adaptive surface related multiple elimination the major amount of multiple energy in seismic data is related to the large reflectivity of the surface a method is proposed for the elimination of all surface related multiples by means of a process that removes the influence of the surface reflectivity from the data an important property of the proposed multiple elimination process is that no knowledge of the subsurface is required on the other hand the source signature and the surface reflectivity do need to be provided as a consequence the proposed process has been implemented adaptively meaning that multiple elimination is designed as an inversion process where the source and surface reflectivity properties are estimated and where the multiple free data equals the inversion residue results on simulated data and field data show that the proposed multiple elimination process should be considered as one of the key inversion steps in stepwise seismic inversion

wavenumber based filtering of marine point source data in seismic processing plane wave decomposition has played a fundamental role serving as a basis for developing sophisticated processing techniques valid for depth dependent models by comparing analytical expressions for the decomposed wavefields we review several processing algorithms of interest for the geophysicist the algorithms may be applied to marine point source data acquired over a horizontally layered viscoelastic and anisotropic medium the plane wave decomposition is based on the fourier transform integral for line source data and the hankel transform integral for point source data the involved wavenumber integrals of the cosine or bessel function type are difficult to evaluate accurately and efficiently however a number of the processing techniques can easily be run as a filtering operation in the spatial domain without transforming to the wavenumber domain the mathematical expressions for the spatial filters are derived using plane wave analysis with numerical examples we demonstra

adaptive multiple subtraction with wavelet based complex unary wiener filters abstractadaptive subtraction is a key element in predictive multiple suppression methods it minimizes misalignments and amplitude differences between modeled and actual multiples and thus reduces multiple contamination in the data set after subtraction due to the high crosscorrelation between their waveforms the main challenge resides in attenuating multiples without distorting primaries as they overlap on a wide frequency range we split this wide band problem into a set of more tractable narrow band filter designs using a 1d complex wavelet frame this decomposition enables a single pass adaptive subtraction via complex single sample unary wiener filters consistently estimated on overlapping windows in a complex wavelet transformed domain each unary filter compensates for amplitude differences within its frequency support and can correct small and large misalignment errors through phase and integer delay corrections this approach greatly simplifies the matching filter estimation and despit

an inverse scattering series method for attenuating multiples in seismic reflection data we present a multidimensional multiple attenuation method that does not require any subsurface information for either surface or internal multiples to derive these algorithms we start with a scattering theory description of seismic data we then introduce and develop several new theoretical concepts concerning the fundamental nature of and the relationship between forward and inverse scattering these include 1 the idea that the inversion process can be viewed as a series of steps each with a specific task 2 the realization that the inverse scattering series provides an opportunity for separating out subseries with specific and useful tasks 3 the recognition that these task specific subseries can have different and more favorable data requirements convergence and stability conditions than does the original complete inverse series and most importantly 4 the development of the first method for physically interpreting the contribution that individual terms and pieces of terms in the inv

the second pinching theorem for hypersurfaces with constant mean curvature in a sphere we generalize the second pinching theorem for minimal hypersurfaces in a sphere due to peng terng wei xu zhang and ding xin to the case of hypersurfaces with small constant mean curvature let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exist two positive constants gamma n and delta n depending only on n such that if h le gamma n and beta n h le s le beta n h delta n then s equiv beta n h and m is one of the following cases i s k big sqrt frac k n big times s n k big sqrt frac n k n big 1 le k le n 1 ii s 1 big frac 1 sqrt 1 mu 2 big times s n 1 big frac mu sqrt 1 mu 2 big here beta n h n frac n 3 2 n 1 h 2 frac n n 2 2 n 1 sqrt n 2h 4 4 n 1 h 2 and mu frac n h sqrt n 2h 2 4 n 1 2

interferometry by deconvolution part 1 theory for acoustic waves and numerical examples interferometry allows for synthesis of data recorded at any two receivers into waves that propagate between these receivers as if one of them behaves as a source this is accomplished typically by crosscorrelations based on perturbation theory and representation theorems we show that interferometry also can be done by deconvolutions for arbitrary media and multidimensional experiments this is important for interferometry applications in which 1 excitation is a complicated source time function and or 2 when wavefield separation methods are used along with interferometry to retrieve specific arrivals unlike using crosscorrelations this method yields only causal scattered waves that propagate between the receivers we offer a physical interpretation of deconvolution interferometry based on scattering theory here we show that deconvolution interferometry in acoustic media imposes an extra boundary condition which we refer to as the free point or clamped point boundary condition depending on the measured field quantity this boundary condition generates so called free point scattering interactions which are described in detail the extra boundary condition and its associated artifacts can be circumvented by separating the reference waves from scattered wavefields prior to interferometry three wavefield separation methods that can be used in interferometry are direct wave interferometry dual field interferometry and shot domain separation each has different objectives and requirements

