robust inversion of seismic data using the huber norm the huber function or huber norm is one of several robust error measures which interpolates between smooth l2 treatment of small residuals and robust l1 treatment of large residuals since the huber function is differentiable it may be minimized reliably with a standard gradient based optimizer we propose to minimize the huber function with a quasi newton method that has the potential of being faster and more robust than conjugate gradient methods when solving nonlinear problems tests with a linear inverse problem for velocity analysis with both synthetic and field data suggest that the huber function gives far more robust model estimates than does a least squares fit with or without damping

high order finite difference simulations of marine csem surveys using a correspondence principle for wave and diffusion fields the computer time required to solve a typical 3d marine controlled source electromagnetic surveying csem simulation can be reduced by more than one order of magnitude by transforming low frequency maxwell equations in the quasi static or diffusive limit to a hyperbolic set of partial differential equations that give a representation of electromagnetic fields in a fictitious wave domain the dispersion and stability analysis can be made equivalent to that of other types of wave simulation problems such as seismic acoustic and elastic modeling second order to eighth order spatial derivative operators are implemented for flexibility fourth order and sixth order methods are the most numerically efficient implementations for this particular scheme an implementation with high order operators requires that both electric and magnetic fields are extrapolated simultaneously into the air layer the stability condition given for high order staggered derivative operators here should be equally valid for seismic wave simulation the bandwidth of recovered fields in the diffusive domain is independent of the bandwidth of the fields in the fictitious wave domain the fields in the fictitious wave domain do not represent observable fields propagation paths and interaction reflection amplitudes are not altered by the transform from the fictitious wave domain to the diffusive frequency domain however the transform contains an exponential decay factor that damps down late arrivals in the fictitious wave domain the propagation paths that contribute most to the diffusive domain fields are airwave shallow water plus typically postcritical events such as refracted and guided waves the transform from the diffusive frequency domain to the fictitious wave domain is an ill posed problem the transform is nonunique this gives a large degree of freedom in postulating temporal waveforms for boundary conditions in the fictitious wave domain that reproduce correct diffusive frequency domain fields

full waveform inversion using a nonlinearly smoothed wavefield conventional full waveform inversion fwi based on the least squares misfit function faces problems in converging to the global minimum when using gradient methods because of the cycle skipping phenomena an initial model producing data that are at most a half cycle away from the observed data is needed for convergence to the global minimum low frequencies are helpful in updating low wavenumber components of the velocity model to avoid cycle skipping however low enough frequencies are usually unavailable in field cases the multiplication of wavefields of slightly different frequencies adds artificial low frequency components in the data which can be used for fwi to generate a convergent result and avoid cycle skipping we generalize this process by multiplying the wavefield with itself and then applying a smoothing operator to the multiplied wavefield or its square to derive the nonlinearly smoothed wavefield which is rich in low frequencies the global correlation norm based objective function can

multiscattering inversion for low model wavenumbers abstracta successful full waveform inversion implementation updates the low wavenumber model components first for a proper description of the wavefield propagation and slowly adds the high wavenumber potentially scattering parts of the model the low wavenumber components can be extracted from the transmission parts of the recorded wavefield emanating directly from the source or the transmission parts from the single or double scattered wavefield computed from a predicted scatter field acting as secondary sources we use a combined inversion of data modeled from the source and those corresponding to single and double scattering to update the velocity model and the component of the velocity perturbation responsible for the single and double scattering the combined inversion helps us access most of the potential model wavenumber information that may be embedded in the data a scattering angle filter is used to divide the gradient of the combined inversion so initially the high wavenumber low scatterin

rigid fuchsian systems in 2 dimensional conformal field theories we investigate fuchsian equations arising in the context of 2 dimensional conformal field theory cft and we apply the katz theory of fucshian rigid systems to solve some of these equations we show that the katz theory provides a precise mathematical framework to answer the question whether the fusion rules of degenerate primary fields are enough for determining the differential equations satisfied by their correlation functions we focus on the case of mathcal w 3 toda cft we argue that the differential equations arising for four point conformal blocks with one nth level semi degenerate field and a fully degenerate one in the fundamental sl3 representation are associated to fuchsian rigid systems we show how to apply katz theory to determine the explicit form of the differential equations the integral expression of solutions and the monodromy group representation the theory of twisted homology is also used in the analysis of the integral expression the computation of the connection coefficients is done for the first time in the case of a katz system with multiplicities thus extending the work done by oshima in the multiplicity free case this approach allows us to construct the corresponding fusion matrices and to perform the whole bootstrap program new explicit factorization of mathcal w 3 correlation functions as well as shift relations between structure constants for general toda theories are also provided

