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

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

the coupled yang mills higgs flow in this paper we discuss some properties of the coupled yang mills higgs flow including the energy inequality bochner type inequality and monotonicity of certain quantities we also discuss the asymptotic behavior of a regular coupled yang mills higgs flow

a comparison of imaging conditions for wave equation shot profile migration the application of a deconvolution imaging condition in wave equation shot profile migration is important to provide illumination compensation and amplitude recovery particularly if the aim is to successfully recover a measure of the medium reflectivity an imaging condition that destroys amplitudes is unacceptable we study a set of imaging conditions with illumination compensation the imaging conditions are evaluated by the quality of the output amplitudes and artifacts produced in numerical experiments using a vertically inhomogeneous velocity model the best of all imaging conditions we tested is the one that divides the crosscorrelation of upgoing and downgoing wavefields by the autocorrelation of the downgoing wavefield also known as the illumination map in an application to marmousi data unconditional division by autocorrelation turned out to be unstable effective stabilization was achieved by smoothing the illumination map

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

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

compactification of the moduli spaces of vortices and coupled vortices vortices and coupled vortices arise from yang mills higgs theories and can be viewed as generalizations or analogues to yang mills connections and in particular hermitianyang mills connections we proved an analytic compactification of the moduli spaces

theory of true amplitude one way wave equations and true amplitude common shot migration one way wave operators are powerful tools for forward modeling and migration here we describe a recently developed true amplitude implementation of modified one way operators and present some numerical examples by true amplitude one way forward modeling we mean that the solutions are dynamically correct as well as kinematically correct that is ray theory applied to these equations yields the upward and downward traveling eikonal equations of the full wave equation and the amplitude satisfies the transport equation of the full wave equation the solutions of these equations are used in the standard wave equation migration imaging condition the boundary data for the downgoing wave is also modified from the one used in the classic theory because the latter data is not consistent with a point source for the full wave equation when the full wave form solutions are replaced by their ray theoretic approximations the imaging formula reduces to the common shot kirchhoff inversion formula in this sense

estimation of primaries and near offset reconstruction by sparse inversion marine data applications ost wave equation based multiple removal algorithms are based on prediction and subtraction of multiples especially for shallow water the prediction strongly relies on a correct interpolation of the missing near offsets the subtraction of predicted multiples from the data can easily lead to the distortion of primaries if primaries and multiples overlap recently a new approach for surface related multiple removal was proposed the estimation of primaries by sparse inversion epsi which is based on a full waveform inversion approach epsi is based on the same primary multiple model as surface related multiple elimination srme and does not require a subsurface model in contrast to srme epsi estimates the primaries as unknowns in a multidimensional inversion process rather than a subtraction process the multidimensional primary impulse responses are parameterized by band limited spikes which are estimated such that they along with their corresponding multiples match the input data an interesting aspect of the epsi method is that it produces a residual which is the part of the input data not explained by primaries and multiples this residual can be analyzed and may provide useful information on the primary estimation process furthermore it has been demonstrated that epsi is also capable of reconstructing the missing near offsets from the multiples the proposed method is applied to a field data set with moderate water depth where it is demonstrated that the results are comparable with srme this data set is used to illustrate the residual for a shallow water field data set it is shown that epsi gives a better result than the standard srme result caused by epsi s capability to reconstruct the missing near offsets

on l sup 1 optimal control it is shown that the problem of minimizing the effect of the worst disturbance on the output where the signals have finite one norms is equivalent to the same problem with bounded disturbances the following situations provide the motivation to study such a problem 1 disturbances are generated by a fixed bounded input bounded output bibo stable linear time invariant lti system with unknown initial conditions and zero input the output of this fixed system is the disturbance input to the original control system 2 disturbances are generated by an unknown bibo stable lti system whose fixed input consists of the unit impulse this is the dual of 1 in both of these situations the disturbances will have a finite one norm

feedback and optimal sensitivity model reference transformations multiplicative seminorms and approximate inverses in this paper the problem of sensitivity reduction by feedback is formulated as an optimization problem and separated from the problem of stabilization stable feedback schemes obtainable from a given plant are parameterized salient properties of sensitivity reducing schemes are derived and it is shown that plant uncertainty reduces the ability of feedback to reduce sensitivity the theory is developed for input output systems in a general setting of banach algebras and then specialized to a class of multivariable time invariant systems characterized by n times n matrices of h infty frequency response functions either with or without zeros in the right half plane the approach is based on the use of a weighted seminorm on the algebra of operators to measure sensitivity and on the concept of an approximate inverse approximate invertibility of the plant is shown to be a necessary and sufficient condition for sensitivity reduction an indicator of approximate invertibility called a measure of singularity is introduced the measure of singularity of a linear time invariant plant is shown to be determined by the location of its right half plane zeros in the absence of plant uncertainty the sensitivity to output disturbances can be reduced to an optimal value approaching the singularity measure in particular if there are no right half plane zeros sensitivity can be made arbitrarily small the feedback schemes used in the optimization of sensitivity resemble the lead lag networks of classical control design some of their properties and methods of constructing them in special cases are presented

