rayleigh marchenko redatuming for target oriented true amplitude imaging abstractmarchenko redatuming is a revolutionary technique to estimate green s functions from virtual sources in the subsurface using only data measured at the earth s surface without having to place either sources or receivers in the subsurface this goal is achieved by crafting special wavefields so called focusing functions that can focus energy at a chosen point in the subsurface despite its great potential strict requirements on the reflection response such as knowledge and accurate deconvolution of the source wavelet including absolute scaling factor and co location of sources and receivers have so far challenged the application of marchenko redatuming to real world scenarios i combine the coupled marchenko equations with a one way version of the rayleigh integral representation to obtain a new redatuming scheme that handles internal as well as free surface multiples using dual sensor band limited seismic data with an unknown source signature from any acquisition system that presents arbit

a perspective on 3d surface related multiple elimination surface related multiple elimination srme is an algorithm that predicts all surface multiples by a convolutional process applied to seismic field data only minimal preprocessing is required once predicted the multiples are removed from the data by adaptive subtraction unlike other methods of multiple attenuation srme does not rely on assumptions or knowledge about the subsurface nor does it use event properties to discriminate between multiples and primaries in exchange for this freedom from the subsurface srme requires knowledge of the acquisition wavelet and a dense spatial distribution of sources and receivers although a 2d version of srme sometimes suffices most field data sets require 3d srme for accurate multiple prediction all implementations of 3d srme face a serious challenge the sparse spatial distribution of sources and receivers available in typical seismic field data sets does not conform to the algorithmic requirements there are several approaches to implementing 3d srme that address the data sparseness problem among those approaches are pre srme data interpolation on the fly data interpolation zero azimuth srme and true azimuth srme field data examples confirm that 1 multiples predicted using true azimuth 3d srme are more accurate than those using zero azimuth 3d srme and 2 on the fly interpolation produces excellent results

renormalization group and asymptotics of solutions of nonlinear parabolic equations we present a general method for studying long time asymptotics of nonlinear parabolic partial differential equations the method does not rely on a priori estimates such as the maximum principle it applies to systems of coupled equations to boundary conditions at infinity creating a front and to higher possibly fractional differential linear terms we present in detail the analysis for nonlinear diffusion type equations with initial data falling off at infinity and also for data interpolating between two different stationary solutions at infinity in an accompanying paper 5 the method is applied to systems of equations where some variables are slaved such as the complex ginzburg landau equation c 1994 john wiley sons inc

a m isometric operators in semi hilbertian spaces abstract in this work the concept of m isometry on a hilbert space are generalized when an additional semi inner product is considered this new concept is described by means of oblique projections

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

identification of image artifacts from internal multiples first order internal multiples are a source of coherent noise in seismic images because they do not satisfy the single scattering assumption fundamental to most seismic processing there are a number of techniques to estimate internal multiples in data in many cases these algorithms leave some residual multiple energy in the data this energy produces artifacts in the image and the location of these artifacts is unknown because the multiples wereestimatedinthedatabeforetheimagewasformed toavoid this problem we propose a method by which the artifacts caused by internal multiples are estimated directly in the image we use ideas from the generalized bremmer series and the lippmannschwinger scattering series to create a forward scattering series to model multiples and an inverse scattering series to describe

data driven wavefield focusing and imaging with multidimensional deconvolution numerical examples for reflection data with internal multiples standard imaging techniques rely on the single scattering assumption this requires that the recorded data do not include internal multiples i e waves that have bounced multiple times between reflectors before reaching the receivers at the acquisition surface when multiple reflections are present in the data standard imaging algorithms incorrectly image them as ghost reflectors these artifacts can mislead interpreters in locating potential hydrocarbon reservoirs recently we introduced a new approach for retrieving the green s function recorded at the acquisition surface due to a virtual source located at depth we refer to this approach as data driven wavefield focusing additionally after applying source receiver reciprocity this approach allowed us to decompose the green s function at a virtual receiver at depth in its downgoing and upgoing components these wavefields were then used to create a ghost free image of the medium with either crosscorrelation or multidimensional deconvolution presenting an advantage over standard prestack migration we tested the robustness of our approach when an erroneous background velocity model is used to estimate the first arriving waves which are a required input for the datadriven wavefield focusing process we tested the new method with a numerical example based on a modification of the amoco model

diffusive stability of oscillations in reaction diffusion systems we study nonlinear stability of spatially homogeneous oscillations in reaction diffusion systems assuming absence of unstable linear modes and linear diffusive behavior for the neutral phase we prove that spatially localized perturbations decay algebraically with the diffusive rate t n 2 in space dimension n we also compute the leading order term in the asymptotic expansion of the solution and show that it corresponds to a spatially localized modulation of the phase our approach is based on a normal form transformation in the kinetics ode which partially decouples the phase equation at the expense of making the whole system quasilinear stability is then obtained by a global fixed point argument in temporally weighted sobolev spaces corresponding author arnd scheel