fuzzy logic based algorithms for maximum covering location problems this paper concerns a class of maximum covering location problems in networks in uncertain environments it is assumed that a relative weights of demand nodes are either deterministic or imprecise described by linguistic expressions and b potential facility site locations are limited to network nodes the concept of coverage is extended to include a degree of node coverage which means that the borders between the subset of covered demand nodes and the subset of uncovered demand nodes are inexact the acceptable service distance travelling times from a facility site to demand nodes are modelled by fuzzy sets three new algorithms for choosing the best facility locations are developed which assume that 1 demands at all nodes are equally important 2 relative weights of demand at nodes are deterministic and 3 weights of demand at nodes are imprecise and described by linguistic terms respectively the algorithms are based on searching among potential facility nodes by applying comparison operations on discrete fuzzy sets it is shown how to extend the proposed algorithms from one site to multi site covering problems illustrative examples of selecting locations for logistics centres in a distribution company are given

seismic inversion based on proximal objective function optimization algorithm abstractseismic impedance inversion has become a common approach in reservoir prediction at present the critical issue in the application of seismic inversion is its low computational efficiency especially in 3d to improve the computational efficiency we have developed an inversion method derived from the proximal objective function optimization algorithm our inversion method calculates each unknown parameter in the model vector one by one during iteration compared with routine gradient dependent inversion algorithms such as the iteratively reweighted least squares irls algorithm our inversion method has lower computational complexity as well as higher efficiency in addition to obtain a sparse reflectivity series a long tailed cauchy distribution is used as the a priori constraint the weak nonlinear problem owing to the introduction of cauchy sparse constraint is addressed by taking advantage of reweighting strategy results of synthetic and real data tests illustrate that the proposed inv

moveout analysis of wave equation extended images conventionalvelocityanalysisappliedtoimagesproducedby wave equationmigrationwithacrosscorrelationimagingconditionusesmoveoutinformationfromspacelagsorfocusinginformationfromtimelag however morerobustvelocity estimation methodscanbedesignedtosimultaneouslytakeadvantageofthe semblanceandfocusinginformationprovidedbymigratedimages such a velocity estimation requires characterization of the moveout surfaces defined jointly for space and time lags extended images the analytic solutions to the moveout surfaces can be derived by solving the system of equations representing the shifted source and receiver wavefields the superposition of the surfaces from many experiments shots is equivalent to the envelope for the family of the individual surface the envelope forms a shape that can be characterized as a cone in the extended space of depth space lag and time lag when imaged with the correct velocity the apex of the cone is located at the correct reflectiondepthandatzerospaceandtimelags whenimagedwith the incorrect velocity the apex of the cone shifts in the depth direction and along the time lag axis the characteristics of the cones are directly related to the quality of the velocity model thus their analysis provides a rich source of information for velocity model building synthetic examples verify the derived formulas characterizing the moveout surfaces the analytic formulas match the numeric experiments well demonstrating the accuracy of the formulas based on information provided by the extended imaging condition future application for velocity updates can benefit from the robustness of the depth focusing analysisandofthehighresolutionofthesemblanceanalysis

full waveform inversion via source receiver extension abstractfull waveform inversion produces highly resolved images of the subsurface and quantitative estimation of seismic wave velocity provided that its initial model is kinematically accurate at the longest data wavelengths if this initialization constraint is not satisfied iterative model updating tends to stagnate at kinematically incorrect velocity models producing suboptimal images the source receiver extension overcomes this cycle skip pathology by modeling each trace with its own proper source wavelet permitting a good data fit throughout the inversion process because source wavelets should be constant or vary systematically across a shot gather a measure of source trace dependence for example the mean square of the signature deconvolved wavelet scaled by time lag can be minimized to update the velocity model for kinematically simple data such measures of wavelet variance are mathematically equivalent to traveltime misfit thus the model obtained by source receiver extended inversi

a discrete location model with fuzzy accessibility measures abstract in crisply defined discrete location problems a number of facilities are to be located at specific points within an area according to precisely quantified criteria however in many location problems especially those associated with social policies non crisply defined criteria are used such as how near or accessible a facility is or how important certain issues are etc in these cases a fuzzy sets approach is more appropriate this paper presents an application of the set partitioning set covering with equality constraints type of integer programming formulation to a discrete location problem with fuzzy accessibility criteria the solution method suggested uses the symmetry of the objectives and the constraints introduced by bellman and zadeh

