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

predictor corrector halley method for nonlinear equations abstract in this paper we suggest and analyze a new two step predictor corrector type iterative methods for solving nonlinear equations of the type f x 0 by using the technique of updating the solution this method can be viewed as a predictor corrector iterative halley s method we also consider the convergence analysis of the proposed method to illustrate the efficiency of this new method we give several examples and a comparison with the method of grau and diaz barrero m grau j l diaz barrero an improvement to ostrowski root finding method appl math comput 173 2006 450 456 is given this method can be considered as a refinement and improvement of the halley method

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

fast acquisition aperture correction in prestack depth migration using beamlet decomposition wave equation based acquisition aperture correction in the local angle domain can improve image amplitude significantly in prestack depth migration however its original implementation is inefficient because the wavefield decomposition uses the local slant stack lss which is demanding computationally we propose a faster method to obtain the image and amplitude correction factor in the local angle domain using beamlet decomposition in the local wavenumber domain for a given frequency the image matrix in the local wavenumber domain for all shots can be calculated efficiently we then transform the shot summed image matrix from the local wavenumber domain to the local angle domain lad the lad amplitude correction factor can be obtained with a similar strategy having a calculated image and correction factor one can apply similar acquisition aperture corrections to the original lss based method for the new implementation we compare the accuracy and efficiency of two beamlet decompositions gabor daubechies frame gdf and local exponential frame lef with both decompositions our method produces results similar to the original lss based method however our method can be more than twice as fast as lss and cost only twice the computation time of traditional one way wave equation based migrations the results from gdf decomposition are superior to those from lef decomposition in terms of artifacts although gdf requires a little more computing time

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 widely convergent generalized pulse spectrum technique for the inversion of two dimensional acoustic wave equation an algorithm of the generalized pulse spectrum technique gpst is used for the inverse problems of two dimensional acoustic wave equation and a skill of saving computational cost is designed in choosing regularizing parameters to overcome the defect of the local convergence of the gpst a widely convergent gpst wcgpst is constructed numerical simulations are carried out to test the feasibility and to study the general characteristics of the wcgpst it is found that the wcgpst is not only as robust as the standard gpst but also possessing widely convergent region

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

a computer implementation of 2 5 d common shot inversion the computer implementation of a two and one half dimensional 2 5 d constant density prestack inversion formalism with laterally and depth dependent background propagation speed is a kirchhoff type inversion summing data from a line of receivers over traveltime curves in the depth dependent background medium with weights determined from born kirchhoff inversion theory this theory predicts that the output will be a reflector map with peak amplitudes on each reflector being in known proportion to the angularly dependent geometrical optics reflection coefficient the 2 5 d feature provides for out of plane spreading correction consistent with the prescribed background medium the method is applied to a synthetic data set and to a physically modeled data set generated at the seismic acoustic laboratory the graphical output demonstrates the validity of the formalism as a kirchhoff migration parameter estimation for the synthetic data confirmed the theory parameter estimation for the experimental data wa

variable background born inversion by wavefield backpropagation the inverse scattering problem for an acoustic medium is formulated by using the variable background born approximation a constant density acoustic medium is probed by a wide band point source and the scattered field is observed along a curved receiver array located outside the region where the medium velocity is different from the assumed background velocity function the solution proposed relies on the introduction of a backpropagated field this field is obtained by using a finite difference scheme backward in time to backpropagate into the medium the scattered field observed along the receiver array the backpropagated field is imaged at the source travel times giving an image of the same type as obtained by reverse time finite difference migration techniques the gradient of this image is then taken along rays linking the source to points in the medium and after scaling this gives the reconstructed potential to relate the reconstructed potential to the true scattering potential high frequency

