the resolving power of seismic amplitude data an anisotropic inversion migration approach a description of the theory and numerical implementation of a 3 d linearized asymptotic anisotropic inversion method based on the generalized radon transform is given we discuss implementation aspects including 1 the use of various coordinate systems 2 regularization by both spectral and bayesian statistical techniques and 3 the effects of limited acquisition apertures on inversion we give applications of the theory in which well resolved parameter combinations are determined for particular experimental geometries and illustrate the interdependence of parameter and spatial resolutions procedures for evaluating uncertainties in the parameter estimates that result from the inversion are derived and demonstrated

one dimensional electromagnetic inverse reflection problem formulation as a riemann hilbert problem and imaging of discontinuities this article deals with the one dimensional electromagnetic inverse problem of geophysics the problem is shown to have the structure of a riemann hilbert problem analogously to the quantum mechanical inverse scattering problem a trace type formula for the electric conductivity is derived further it is shown that the born approximation of this formula contains all the information of the possible discontinuities of the conductivity profile the proof is based on the idea of singularity expansions of pseudodifierential operators

the paraxial ray method the paraxial ray method is an economical way of computing approximate green s functions in heterogeneous media the method uses information from the standard dynamic ray tracing method to extrapolate the seismic wave field at receivers in the neighborhood of a ray so that two point ray tracing is not required applicability conditions are explicit they define where asymptotic high frequency methods are valid and how far away from the ray the extrapolation remains accurate increasing the density of the ray fan improves accuracy but increases computation time however since reasonable accuracy is obtained with relatively few rays the method yields results similar to the two point ray tracing method but at a fraction of the cost examples of wave field extrapolation from a ray to neighboring receivers show that traveltime extrapolation is more accurate than amplitude extrapolation accuracy robustness and efficiency tests comparing paraxial ray synthetic seismograms with acoustic finite difference

iterative multiparameter waveform inversion of precritical reflection data using prestack time kirchhoff approximation abstractwe have studied elastic wave scattering and iterative inversion in the context of the kirchhoff approximation the approach is more consistent with the weak contrast reflectivity functions of zoeppritz equations as compared to the born approximation to reduce the computational cost associated with inversion we demonstrated the use of amplitude variation with offset avo analysis prestack time migrations pstms and the corresponding forward modeling in an iterative scheme forward modeling and migration inversion operators are based on the double square root dsr equations of pstm and linearized reflectivity functions all operators involved in the inversion including the background model for dsr and avo are defined in p to p traveltime and are updated at each iteration our method is practical for real data applications because all operators of the inversion are known to be applicable for standard methods we have evaluated the inversion on synthetic and real data using the waveform chara

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

generalised operator reduction formula for multiple hypergeometric series nf x1 xn the reduction formula obtained in a previous paper niukkanen 1983 has been generalised as follows if nf and nf are generalised hypergeometric series of n variables ghs n then the operation on nf x1t1 xntn nf delta delta t1 delta delta tn gives at t1 t2 tn 0 a function nf x1 xn which is again a ghs n the differentiation procedure can be regarded as an algebraic omega multiplication which gives rise to a group theoretical interpretation of the method a concept of omega equivalent relations has been introduced which allows systematisation of numerous results obtained in special functions theory as the functions nf comprise a number of physically interesting series the operator factorisation method seems to be applicable to many physical problems providing a possibility of reducing any nf to simpler functions of the same class

to a unified approach to 3 d seismic reflection imaging part ii theory m tygel j schleicher and p hubral geophysics 61 759 775 it came to our attention that in the paper of tygel et al 1996 formula a 5 is incorrect first it is different from the cited equation 17 of tygel et al 1995 second the derivation of formula 17 in appendix a of tygel et al 1995 is invalid for the purposes of tygel et al 1996 because in the 1995 paper a rotation is treated whereas in tygel et al 1996 a projection matrix is needed further investigation of the subject was needed to resolve these inconsistencies

approximate linearized inversion by optimal scaling of prestack depth migration linearized inversion provides one possible sense of image amplitude correctness an image or bandlimited model perturbation has correct amplitudes if it is an approximate inversion that is if linearized modeling demigration with the image as input reproduces the data approximately the theory of linearized acoustic inverse scattering with slowly varying background or macromodel shows that an approximate inversion may be recovered from the output of prestack depth migration by a combination of scaling and filtering the necessary filter is completely specified by the theory and the scale factor may be estimated via filtering linearized modeling a second migration and the solution of a small auxiliary inverse problem

on the imaging of reflectors in the earth in this paper i present a modification of the beylkin inversion operator this modification accounts for the band limited nature of the data and makes the role of discontinuities in the sound speed more precise the inversion presented here partially dispenses with the small parameter constraint of the born approximation this is shown by applying the proposed inversion operator to upward scattered data represented by the kirchhoff approximation using the angularly dependent geometrical optics reflection coefficient a fully nonlinear estimate of the jump in sound speed may be extracted from the output of this algorithm interpreted in the context of these kirchhoff approximate data for the forward problem the inversion of these data involves integration over the source receiver surface the reflecting surface and frequency the spatial integrals are computed by the method of stationary phase the output is asymptotically a scaled singular function of the reflecting surface the singular function of a

matrix cyclization over complex polynomials abstract let a be an n n b an n m complex polynomial matrix the following open problem is investigated does there exist a complex polynomial m n matrix f such that a bf has a cyclic vector in the image of b an explicit solution of the problem is given for the following generic situation n 4 b ab a n 1 b is rightinvertible necessary the entries of a specific normalized form of a b do not satisfy certain polynomial equations and one specific entry has at least one simple zero outside a certain finite set the exceptional equations are given explicitely and can easily be checked for

permittivity recovery from multiple characteristic electric waves we use the calculus of fourier integral operators with singular symbols to reconstruct electrical permittivity singularities from remote measurements of scattered electric waves the reconstruction is microlocal and is uniformly valid in the neighborhood of conically refracted waves

