

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

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

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

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

wave field extrapolation techniques in seismic migration a tutorial the objective of this paper is to provide a general view on methods of wave field extrapolation as used in seismic modeling and seismic migration i e the kirchhoff summation approach the plane wave method k f method and the finite difference technique particular emphasis is given to the relationship between the different methods by formulating the problem in the space frequency domain x y omega domain a systems approach can be adopted which results in simple and concise expressions these expressions clearly show that forward extrapolation is described by a spatial convolution procedure and inverse extrapolation is described by a spatial deconvolution procedure in the situation of lateral velocity variations the de convolution procedure becomes space variant the space frequency domain is most suitable for recursive depth migration in addition frequency dependent properties such as absorption dispersion and spatial bandwidth can be handled easily it is shown that all extrapolation methods are based on two equations taylor series and wave equation in the kirchhoff summation approach all terms of the taylor series are summed to an exact analytical expression the kirchhoff integral for plane surfaces it formulates the extrapolation procedure in terms of a spatial convolution integral which must be discretized in practical applications the fourier transformed version of the kirchhoff integral is used in the plane wave method k f method this actually means that spatial de convolution in the x y omega domain is translated into multiplication in the k x k y omega domain of course this is not allowed if the extrapolation operators are space variant in explicit finite difference techniques a truncated version of the taylor series is used with some optimum adjustments of the coefficients for only one or two terms in the taylor series a spatial low pass filter must be applied to compensate for the amplitude errors at high tilt angles explicit methods are simple and most suitable for three dimensional 3 d applications in implicit finite difference schemes the wave field extrapolator is written in terms of an explicit forward extrapolator and an explicit inverse extrapolator properly designed implicit schemes do not show amplitude errors and therefore amplitude correction filters need not be applied in comparison with explicit schemes implicit schemes are more sensitive to improper boundary conditions at both ends of the data file it is shown that the forward seismic model can be elegantly described by a matrix equation using separate operators for downward and upward traveling waves using this model inverse extrapolation involves one matrix inversion procedure to compensate for the downward propagation effects and one matrix inversion procedure to compensate for the upward propagation effects

an explicit analytical solution for sound propagation in a three dimensional penetrable wedge with small apex angle a problem of sound propagation in a shallow water waveguide with a weakly sloping penetrable bottom is considered the adiabatic mode parabolic equations are used to approximate the solution of the three dimensional 3d helmholtz equation by modal decomposition of the acoustic pressure field the mode amplitudes satisfy parabolic equations that admit analytical solutions in the special case of the 3d wedge using the analytical formula for modal amplitudes an explicit and remarkably simple expression for the acoustic pressure in the wedge is obtained the proposed solution is validated by the comparison with a solution of the 3d penetrable wedge problem obtained using a fully 3d parabolic equation that includes a leading order cross term correction

applications and time domain solution of higher order parabolic equations in underwater acoustics a higher order parabolic equation and the corresponding higher order time domain parabolic equation are derived from a pade series and solved numerically these models provide accurate solutions for problems involving very wide angle propagation e g propagation in the nearfield or over a hard ocean bottom propagation in domains in which sound speed variations are large e g propagation in deep water deep within the ocean bottom in high speed ocean bottoms and possibly of different wave types and propagation out to very long ranges the possibility of modeling elastic wave propagation with a similar approach is considered

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

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

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

approximate potential symmetries for partial differential equations the method of approximate potential symmetries for partial differential equations with a small parameter is introduced by writing a given perturbed partial differential equation r in a conserved form an associated system s with potential variables as additional variables is obtained approximate lie point symmetries admitted by s induce approximate potential symmetries of r as applications of the theory approximate potential symmetries for a perturbed wave equation with variable wave speed and a nonlinear diffusion equation with perturbed convection terms are obtained the corresponding approximate group invariant solutions are also derived

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

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

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

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

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

effective boundary conditions resulting from anisotropic and optimally aligned coatings the two dimensional case we discuss a thermally conducting body insulated by a thin anisotropically conducting coating the coating is optimally aligned in the sense that the normal vector inside the coating is always an eigenvector of the thermal tensor we study the effects of the coating by investigating the limiting behavior of solutions u of the heat equation with either dirichlet or neumann boundary conditions imposed on the outer boundary of the coating as the thickness of the coating shrinks to zero in the two dimensional case we find the complete list of effective boundary conditions satisfied by the limit of u on the boundary of the uncoated body this list contains not only the usual dirichlet neumann and robin boundary conditions but also some new and even nonlocal ones involving the dirichlet to neumann mapping and the hilbert transform on the circle we also prove that u converges to its limit in various norms that include the l 2 the sobolev and the holder ones during the course of this study we establish a schauder theory for the regularity of weak solutions of general second order parabolic equations near an interface where the transmission condition is satisfied

highly accurate absorbing boundary conditions for wide angle wave equations we develop a new class of absorbing boundary conditions abcs to prevent unwanted artifacts and wraparounds associated with aperture truncation in migration modeling using high order one way wave equations the fundamental approach behind the proposed development is the efficient discretization of the half space beyond the boundary of interest using midpoint integrated imaginary finite elements an idea recently utilized in the development of effective one way wave equations the proposed absorbing boundary conditions essentially add absorbing layers at the aperture truncation points we derive the absorbing boundary conditions analyze their properties and develop a stable explicit finite difference scheme to solve the downward continuation problem modified by these boundary conditions with the help of numerical examples we conclude that with as few as three absorbing layers i e two additional gridpoints the waves can be absorbed completely thus preventing associated artifacts

parabolic wave equation approximations in heterogenous media the properties of different variants of parabolic approximations of scalar wave equations are analyzed these equations are of general form which includes those used in seismology underwater acoustics and other applications a new version of the parabolic approximation is derived for heterogeneous media it has optimal properties with respect to wave reflection at material interfaces the amplitudes of the reflected and transmitted waves depend continuously on the interface existence uniqueness and energy estimates are proved