isotropic angle domain elastic reverse time migration multicomponent data usually are not processed with specifically designed procedures but with procedures analogous to those used for single component data in isotropic media the vertical and horizontal components of the data commonly are taken as proxies for the p and s wave modes which are imaged independently with the acoustic wave equations this procedure works only if the vertical and horizontal components accurately represent p and s wave modes which generally is not true therefore multicomponent images constructed with this procedure exhibit artifacts caused by incorrect wave mode separation at the surface an alternative procedure for elastic imaging uses the full vector fields for wavefield reconstruction and imaging thewavefieldsarereconstructedusingthemulticomponentdata as a boundary condition for a numerical solution to the elastic wave equation the key component for wavefield migration is theimagingcondition whichevaluatesthematchbetweenwavefields reconstructed from sources and receivers for vector wave fields a simple component by component crosscorrelation between two wavefields leads to artifacts caused by crosstalk between the unseparated wave modes we can separate elastic wavefields after reconstruction in the subsurface and implement theimagingconditionascrosscorrelationofpurewavemodesinstead of the cartesian components of the displacement wavefield thisapproachleadstoimagesthatareeasiertointerpretbecause they describe reflectivity of specified wave modes at interfaces of physical properties as for imaging with acoustic wavefields the elastic imaging condition can be formulated conventionally crosscorrelation with zero lag in space and time and extendedtononzerospaceandtimelags theelasticimagesproduced by an extended imaging condition can be used for angle decomposition of primary pp or ss and converted ps or sp reflectivity angle gathers constructed with this procedure have applicationsformigrationvelocityanalysisandamplitude variation with angleanalysis

wave equation processing using finite difference propagators part 1 wavefield dissection and imaging of marine multicomponent seismic data abstractmethods for wavefield injection are used in for instance reverse time extrapolation of shot gathers in reverse time migration for correct injection of recorded data without any ambiguity of the propagation direction the wavefield injection methodology requires pressure and particle velocity data such as multicomponent towed marine or seabed seismic recordings we discovered that by carefully considering the models medium parameters and boundary conditions for injection wavefield injection of multicomponent data can also be used to solve several long standing challenges in marine seismic data processing by means of conventional time space domain finite difference propagators we outlined and demonstrated several of these important applications including up down separation of wavefields deghosting direct wave removal source signature estimation multiple removal and imaging using primaries and multiples only acoustic models are considered but the concepts are straightforward to general

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

green s function representations for seismic interferometry the term seismic interferometry refers to the principle of generating new seismic responses by crosscorrelating seismic observations at different receiver locations the first version of this principle was derived by claerbout 1968 who showed that the reflection response of a horizontally layered medium can be synthesized from the autocorrelation of its transmission response for an arbitrary 3d inhomogeneous lossless medium it follows from rayleigh s reciprocity theorem and the principle of time reversal invariance that the acoustic green s function between any two points in the medium can be represented by an integral of crosscorrelations of wavefield observations at those two points the integral is along sources on an arbitrarily shaped surface enclosing these points no assumptions are made with respect to the diffusivity of the wavefield the rayleigh betti reciprocity theorem leads to a similar representation of the elastodynamic green s function when a part of the enclosing surface is the earth s free surface the integral needs only to be evaluated over the remaining part of the closed surface in practice not all sources are equally important the main contributions to the reconstructed green s function come from sources at stationary points when the sources emit transient signals a shaping filter can be applied to correct for the differences in source wavelets when the sources are uncorrelated noise sources the representation simplifies to a direct crosscorrelation of wavefield observations at two points similar as in methods that retrieve green s functions from diffuse wavefields in disordered media or in finite media with an irregular bounding surface

source deghosting by depth apparition abstractmarine seismic data are distorted by ghosts as waves propagating upward reflect downward from the sea surface ghosts appear on the source side and the receiver side however whereas the receiver side ghost problem has been studied in detail and many different solutions have been proposed and implemented commercially the source side ghost problem has remained largely unsolved with few satisfactory solutions available we have developed a new and simple method to remove sea surface ghosts that is related to the recently introduced concept of signal apparition as opposed to the temporal spatial source signature modulation functions used in the original signal apparition theory our source deghosting method relies on using sources at different depths but not at the same lateral positions the new method promises to be particularly suitable for 3d applications on sparse or incomplete acquisition geometries

n dimensional area of minimal rotational hypersurfaces in spheres abstract let m r n denote the set of compact minimal rotational hypersurfaces in the n 1 dimensional sphere s n 1 for any compact hypersurface m s n 1 let us denote by m its n dimensional area in this paper we show that for any integer n 2 the set of numbers a n m m m r n is a discrete set we also show that if m m r 3 then either m s 3 or m m 0 3 or m m 0 3 where m 0 n s n 1 n 1 n s 1 1 n finally for 2 n 100 we show numerically evidence that if m m r n then either m s n or m m 0 n or m 2 m 0 n we strongly believe that for any m m r n either m s n or m m 0 n or m 2 m 0 n for all n

an effective linear approximation method for separable programming problems among the numerous applications of piecewise linearization methods include data fitting network analysis logistics and statistics in the early 1950s a concave function was found to be able to be linearized by introducing 0 1 variables most textbooks in operations research offer such methods for expressing linear approximations various methods of linearization have also been developed in recent literature nevertheless the transformed linear scheme has a severe shortcoming most standard procedures for linearizing typically involve a large increase in the number of binary variables consequently the gains to be derived from dealing with linear functions are quite likely to be nullified by the increase in the size of the problem conventional methods for linearizing a concave function with m break points require m 1 binary variables however when m becomes large the computation will be very time consuming and may cause a heavy computational burden this study proposes an effective approach in which only log 2 m 1 binary variables are used the proposed method has the following features i it offers more convenient and efficient means of expressing a piecewise linear function ii fewer 0 1 variables are used iii the computational results show that the proposed method is much more efficient and faster than the conventional one especially when the number of break points becomes large