3d ray born migration inversion part 1 theory prestack ray born migration inversion can be split in two steps the computation of common image gathers cigs and their weighted stack the migration stack the choice of the domain for the cigs shot offset angle etc has a direct impact on the resolution of the migration stack this resolution can be studied easily in the frame of ray born migration inversion theory resulting into improved migration inversion formulas according to the acquisition geometry this paper is devoted to this analysis in the cases of a simple 2d acquisition and of a 3d swath acquisition both corresponding to classical data sets from the seg eage 3d overthrust experiment we show that the migration formula originally designed for 3d marine acquisition is not adaptable to the 3d swath acquisition finally we propose a new formula for this specific acquisition which improves the resolution of the final migrated image the relevance of this new formula is illustrated in the frame of the seg eage experiment in the companion paper

transformation of certain generalized kampe de feriet functions new transformations of generalized kampe de feriet functions of several variables which occur in physical and quantum chemical contexts are given unlike many of the known results for these functions the variables are not fixed

an alternative formula for approximate extended born inversion abstractvarious modifications of reverse time migration rtm provide asymptotic inverses to the subsurface offset extended born modeling operator for constant density acoustics these approximate inverses have the same quality asymptotic accuracy as do generalized radon transform pseudoinverses but they can be computed without any ray tracing whatsoever we have developed an approximate inverse of this type whose additional computational cost above that of subsurface offset extended rtm is negligible

cyclic accessibility of reachable states characterizes von neumann regular rings abstract the class of commutative von neumann regular rings is characterized by a generalization of the feedback cyclization property to non necessarily reachable systems for any system a b there exist a matrix k and a vector u such that a b and the single input system a bk bu have the same submodule of reachable states an explicit algorithm is presented to obtain k u for a given system a b

a brief comparison of some kirchhoff integral formulas for migration and inversion kirchhoff migration has traditionally been viewed as an imaging procedure usually few claims are made regarding the amplitudes in the imaged section in recent years a number of inversion formulas similar in form to those of kirchhoff migration have been proposed a kirchhoff type inversion produces not only an image but also an estimate of velocity variations or perhaps reflection coefficients the estimate is obtained from the peak amplitudes in the image in this paper prestack kirchhoff migration and inversion formulas for the one parameter acoustic wave equation are compared following a heuristic approach based on the imaging principle a migration formula is derived which turns out to be identical to one proposed by bleistein for inversion prestack kirchhoff migration and inversion are thus seen to be the same both in terms of the image produced and the peak amplitudes of the output

a b cyclic submodules abstract an a b cyclic submodule m is generated by the states of one single trajectory of a linear control system whose parameters come from a commutative ring m is finite when it is generated by the states of a deadbeat control process motivations and basic properties of such modules are given and among several further results it is shown that the family of finite a b cyclic submodules is an invariant which e g over polynomials can be determined by an appropriate grobner basis computation

a unified approach to 3 d seismic reflection imaging part ii theory diffraction stack and isochrone stack integrals are quantitatively described and employed they constitute an asymptotic transform pair both integrals are the key tools of a unified approach to seismic reflection imaging that can be used to solve a multitude of amplitude preserving target oriented seismic imaging or image transformation problems these include for instance the generalizations of the kinematic map transformation examples discussed in part i all image transformation problems can be addressed by applying both stacking integrals in sequence whereby the macro velocity model the measurement configuration or the ray code of the considered elementary reflections may change from step to step this leads to weighted kirchhoff or generalized radon type summations along certain stacking surfaces or inplanats for which true amplitude ta weights are provided to demonstrate the value of the proposed imaging theory which is based on analytically chaining the two stacking integrals and using certain inherent dualities we examine in detail the amplitude preserving configuration transform and remigration for the case of a 3 d laterally inhomogeneous velocity medium

a generalized diffraction tomography algorithm available diffraction tomography algorithms are based on fourier transform techniques and require either plane wave illumination in a uniform background medium or far field illumination combined with paraxial approximations in this paper a generalized diffraction tomography algorithm is introduced that can handle both irregularly spaced measurement data nonuniform background models and general aquisition geometries using data from water tank experiments the method s ability to yield high quality reconstructions of geometry and velocity as long as the weak scattering assumption is satisfied is demonstrated

accommodating lateral velocity changes in kirchhoff migration by means of fermat s principle when velocity varies laterally as well as with depth an exact kirchhoff depth migration requires that rays be traced from each depth point in the section to each source receiver location because such a procedure is prohibitively expensive kirchhoff migration is usually carried out by using a velocity function that depends only on depth this paper introduces a new method based on fermat s principle which is a compromise between these two extremes the slowness reciprocal velocity function is written as the sum of two functions the first of which is large and depends only on depth while the other is small and varies both with depth and position along the line raypaths are traced for the first slowness function and are used to calculate migration curves for each depth point these same raypaths are used to calculate traveltime perturbations due to the laterally varying part of the slowness the traveltime perturbations are added to the migration curve to obtain an approximation to the exact migrat