a discrepancy based penalty method for extended waveform inversion abstractextended waveform inversion globalizes the convergence of seismic waveform inversion by adding nonphysical degrees of freedom to the model thus permitting it to fit the data well throughout the inversion process these extra degrees of freedom must be curtailed at the solution for example by penalizing them as part of an optimization formulation for separable partly linear models a natural objective function combines a mean square data residual and a quadratic regularization term penalizing the nonphysical linear degrees of freedom the linear variables are eliminated in an inner optimization step leaving a function of the outer nonlinear variables to be optimized this variable projection method is convenient for computation but it requires that the penalty weight be increased as the estimated model tends to the physical solution we describe an algorithm based on discrepancy that is maintaining the data residual at the inner optimum within a prescribed range to control the pena

from migration to inversion velocity analysis abstractmigration velocity analysis mva is a technique defined in the image domain to determine the background velocity model controlling the kinematics of wave propagation in the presence of discontinuous interfaces the velocity gradient used to iteratively update the velocity model exhibits spurious oscillations for more stable results we replace the migration part by an inversion scheme by definition migration is the adjoint of the born modeling operator whereas inversion is its asymptotic inverse we have developed new expressions in 1d and 2d cases based on two way wave equation operators the objective function measures the quality of the images obtained by inversion in the extended domain depending on the subsurface offset in terms of implementation the new approach is very similar to classic mva a 1d analysis found that oscillatory terms around the interface positions can be removed by multiplying the inversion result with the velocity at a specific power before evaluating the objectiv

adaptive waveform inversion theory abstractconventional full waveform seismic inversion attempts to find a model of the subsurface that is able to predict observed seismic waveforms exactly it proceeds by minimizing the difference between the observed and predicted data directly iterating in a series of linearized steps from an assumed starting model if this starting model is too far removed from the true model then this approach leads to a spurious model in which the predicted data are cycle skipped with respect to the observed data adaptive waveform inversion awi provides a new form of full waveform inversion fwi that appears to be immune to the problems otherwise generated by cycle skipping in this method least squares convolutional filters are designed that transform the predicted data into the observed data the inversion problem is formulated such that the subsurface model is iteratively updated to force these wiener filters toward zero lag delta functions as that is achieved the predicted data evolve toward the observed

joint migration inversion simultaneous determination of velocity fields and depth images using all orders of scattering the future in seismic exploration and seismic monitoring is the inclusion of all orders of scattering in the imaging and inversion algorithms using the full wavefield in an inversion process allows us to surpass imprints provided by incomplete acquisition and provide more accurate information on the subsurface as noise becomes signal we describe an inversion process that does not use the standard parameters from finite difference type modeling being local velocity and density but that describes the seismic data in terms of elastic reflectivity operators and propagation operators with these operators our full wavefield modeling process builds the complete two way seismic response a major advantage of our alternative to the traditional full waveform inversion methodology is that reflectivity operators are traveltime free and propagation operators are scattering free making the inversion problem significantly more linear a major advantage of our algorithm is 1 that multiples are utilized to impro

efficient calculation of a partial derivative wavefield using reciprocity for seismic imaging and inversion linearized inversion of surface seismic data for a model of the earth s subsurface requires estimating the sensitivity of the seismic response to perturbations in the earth s subsurface this sensitivity or jacobian matrix is usually quite expensive to estimate for all but the simplest model parameterizations we exploit the numerical structure of the finite element method modern sparse matrix technology and source receiver reciprocity to develop an algorithm that explicitly calculates the jacobian matrix at only the cost of a forward model solution furthermore we show that we can achieve improved subsurface images using only one inversion iteration through proper scaling of the image by a diagonal approximation of the hessian matrix as predicted by the classical gauss newton method our method is applicable to the full suite of wave scattering problems amenable to finite element forward modeling we demonstrate our method through some simple 2 d synthetic examples

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

topological theory for selberg type integral associated with rigid fuchsian systems using the analytic realization of middle convolution due to dettweiler and reiter we show that any rigid fuchsian system can be obtained as a subsystem of some generating system which has an integral representation of solutions of selberg type twisted homology groups and twisted cohomology groups associated with such integrals are studied in particular contiguity relations and twisted cycles which realize local exponents are obtained

time domain full waveform inversion of exponentially damped wavefield using the deconvolution based objective function full waveform inversion fwi suffers from the cycle skipping problem when the available frequency band of data is not low enough we apply an exponential damping to the data to generate artificial low frequencies which helps fwi avoid cycle skipping in this case the least square misfit function does not properly deal with the exponentially damped wavefield in fwi because the amplitude of traces decays almost exponentially with increasing offset in a damped wavefield thus we use a deconvolution based objective function for fwi of the exponentially damped wavefield the deconvolution filter includes inherently a normalization between the modeled and observed data thus it can address the unbalanced amplitude of a damped wavefield we specifically normalize the modeled data with the observed data in the frequency domain to estimate the deconvolution filter and selectively choose a frequency band for normalization that mainly includes the artificial low frequencies we calculate the gradient of the o