full waveform inversion based on kaniadakis statistics full waveform inversion fwi is a wave equation based methodology to estimate the subsurface physical parameters that honor the geologic structures classically fwi is formulated as a local optimization problem in which the misfit function to be minimized is based on the least squares distance between the observed data and the modeled data residuals or errors from a probabilistic maximum likelihood viewpoint the minimization of the least squares distance assumes a gaussian distribution for the residuals which obeys gauss s error law however in real situations the error is seldom gaussian and therefore it is necessary to explore alternative misfit functions based on non gaussian error laws in this way starting from the ensuremath kappa generalized exponential function we propose a misfit function based on the ensuremath kappa generalized gaussian probability distribution associated with the kaniadakis statistics or ensuremath kappa statistics which we call ensuremath kappa fwi in this study we perform numerical simulations on a realistic acoustic velocity model considering two noisy data scenarios in the first one we considered gaussian noisy data while in the second one we considered realistic noisy data with outliers the results show that the ensuremath kappa fwi outperforms the least squares fwi providing better parameter estimation of the subsurface especially in situations where the seismic data are very noisy and with outliers independently of the ensuremath kappa parameter although the ensuremath kappa parameter does not affect the quality of the results it is important for the fast convergence of fwi

on seismic deghosting using green s theorem abstractwe have examined theoretically how receiver side deghosting of pressure measurements can be derived from the green s theorem method we split the green s function that obeys dirichlet boundary conditions on the sea surface and at the receiver plane into two contributions the first emitting energy downward only from its source location and the other emitting energy only upward using the normal derivative of the source side downgoing green s function in the green s theorem evaluation over the receiver plane the upgoing part of the pressure field is predicted this is the receiver side deghosted field by inserting the source side upgoing normal derivative green s function in green s theorem its evaluation over the receiver plane predicts the downgoing part of the pressure field for a plane horizontal receiver surface the required green s function can be derived using the image series expansion method to display the fundamental frequencies of this green s function we have applied a fourier se

rough sea deghosting of streamer seismic data using pressure gradient approximations a new method is presented for seismic deghosting of towed streamer data acquired in rough seas the deghosting scheme combines pressure recordings along one or several cables with an estimate of the vertical pressure gradient or the vertical component of the particle velocity the estimation of the vertical pressure gradient requires continuous elevation measurements of the wave height directly above the receivers the vertical pressure gradient estimate is obtained by spatially weighting the pressure field each spatial weight generally is the product of two weight functions the first is a function of partial derivatives acting solely along the horizontal cartesian coordinates it can be implemented by finite difference or fourier derivative operations the second is a function of the vertical cartesian coordinate and accounts for the varying sea state this weight can be changed from one receiver to the next making the deghosting a local process integrated with the measured pressure field the esti

exact wave field simulation for finite volume scattering problems an exact boundary condition is presented for scattering problems involving spatially limited perturbations of arbitrary magnitude to a background model in generally inhomogeneous acoustic media the boundary condition decouples the wave propagation on a perturbed domain while maintaining all interactions with the background model thus eliminating the need to regenerate the wave field response on the full model the method which is explicit relies on a kirchhoff type integral extrapolation to update the boundary condition at every time step of the simulation the green s functions required for extrapolation through the background model are computed efficiently using wave field interferometry

the gauss kronecker curvature of minimal hypersurfaces in four dimensional space forms let mathbb q 4 c be a four dimensional space form of constant curvature c in this paper we show that the infimum of the absolute value of the gauss kronecker curvature of a complete minimal hypersurface in mathbb q 4 c c leq 0 whose ricci curvature is bounded from below is equal to zero further we study the connected minimal hypersurfaces m 3 of a space form mathbb q 4 c with constant gauss kronecker curvature k for the case c 0 we prove by a local argument that if k is constant then k must be equal to zero we also present a classification of complete minimal hypersurfaces of mathbb q 4 c with k constant

a new characterization of the clifford torus via scalar curvature pinching abstract let m n be a compact hypersurface with constant mean curvature h in s n 1 denote by s the squared norm of the second fundamental form of m we prove that there exists an explicit positive constant n depending only on n such that if h n and n h s n h n 23 then s n h and m is one of the following cases i s k k n s n k n k n 1 k n 1 ii s 1 1 1 2 s n 1 1 2 here n h n n 3 2 n 1 h 2 n n 2 2 n 1 n 2 h 4 4 n 1 h 2 and n h n 2 h 2 4 n 1 2 for n 6 12 24 we give examples to show that the assumption h n cannot be removed