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

pacemakers in large arrays of oscillators with nonlocal coupling we model pacemaker effects of an algebraically localized heterogeneity in a 1 dimensional array of oscillators with nonlocal coupling we assume the oscillators obey simple phase dynamics and that the array is large enough so that it can be approximated by a continuous nonlocal evolution equation we concentrate on the case of heterogeneities with positive average and show that steady solutions to the nonlocal problem exist in particular we show that these heterogeneities act as a wave source this effect is not possible in 3 dimensional systems such as the complex ginzburg landau equation where the wavenumber of weak sources decays at infinity to obtain our results we use a series of isomorphisms to relate the nonlocal problem to the viscous eikonal equation we then use fredholm properties of the laplace operator in kondratiev spaces to obtain solutions to the eikonal equation and by extension to the nonlocal problem

generic instability of spatial unfoldings of almost homoclinic periodic orbits we consider spatially homogeneous time periodic solutions of general partial differential equations we prove that when such a solution is close enough to a homoclinic orbit or a homoclinic bifurcation for the differential equation governing the spatially homogeneous solutions of the pde then it is generically unstable with respect to large wavelength perturbations moreover the instability is of one of the two following types either the well known kuramoto phase instability corresponding to a floquet multiplier becoming larger than 1 or a fundamentally different kind of instability occurring with a period doubling at an intrinsic finite wavelength and corresponding to a floquet multiplier becoming smaller than 1

solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time abstract we establish maximal regularity of type lp for a parabolic evolution equation u t a t u t f t with a c 0 t l d a 0 x and construct the corresponding evolution family on the underlying banach space x our proofs are based on the operator sum method and the use of evolution semigroups the results are applied to parabolic partial differential equations with continuous coefficients

estimation of multiple scattering by iterative inversion part i theoretical considerations a review has been given of the surface related multiple problem by making use of the so called feedback model from the resulting equations it has been concluded that the proposed solution does not require any properties of the subsurface however source detector and reflectivity properties of the surface need be specified those properties have been quantified in a surface operator and this operator is estimated as part of the multiple removal problem the surface related multiple removal algorithm has been formulated in terms of a neumann series and in terms of an iterative equation the neumann formulation requires a nonlinear optimization process for the surface operator while the iterative formulation needs a number of linear optimizations the iterative formulation also has the advantage that it can be integrated easily with another multiple removal method an algorithm for the removal of internal multiples has been proposed as well this algorithm is an extension of the surface related method removal of internal multiples requires knowledge of the macro velocity model between the surface and the upper boundary of the multiple generating layer in part ii also published in this issue the success of the proposed algorithms has been demonstrated on numerical experiments and field data examples

partial isometries in semi hilbertian spaces in this work the concepts of isometry unitary and partial isometry on a hilbert space are generalized when an additional semi inner product is considered these new concepts are described by means of oblique projections

on the nonlinear stability of plane waves for the ginzburg landau equation i consider the nonlinear stability of plane wave solutions to a ginzburg landau equation with additional fifth order terms and cubic terms containing spatial derivatives i show that under the constraint that the diffusion coefficient be real these waves are stable furthermore it is shown that the radial component of the perturbation decays at a faster rate than the phase component of the perturbation as t the result is also applicable to the classical ginzburg landau equation 1994 john wiley sons inc

renormalization group and the ginzburg landau equation we use renormalization group methods to prove detailed long time asymptotics for the solutions of the ginzburg landau equations with initial data approaching asx different spiraling stationary solutions a universal pattern is formed depending only on this asymptotics at spatial infinity

solutions without phase slip for the ginsburg landau equation we consider the ginsburg landau equation for a complex scalar field in one dimension and consider initial data which have two different stationary solutions as their limits in space asx if these solutions are not very different then we show that the initial data will evolve to a stationary solution by a phase melting process which avoids phase slips i e which does not go through zero amplitude