extended isochron rays in prestack depth map migration many processes in seismic data analysis and seismic imaging can be identified with solution operators of evolution equations these include data downward continuation and velocity continuation we have addressed the question of whether isochrons defined by imaging operators can be identified with wavefronts of solutions to an evolution equation rays associated with this equation then would provide a natural way of implementing prestack map migration assuming absence of caustics we have developed constructive proof of the existence of a hamiltonian describing propagation of isochrons in the context of common offset depth migration in the presence of caustics one should recast to a sinking survey migration framework by manipulating the double square root operator we obtain an evolution equation that describes sinking survey migration as a propagation in two way time with surface data being a source function this formulation can be viewed as an extension of the exploding reflector concept from zero o

fast 2 d ray born migration inversion in complex media in this paper we evaluate the capacity of a fast 2 d ray born migration inversion algorithm to recover the true amplitude of the model parameters in 2 d complex media the method is based on a quasi newtonian linearized inversion of the scattered wavefield asymptotic green s functions are computed in a smooth reference model with a dynamic ray tracing based on the wavefront construction method the model is described by velocity perturbations associated with diffractor points both the first traveltime and the strongest arrivals can be inverted the algorithm is implemented with several numerical approximations such as interpolations and aperture limitation around common midpoints to speed the algorithm both theoritical and numerical aspects of the algorithm are assessed with three synthetic and real data examples including the 2 d marmousi example comparison between logs extracted from the exact marmousi perturbation model and the computed images shows that the amplitude of the velocity perturbations

a geometrical optics based numerical method for high frequency electromagnetic fields computations near fold caustics part i this paper presents an eulerian numerical method for the computation of a bi valued solution of hamilton jacobi type equation in a particular geometric setting more precisely we consider high frequency electromagnetic fields in the vicinity of fold caustics

a unified approach to 3 d seismic reflection imaging part i basic concepts given a 3 d seismic record for an arbitrary measurement configuration and assuming a laterally and vertically inhomogeneous isotropic macro velocity model a unified approach to amplitude preserving seismic reflection imaging is provided this approach is composed of 1 a weighted kirchhoff type diffraction stack integral to transform migrate seismic reflection data from the measurement time domain into the model depth domain and of 2 a weighted kirchhoff type isochrone stack integral to transform demigrate the migrated seismic image from the depth domain back into the time domain both the diffraction stack and isochrone stack integrals can be applied in sequence i e they can be chained for different measurement configurations or different velocity models to permit two principally different amplitude preserving image transformations these are 1 the amplitude preserving transformation directly in the time domain of one 3 d seismic record section into another one pertaining to a different

the applicability of dip moveout azimuth moveout in the presence of caustics reflection seismic data continuation is the computation of data at source and receiver locations that differ from those in the original data using whatever data are available we develop a general theory of data continuation in the presence of caustics and illustrate it with three examples dip moveout dmo azimuth moveout amo and offset continuation this theory does not require knowledge of the reflector positions we construct the output data set from the input through the composition of three operators an imaging operator a modeling operator and a restriction operator this results in a single operator that maps directly from the input data to the desired output data we use the calculus of fourier integral operators to develop this theory in the presence of caustics for both dmo and amo we compute impulse responses in a constant velocity model and in a more complicated model in which caustics arise this analysis reveals errors that can be introduced by assuming for example a model with a constant vertical velocity gradient when the true model is laterally heterogeneous data continuation uses as input a subset common offset common angle of the available data which may introduce artifacts in the continued data one could suppress these artifacts by stacking over a neighborhood of input data using a small range of offsets or angles for example we test data continuation on synthetic data from a model known to generate imaging artifacts we show that stacking over input scattering angles suppresses artifacts in the continued data

imaging of velocity singularities with multiscale operators in this paper by using the multiscale operator and based on the beylkin s formula j math phys 26 99 108 1985 the inversion method of velocity singularities is presented and by applying the multiscale operator to upward scattered data represented by the kirchhoff approximation the multiscale imaging formula is shown for the band limited inverse problem based on the multiscale imaging formula in this paper the behavior of the velocity singularities can be analyzed across the multiscale and the formula to reconstruct the velocity is obtained from the output of the multiscale operator interpreted in the terms of the kirchhoff approximate data particularly at wide offset and even postcritical case for the full band input data the same result with beylkin s in the above reference about the velocity discontinuities is obtained our method in this paper can suppress the effects of the noise in the real world data and analyze the effect produced by band limited input data

exponential asymptotically bounded velocity model part ii ray tracing a ray tracing procedure is derived for the new exponential asymptotically bounded eab velocity model introduced in part i of this paper the model inherits the properties of a medium with linear velocity variation in depth in the shallow zone and of a medium with constant velocity in the deep zone two types of rays departing from the source point on the earth s surface exist in this model depending on the takeoff angles the rays of the first kind are symmetric arcs that return up to the earth s surface and have a limited maximum depth of propagation the rays of the second kind propagate down to infinite depth in the shallow region they are curved lines but at large depth they become asymptotically straight the form of the ray is governed by the takeoff angle at the source point where a critical angle splits the two kinds of rays this critical angle depends only on the ratio between the velocity at the source point and the asymptotic velocity of the eab model we derive the formulae required to calculate the two kinds of rays and solve the inverse problem of two point ray tracing finally we construct the 2d and 3d isochron surfaces for a finite offset