the optimized expansion based low rank method for wavefield extrapolation abstractspectral methods are fast becoming an indispensable tool for wavefield extrapolation especially in anisotropic media because it tends to be dispersion and artifact free as well as highly accurate when solving the wave equation however for inhomogeneous media we face difficulties in dealing with the mixed space wavenumber domain extrapolation operator efficiently to solve this problem we evaluated an optimized expansion method that can approximate this operator with a low rank variable separation representation the rank defines the number of inverse fourier transforms for each time extrapolation step and thus the lower the rank the faster the extrapolation the method uses optimization instead of matrix decomposition to find the optimal wavenumbers and velocities needed to approximate the full operator with its explicit low rank representation as a result we obtain lower rank representations compared with the standard low rank method within reasonable accuracy and thus cheaper extrapola

full waveform inversion guided by travel time tomography full waveform inversion fwi is a process in which seismic numerical simulations are fit to observed data by changing the wave velocity model of the medium under investigation the problem is nonlinear and therefore optimization techniques have been used to find a reasonable solution to the problem the main problem in fitting the data is the lack of low spatial frequencies this deficiency often leads to a local minimum and to nonplausible solutions in this work we explore how to obtain low frequency information for fwi our approach involves augmenting fwi with travel time tomography which has low frequency features by jointly inverting these two problems we enrich fwi with information that can replace low frequency data in addition we use high order regularization in a preliminary inversion stage to prevent high frequency features from polluting our model in the initial stages of the reconstruction this regularization also promotes the nondominant low frequency modes that exist in the fwi sen

inversion of normal incidence seismograms the well known instability of kunetz s 1963 inversion algorithm can be explained by the progressive manner in which the calculations are done descending from the surface and by the fact that completely different impedances can yield indistinguishable synthetic seismograms those difficulties can be overcome by using an iterative algorithm for the inversion of the one dimensional 1 d wave equation together with a stabilizing constraint on the sums of the jumps of the desired impedance for computational efficiency the synthetic seismogram is computed by the method of characteristics and the gradient of the error criterion is computed by optimal control techniques adjoint state equation the numerical results on simulated data confirm the expected stability of the algorithm in the presence of measurement noise tests include noise levels of 50 percent the inversion of two field sections demonstrates the practical feasibility of the method and the importance of taking into account all internal a

simultaneous inversion of full data bandwidth by tomographic full waveform inversion abstractthe convergence of full waveform inversion can be improved by extending the velocity model along either the subsurface offset axis or the time lag axis the extension of the velocity model along the time lag axis enables us to linearly model large time shifts caused by velocity perturbations this linear modeling was based on a new linearization of the scalar wave equation in which perturbation of the extended slowness squared was convolved in time with the second time derivative of the background wavefield the linearization was accurate for reflected events and transmitted events we determined that it can effectively model conventional reflection data as well as modern long offset data containing diving waves it also enabled the simultaneous inversion of reflections and diving waves even when the starting velocity model was far from being accurate we solved the optimization problem related to the inversion with a nested algorithm the inner iterations were based on the proposed linearization

elastic least squares reverse time migration abstractwe use elastic least squares reverse time migration lsrtm to invert for the reflectivity images of p and s wave impedances elastic lsrtm solves the linearized elastic wave equations for forward modeling and the adjoint equations for backpropagating the residual wavefield at each iteration numerical tests on synthetic data and field data reveal the advantages of elastic lsrtm over elastic reverse time migration rtm and acoustic lsrtm for our examples the elastic lsrtm images have better resolution and amplitude balancing fewer artifacts and less crosstalk compared with the elastic rtm images the images are also better focused and have better reflector continuity for steeply dipping events compared to the acoustic lsrtm images similar to conventional least squares migration elastic lsrtm also requires an accurate estimation of the p and s wave migration velocity models however the problem remains that when there are moderate errors in the velocity model and strong multiples lsrtm

multiscale adjoint waveform difference tomography using wavelets abstractfull waveform seismic inversions based on minimizing the distance between observed and predicted seismograms are in principle able to yield better resolved earth models than those minimizing misfits derived from traveltimes alone adjoint based methods provide an efficient way of calculating the gradient of the misfit function via a sequence of forward modeling steps which using spectral element codes can be carried out in realistically complex media convergence and stability of full waveform difference adjoint schemes are greatly improved when data and synthetics are progressively presented to the algorithms in a constructive multiscale approximation using a bi orthogonal wavelet transform wavelets provide the nonredundant spectral decomposition that paves the way for the inversion to proceed successively from long wavelength fitting to detailed exploration of the phases in the seismogram the choice of wavelet class and type the initial depth of the multiscale decomposition and the min