hierarchical clustering problems and analysis of fuzzy proximity relation on granular space in this paper based on granular space some hierarchical clustering problems and analysis for fuzzy proximity relation are developed by using rigorous mathematical descriptions and four results are obtained first the granular representation of a fuzzy proximity relation is studied an algorithm is obtained to compute the granular space or hierarchical clustering structure and the min transitive closure derived from the same fuzzy proximity relation on a finite universe and the consistent clustering properties are discussed second a consistency index that is based on granular space is developed and a global optimizing mathematical model is established to obtain the optimal clustering of a fuzzy proximity relation furthermore for given two fuzzy proximity relations on different granulations of the same universe a collaborative clustering method is studied by the intersection operation and this method can provide a better hierarchical clustering structure finally the theory for hierarchical clustering analysis of fuzzy proximity relations on granular space is established for given two fuzzy proximity relations on the same universe two sufficient conditions theorems 6 1 and 6 2 of isomorphism are given and the necessary condition theorem 6 3 and the sufficient condition theorem 6 4 for e similarity are obtained we also illustrate the effectiveness of these theories and methods with some practical examples these results may help form a comprehensive theory and methodology for related potential application and will help us to obtain a deeper understanding of the essence of hierarchical clustering procedures

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

on extended fuzzy relational model with proximity relations in this paper shenoi and melton s model of fuzzy relational database is considered proximity relation being an important tool of the model is characterized theoretically an optimal distribution in 0 1 and consequently the domain partition are defined and studied with examples on fuzzy queries

on a steffensen hermite method of order three in this paper we study a third order steffensen type method obtained by controlling the interpolation nodes in the hermite inverse interpolation polynomial of degree 2 we study the convergence of the iterative method and we provide new convergence conditions which lead to bilateral approximations for the solution it is known that the bilateral approximations have the advantage of offering a posteriori bounds of the errors the numerical examples confirm the advantage of considering these error bounds

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

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

on iterative methods for nonlinear equations several iterative methods have been suggested and analyzed for solving nonlinear equation f x 0 all these methods can be classified as one step and two step methods in this paper we have suggested and analyzed several two step methods for solving nonlinear equations we have also shown that these two step methods are quadratically convergent several numerical examples are given to illustrate this comparison our results can be viewed as important refinement and improvement of the previously known results

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

velocity inversion using a stratified reference we present an inversion algorithm for backscattered stacked seismic data which will reconstruct the velocity profile in realistic earth conditions the basic approach follows that of the original constant reference speed cohen and bleistein paper 1979a in that hig hfrequency asymptotics and perturbation methods are used however we use a reference speed which may vary with depth and this greatly enhances the validity of the perturbation assumption and hence the inversion results the new algorithm enjoys the same economies and stability properties of the original algorithm making it very competitive with current migration schemes four major assumptions are made 1 the acoustic wave equation is an adequate model 2 stacked data have amplitude information worth preserving fairly accurately 3 the actual reflectivity coefficients can be adequately modeled as perturbations from a continuous reference velocity which depends only on the depth variable and 4 the subsurface can be adequately

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

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

a new modified halley method without second derivatives for nonlinear equation in a recent paper noor and noor k inayat noor m aslam noor predictor corrector halley method for nonlinear equations appl math comput in press doi 10 1016 j amc 11 023 have suggested and analyzed a predictor corrector method halley method for solving nonlinear equations in this paper we modified this method by using the finite difference scheme which has a quintic convergence we have compared this modified halley method with some other iterative of fifth orders convergence methods which shows that this new method is a robust one several examples are given to illustrate the efficiency and the performance of this new method

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

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

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

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

composite boundary valued solution of the 2 5 d green s function for arbitrary acoustic media theoretically the green s function can be used to calculate the wavefield response of a specified source and the frechet derivative with respect to the model parameters for crosshole seismic full waveform inversion in this paper we apply the finite element method to numerically compute the 2 5 d green s function for an arbitrary acoustic medium by solving a composite boundary valued problem in the wavenumber frequency domain the composite boundary condition consists of a 2 5 d absorbing boundary condition for the propagating wave field and a mixed boundary condition for the evanescent field in inhomogeneous media modeling a numerical experiment performed for a uniform earth having a known exact solution shows the accuracy of the computation in the frequency and time domain an inhomogeneous medium test involving an embedded low velocity layer demonstrates that the permissible range of ky at each frequency can be determined rationally from the critical wavenumber value of the medium around the sou