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

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

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

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

one pass 3 d seismic extrapolation with the 45 wave equation one pass 3 d modeling and migration for poststack seismic data may be implemented by replacing the traditional 45 one way wave equation a third order partial differential equation with a pair of second and first order partial differential equations except for an extra correction term the resulting second order equation has a form similar to the claerbout 15 one way wave equation which is known to have a nearly circular impulse response in this approach there is no need to compensate for splitting errors numerical tests on synthetic data show that this algorithm has the desirable attributes of being second order in accuracy and economical to solve a modification of the crank nicholson implementation maintains stability

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

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

rational operators for filtering the parabolic equation pe method has been applied to approximate the wave equation construct radiation conditions solve scattering problems construct initial conditions and derive energy flux conditions a new application of the pe method filters based on rational approximations of depth operators is described and tested the rational approximation is designed to act as the identity operator on the desired part of the spectrum of the depth separated wave equation and to annihilate other parts of the spectrum this approach does not require explicit knowledge of the spectrum the applications of the filter include directly solving eigenvalue problems annihilating components of the wave number spectrum generating initial conditions at the source range and eliminating gibbs oscillations that arise in energy conserving pe solutions

group theoretical framework for potential symmetries of evolution equations we develop algebraic approach to the problem of classification of potential symmetries of nonlinear evolution equations it is essentially based on the recently discovered fact r zhdanov j math phys 50 053522 2009 that any such symmetry is mapped into a contact symmetry the approach enables using the classical results on classification of contact symmetries of nonlinear evolution equations by sokolov and magadeev to classify evolution equations admitting potential symmetries we construct several examples of new nonlinear fourth order evolution equations admitting potential symmetries since the symmetries obtained depend on nonlocal variables they cannot be derived by the infinitesimal lie approach

the effective boundary conditions and their lifespan of the logistic diffusion equation on a coated body abstract we consider the logistic diffusion equation on a bounded domain which has two components with a thin coating surrounding a body the diffusion tensor is isotropic on the body and anisotropic on the coating the size of the diffusion tensor on these components may be very different within the coating the diffusion rates in the normal and tangent directions may be in different scales we find effective boundary conditions ebcs that are approximately satisfied by the solution of the diffusion equation on the boundary of the body we also prove that the lifespan of each ebc which measures how long the ebc remains effective is infinite the ebcs enable us to see clearly the effect of the coating and ease the difficult task of solving the pde in a thin region with a small diffusion tensor the motivation of the mathematics includes a nature reserve surrounded by a buffer zone

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

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

transparent boundary conditions for split step pade approximations of the one way helmholtz equation in this paper we generalize the nonlocal discrete transparent boundary condition introduced by f schmidt and p deuflhard 1995 comput math appl 29 53 76 and by f schmidt and d yevick 1997 j comput phys 134 96 107 to propagation methods based on arbitrary pade approximations of the two dimensional one way helmholtz equation our approach leads to a recursive formula for the coefficients appearing in the nonlocal condition which then yields an unconditionally stable propagation method

a new algorithm for finite difference migration of steep dips one way wave propagation is formulated in the x t domain as an integro differential equation and is applied to the migration of stacked data a key step is to replace the phase shift square root in the frequency domain representation by an integral of a rational function the resulting expression is interpreted in the space time domain approximating the integral by a finite sum leads to a number of practical approximations the lowest order being claerbout s 15 degree equation and others being various high order equations in the literature optimal mth order quadrature formulas based upon chebychev criteria suggest a second order approximation which takes 20 percent more time than the 15 degree equation but is accurate to over 50 degrees

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

higher order paraxial wave equation approximations in heterogenous media a new family of paraxial wave equation approximations is derived these approximations are of higher order accuracy than the parabolic approximation and they can be applied to the same computationa

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

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

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

energy conserving and reciprocal solutions for higher order parabolic equations conservation of sound energy in nonabsorbing media and acoustical reciprocity in stationary fluids at rest are the fundamental properties of the acoustic field preserving these properties in parabolic approximations is essential for accurate modeling of sound propagation in shallow water environments recently godin proposed a family of wide angle parabolic equations pe s for motionless fluid which are energy conserving reciprocal and allow improved description of mode coupling without adversely affecting phase accuracy for moving media a generalized claerbout pe was also derived its solutions preserve the acoustic energy in the absence of dissipation and satisfy the flow reversal theorem frt which is an extension of reciprocity principle to moving media the report presents a numerical scheme for solving the proposed pe s the algorithm preserves the conservational properties of the original pseudo differential equations implicit finite difference solutions satisfy discrete analogs of the e

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

potential symmetry generators and associated conservation laws of perturbed nonlinear equations some recent results on approximate lie group methods and previously developed concepts on potential symmetries are extended and applied to mainly nonlinear systems perturbed by a small parameter the potential or auxilliary form of the perturbed system necessarily requires a knowledge of an approximate conservation law of the system we then also use knowledge of approximate potential symmetries to calculate new approximate potential conservation laws