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

waveform inversion by one way wavefield extrapolation one way riemannian wavefield extrapolation rwe on computational meshes conforming to the direction of turning wave propagation is an alternative forward modeling procedure for waveform inversion forward modeling tests demonstrate that the rwe approach may be a sufficiently accurate approximation for calculating the wavefield phases important for early arrival waveform inversion initial results indicate that rwe waveforms are well matched at wide offsets to finite difference data and can be used in a waveform inversion scheme to invert synthetic data for moderate 1d velocity perturbations

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

constrained full waveform inversion by model reparameterization the waveform inversion problem is inherently ill posed traditionally regularization schemes are used to address this issue for waveform inversion where the model is expected to have many details reflecting the physical properties of the earth regularization and data fitting can work in opposite directions the former smoothing and the latter adding details to the model we propose constraining estimated velocity fields by reparameterizing the model this technique also called model space preconditioning is based on directional laplacian filters it preserves most of the details of the velocity model while smoothing the solution along known geological dips preconditioning also yields faster convergence at early iterations the laplacian filters have the property to smooth or kill local planar events according to a local dip field by construction these filters can be inverted and used in a preconditioned waveform inversion strategy to yield geologically meaningful models we illustrate with 2d syn

cascaded wavefield tomography and inversion using extended common image point gathers a case study abstractimage domain wavefield tomography wt exploits focusing characteristics of extended images for updating the velocity field to make good use of this information one must understand how such images behave if the migration velocity is accurate this is not trivial because focusing depends on not only the model error but also on the acquisition setup the data bandwidth and illumination variation caused by the overburden we address this problem by constructing penalty functions based on the point spread functions of the imaging operator that characterize focusing in extended images moreover instead of sampling the extended images at preset distances along the surface we sample the image by constructing common image point gathers which are more economical from a computational point of view and also measure the spatial and temporal focusing of the wavefield coupled with image residuals exploiting illumination based penalty functions we construct robust wavefield tomographic updates from spar

a partial low rank method for solving acoustic wave equation abstract numerical solutions of the acoustic wave equation especially in anisotropic media is crucial to seismic modeling imaging and inversion as it provides efficient practical and stable approximate representation of the medium however a clean implementation free of shear wave artifacts and dispersion of wave propagation especially in anisotropic media requires an integral operator the direct evaluation of which is extremely expensive recently the low rank method was proposed to provide a good approximation to the integral operator utilizing fourier transforms thus we propose to split the integral operator into two terms the first term provides a differential operator that approximates that can be approximated with a standard finite difference method we then apply the low rank approximation on the residual term of the finite difference operator we implement the two terms in two complementing steps in which the spectral step corrects for any errors admitted by the finite difference step even though we utilize finite difference approximations the resulting algorithm admits spectral accuracy also through the finite difference step the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner numerical examples show that the method is of high accuracy and efficiency

correlation based reflection full waveform inversion because modeling for full waveform inversion fwi cannot produce reflections unless the velocity model has the scattering potential high wavenumbers using a migration demigration process to generate modeling data which is a key step in what is now known as reflection fwi rfwi is a credible alternative to tackle the reflection nonlinearity associated with fwi however because rfwi depends on a conventional data residual or zero lag correlation objective function high nonlinearity can still exist when the true amplitude migration is not used as well as at far offsets due to cycle skipping to avoid the cycle skipping and the need for a true amplitude migration we have developed a correlation based reflection full waveform inversion method to update the low wavenumber components of the velocity model the success of this method relies on a sensitivity kernel decomposition and a correlation based objective function the sensitivity kernel decomposition makes it possible to separate out the contributions of different subkernels and to smear the reflected wave residuals along the rabbit ear wavepath to obtain middle and deep background model estimates the correlation based objective function measures differences in kinematic information and behaves in a more linear way than the traditional waveform residual misfit moreover our approach is less sensitive to the frequency content and amplitude information of the seismic data enabling reliable background velocity estimates to be obtained without the need for low frequencies and full physics modeling because the kinematic features of reflected waves are described correctly the inversion result of the proposed method can be used as a migration model or an initial model for conventional fwi to achieve a correct high wavenumber model update