crossline wavefield reconstruction from multicomponent streamer data part 2 joint interpolation and 3d up down separation by generalized matching pursuit computation of the 3d upgoing downgoing separated wavefield at any desired position within a marine streamer spread is enabled by multicomponent streamers that can measure the crossline and vertical components of water particle motion in addition to the pressure we introduce the concept of simultaneous interpolation and deghosting and describe a new technique generalized matching pursuit gmp to achieve this this method is based on the matching pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions in the gmp method the basis functions describing the unknown 3d upgoing wavefield are filtered by appropriate forward ghost operators before being matched to the multicomponent measurements as a data dependent method gmp can operate on data samples that are highly aliased in the crossline direction without relying on assumptions about seismic events such as linearity the technique is naturally suitable for data with only a small number of samples that may be irregularly spaced we demonstrate the efficacy and robustness of the gmp method on several synthetic data sets of increasing complexity and in the presence of noise

ancient solutions to mean curvature flow for isoparametric submanifolds mean curvature flow for isoparametric submanifolds in euclidean spaces and spheres was studied in liu and terng duke math j 147 1 157 179 2009 in this paper we will show that all these solutions are ancient solutions and study their limits as time goes to negative infinity we also discuss rigidity of ancient mean curvature flows for hypersurfaces in spheres and its relation to the chern s conjecture on the norm of the second fundamental forms of minimal hypersurfaces in spheres

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

gap theorems for minimal submanifolds of a hyperbolic space abstract this paper provides some gap theorems for complete immersed minimal submanifolds of dimension no less than five in a hyperbolic space namely we show that an n 5 dimensional complete immersed minimal submanifold m in a hyperbolic space is totally geodesic if the l 2 norm of a on geodesic balls centered at some point p m has less than quadratic growth and if either sup x m a 2 x is not too large or the l n norm of a on m is small here a is the second fundamental form of m

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

true amplitude vector acoustic imaging application of gaussian beams abstractacoustic gaussian beam migration is an attractive imaging method because it is flexible with input geometry efficient and accurate in imaging multipath arrivals however one of the hurdles that this method must overcome in production processing is its extension to use multimeasurement data as recently allowed by novel acquisition technologies this is inevitable when the compensation of the ghost effect is best corrected within a true amplitude imaging process a necessity for amplitude variation with offset work for this purpose i introduced a novel formalism for vector acoustic imaging based on green s function theory which can remove the ghost effect and produce amplitudes on reflectors that are proportional to the reflection coefficients i established a theoretical framework with gaussian beam representations of green s functions including the weighted beam stacking approach that reduced the cost of computation i extended my formulas to use the steep descent i e stationary phase

crossline wavefield reconstruction from multicomponent streamer data part 1 multichannel interpolation by matching pursuit mimap using pressure and its crossline gradient we introduce a technique that uses multicomponent seismic measurements to reconstruct the seismic wavefield at any desired crossline position between towed streamers this method called multichannel interpolation by matching pursuit mimap operates on pressure and crossline particle motion measurements it is based on the matching pursuit technique and iteratively reconstructs the signal as a combination of optimal basis functions being a data dependent technique mimap can interpolate severely aliased data without assumptions about seismic events such as linearity or the model related to the seismic wavefield mimap has the capability to perform well in the presence of irregular sampling and is robust when only a small number of samples are available using synthetic data examples we show that the new method has the potential to interpolate signals that are sampled at realistic crossline streamer spacing and in the presence of noise

nonlinear extended images via image domain interferometry wave equation finite frequency imaging and inversion still face many challenges in addressing the inversion of highly complexvelocitymodelsaswellasindealingwithnonlinearimaging e g migration of multiples amplitude preserving migration extended images eis are particularly important for designing image domain objective functions aimed at addressing standing issues in seismic imaging such as two way migration velocity inversion or imaging inversion using multiples general oneand two way representations for scattered wavefields can describe and analyze eis obtained in wave equation imaging we have developed a formulation that explicitly connects the wavefield correlations done in seismic imaging with the theory and practiceofseismicinterferometry inlightofthisconnection we define eis as locally scattered fields reconstructed by model dependent image domain interferometry because they incorporate the same one and two way scattering representations used for seismic interferometry the reciprocity based eis can in principle account for all possible nonlinear effects in the imaging process i e migrationofmultiplesandamplitudecorrections in this case the practice of two way imaging departs considerably from the one way approach we have studied the differences betweentheseapproachesinthecontextofnonlinearimaging analyzingthedifferencesinthewavefieldextrapolationstepsaswell as in imposing the extended imaging conditions when invoking single scatteringeffectsandignoringamplitudeeffectsingenerating eis the one and two way approaches become essentially the same as those used in today s migration practice with the straightforwardadditionofspaceandtimelagsinthecorrelationbased imaging condition our formal description of the eis and theinsightthattheyarescatteredfieldsintheimagedomainmay beusefulinfurtherdevelopmentofimagingandinversionmethodsinthecontextoflinear migration basedvelocityinversionor inmoresophisticatedimage domainnonlinearinversescattering approaches

a dual ascent procedure for large scale uncapacitated network design the fixed charge network design problem arises in a variety of problem contexts including transportation communication and production scheduling we develop a family of dual ascent algorithms for this problem this approach generalizes known ascent procedures for solving shortest path plant location steiner network and directed spanning tree problems our computational results for several classes of test problems with up to 500 integer and 1 98 million continuous variables and constraints show that the dual ascent procedure and an associated drop add heuristic generate solutions that in almost all cases are guaranteed to be within 1 to 4 of optimality moreover the procedure requires no more than 150 seconds on an ibm 3083 computer the test problems correspond to dense and sparse networks including some models that arise in freight transport