generalised hypergeometric series nf x1 xn arising in physical and quantum chemical applications functions nf x1 xn which are a straightforward generalisation of standard hypergeometric functions of n variables are introduced a convenient operator representation is established for these functions which permits one to consider in a uniform and simple way standard hypergeometric functions generalised hypergeometric series of one variable appell kampe de feriet and lauricella functions etc as well as many other nameless hypergeometric series arising in physical and quantum chemical applications generating functions generalisations of bailey 1935 and chaundy 1943 expansions recurrence relations reduction rules and some other topics are discussed

asymptotic pseudounitary stacking operators stacking operators are widely used in seismic imaging and seismic data processing examples include kirchhoff datuming migration offset continuation dip moveout and velocity transform two primary approaches exist for inverting such operators the first approach is iterative least squares optimization which involves the construction of the adjoint operator the second approach is asymptotic inversion where an approximate inverse operator is constructed in the high frequency asymptotics adjoint and asymptotic inverse operators share the same kinematic properties but their amplitudes weighting functions are defined differently this paper describes a theory for reconciling the two approaches i introduce a pair of asymptotic pseudounitary operators which possess both the property of being adjoint and the property of being asymptotically inverse the weighting function of the asymptotic pseudounitary stacking operators is shown to be completely defined by the derivatives of the operator kinematics i exemplify the general theory by considering several particular examples of stacking operators simple numerical experiments demonstrate a noticeable gain in efficiency when the asymptotic pseudounitary operators are applied for preconditioning iterative least squares optimization

the inversion problem and applications of the generalized radon transform we prove that under certain conditions the inversion problem for the generalized radon transform reduces to solving a fredholm integral equation and we obtain the asymptotic expansion of the symbol of the integral operator in this equation we consider applications of the generalized radon transform to partial differential equations with variable coefficients and provide a solution to the inversion problem for the attenuated and exponential radon transforms

analytical and numerical analysis of tomographic resolution with band limited signals we investigate the resolution of band limited seismic signals using analytical arguments based on the rytov approximation of the single scattering problem and numerical modeling of the full wave equation the experimental framework is seismic transmission tomography contrary to intuition rytov s theory predicts that time delays carry only limited information about the velocity of the medium along the shortest path mathematical ray the information loss being more severe in 3d than in 2d geometries we show that signal amplitude provides the information missing in time delay and we argue that a complete imaging should invert both delays and amplitudes we also address the question of picking time delays and show that this should be done on the signal peak when using the rytov approximation for data inversion finally we test the case of extended velocity perturbations and we show that the rytov approximation handles fairly well a thin high velocity layer including both source and receiver while i

an approximate inverse to the extended born modeling operator abstractgiven a correct data consistent velocity model reverse time migration rtm correctly positions reflectors but generally with incorrect amplitudes and wavelets iterative least squares migration lsm corrects the amplitude and wavelet by fitting data in the sense of born modeling that is replacing migration by born inversion however lsm also requires a correct velocity model and it may require many migration demigration cycles we modified rtm in the subsurface offset domain to create an asymptotic high frequency approximation to extended lsm this extended born inversion operator outputs extended reflectors depending on the subsurface offset and position in the earth with correct amplitude and phase in the sense that similarly extended born modeling reproduces the data to good accuracy although the theoretical justification of the inversion property relies on ray tracing and stationary phase application of the weight operators does not require any computational ray tracing the co

multiparameter inversion for acoustic and elastic media inversion and imaging of an acoustic two parameter medium and for a three parameter isotropic elastic medium from seismic data are discussed the inverse problem is linearized by considering the actual medium as a perturbation from a known background model by using the single scattering approximation integral equations relating the singly scattered field linearly to the unknown parameters are obtained asymptotic solutions of the linearized inverse problem are derived using the generalized radon transform this method is closely related to methods of seismic migration spatially varying background parameters and an almost arbitrary source receiver configuration are allowed the computation is performed in the time domain all available data are used even if redundant combinations of the parameters which can be reconstructed best and worst are identified the data may be read just once and in an arbitrary order

microlocal analysis of seismic inverse scattering in anisotropic elastic media seismic data is modeled in the high frequency approximation using the techniques of microlocal analysis we consider general anisotropic elastic media our methods are designed to allow for the formation of caustics the data is modeled in two ways first we give a microlocal treatment of the kirchhoff approximation where the medium is assumed to be piecewise smooth and reflection and transmission occur at interfaces second we give a refined view on the born approximation based upon a linearization of the scattering process in the medium parameters around a smooth background medium the joint formulation of born and kirchhoff scattering allows us to take into account general scatterers as well as the nonlinear dependence of reflection coefficients on the medium parameters the latter allows the treatment of scattering up to grazing angles the outcome of the analysis is a characterization of the singular part of seismic data we obtain a set of pseudodifferential operators that annihilate the data in the process we construct a fourier integral operator and a reflectivity function such that the data can be represented by this operator acting on the reflectivity function in our construction this fourier integral operator becomes invertible we give the conditions for invertibility for general acquisition geometry the result is also of interest for inverse scattering in acoustic media c 2002 john wiley sons inc