seismic envelope inversion and modulation signal model abstractwe recognized that the envelope fluctuation and decay of seismic records carries ultra low frequency ulf i e the frequency below the lowest frequency in the source spectrum signals that can be used to estimate the long wavelength velocity structure we then developed envelope inversion for the recovery of low wavenumber components of media smooth background so that the initial model dependence of waveform inversion can be reduced we derived the misfit function and the corresponding gradient operator for envelope inversion to understand the long wavelength recovery by the envelope inversion we developed a nonlinear seismic signal model the modulation signal model as the basis for retrieving the ulf data and studied the nonlinear scale separation by the envelope operator to separate the envelope data from the wavefield data envelope extraction a demodulation operator envelope operator was applied to the waveform data numerical tests using synthetic data for the marmousi model pro

inversion gradients for acoustic vti wavefield tomography abstractwavefield tomography can handle complex subsurface geology better than ray based techniques and ultimately provide a higher resolution here we implement forward and adjoint wavefield extrapolation for vti transversely isotropic with a vertical symmetry axis media using a generalized pseudospectral operator based on a separable approximation for the p wave dispersion relation this operator is employed to derive the gradients of the differential semblance optimization dso and modified image power objective functions we also obtain the gradient expressions for a data domain objective function that can more easily incorporate borehole information necessary for stable vti velocity analysis these gradients are similar to the ones obtained with a space time finite difference fd scheme for a system of coupled wave equations but the pseudospectral method is not hampered by the imprint of the shear wave artifact numerical examples also show the potential advantages of the modified image power

one step extrapolation method for reverse time migration we have proposed a new method a one step extrapolation algorithm to solve the acoustic wave equation by introducing a square root operator the two way wave equation can be formulated as a first order partial differential equation in time which is similar to the one way wave equation to solve the new wave equation we used a stable explicit extrapolation method in the time direction and handled lateral velocity variations in the space and wavenumber domains unlike the conventional explicit finite difference schemes the new method does not suffer from numerical instability or numerical dispersion problems it can be used to design cost effective and high quality reverse time migration or modeling code

simultaneous inversion of the background velocity and the perturbation in full waveform inversion abstractthe gradient of standard full waveform inversion fwi attempts to map the residuals in the data to perturbations in the model such perturbations may include smooth background updates from the transmission components and high wavenumber updates from the reflection components however if we fix the reflection components using imaging the gradient of what is referred to as reflected waveform inversion rwi admits mainly transmission background type updates the drawback of existing rwi methods is that they lack an optimal image capable of producing reflections within the convex region of the optimization because the influence of velocity on the data was given mainly by its background propagator and perturbed reflectivity components we have optimized both components simultaneously using a modified objective function specifically we used an objective function that combined the data generated from a source using the background velocity and that by the perturbed velocity through born modeli

velocity model building by 3d frequency domain full waveform inversion of wide aperture seismic data we assessed 3d frequency domain fd acoustic fullwaveforminversionfwidataasatooltodevelophigh resolution velocity models from low frequency global offset data the inverse problem was posed as a classic leastsquares optimization problem solved with a steepest descent method inversion was applied to a few discrete frequencies allowing management of a limited subset of the 3d data volume the forward problem was solved with a finite difference frequency domain method based on a massively paralleldirectsolver allowingefficientmultiple shotsimulations the inversion code was fully parallelized for distributedmemory platforms taking advantage of a domain decomposition of the modeled wavefields performed by the direct solver after validation on simple synthetic tests fwi was applied to two targets channel and thrust system of the 3d seg eageoverthrustmodel correspondingto3ddomains of 78 752 25 km and 13 513 54 65 km respectively the maximum inverted frequencies are 15 and 7 hz for the two applications amaximum of 30 dual core biprocessor nodes with 8 gb of shared memory per node were used for the second target the main structures were imaged successfully at a resolution scale consistent with the inverted frequencies our study confirms the feasibility of 3d frequency domain fwi of global offset data on large distributed memory platforms to develop high resolution velocity models these high velocity models may provide accurate macromodelsforwave equationprestackdepthmigration