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

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

concepts of probabilistic sets abstract in the field of pattern recognition or decision making theory the important subjects are as follows 1 ambiguity of property of objects 2 variety of character of objects 3 subjectivity of observers 4 evolution of knowledge of observers i e learning considering these points the concept of probabilistic set is proposed it is based on both probability theory and fuzzy concepts a probabilistic set on a total space is defined by a point wise measurable function from a parameter space which is a probability space to a characteristic space which is a measurable space it is shown that the family of all probabilistic sets constitutes a complete pseudo boolean algebra moment analysis is possible by using a probability measure of the parameter space other useful concepts are also mentioned such as probabilistic mappings and expected cardinal numbers

an improvement to ostrowski root finding method an improvement to the iterative method based on the ostrowski one to compute nonlinear equation solutions which increases the local order of convergence is suggested the adaptation of a strategy presented here gives a new iteration function with an additional evaluation of the function it also shows a smaller cost if we use adaptive multi precision arithmetic the numerical results computed using this system with a floating point system representing 200 decimal digits support this theory

imaging method utilizing attenuation and speed parameters in inverse scattering techniques methods for imaging the internal structures of an object using an acoustic wave field are provided in one aspect for example a method of imaging internals of a physical object using acoustic waves may include transmitting an acoustic wave field toward the object receiving a resultant acoustic wave field with a receiver where the resultant acoustic wave field is in response to the transmitted acoustic wave field reflected from or transmitted through the object and determining a predicted resultant acoustic wave field derived from a model of the object the method may also include determining a residual between the predicted resultant acoustic wave field and the resultant acoustic wave field and back propagating the residual to determine corrections to the model of the object in another aspect the above recited steps may be further iterated to successively refine the model of the object over a number of iterations until a predefined condition is reached

an iterative method with cubic convergence for nonlinear equations in this paper we suggest and analyze a new three step iterative method for solving nonlinear equations we show that this new iterative method has third order convergence several numerical examples are given to illustrate the efficiency and performance of this new method new method can be viewed as an improvement of the previously known iterative methods

an analytical approach to migration velocity analysis prestack depth migration provides a powerful tool for velocity analysis in complex media both prominent approaches to velocity analysis depth focusing analysis and residual curvature analysis rely on approximate formulas to update velocity generally these formulas are derived under the assumptions of horizontal reflector lateral velocity homogeneity or small offset therefore the conventional methods for updating velocity lack accuracy and computational efficiency when velocity has large lateral variations here based on ray theory i find the analytic representation for the derivative of imaged depths with respect to migration velocity this derivative function characterizes a general relationship between residual moveout and residual velocity using the derivative function and the perturbation method i derive a new formula to update velocity from residual moveout in the derivation i impose no limitation on offset dip or velocity distribution consequently i revise the residual curvature a

efficient 2 5 d true amplitude migration kirchhoff depth migration is a widely used algorithm for imaging seismic data in both two and three dimensions to perform the summation at the heart of the algorithm standard kirchhoff migration requires a traveltime map for each source and receiver true amplitude kirchhoff migration in 2 5 d x z media additionally requires maps of amplitudes out of plane spreading factors and takeoff angles these quantities are necessary for calculating the true amplitude weight term in the summation the increased input output i o and computational expense of including the true amplitude weight term is often not justified by significant improvement in the final muted and stacked image for this reason some authors advocate neglecting the weight term in the kirchhoff summation entirely for most everyday imaging purposes we demonstrate that for nearly the same expense as ignoring the weight term a much better solution is possible we first approximate the true amplitude weight term by the weight term for co