representation theorems and green s function retrieval for scattering in acoustic media reciprocity theorems for perturbed acoustic media are provided in the form of convolution and correlation type theorems these reciprocity relations are particularly useful in the general treatment of both forward and inverse scattering problems using green s functions to describe perturbed and unperturbed waves in two distinct wave states representation theorems for scattered waves are derived from the reciprocity relations while the convolution type theorems can be manipulated to obtain scattering integrals that are analogous to the lippmann schwinger equation the correlation type theorems can be used to retrieve the scattering response of the medium by cross correlations unlike previous formulations of green s function retrieval the extraction of scattered wave responses by cross correlations does not require energy equipartitioning allowing for uneven energy radiation brings experimental advantages to the retrieval of fields scattered by remote lossless and or attenuative scatterers these concepts are illustrated with a number of examples including analytic solutions to a one dimensional scattering problem and a numerical example in the context of seismic waves recorded on the ocean bottom

a differential semblance algorithm for the inverse problem of reflection seismology abstract this paper presents an analysis of stability and convergence for a special case of differential semblance optimization dso this approach to model estimation for reflection seismology is a variant of the output least squares inversion of seismograms enjoying analytical and numerical properties superior to those of more straightforward versions we study a specialization of dso appropriate to the inversion of convolutional approximation planewave seismograms over layered constant density acoustic media we prove that the differential semblance variational principle is locally convex in suitable model classes for a range of data noise moreover the structure of the convexity estimates suggest a family of quasi newton algorithms we describe an implementation of one of these algorithms and present some numerical results

velocity inversion a case study in infinite dimensional optimization the goal of seismic velocity inversion is the estimation of seismic wave velocities inside the earth by attempting to predict in a least error sense seismic waveforms measured at its surface we present velocity inversion as a case study in the various infinite dimensional pathologies which may afflict practically important problems of distributed parameter identification treated as optimization problems in function spaces these features differentiate various problem formulations far beyond the degree one would expect for finite small dimensional problems we illustrate this differentiation by comparing the characteristics of three different least squares formulations of velocity inversion

an optimal true amplitude least squares prestack depth migration operator in order to define an optimal true amplitude prestack depth migration for multishot and multitrace data we develop a general methodology based on the least squares data misfit function associated with a forward model the amplitude of the migrated events are restored at best for any given geometry and any given preliminary filtering and amplitude correction of the data the migrated section is then the gradient of the cost function multiplied by a weight matrix a study of the hessian associated with this data misfit shows how efficiently to find a good weight matrix via the computation of only few elements of this hessian thanks to this matrix the resulting migration operator is optimal in the sense that it ensures the best possible restoration of the amplitudes among the large class of least squares migrations applied to a forward model based on born ray tracing and diffracting points approximation this optimal migration outperforms or at least equals the classic kirchhoff formula since the latter belongs to the class of least squares migrations and is only optimal for one shot and an infinite aperture numerical results illustrate this construction and confirm the above expectations

three dimensional born inversion with an arbitrary reference recent work of g beylkin helped set the stage for very general seismic inversions we have combined these broad concepts for inversion with classical high frequency asymptotics and perturbation methods to bring them closer to practically implementable algorithms applications include inversion schemes for both stacked and unstacked seismic data basic assumptions are that the data have relative true amplitude and that a reasonably accurate background velocity c x y z is available the perturbation from this background is then sought since high frequency approximations are used throughout the resulting algorithms essentially locate discontinuities in velocity an expression for a full 3 d velocity inversion can be derived for a general data surface in this degree of generality the formula does not represent a computationally feasible algorithm primarily because a key jacobian determinant is not expressed in practical terms in several important cases however this shortcoming can be overcome and

strong feedback cyclization for systems over rings abstract for nonnecessarily reachable systems over a commutative ring r we define a strong form of feedback cyclization fc with this natural generalization of the fc property we obtain a feedback reduced form for systems over strong fc rings i e rings for which every system verifies the fc property in the particular case of reachable single input systems this gives the usual control canonical form of bumby et al remarks on the pole shifting problem over rings j pure appl algebra 20 1981 113 127 also it is proved that a commutative ring with 1 in its stable range has the strong fc property if and only if it has the ucu property which is the natural parallel form of the result given by brewer et al pole assignability in polynomial rings power series rings and prufer domains j algebra 106 1987 265 286 for the reachable case many classes of rings which were known to be fc rings are in fact strong fc rings but there are fc rings which are not strong fc rings

a new slant on seismic imaging migration and integral geometry a new approach to seismic migration formalizes the classical diffraction or common tangent stack by relating it to linearized seismic inversion and the generalized radon transform this approach recasts migration as the problem of reconstructing the earth s acoustic scattering potential from its integrals over isochron surfaces the theory rests on a solution of the wave equation with the geometrical optics green function and an approximate inversion formula for the generalized radon transform the method can handle both complex velocity models and nearly arbitrary configurations of sources and receivers in this general case the method can be implemented as a weighted diffraction stack with the weights determined by tracing rays from image points to the experiment s sources and receivers when tested on a finite difference simulation of a deviated well vertical seismic profile a hybrid experiment which is difficult to treat with conventional wave equation methods the algorithm accurately reconstructed faulted earth models analytical reconstruction formulas are derived from the general formula for zero offset and fixed offset surface experiments in which the background velocity is constant the zero offset inversion formula resembles standard kirchhoff migration our analysis provides a direct connection between the experimental setup source and receiver positions source wavelet background velocity and the spatial resolution of the reconstruction synthetic examples illustrate that the lateral resolution in seismic images is described well by the theory and is improved greatly by combining surface data and borehole data the best resolution is obtained from a zero offset experiment that surrounds the region to be imaged