a comparison of mixed integer programming models for nonconvex piecewise linear cost minimization problems we study a generic minimization problem with separable nonconvex piecewise linear costs showing that the linear programming lp relaxation of three textbook mixed integer programming formulations each approximates the cost function by its lower convex envelope we also show a relationship between this result and classical lagrangian duality theory

depth migration by the gaussian beam summation method seismic depth migration aims to produce an image of seismic reflection interfaces ray methods are suitable for subsurface target oriented imaging and are less costly compared to two way wave equation based migration but break down in cases when a complex velocity structure gives rise to the appearance of caustics ray methods also have difficulties in correctly handling the different branches of the wavefront that result from wave propagation through a caustic on the other hand migration methods based on the two way wave equation referred to as reverse time migration are known to be capable of dealing with these problems however they are very expensive especially in the 3d case it can be prohibitive if many iterations are needed such as for velocity model building our method relies on the calculation of the green functions for the classical wave equation by per forming a summation of gaussian beams for the direct and back propagated wavefields the subsurface image is obtained by cal culating

fast l p solution of large sparse linear systems application to seismic travel time tomography least squares inversion tends to be sensitive to small changes in the assumptions in the statistical lexicon it is not robust least squares solutions of noisy ill conditioned systems often exhibit unphysical high frequency oscillations which have to be damped or smoothed in some fashion the iteratively reweighted least squares irls algorithm provides a means of computing approximate lp solutions 1 p we will show that irls can be combined with the preconditioned conjugate gradient algorithm in order to solve large sparse rectangular systems of linear algebraic equations very efficiently further for p 1 the resulting algorithm is extremely stable it seems to give reasonably smooth solutions without minute adjustment of the parameters and does not become unstable if an overly optimistic error estimate is used it also has the ability to reject or at least significantly diminish the influence of outliers in the data and does not require the use of damping in addition we will show how to efficiently include diagonal weighting of the observations and parameters in order to incorporate a priori information into the inversion this combined inversion algorithm has wide applicability in real world inverse theory parameter estimation filtering etc where data are noisy and not uniformly reliable numerical examples of the l1 inversion of noisy seismic travel time data will be shown

approximate solutions of acoustic 3d integral equation and their application to seismic modeling and full waveform inversion abstract over the recent decades a number of fast approximate solutions of lippmann schwinger equation which are more accurate than classic born and rytov approximations were proposed in the field of electromagnetic modeling those developments could be naturally extended to acoustic and elastic fields however until recently they were almost unknown in seismology this paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media we evaluated the numerical merits of those methods and provide an estimation of their numerical complexity in our numerical realization we use the matrix free implementation of the corresponding integral operator we study the accuracy of those approximate solutions and demonstrate that the quasi analytical approximation is more accurate than the born approximation further we apply the quasi analytical approximation to the solution of the inverse problem it is demonstrated that this approach improves the estimation of the data gradient comparing to the born approximation the developed inversion algorithm is based on the conjugate gradient type optimization numerical model study demonstrates that the quasi analytical solution significantly reduces computation time of the seismic full waveform inversion we also show how the quasi analytical approximation can be extended to the case of elastic wavefield

migration inversion for incident waves synthesized from common shot data gathers wavefield synthesis is a process for producing reflection responses from more general sources or from prescribed incident waves by combining common shot data gathers synthesis can provide surveywide data sets similar in that regard to common offset data gathers but with the added advantage that each synthesized data set is a solution to a single wave equation a common offset data set does not have this last feature thus synthesized data sets can be processed by true amplitude wave equation migration the output is then known to be true amplitude in the same sense as is the output of kirchhoff inversion that is the peak amplitude is proportional to the ray theoretic reflection coefficient at a determinable specular incidence angle multiplied bythe area under the frequency domain source signature and scaled by 1 2 alternatively the kirchhoff inversion of synthesized data has a beylkin determinant that is expressed in terms of the ray theoretic green s function amplitude this is in contrast to 3

on chern s conjecture for minimal hypersurfaces and rigidity of self shrinkers using the generalized yau parameter method and the sylvester theory we verify that if m is a compact minimal hypersurface in sn 1 whose squared length of the second fundamental form satisfies 0 a 2 n n22 then a 2 n and m is a clifford torus moreover we prove that if m is a complete self shrinker with polynomial volume growth in rn 1 and if the squared length of the second fundamental form of m satisfies 0 a 2 1 121 then a 2 1 and m is a round sphere or a cylinder our results improve the rigidity theorems due to ding and xin 21 22

greedy facility location algorithms analyzed using dual fitting with factor revealing lp in this article we will formalize the method of dual fitting and the idea of factor revealing lp this combination is used to design and analyze two greedy algorithms for the metric uncapacitated facility location problem their approximation factors are 1 861 and 1 61 with running times of o m log m and o n3 respectively where n is the total number of vertices and m is the number of edges in the underlying complete bipartite graph between cities and facilities the algorithms are used to improve recent results for several variants of the problem