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

nonlinear scattering based imaging in elastic media theory theorems and imaging conditions with the more widespread introduction of multicomponent recording devices in land and marine ocean bottom seismic acquisition elastic imaging may become mainstream in coming years we have derived new nonlinear elastic imaging conditions a correlation type representation theorem for perturbed elastic media commonly used in seismic interferometry to explain how a scattered wave response between two receivers sources may be predicted given a boundary of sources receivers can be considered as a starting point for the derivation here we use this theorem to derive and interpret imaging conditions for elastic migration by wavefield extrapolation e g elastic reverse time migration some approximations lead to a known heuristically derived imaging condition that crosscorrelates p and s wave potentials that are separated in the subsurface after full wavefield extrapolation this formal connection reveals that the nonapproximated correlation type representation theorem can be interpreted as a nonlinear imaging condition that accounts also for multiply scattered and multiply converted waves properly focusing such energy at each image point we present a synthetic data example using either an ideal acquisition on a full closed boundary or a real partial boundary seismic exploration survey and we demonstrate the importance of nonlinearities in pure and converted mode imaging in pp imaging they result in better illumination and artifact reduction whereas in ps imaging they show how zero time lag and zero space lag crosscorrelation imaging conditions are not ideal for imaging of converted mode waves because no conversion arises from zero offset experiments

shot profile true amplitude crosscorrelation imaging condition the u d imaging condition for shot profile migration can be used to estimate the angle dependent reflection coefficient but is difficult to implement numerically because of the spectral division involved most techniques for stabilizing the division require a damping factor which might be difficult to estimate and which also introduces bias into the final result a stable result can be achieved by approximating the imaging condition with a crosscorrelation of the up and downgoing wavefields at zero time lag but this will lead to incorrect amplitudeversus angle ava behavior of the estimated reflection coefficient we use a simple model for wave propagation of primary reflections in the wavenumber frequency domain and invert the model with respect to the reflection coefficient by using the properties of wavefield extrapolators it can then be shown that the reflection coefficients can be estimated by crosscorrelation of the upgoing wavefield and a downgoing wavefield where the initial wavefield is the inverse of the wavefield generated by a point source the new imaging condition gives the correct ava behavior for horizontal reflectors for dipping reflectors it is shown that a postmigration correction factor can be used to recover the correct angle behavior of the reflection coefficient the new imaging condition is numerically stable does not involve damping factors is simple to implement numerically and is a simple modification of the classical crosscorrelation imaging condition numerical examples confirm the correct ava behavior of the new imaging condition

accounting for free surface multiples in marchenko imaging imagine placing a receiver at any location in the earth and recording the response at that location to sources on the surface in such a world we could place receivers around our reservoir to better image the reservoir and understand its properties realistically this is not a feasible approach for understanding the subsurface we have developed an alternative and realizable approach to obtain the response of a buried virtual receiver for sources at the surface our method is capable of retrieving the green s function for a virtual point in the subsurface to the acquisition surface in our case a physical receiver is not required at the subsurface point instead we require the reflection measurements for sources and receivers at the surface of the earth and a macromodel of the velocity no small scale details of the model are necessary we can interpret the retrieved green s function as the response to sources at the surface for a virtual receiver in the subsurface we obtain this green s function by solving the marchenko equation an integral equation pertinent to inverse scattering problems our derivation of the marchenko equation for the green s function retrieval takes into account the free surface reflections present in the reflection response previous work considered a response without free surface multiples we decompose the marchenko equation into up and downgoing fields and solve for these fields iteratively the retrieved green s function not only includes primaries and internal multiples as do previous methods but it also includes freesurface multiples we use these up and downgoing fields to obtain a 2d image of our area of interest in this case below a synclinal structure

l 1 optimal feedback controllers for mimo discrete time systems the problem considered in this paper is the design of a closed loop system consisting of a mimo discrete time plant and compensator in such a way that the system is internally stable and optimally tracks all persistent bounded inputs the solution consists of two parts first the calculation of the minimum value of the performance index a weighted transfer function which is done by solving a linear programming problem and second construction of the optimal transfer function by solving a set of linear equations it is shown that in general the discrete time problem will have a rational solution and thus appears to be of considerable practical significance