direct computation of multivalued phase space solutions for hamilton jacobi equations a method is presented for the exact and complete resolution of hamiltonian systems only using the corresponding hamilton jacobi equation we show that it possible to get the ode solution from the corresponding pde

discrete radon transform this paper describes the discrete radon transform drt and the exact inversion algorithm for it similar to the discrete fourier transform dft the drt is defined for periodic vector sequences and studied as a transform in its own right casting the forward transform as a matrix vector multiplication the key observation is that the matrix although very large has a block circulant structure this observation allows construction of fast direct and inverse transforms moreover we show that the drt can be used to compute various generalizations of the classical radon transform rt and in particular the generalization where straight lines are replaced by curves and weight functions are introduced into the integrals along these curves in fact we describe not a single transform but a class of transforms representatives of which correspond in one way or another to discrete versions of the rt and its generalizations an interesting observation is that the exact inversion algorithm cannot be obtained directly from radon s inversion formula given the fact that the rt has no nontrivial one dimensional analog exact invertibility makes the drt a useful tool geared specifically for multidimensional digital signal processing exact invertibility of the drt flexibility in its definition and fast computational algorithm affect present applications and open possibilities for new ones some of these applications are discussed in the paper

straight rays redatuming a fast and robust alternative to wave equation based datuming wave equation based redatuming is expensive and requires a detailed knowledge of the shallow velocity field we derive the analytical expression of a new prestack wavefield extrapolation operator the topographic datuming operator tdo which applies redatuming based on straight rays approximation above and below a chosen datum this redatuming operator is directly applied to common source gathers to downward continue the source and the receivers simultaneously to the datum level without resorting to common receiver gathers as a result the method is far more efficient and robust than the conventional wave equation based redatuming and does not require an accurate depth domain interval velocity model in addition tdo unlike wave equation based redatuming requires effective velocities above datum and thus can be applied using attributes valid for static correction methods effective velocities beneath the datum permit us to replace the surface integral which is needed for wave equation redatuming

imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized radon transform this paper treats the linearized inverse scattering problem for the case of variable background velocity and for an arbitrary configuration of sources and receivers the linearized inverse scattering problem is formulated in terms of an integral equation in a form which covers wave propagation in fluids with constant and variable densities and in elastic solids this integral equation is connected with the causal generalized radon transform grt and an asymptotic expansion of the solution of the integral equation is obtained using an inversion procedure for the grt the first term of this asymptotic expansion is interpreted as a migration algorithm as a result this paper contains a rigorous derivation of migration as a technique for imaging discontinuities of parameters describing a medium also a partial reconstruction operator is explicitly derived for a limited aperture when specialized to a constant background velocity and specific source receiver geometries our results are directly related to some known migration algorithms

the link of kirchhoff migration and demigration to kirchhoff and born modeling the kirchhoff approximation provides a representation of seismic data as a summation of imaged data along isochron surfaces demigration the asymptotic inversion of this representation provides a migration as a summation of seismic data along diffraction surfaces we replace born inversion techniques with kirchhoff inversion techniques and further show the link between the kirchhoff and born representations after the born linearized reflection coefficient is replaced by the kirchhoff reflection coefficient

a geometric approach to dynamic feedback cyclization over commutative rings a different geometric point of view is introduced in the study of dynamic feedback cyclization over rings this considerably simplifies the theory in the free case in the sense that the main results can now be obtained by simple proofs the non free case remains to be examined

computation of maximal reachability submodules abstract a new and conceptually simple procedure is derived for the computation of the maximal reachability submodule of a given submodule of the state space of a linear discrete time system over a noethenian ring r the procedure is effective if r is effective and if kernels and intersections can be computed the procedure is compared with a rather different procedure by assan e a published recently

wave field reciprocity and optimization in remote sensing a unified approach to local optimization techniques and wave field reciprocity as applied to constructing solutions to remote sensing imaging and inversion problems in acoustic elastic and electromagnetic wave theory is presented the starting point is a system of linear first order partial differential equations in space time of the class to which the indicated wave phenomena gives rise for this system three types of remote sensing problems the inverse source the inverse scattering and the inverse transducted wave field problems are formulated and the construction of their solutions via local optimization techniques is discussed emphasis is placed on iterative algorithms that are based on a guaranteed decrease in the mismatch between modelled and observed data at each update of the medium subsequently the wave field reciprocity theorems of the time convolution and the time correlation types are derived and their occurrence in the optimization procedures is discussed also attention is paid to approximate methods in particular to the rayleigh gans born approximation approximations of this sort provide the means to invoke the method of preconditioning in the process of inverting the operator equations exotic media for example chiral media in electromagnetics are included in the analysis