solving the uncapacitated network design problem by a lagrangean heuristic and branch and bound the network design problem is a multicommodity minimal cost network flow problem with fixed costs on the arcs i e a structured linear mixed integer programming problem it has various applications such as construction of new links in transportation networks topological design of computer communication networks and planning of empty freight car transportation on railways we present a lagrangean heuristic within a branch and bound framework as a method for finding the exact optimal solution of the uncapacitated network design problem with single origins and destinations for each commodity the simplest problem in this class but still np hard the lagrangean heuristic uses a lagrangean relaxation as subproblem solving the lagrange dual with subgradient optimization combined with a primal heuristic the benders subproblem yielding primal feasible solutions computational tests on problems of various sizes up to 1000 arcs 70 nodes and 138 commodities or 40 nodes and 600 commodities and of several different structures lead to the conclusion that the method is quite powerful outperforming for example a state of the art mixed integer code both with respect to problem size and solution time

robust iterative inversion for the one dimensional acoustic wave equation seismic inversion can be formulated by considering a linearized integral relation which is deduced from the wave equation this born inversion approach is equivalent to linear least squares inversion for a particular parameterization of the medium the least squares solution is a member of a family of generalized lp norm solutions which are deduced from a maximum likelihood formulation this formulation allows design of various statistical inversion solutions we present two iterative solutions to the one dimensional 1 d seismic inverse problem the iterative least squares ils and the iterative reweighted least squares irls methods the ils method involves solving a distorted background velocity problem after the initial least squares solution is obtained the irls method is used as a robust regression technique which is better suited for dealing with certain types of noise and is computationally faster than ils several numerical examples demonstrate that the irls method accurately estimates impeda

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

on seismic deghosting by spatial deconvolution abstractreceiver side deghosting can be derived and implemented in the frequency domain as spatial deterministic deconvolution of marine pressure recordings the deghosting deconvolution operator is found analytically as the inverse fourier transform of the wavenumber domain wave equation deghosting function for a sea surface reflection coefficient of 1 the wavenumber domain deghosting function has well known poles at fundamental frequencies equal to an integer multiple of a function of the receiver depth and the plane wave dip angle relative to the depth axis the first singularity is always at 0 hz the spatial deghosting operator has singularities at fundamental frequencies equal to an integer multiple of a function of the receiver depth independent of its lateral coordinates the first singularity is again at 0 hz in addition the deghosting operator that is applied to 3d data has singularity when its lateral coordinate is zero a simple numerical example demonstrates the method

a mobius rigidity of compact willmore hypersurfaces in sn 1 abstract let x m n s n 1 be an immersed hypersurface without umbilical point then one can define the mobius metric g the mobius second fundamental form b and the blaschke tensor a on the hypersurface m n which are invariant under the mobius transformation group of s n 1 a hypersurface is called a willmore hypersurface if it is the critical point of the volume functional of m n with respect to the mobius metric g in this paper we prove that if the hypersurface x is a compact willmore hypersurface without umbilical point then m n a b 2 n 1 n d m g 0 the equality holds if and only if the hypersurface m n is mobius equivalent to one of the willmore tori w n m s m n m n s n m m n s n 1 1 m n 1 where the tensor a a b 2 1 n t r a n 1 n g

ddvv type inequality for skew symmetric matrices and simons type inequality for riemannian submersions abstract in this paper we will first derive a ddvv type optimal inequality for real skew symmetric matrices then we apply it to establish a simons type integral inequality for riemannian submersions with totally geodesic fibres and yang mills horizontal distributions in this way we show phenomenons of duality between submanifold geometry and riemannian submersion particularly between second fundamental form of a submanifold and integrability tensor of a riemannian submersion

closed willmore minimal hypersurfaces with constant scalar curvature in s5 1 are isoparametric abstract in this paper we will prove that any closed minimal willmore hypersurface m 4 of s 5 with constant scalar curvature must be isoparametric to be precise m 4 is either an equatorial 4 sphere a product of sphere s 2 2 2 s 2 2 2 or a cartan s minimal hypersurface in particular the value of the second fundamental form s can only be 0 4 12 this result strongly supports chern s conjecture

kirchhoff scattering series insight into the multiple attenuation method there are two basic integral equations used to represent wavefields in theoretical seismology the lippmann schwinger integral equation and the representation theorem the born scattering series currently used for attenuating free surface multiples has been derived from the lippmann schwinger integral equation similarly we have used the representation theorem here to derive a kirchhoff scattering series for attenuating free surface multiples in towed streamer data the kirchhoff series for attenuating free surface multiples is in theory equivalent to the born series most important like the born series it does not require any knowledge of the subsurface however it still provides useful insight into the multiple attenuation methods because the form of some quantities involved in the kirchhoff series is different from the form in the born series for example in towed streamer seismic data the kirchhoff series requires measurements of the vertical derivative of the pressure field in addition to the

on chernʼs problem for rigidity of minimal hypersurfaces in the spheres abstract for a compact minimal hypersurface m in s n 1 with the squared length of the second fundamental form s we confirm that there exists a positive constant n depending only on n such that if n s n n then s n i e m is a clifford minimal hypersurface in particular when n 6 the pinching constant n n 23