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

i v fuzzy relations and sanchez s approach for medical diagnosis abstract this paper introduces the concept of i v fuzzy relations i v fuzzy tolerance relations composition of i v fuzzy relations etc several useful propositions are proved finally sanchez s approach for medical diagnosis is considered and discussed using the idea of i v fuzzy relations

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

a new iteration method for solving algebraic equations in this paper a nonlinear algebraic equation f x 0 by applying taylor s theorem is converted into a coupled iteration systeml x n 1 g x n 0andg x n f x n l x n wherel x ax 2 bx c the formula is of high convergence some examples are given

iterative methods improving newton s method by the decomposition method in this paper we present a sequence of iterative methods improving newton s method for solving nonlinear equations the adomian decomposition method is applied to an equivalent coupled system to construct the sequence of the methods whose order of convergence increases as it progresses the orders of convergence are derived analytically and then rederived by applying symbolic computation of maple some numerical illustrations are given

born theory of wave equation dip moveout wave equation dip moveout dmo addresses the dmo amplitude problem of finding an algorithm which faithfully preserves angular reflectivity while processing data to zero offset only three fundamentally different theoretical approaches to the dmo amplitude problem have been proposed 1 mathematical decomposition of a prestack migration operator 2 intuitively accounting for specific amplitude factors and 3 cascading operators for prestack migration or inversion and zero offset forward modeling pursuing the cascaded operator method wave equation dmo for shot profiles has been developed in this approach a prestack common shot inversion operator is combined with a zero offset modeling operator both integral operators are theoretically based on the born asymptotic solution to the point source scalar wave equation this total process termed born dmo simultaneously accomplishes geometric spreading corrections nmo and dmo in an amplitude preserving manner the theory is for constant velocity a

on fuzzy equivalence i fuzzy reflexive symmetric and transitive relations on fuzzy subsets are studied various types of reflexivities and corresponding equivalences are considered the possible fuzzy partitioning of a fuzzy subset is investigated

extending the fuzzy database with fuzzy numbers abstract the fuzzy relational database model originated by the authors permits fuzzy domain values from a discrete finite universe the model is extended here by demonstrating that fuzzy numbers may be employed as domain values without loss of consistency with respect to representation or the relational algebra where equivalence is required in an ordinary relational database similarity is employed in a fuzzy relational database for discrete finite universes similarity between atomic elements is described via a fuzzy similarity relation with max min transitivity two or more fuzzy numbers are defined to be similar if their union forms a continuous level set over the real line this convention effects the partitioning of fuzzy number domains that is necessary to assure the well definedness of the fuzzy relational algebra

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

fifth order iterative methods for solving nonlinear equations abstract in this paper we suggest and analyze a new two step iterative method for solving nonlinear equation f x 0 by rewriting the given nonlinear equation as a coupled system of equations and using the taylor series it is shown that this new iterative method is of fifth order several examples are given to illustrate its performance and efficiency comparison with other methods is also given this method can be viewed as an improvement of the halley s method

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

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

crosshole seismic inversion with normalized full waveform amplitude data we investigate a simple scheme for full waveform amplitude spectrum inversion of crosshole seismic data with an unknown source wavelet the method is based on our 2d 2 5d finite element method of acoustic wave modeling the normalized amplitude data defined as the spectral ratio of the original trace amplitude to the average amplitude for the entire common shot gather are used for full waveform inversion in the frequency domain in essence the normalization of amplitudes removes the source wavelet and is easily carried out in the time domain or frequency domain from crosshole seismic surveying two synthetic models simulating respectively a dipping fracture model and a complicated sedimentary structure are examined with the inversion scheme the numerical results show that clear images of the targets can be obtained by using the single and multiple frequency data for comparison two other amplitude inversion schemes in which the source signatures are known and estimated with an iterative procedure were carried out for the same models the results show that the iteratively source estimated procedure also produces satisfactory images of the velocity structure and yields an approximate amplitude of the source wavelet with the multifrequency data three such schemes yield very competitive results

some higher order modifications of newton s method for solving nonlinear equations in this paper we consider constructing some higher order modifications of newton s method for solving nonlinear equations which increase the order of convergence of existing iterative methods by one or two or three units this construction can be applied to any iteration formula and per iteration the resulting methods add only one additional function evaluation to increase the order some illustrative examples are provided and several numerical results are given to show the performance of the presented methods