multicomponent kirchhoff migration this paper presents a method for elastic and viscoelastic imaging of multicomponent seismic data the method is based on claerbout s survey sinking concept and the visco elastic kirchhoff integral for the displacement field assuming a multishot and multireceiver experiment the migration process is formulated as a wavefield reconstruction problem using the visco elastic kirchhoff integral twice first the receiver coordinates are downward continued second the source coordinates are downward continued the multicomponent seismic data are treated as a vector wavefield in which the data measurements may be displacement velocity or traction pressure the theoretical formulation is based on the viscoelastic hooke s law and newton s equation of motion as the physical model for seismic wave propagation it is valid for linear viscoelastic media with any anisotropic symmetry when the lowest order ray approximation is introduced the migration equation takes a form similar to conventional kirchhoff migra

fast computation of fourier integral operators we introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation general hyperbolic equations and curvilinear tomography the problem is to numerically evaluate a so called fourier integral operator fio of the form int e 2 pi i phi x xi a x xi hat f xi d xi at points given on a cartesian grid here xi is a frequency variable hat f xi is the fourier transform of the input f a x xi is an amplitude and phi x xi is a phase function which is typically as large as xi hence the integral is highly oscillatory because a fio is a dense matrix a naive matrix vector product with an input given on a cartesian grid of size n by n would require o n 4 operations this paper develops a new numerical algorithm which requires o n 2 5 log n operations and as low as o sqrt n in storage space the constants in front of these estimates are small it operates by localizing the integral over polar wedges with small angular aperture in the frequency plane on each wedge the algorithm factorizes the kernel e 2 pi i phi x xi a x xi into two components 1 a diffeomorphism which is handled by means of a nonuniform fft and 2 a residual factor which is handled by numerical separation of the spatial and frequency variables the key to the complexity and accuracy estimates is the fact that the separation rank of the residual kernel is provably independent of the problem size several numerical examples demonstrate the numerical accuracy and low computational complexity of the proposed methodology we also discuss the potential of our ideas for various applications such as reflection seismology

prestack kirchhoff inversion of common offset data in trying to resolve complex geologic structures the pitfalls in employing the cdp method become evident additionally stacking multioffset traces corrupts the amplitudes necessary for stratigraphic analysis in order to preserve whatever structural and amplitude information is in the data prestack processing should be performed given common offset data and the velocity above a reflector prestack acoustic kirchhoff inversion resolves the location of the interface when amplitude information has been preserved in the data the method additionally calculates the reflection coefficient at each interface point for band limited seismic data the inversion operator produces a sinc like picture of the reflector with the peak amplitude of this band limited singular function equal to the angularly dependent reflection coefficient the inversion development is based upon high frequency kirchhoff data which are inserted into a general 3 d inversion operator asymptotically evaluating the four resulting integrals by the method of four dimensional stationary phase permits an inversion amplitude function to be chosen so that the inversion operator produces a singular function of support on the reflector weighted by the reflection coefficient specializing the three dimensional inversion operator to two and one half dimensions allows for processing of single lines of common offset data synthetic examples illustrate the accuracy of the method for constant velocity kirchhoff data as well as the problems in applying constant velocity data to multivelocity models

offset continuation stacking theory and proof of concept abstractthe offset continuation operation oco is a seismic configuration transform designed to simulate a seismic section as if obtained with a certain source receiver offset using the data measured with another offset based on this operation we have introduced the oco stack which is a multiparameter stacking technique that transforms 2d 2 5d prestack multicoverage data into a stacked common offset co section similarly to common midpoint and common reflection surface stacks the oco stack does not rely on an a priori velocity model but provided velocity information itself because oco is dependent on the velocity model used in the process the method can be combined with trial stacking techniques for a set of models thus allowing for the extraction of velocity information the algorithm consists of data stacking along so called oco trajectories which approximate the common reflection point trajectory i e the position of a reflection event in the multicoverage data as a function of source rec

migration velocity analysis using residual diffraction moveout in the poststack depth domain abstractdiffraction events contain more direct information on the medium velocity than reflection events we have developed a method for migration velocity improvement and diffraction localization based on a moveout analysis of over or undermigrated diffraction events in the depth domain the method uses an initial velocity model as input it provides an update to the velocity model and diffraction locations in the depth domain as a result the algorithm is based on the focusing of remigration trajectories from incorrectly migrated diffraction curves these trajectories are constructed by applying a ray tracing like approach to the image wave equation for velocity continuation the starting points of the trajectories are obtained from fitting an ellipse or hyperbola to the picked uncollapsed diffraction events in the depth migrated domain focusing of the remigration trajectories points out the approximate location of the associated diffractor as well as local velocity attributes apart from the migrati

dynamic feedback a b invariant submodules for linear systems over commutative noetherian domains abstract in this paper some basic characterizations of a b invariant submodules for linear systems over commutative noetherian domains are studied first a notion of dynamic feedback a b invariant submodules for such systems is introduced and then it is shown that this notion is equivalent to a b invariance