prestack gaussian beam depth migration kirchhoff migration is the most popular method of three dimensional prestack depth migration because of its flexibility and efficiency its effectiveness can become limited however when complex velocity structure causes multipathing of seismic energy an alternative is gaussian beam migration which is an extension of kirchhoff migration that overcomes many of the problems caused by multipathing unlike first arrival and most energetic arrival methods which retain only one traveltime this alternative method retains most arrivals by the superposition of gaussian beams this paper presents a prestack gaussian beam migration method that operates on common offset gathers the method is efficient because the computation of beam superposition isolates summations that do not depend on the seismic data and evaluates these integrals by considering their saddle points gaussian beam migration of the two dimensional marmousi test data set demonstrates the method s effectiveness for structural imaging in a case w

on seismic deghosting using integral representation for the wave equation use of green s functions with neumann or dirichlet boundary conditions abstractthe recent interest in broadband seismic technology has spurred research into new and improved seismic deghosting solutions one starting point for deriving deghosting methods is the representation theorem which is an integral representation for the wave equation recent research results show that by using green s functions with dirichlet boundary conditions in the representation theorem source side deghosting of already receiver side deghosted wavefields can be achieved we found that the choice of green s functions with neumann boundary conditions on the sea surface and the plane that contains the sources leads to an identical but simpler solution with fewer processing steps in addition we found that pressure data can be receiver side deghosted by introducing green s functions with dirichlet boundary conditions on the sea surface and the plane containing the receivers into a modified representation theorem the deghosting methods derived from the representation theorem are wave theoretic alg

spatial filtering in ambient noise interferometry theoretically the empirical green s function between a pair of receivers can be extracted from the cross correlation of the received diffuse noise the diffuse noise condition rarely is met in the ocean and directional sources may bias the green s function here matrix based spatial filters are used for removing unwanted contributions in the cross correlations two methods are used for solving the matrix filter design problem first a matrix least square problem is solved with a low rank approximation of the pseudo inverse here derived for linear and planar arrays second a convex optimization approach is used to solve the design problem reformulated with ad hoc constraints the spatial filter is applied to real data cross correlations of elements from a linear array to attenuate the contribution of a discrete interferer in the case of a planar array and simulated data a spatial filter enables a passive upgoing downgoing wavefield separation along with an efficient rejection of horizontally propagating noise the impact of array size and frequency band on the filtered cross correlations is discussed

wave equation processing using finite difference propagators part 2 deghosting of marine hydrophone seismic data abstractwe have developed a new and simple method for deghosting of conventional hydrophone streamer data towed at arbitrary variable depths the method uses a time space domain finite difference fd solution to the wave equation with pressure field boundary conditions to predict and remove ghosts because it operates in the time domain our method is unaffected by any number of notches in the frequency spectrum of the data and therefore will deghost through notches apart from the acquired hydrophone data the method relies on the depth profile of the streamer recording the data beneath a sea surface with a known reflection coefficient as well as the propagation velocity in the water above the streamer the method was applied to simple and more complex synthetic data which illustrated its ability to deal with complex data and any acquisition geometry

global minimization of large scale constrained concave quadratic problems by separable programming the global minimization of a large scale linearly constrained concave quadratic problem is considered the concave quadratic part of the objective function is given in terms of the nonlinear variablesx r n while the linear part is in terms ofy r k for large scale problems we may havek much larger thann the original problem is reduced to an equivalent separable problem by solving a multiple cost row linear program with 2n cost rows the solution of one additional linear program gives an incumbent vertex which is a candidate for the global minimum and also gives a bound on the relative error in the function value of this incumbent ana priori bound on this relative error is obtained which is shown to be 0 25 in important cases if the incumbent is not a satisfactory approximation to the global minimum a guaranteede approximate solution is obtained by solving a single liner zero one mixed integer programming problem this integer problem is formulated by a simple piecewise linear underestimation of the separable problem

scalar curvature of minimal hypersurfaces in a sphere abstract we improve the well known scalar curvature pinching theorem due to peng terng for n n 5 dimensional minimal hypersurfaces to the case of arbitrary n precisely if m is a closed and minimal hypersurface in a unit sphere s n 1 then there exists a positive constant n depending only on n such that if n s n n then s n i e m is a clifford torus s k k n s n k n k n k 1 2 n 1

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

inverse extrapolation of primary seismic waves forward wave field extrapolation operators simulate propagation effects from one depth level to another inverse wave field extrapolation operators eliminate those propagation effects since forward wave field extrapolation can be described in terms of spatial convolution inverse wave field extrapolation can be described in terms of spatial deconvolution a simple approximation to a stable spatially band limited deconvolution operator is obtained by taking the complex conjugate of the convolution operator a one way version of the kirchhoff integral that contains the conjugate complex green s function is derived unlike the situation with respect to the forward problem the modification of the closed surface integral into an open boundary integral involves an approximation that is identical to the approximation in the conjugate complex deconvolution approach this approximation neglects the evanescent field and causes a second order amplitude error for a plane acquisition surface the one way kirchhoff