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

seismic waveform inversion in the frequency domain part 1 theory and verification in a physical scale model seismic waveforms contain much information that is ignored under standard processing schemes seismic waveform inversion seeks to use the full information content of the recorded wavefield in this paper i present apply and evaluate a frequency space domain approach to waveform inversion the method is a local descent algorithm that proceeds from a starting model to refine the model in order to reduce the waveform misfit between observed and model data the model data are computed using a full wave equation viscoacoustic frequency domain finite difference method ray asymptotics are avoided and higher order effects such as diffractions and multiple scattering are accounted for automatically the theory of frequency domain waveform wavefield inversion can be expressed compactly using a matrix formalism that uses finite difference finite element frequency domain modeling equations expressions for fast local descent inversion using back propagation techniques then follow naturally implementation of

elastic reflection based waveform inversion with a nonlinear approach abstractfull waveform inversion fwi is a highly nonlinear problem due to the complex reflectivity of the earth and this nonlinearity only increases under the more expensive elastic assumption in elastic media we need a good initial p wave velocity and even better initial s wave velocity models with accurate representation of the low model wavenumbers for fwi to converge however inverting for the low wavenumber components of p and s wave velocities using reflection waveform inversion rwi with an objective to fit the reflection shape rather than produce reflections may mitigate the limitations of fwi because fwi performing as a migration operator is preferred of the high wavenumber updates along reflectors we have developed an elastic rwi that inverts for the low wavenumber and perturbation components of the p and s wave velocities to generate the full elastic reflection wavefields we derive an equivalent stress source made up by the inverted model perturbations and incident wavefields w

elastic least squares reverse time migration via linearized elastic full waveform inversion with pseudo hessian preconditioning abstracttime domain elastic least squares reverse time migration lsrtm is formulated as a linearized elastic full waveform inversion problem the elastic born approximation and elastic reverse time migration rtm operators are derived from the time domain continuous adjoint state method our approach defines p and s wave impedance perturbations as unknown elastic images our algorithm is obtained using continuous functional analysis in which the problem is discretized at the final stage optimize before discretize approach the discretized numerical versions of the elastic born operator and its adjoint elastic rtm operator pass the dot product test the conjugate gradient least squares method is used to solve the least squares migration quadratic optimization problem in other words the hessian operator for elastic lsrtm is implicitly inverted via a matrix free algorithm that only requires the action of forward and adjoint operators on vectors the diagonal of the pseudo hessian operator is used t

an adaptive multiscale algorithm for efficient extended waveform inversion abstractsubsurface offset extended full waveform inversion fwi may converge to kinematically accurate velocity models without the low frequency data accuracy required for standard data domain fwi however this robust alternative approach to waveform inversion suffers from a very high computational cost resulting from its use of nonlocal wave physics the computation of strain from stress involves an integral over the subsurface offset axis which must be performed at every space time grid point we found that a combination of data fit driven offset limits grid coarsening and low pass data filtering can reduce the cost of extended inversion by one to two orders of magnitude

wave equation tomography the relation between ray trace and diffraction tomography is usually obscured by formulation of the two methods in different domains the former in space the latter in wavenumber here diffraction tomography is reformulated in the space domain under the title of wave equation tomography with this transformation wave equation tomography projects monochromatic scattered wavefields back over source receiver wavepaths just as ray trace tomography projects traveltime delays back over source receiver raypaths derived under the born approximation these wavepaths are wave theoretic back projection patterns for reflected energy derived under the rytov approximation they are wave theoretic back projection patterns for transmitted energy differences between ray trace and wave equation tomography are examined through comparison of wavepaths and raypaths followed by their application to a transmission geometry synthetic data set rytov wave equation tomography proves superior to ray trace tomography in de

facility location allocation problem in random fuzzy environment using α β cost minimization model under the hurewicz criterion facility location allocation fla which has been proved to be a valuable method in siting service facility is widely used in real life such as emergency service systems telecommunication net works public services etc many researchers have studied the fla problem in a deterministic stochastic or fuzzy environment however those models cannot satisfy various customers demands in some cases thus this paper considers the fla problem under random fuzzy environment using a b cost minimization model under the hurwicz criterion it will be proved that this model can deal with various fla problems in random fuzzy and random fuzzy environments by varying the value l it can balance the optimistic level of the decision makers for solving the random fuzzy model efficiently the simplex algorithm random fuzzy simulation and genetic algorithm are integrated to produce a hybrid intelligent algorithm finally a numerical example is presented for illustration

migration based traveltime waveform inversion of 2 d simple structures a synthetic example migration based traveltime mbtt formulation provides algorithms for automatically determining background velocities from full waveform surface seismic reflection data using local optimization methods in particular it addresses the difficulty of the nonconvexity of the least squares data misfit function the method consists of parameterizing the reflectivity in the time domain through a migration step and providing a multiscale representation for the smooth background velocity we present an implementation of the mbtt approach for a 2 d finite difference fd full wave acoustic model numerical analysis on a 2 d synthetic example shows the ability of the method to find much more reliable estimates of both long and short wavelengths of the velocity than the classical least squares approach even when starting from very poor initial guesses this enlargement of the domain of attraction for the global minima of the least squares misfit has a price each evaluation of the new objective function requires besides the usual fd full wave forward modeling an additional full wave prestack migration hence the fd implementation of the mbtt approach presented in this paper is expected to provide a useful tool for the inversion of data sets of moderate size