connections with l p bounds on curvature we show by means of the implicit function theorem that coulomb gauges exist for fields over a ball inr n when the integrall n 2 field norm is sufficiently small we then are able to prove a weak compactness theorem for fields on compact manifolds withl p integral norms bounded p n 2

smoothing imaging condition for shot profile migration amplitudes in shot profile migration can be improved if the imaging condition incorporates a division deconvolution in the time domain of the upgoing wavefield by the downgoing wavefield this division can be enhanced by introducing an optimal wienerfilterwhichassumesthatthenoisepresentinthedatahas a white spectrum this assumption requires a damping parameter related to the signal to noise ratio often chosen by trial and error in practice the damping parameter replaces the small valuesofthespectrumofthedowngoingwavefieldandavoidsdivision by zero the migration results can be quite sensitive to the damping parameter and in most applications the upgoing and downgoing wavefields are simply multiplied alternatively the division can be made stable by filling the small values of the spectrumwithanaverageoftheneighboringpoints thisaveraging is obtained by running a smoothing operator on the spectrum ofthedowngoingwavefield thisoperationcalledthesmoothing imaging condition our results show that where the spectrum of the downgoing wavefield is high the imaging condition with dampingandsmoothingyieldssimilarresults thuscorrectingfor illumination effects where the spectrum is low the smoothing imaging condition tends to be more robust to the noise level present in the data thus giving better images than the imaging condition with damping in addition our experiments indicate that the parameterization of the smoothing imaging condition i e choice of window size for the smoothing operator is easy and repeatable from one data set to another making it a valuable additiontoourimagingtoolbox

optimal rejection of persistent bounded disturbances in this paper we formulate the problem of optimal disturbance rejection in the case where the disturbance is generated as the output of a stable system in response to an input which is assumed to be of unit amplitude but is otherwise arbitrary the objective is to choose a controller that minimizes the maximum amplitude of the plant output in response to such a disturbance mathematically this corresponds to requiring uniformly good disturbance rejection over all time since the problem of optimal tracking is equivalent to that of optimal disturbance rejection if a feedback controller is used see 7 sect 5 6 the theory presented here can also be used to design optimal controllers that achieve uniformly good tracking over all time rather than a tracking error whose l 2 norm is small as is the case with the currently popular h infty theory the present theory is a natural counterpart to the existing theory of optimal disturbance rejection the so called h infty theory which is based on the assumption that the disturbance to be rejected is generated by a stable system whose input is square integrable and has unit energy it is shown that the problem studied here has quite different features from its predecessor complete solutions to the problem are given in several important cases including those where the plant is minimum phase or when it has only a single unstable zero in other cases procedures are given for obtaining bounds on the solution and for obtaining suboptimal controllers

anti self dual yang mills connections over complex algebraic surfaces and stable vector bundles on presente une correspondance entre la geometrie algebrique et la geometrie differentielle des fibres vectoriels soit une surface algebrique projective x qui a un plongement donne x cp n et soit une metrique de kahler sur x dont la classe de cohomologie associee est duale a la classe de section d hyperplan h un fibre sur x est stable par rapport au plongement projectif si et seulement si il admet une connexion irreductible d hermite einstein par rapport a la metrique cette connexion est alors unique

strategies for imaging with marchenko retrieved green s functions abstractrecent papers show that imaging with the retrieved green s function constructed by the marchenko equations called marchenko imaging reduces artifacts from internal and free surface multiples compared with standard imaging techniques even though artifacts are reduced they can still be present in the image depending on the imaging condition used we have found that when imaging with the up and downgoing green s functions the multidimensional deconvolution mdd imaging condition yields better images than correlation and deconvolution better in this case means improved resolution fewer artifacts and a closer match with the true reflection coefficient of the model we have determined that the mdd imaging condition only uses primaries to construct the image whereas multiples are implicitly subtracted in the imaging step consequently combining the first arrival of the downgoing green s function with the complete upgoing green s function produces superior or at least equivalent images th

l 1 optimal compensators for continuous time systems previous work has been concerned with minimizing the l 1 norm of an error pulse response for discrete time siso 1 and mimo 2 systems in this paper we study the problem of minimizing the l1 norm of the error impulse response for siso continuous time systems this problem is quite different from the discrete time problem in that irrational solutions are obtained even when the problem data are rational two methods are suggested for the solution of the continuous time problem an exact method which leads to a finite dimensional nonlinear programming problem and an approximate method which leads to a linear programming problem