relating source receiver interferometry to an inverse scattering series to derive a new method to estimate internal multiples we have evaluated an explicit relationship between the representations of internal multiples by source receiver interferometry and an inverse scattering series this provides a new insight into the interaction of different terms in each of these internal multiple prediction equations and explains why amplitudes of estimated multiples are typically incorrect a downside of the existing representations is that their computational cost is extremely high which can be a precluding factor especially in 3d applications using our insight from source receiver interferometry we have developed an alternative computationally more efficient way to predict internal multiples the new formula is based on crosscorrelation and convolution two operations that are computationally cheap and routinely used in interferometric methods we have compared the results of the standard and the alternative formulas qualitatively in terms of the constructed wavefields and quantitatively in terms of the computational cost using examples from a synthetic data set

internal multiple prediction and removal using marchenko autofocusing and seismic interferometry standard seismic processing steps such as velocity analysis and reverse time migration imaging usually assume that all reflections are primaries multiples represent a source of coherent noise and must be suppressed to avoid imaging artifacts many suppression methods are relatively ineffective for internal multiples we show how to predict and remove internal multiples using marchenko autofocusing and seismic interferometry we first show how internal multiples can theoretically be reconstructed in convolutional interferometry by combining purely reflected up and downgoing green s functions from virtual sources in the subsurface we then generate the relevant up and downgoing wavefields at virtual sources along discrete subsurface boundaries using autofocusing then we convolve purely scattered components of up and downgoing green s functions to reconstruct only the internal multiple field which is adaptively subtracted from the measured data crucially this is all possible without detailed modeled information about the earth s subsurface the method only requires surface reflection data and estimates of direct nonreflected arrivals between subsurface virtual sources and the acquisition surface the method is demostrated on a stratified synclinal model and shown to be particularly robust against errors in the reference velocity model used

quasi isometries associated to a contractions abstract given two operators a and t a 0 a 1 on a hilbert space h satisfying t a t a we study the maximum subspace of h which reduces m a 1 2 t to a quasi isometry that is on which the equality m m m 2 m 2 holds in some cases this subspace coincides with the maximum subspace which reduces m to a normal partial isometry for example when a t t and in particular if t is a cohyponormal contraction in this case the corresponding subspace can be completely described in terms of asymptotic limit of the contraction t when m is quasinormal and m m a then the former above quoted subspace reduces to the kernel of a a 2 the case of an arbitrary contraction or particularly of a partial isometry is also considered for which we completely describe the reducing quasi isometric part as well as the invariant quasi isometric part

partial isometries and the conjecture of c k fong and s k tsui abstract we investigate some bounded linear operators t on a hilbert space which satisfy the condition t re t we describe the maximum invariant subspace for a contraction t on which t is a partial isometry to obtain that in certain cases the above condition ensures that t is self adjoint in other words we show that the fong tsui conjecture holds for partial isometries contractive quasi isometries or 2 quasi isometries and brownian isometries of positive covariance or even for a more general class of operators

diffusive mixing of stable states in the ginzburg landau equation the ginzburg landau equation on the real line has spatially periodic steady states of the form with and for we construct solutions which converge for all t 0 to the limiting pattern as these solutions are stable with respect to sufficiently small perturbations and behave asymptotically in time like where is uniquely determined by the boundary conditions this extends a previous result of brk92 by removing the assumption that should be close to zero the existence of the limiting profile is obtained as an application of the theory of monotone operators and the long time behavior of our solutions is controlled by rewriting the system in scaling variables and using energy estimates involving an exponentially growing damping term

modeling of wave propagation in inhomogeneous media we present a methodology providing a new perspective on modeling and inversion of wave propagation satisfying time reversal invariance and reciprocity in generally inhomogeneous media the approach relies on a representation theorem of the wave equation to express the green function between points in the interior as an integral over the response in those points due to sources on a surface surrounding the medium following a predictable initial computational effort green s functions between arbitrary points in the medium can be computed as needed using a simple cross correlation algorithm

adaptive multiple subtraction using regularized nonstationary regression stationary regression is the backbone of seismic data processing algorithms including match filtering which is commonly applied for adaptive multiple subtraction however the assumption of stationarity is not always adequate for describing seismic signals i have developed a general method of nonstationary regression and that applies to nonstationary match filtering the key idea is the use of shaping regularization to constrain the variability of nonstationary regression coefficients simple computational experiments demonstrate advantages of shaping regularization over classic tikhonov s regularization including a more intuitive selection of parameters and a faster iterative convergence using benchmark synthetic data examples i have successfully applied this method to the problem of adaptive subtraction of multiple reflections