accelerating extended least squares migration with weighted conjugate gradient iteration abstractleast squares migration lsm iteratively achieves a mean square best fit to seismic reflection data provided that a kinematically accurate velocity model is available the subsurface offset extension adds extra degrees of freedom to the model thereby allowing lsm to fit the data even in the event of significant velocity error this type of extension also implies additional computational expense per iteration from crosscorrelating source and receiver wavefields over the subsurface offset and therefore places a premium on rapid convergence we have accelerated the convergence of extended least squares migration by combining the conjugate gradient algorithm with weighted norms in range data and domain model spaces that render the extended born modeling operator approximately unitary we have developed numerical examples that demonstrate that the proposed algorithm dramatically reduces the number of iterations required to achieve a given level of fit or gradient reduction compared with conjuga

modeling of seismic data in the downward continuation approach seismic data are commonly modeled by a high frequency single scattering approximation this amounts to a linearization in the medium coefficient about a smooth background the discontinuities are contained in the medium perturbation the high frequency part of the wavefield in the background medium is described by a geometrical optics representation it can also be described by a one way wave equation based on this we derive a downward continuation operator for seismic data this operator solves a pseudodifferential evolution equation in depth the so called double square root equation we consider the modeling operator based on this equation if the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal the singular part of the data is described by the solution to an inhomogeneous double square root equation

two and one half dimensional born inversion with an arbitrary reference multidimensional inversion algorithms are presented for both prestack and poststack data gathered on a single line these algorithms both image the subsurface i e give a migrated section and given relative true amplitude data estimate reflection strength or impedance on each reflector the algorithms are two and one half dimensional 2 5 d in that they incorporate three dimensional 3 d wave propagation in a medium which varies in only two dimensions the use of 3 d sources does not entail any computational penalty and it avoids the serious degradation of amplitude incurred by using the 2 d wave equation our methods are based on the linearized inversion theory associated with the born inversion thus we assume that the sound speed profile is well approximated by a given background velocity plus a perturbation it is this perturbation that we seek to reconstruct we are able to treat the case of an arbitrary continuous background profile however the cost of implementation increases as one

parameter inversion and angle migration in anisotropic elastic media seismic parameter inversion using the inverse generalized radon transform grt was originally formulated as an integral over source and receiver surface coordinates involving a jacobian from angle to surface coordinates by formulating the inversion directly in terms of angle coordinates at the subsurface image point i obtain simplified inversion formulas for point scatterers and subsurface reflectors amplitude preserving angle migration based on the inverse grt gives an estimate of the scattering coefficient in the born integral as a function of scattering angle and azimuth the scattering coefficient is directly related to the linearized plane wave reflection coefficient between two anisotropic media which limits the result to small angles and small contrasts in the elastic parameters i use the kirchhoff modeling integral for anisotropic media to formulate a kirchhoff angle migration integral that is valid for large angles and large parameter contrasts it gives an estimate of the plane wave refle

feedback classification of linear systems over von neumann regular rings abstract it is proved that feedback classification of a linear system over a commutative von neumann regular ring r can be reduced to the classification of a finite family of systems each of which is properly split into a reachable and a non reachable part where the reachable part is in a brunovski type canonical form while the non reachable part can only be altered by similarity if a canonical form is known for similarity of matrices over r then it can be used to construct a canonical form for feedback equivalence an explicit algorithm is given to obtain the canonical form in a computable context together with an example over a finite ring

domain reconstruction using photothermal techniques a numerical method to detect objects buried in a medium by surface thermal measurements is presented we propose a new approach combining the use of topological derivatives and laplace transforms the original optimization problem with time dependent constraints is replaced by an equivalent problem with stationary constraints by means of laplace transforms the first step in the reconstruction scheme consists in discretizing the inversion formula to produce an approximate optimization problem with a finite set of constraints then an explicit expression for the topological derivative of the approximate shape functional is given this formula is evaluated at low cost using explicit expressions of the forward and adjoint fields involved we apply this technique to a simple shape reconstruction problem set in a half space good approximations of the number location and size of the obstacles are obtained the description of their shapes can be improved by more expensive hybrid methods combining time averaging with topological derivative based iterative schemes

elastic least squares reverse time migration abstractleast squares migration can produce images with improved resolution and reduced migration artifacts compared with conventional imaging we have developed a method for elastic least squares reverse time migration lsrtm based on a new perturbation imaging condition that yields scalar images of squared p and s velocity perturbations these perturbation images do not suffer from polarity reversals that are common for more conventional elastic imaging methods we use 2d synthetic and field data examples to demonstrate the proposed lsrtm algorithm using the perturbation imaging condition our results show that elastic lsrtm improves the energy focusing and illumination of the elastic images and it attenuates artifacts resulting for instance from sparseness in the wavefield sampling and crosstalk of the p and s modes compared with rtm images the lsrtm images provide more accurate relative amplitude information that is useful for reservoir characterization

least squares gaussian beam migration abstractleast squares migration lsm can be effective to mitigate the limitation of finite seismic acquisition balance the subsurface illumination and improve the spatial resolution of the image but it requires iterations of migration and demigration to obtain the desired subsurface reflectivity model the computational efficiency and accuracy of migration and demigration operators are crucial for applying the algorithm we have developed a test of the feasibility of using the gaussian beam as the wavefield extrapolating operator for the lsm denoted as least squares gaussian beam migration our method combines the advantages of the lsm and the efficiency of the gaussian beam propagator our numerical evaluations including two synthetic data sets and one marine field data set illustrate that the proposed approach could be used to obtain amplitude balanced images and to broaden the bandwidth of the migrated images in particular for the low wavenumber components