linearized wave equation migration velocity analysis by image warping seismic imaging produces images of contrasts in physical parameters in the subsurface e g velocity or impedance to build such images a background model describing the wave kinematics in the earth is necessary in practice the structural image and background velocity model are unknown and have to be estimated from the acquired data migration velocity analysis deals with estimation of the background model in the framework of seismic migration and relies on two main elements data redundancy and invariance of the structures with respect to different seismic experiments because all the experiments probe the same model the reflectors must be invariant in suitable domains e g shots or reflection angle the semblance principle is the tool used to measure the invariance of a set of multiple images we measure the similarity of the structural features between pairs of single shot migrated images obtained from adjacent experiments by using the estimated warping vector field between two migrated images we construct an image perturbation which describes the difference in reflectivity observed by two shots we derive an expression for the image perturbation that drives a migration velocity analysis procedure based on a linearization of the wave equation with respect to the model parameters synthetic 2d examples show promising results in retrieving errors in the velocity model this methodology can be directly applied to 3d

full waveform inversion with extrapolated low frequency data abstractthe availability of low frequency data is an important factor in the success of full waveform inversion fwi in the acoustic regime the low frequencies help determine the kinematically relevant low wavenumber components of the velocity model which are in turn needed to avoid convergence of fwi to spurious local minima however acquiring data less than 2 or 3 hz from the field is a challenging and expensive task we have explored the possibility of synthesizing the low frequencies computationally from high frequency data and used the resulting prediction of the missing data to seed the frequency sweep of fwi as a signal processing problem bandwidth extension is a very nonlinear and delicate operation in all but the simplest of scenarios it can only be expected to lead to plausible recovery of the low frequencies rather than their accurate reconstruction even so it still requires a high level interpretation of band limited seismic records into individual events each of which can be extr

fuzzy facility location allocation problem under the hurwicz criterion facility location allocation fla problem has been widely studied by operational researchers due to its many practical applications many researchers have studied the fla problem in a deterministic environment however the models they proposed cannot accommodate satisfactorily various customer demands in the real world thus we consider the fla problem with uncertainties in this paper a new model named cost model under the hurwicz criterion is presented with fuzzy demands in order to solve this model the simplex algorithm fuzzy simulations and a genetic algorithm are integrated to produce a hybrid intelligent algorithm finally some numerical examples are presented to illustrate the effectiveness of the proposed algorithm

middle convolution of fuchsian systems and the construction of rigid differential systems abstract in m dettweiler s reiter an algorithm of katz and its application to the inverse galois problem j symbolic comput 30 2000 761 798 a purely algebraic analogon of katz middle convolution functor see n m katz rigid local systems ann of math stud vol 139 princeton university press 1997 is given in this paper we find an explicit riemann hilbert correspondence for this functor this leads to a construction algorithm for differential systems which correspond to rigid local systems on the punctured affine line via the riemann hilbert correspondence

the simple plant location problem survey and synthesis abstract with emphasis on the simple plant location problem splp we consider an important family of discrete deterministic single criterion np hard and widely applicable optimization problems the introductory discussion on problem formulation aspects is followed by the establishment of relationships between splp and set packing set covering and set partitioning problems which all are among those structures in integer programming having the most wide spread applications an extensive discourse on solution properties and computational techniques spanning from early heuristics to the presumably most novel exact methods is then provided other subjects of concern include a subfamily of splp s solvable in polynomial time analyses of approximate algorithms transformability of p center and p median to splp and structural properties of the splp polytope along the way we attempt to synthesize these findings and relate them to other areas of integer programming

relating jack wavefunctions to textrm wa _ k 1 theories the k r admissible jack polynomials recently proposed as many body wavefunctions for non abelian fractional quantum hall systems have been conjectured to be related to some correlation functions of the minimal model of the algebra by studying the degenerate representations of this conformal field theory we provide a proof for this conjecture

an iterative inversion of back scattered acoustic waves the application of the born approximation to the scattered wave field followed by a wkbj and far field approximation on the propagation green s function for a slowly varying background medium leads to a simple integral relation between the density and bulk modulus anomalies superimposed on the background medium and the scattered wave field an iterative inversion scheme based on successive back projections of the wave field is used to reconstruct the two acoustic parameters the scheme when applied to data generated using the direct integral relation shows that the variations of the parameters can be reconstructed the procedure is readily applicable to actual data since every iterative step is essentially a prestack kirchhoff migration followed by the application of the direct born approximation and far field operator