comparison of the least squares criterion and the cauchy criterion in frequency wavenumber inversion one alternative to the least squares inversion tech nique is the use of a cauchy error criterion we show how inversion algorithms of the gauss newton type based on the least squares method can be modified to handle the cauchy norm a criterion for the lower bound of the scale parameter in the cauchy norm is given we compare the least squares and cauchy error criteria by inverting synthetic data corrupted by ran dom noise and weather noise the data are trans formed to the frequency wavenumber domain before the inversion starts the numerical examples show that the algorithm based on the cauchy criterion is more robust in the presence of the noise tested here per iteration the computer costs of the two algorithms are approximately the same

frequency wavenumber inversion of acoustic data we describe an inversion procedure for obtaining velocities densities and layer thicknesses from multioffset reflection seismograms generated by a point source excitation in a stratified acoustic medium the reference seismograms are decomposed into cylindrical waves in the frequency wavenumber domain where the inversion is performed by minimizing the difference between the reference data and modeled data using a least squares technique the modeling algorithm is based on the classical reflectivity method we include the effects of limited temporal and spatial apertures by applying a frequency and wavenumber dependent filter in the modeling process this inversion is suited for jacobian matrix methods where the first derivatives of the model generated data with respect to the model parameters are calculated from analytical expressions together with the modeling the numerical examples show that the inversion algorithm recovers with reasonable accuracy the parameters used to generate the reference data the low frequency trends of both the velocity and the density are restored the convergence of the iterative minimization algorithm is fast

exploiting special structures in constructing a hierarchy of relaxations for 0 1 mixed integer problems a new hierarchy of relaxations is presented that provides a unifying framework for constructing a spectrum of continuous relaxations spanning from the linear programming relaxation to the convex hull representation for linear mixed integer 0 1 problems this hierarchy is an extension of the reformulation linearization technique rlt of sherali and adams 1990 1994a and is particularly designed to exploit special structures specifically inherent special structures are exploited by identifying particular classes of multiplicative factors that are applied to the original problem to reformulate it as an equivalent polynomial programming problem and subsequently this resulting problem is linearized to produce a tighter relaxation in a higher dimensional space this general framework permits us to generate an explicit hierarchical sequence of tighter relaxations leading up to the convex hull representation a similar hierarchy can be constructed for polynomial mixed integer 0 1 problems additional ideas for further strengthening rlt based constraints by using conditional logical implications as well as relationships with sequential lifting are also explored several examples are presented to demonstrate how underlying special structures including generalized and variable upper bounding covering partitioning and packing constraints as well as sparsity can be exploited within this framework for some types of structures low level relaxations are exhibited to recover the convex hull of integer feasible solutions

optimal objective function approximation for separable convex quadratic programming we present an optimal piecewise linear approximation method for the objective function of separable convex quadratic programs the method provides guidelines on how many grid points to use and how to position them for a piecewise linear approximation if the error induced by the approximation is to be bounded a priori

a gap theorem of self shrinkers in this paper we study complete self shrinkers in euclidean space and prove that an n dimensional complete self shrinker with polynomial volume growth in euclidean space r n 1 is isometric to either r n s n n or r n m s m m 1 m n 1 if the squared norm s of the second fundamental form is constant and satisfies s 10 7

source receiver reverse time imaging of dual source vector acoustic seismic data abstractnovel technologies in seismic data acquisition allow for recording full vector acoustic va data pointwise recordings of pressure and its multicomponent gradient excited by pressure only as well as dipole gradient sources building on recent connections between imaging and seismic interferometry we present a wave equation based nonlinear reverse time imaging approach that takes full advantage of dual source multicomponent data the method s formulation relies on source receiver scattering reciprocity thus making proper use of va fields in the wavefield extrapolation and imaging condition steps in a self consistent manner the va imaging method is capable of simultaneously focusing energy from all in and outgoing waves the receiver side up and downgoing receiver ghosts fields are handled by the va receiver extrapolation whereas source side in and outgoing source ghosts arrivals are accounted for when combining dual source data at the imaging condition additionally va imaging handl

multidimensional signature deconvolution and free surface multiple elimination of marine multicomponent ocean bottom seismic data this paper presents a wave equation method for multidimensional signature deconvolution designature and elimination of free surface related multiples demultiple in four component 4c ocean bottom seismic data the designature demultiple method has the following characteristics it preserves primary amplitudes while attenuating free surface related multiples it requires no knowledge of the sea floor parameters and the subsurface it requires information only of the local density and acoustic wave propagation velocity just above the sea floor it accommodates source arrays and no information except location of the physical source array its volume and its radiation characteristics wavelet is required designature is an implicit part of the demultiple process hence the method is capable of transforming recorded reflection data excited by any source array below the sea surface into free surface demultipled data that would be recorded from a point source with any desired signature in addition the incident wavefield is not subtracted from the data prior to free surface demultiple hence separation of incident and scattered fields is not an issue as it is for most other free surface demultiple schemes the designature demultiple algorithm can be divided into two major computational steps first a multidimensional deconvolution operator inversely proportional to the time derivative of the downgoing part of the normal component of the particle velocity just above the sea floor is computed second an integral equation is solved to find any component of the designatured free surface demultipled multicomponent field when the geology is horizontally layered the designature and free surface demultiple scheme greatly simplifies and lends itself toward implementation in the p domain or frequency wavenumber domain as deterministic deconvolution of common shot gathers or common receiver gathers when source array variations are negligible