diffusive stability of spatial periodic solutions of the swift hohenberg equation we are interested in the nonlinear stability of the eckhaus stable equilibria of the swift hohenberg equation on the infinite line with respect to small integrable perturbations the difficulty in showing stability for these stationary solutions comes from the fact that their linearizations possess continuous spectrum up to zero the nonlinear stability problem is treated with renormalization theory which allows us to show that the nonlinear terms are irrelevant i e that the fully nonlinear problem behaves asymptotically as the linearized one which is under a diffusive regime

criteria for the stability of spatial extensions of fixed points and periodic orbits of differential equations in dimension 2 abstract we consider partial differential equations that can be viewed as spatial extensions of two dimensional differential equations with respect to a coupling matrix we provide analytical and geometrical criteria governing the stability of spatially homogeneous stationary solutions and the phase stability of spatially homogeneous time periodic solutions we distinguish cases where the differential equation has a conservative or a dissipative behavior the geometrical criteria roughly speaking relate to the sense of rotation of on one hand the coupling matrix and on the other the flow of the differential equation around the spatially homogeneous solution

autofocus imaging image reconstruction based on inverse scattering theory conventional imaging algorithms assume single scattering and therefore cannot image multiply scattered waves correctly the multiply scattered events in the data are imaged at incorrect locations resulting in spurious subsurface structures and erroneous interpretation this drawback of current migration imaging algorithms is especially problematic for regions where illumination is poor e g subsalt in which the spurious events can mask true structure here we discuss an imaging technique that not only images primaries but also internal multiples accurately using only surfacereflection data and direct arrivals we generate the up and down going wavefields at every image point in the subsurface an imaging condition is applied to these up and downgoing wavefields directly to generate the image because the above algorithm is based on inverse scattering theory the reconstructed wavefields are accurate and contain multiply scattered energy in addition to the primary event as corroborated by our synthetic examples imaging of these multiply scattered energy helps eliminate spurious reflectors in the image other advantages of this imaging algorithm over existing imaging algorithms include more accurate amplitudes target oriented imaging and a highly parallelizable algorithm

adaptive overburden elimination with the multidimensional marchenko equation abstractiterative substitution of the multidimensional marchenko equation has been introduced recently to integrate internal multiple reflections in the seismic imaging process in so called marchenko imaging a macro velocity model of the subsurface is required to meet this objective the model is used to back propagate the data during the first iteration and to truncate integrals in time during all successive iterations in case of an erroneous model the image will be blurred akin to conventional imaging and artifacts may arise from inaccurate integral truncations however the scheme is still successful in removing artifacts from internal multiple reflections inspired by these observations we rewrote the marchenko equation such that it can be applied early in a processing flow without the need of a macro velocity model instead we have required an estimate of the two way traveltime surface of a selected horizon in the subsurface we have introduced an approximation such that adaptive subtracti

implementation of the marchenko method the marchenko method makes it possible to compute subsurface to surface green s functions from reflection measurements at the surface applications of the marchenko method have already been discussed in many papers but its implementation aspects have not yet been discussed in detail solving the marchenko equation is an inverse problem the marchenko method computes a solution of the marchenko equation by an adaptive iterative scheme or by a direct inversion we have evaluated the iterative implementation based on a neumann series which is considered to be the conventional scheme at each iteration of this scheme a convolution in time and an integration in space are performed between a so called focusing update function and the reflection response in addition by applying a time window one obtains an update which becomes the input for the next iteration in each iteration upgoing and downgoing focusing functions are updated with these terms after convergence of the scheme the resulting upgoing and downgoing focusing functions are used to compute the upgoing and downgoing green s functions with a virtual source position in the subsurface and receivers at the surface we have evaluated this algorithm in detail and developed an implementation that reproduces our examples the software fits into the seismic unix software suite of the colorado school of mines

geometric stability analysis for periodic solutions of the swift hohenberg equation in this paper we describe invariant geometrical structures in the phase space of the swift hohenberg equation in a neighborhood of its periodic stationary states we show that in spite of the fact that these states are only marginally stable i e the linearized problem about these states has continuous spectrum extending all the way up to zero there exist finite dimensional invariant manifolds in the phase space of this equation which determine the long time behavior of solutions near these stationary solutions in particular using this point of view we obtain a new demonstration of schneider s recent proof that these states are nonlinearly stable