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

studies in fuzzy relations over fuzzy subsets in this paper a generalisation of fuzzy relations is introduced fuzzy relations are defined on fuzzy subsets properties like ordered reflexivity symmetry transitity transitive closures of such generalised relations and operations on them are discussed

asymptotic elastic green s tensor the green s function for the acoustic wave equation has been an essential ingredient in obtaining frequency inversion formulas in the acoustic limit cohen and bleistein 1979 clayton and stolt 1981 beylkin 1985 cohen et al 1986 bleistein et al 1987 similarly the green s tensor is required for inversions based on the elastic wave equations indeed kuo and dai 1984 and dai and kuo 1986 have already used an approximation to this tensor to derive a migration result for two component seismic data in the case of a homogeneous medium

proximity relations in the fuzzy relational database model abstract the fuzzy relational model of buckles and petry is a rigorous scheme for incorporating non ideal or fuzzy information in a relational database in addition to providing a consistent scheme for representing fuzzy information in the relational structure the model possesses two critical properties that hold for classical relational databases these properties are that no two tuples have identical interpretations and each relational operation has a unique result the fuzzy relational model relies on similarity relations for each scalar domain in the fuzzy database these relations are reflexive symmetric and max min transitive in addition to introducing fuzziness into the relational model each similarity relation induces equivalence classes in its domain it is the existence of these equivalence classes that provides the model with the important properties possessed by classical relational databases in this paper we extend the fuzzy relational database model of buckles and petry to deal with proximity relations for scalar domains since reflexivity and symmetry are the only constraints placed on proximity relations they generalize the notion of similarity relations we show that it is possible to induce equivalence classes from proximity relations thus the nice properties of the fuzzy relational model of buckles and petry are preserved furthermore the removal of the max min transitivity restriction also provides database users with more freedom to express their value structures

improving newton raphson method for nonlinear equations by modified adomian decomposition method in this paper we present some efficient numerical algorithms for solving nonlinear equations based on newton raphson method the modified adomian decomposition method is applied to construct the numerical algorithms some numerical illustrations are given to show the efficiency of algorithms

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

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 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

an efficient hybrid method for solving systems of nonlinear equations this paper focuses on methods of solving systems of nonlinear equations that require fewer computations than those using regular methods the proposed method presents a new technique to approximate the jacobian matrix needed in the process which still retains the good features of newton s type methods but which can reduce the amount of work

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

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

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

modified iterative methods with cubic convergence for solving nonlinear equations abstract in this paper we suggest and analyze two new predictor corrector methods for solving nonlinear equations f x 0 these methods can be considered as two step methods we show that these methods have cubic convergence using this one of the new methods real or complex roots for certain nonlinear equations can be obtained several numerical examples are given to illustrate the efficiency and performance of the new methods our method can be considered as an improvement of the existing methods and can be viewed as an alternative to the existing methods

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

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

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

approximate analytical formulas for kirchhoff migration operator two important tools in seismic processing kirchhoff migration and demigration operators are the basis for many imaging problems solution due to high numerical and computational requirements the use of those tools for three dimensions are very computation costly this fact has motivated us to investigate kirchhoff migration operations for simpler types of media in order to provide faster results to be used as an approximation for more realistic media to obtain results with lower computational effort a convenient environment is the so called 2 5d situation i e considering 3 d wave propagation in a medium that does not vary in the horizontal direction perpendicular to the seismic line in this case 2 d ray tracing is sufficient to describe the 3 d propagation effects particularly geometric spreading in a medium where the parameters depend only on the depth component 1 d situation the imaging operations only require the solution of semi analytical integrals which can be both precisely and immediately implemented for some particular cases of vertical velocity distributions approximate analytical formulas are devised for migration stacking lines and weight functions several imaging algorithms present very efficient computational performance by using those models thus it is possible to establish a set of cases which may be useful for validating the implementation of more complex situations

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

ray kirchhoff migration in inhomogeneous media a kirchhoff method that avoids possible singularities on the surface of integration and is more accurate than previous kirchhoff methods has been developed for seismic migration in laterally inhomogeneous media it is based on a newly derived integral solution to the acoustic wave equation this solution indicates that wave fields in an inhomogeneous medium can be expressed as a summation of ray solutions determined by the transport and extended eikonal equations the extended eikonal equation is in turn solved by an asymptotic series for implementation a perturbation scheme is developed for solving the ray and transport equations in addition to computing higher order terms of the asymptotic series this scheme can be used to avoid most of the ray tracing required for wave extrapolation in a medium where vertical variations may be large but lateral variations are small compared to velocity itself where an analytic ray solution for the reference velocity used in the perturbation scheme exists ray kirchhoff migration in such a medium can be carried out without numerical ray tracing

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

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

a fifth order iterative method for solving nonlinear equations in this paper we present a new fifth order method for solving nonlinear equations per iteration the new method requires two function and two first derivative evaluations it is shown that the new method is fifth order convergent several numerical examples are given to illustrate the performance of the presented method

some developments in general variational inequalities general variational inequalities provide us with a unified natural novel and simple framework to study a wide class of equilibrium problems arising in pure and applied sciences in this paper we present a number of new and known numerical techniques for solving general variational inequalities using various techniques including projection wiener hopf equations updating the solution auxiliary principle inertial proximal penalty function dynamical system and well posedness we also consider the local and global uniqueness of the solution and sensitivity analysis of the general variational inequalities as well as the finite convergence of the projection type algorithms our proofs of convergence are very simple as compared with other methods our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems since the general variational inequalities include quasi variational inequalities and quasi implicit complementarity problems as special cases results presented here continue to hold for these problems several open problems have been suggested for further research in these areas

a note on fifth order iterative methods for solving nonlinear equations abstract in noor and noor 2006 m a noor k i noor fifth order iterative methods for solving nonlinear equations appl math comput in press doi 10 1016 j amc 2006 10 007 a new two step iterative method with fifth order for solving nonlinear equations has been proposed in this note we show that the iterative method given by the authors has third order convergence not the fifth order one as claimed in their work

migration velocity analysis theory and an iterative algorithm imaging complex structures inside the earth requires reasonable velocities that can be provided by applying prestack depth migration to multichannel seismic data migration velocity analysis is based on the principle that the images in the migrated data will be distorted when an erroneous velocity is used and the difference of the imaged depths residual moveout at a common image gather is a measure of the error in the velocity the imaging equations that we derive from snell s law describe a general quantitative relationship between migration images and migration velocity based on the imaging equations we analyze properties of common image gathers and derive analytical formulas to represent residual moveout in some cases these formulas show what factors affect the sensitivity of velocity analysis which is useful to assess errors involved in velocity estimates in addition we develop a simple iteration algorithm to correct the layer velocities from residual moveout the algorithm presented here is applicable to a medium that consists of constant velocity layers separated by arbitrary smooth interfaces some computer implementations are presented for both synthetic data and physical tank data they demonstrate the effectiveness of our velocity analysis approach

how to make t transitive a proximity relation three ways to approximate a proximity relation r i e a reflexive and symmetric fuzzy relation by a t transitive one where t is a continuous archimedean t norm are given the first one aggregates the transitive closure r macr of r with a maximal t transitive relation b contained in r the second one computes the closest homotecy of r macr or b to better fit their entries with the ones of r the third method uses nonlinear programming techniques to obtain the best approximation with respect to the euclidean distance for t the lukasiewicz or the product t norm the previous methods do not apply for the minimum t norm an algorithm to approximate a given proximity relation by a min transitive relation a similarity is given in the last section of the paper