wave field separation in two dimensional anisotropic media until recently the term elastic usually implied two dimensional 2 d and isotropic in this limited context the divergence and curl operators have found wide use as wave separation operators for example mora 1987 used them in his inversion method to allow separate correlation of p and s arrivals although the separation is buried in the math and not obvious clayton 1981 used them explicitly in several modeling and inversion methods devaney and oristaglio 1986 used closely related operators to separate p and s arrivals in elastic vsp data

2d and 3d elastic wavefield vector decomposition in the wavenumber domain for vti media a pragmatic decomposition of a vector wavefield into p and s waves is based on the helmholtz theory and the christoffel equation it is applicable to vti media when the plane wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber except for the kiss singularities on the vti symmetry axis unlike divergence and curl which separate the wavefield into a scalar and a vector field the decomposed p and s wavefields are both vector fields with correct amplitude phase and physical units if the vector components of decomposed wavefields are added they reconstruct those of the original input wavefield wavefield propagation in any portions of a vti medium that have the same polarization distribution i e the same eigenvector in the wavenumber domain have the same decomposition operators and can be recon structed with a single 3d fourier transform for each operator e g one for p waves and one for s waves this applies to isotropic wavefields a

fast algorithms for elastic wave mode separation and vector decomposition using low rank approximation for anisotropic media abstractwave mode separation and vector decomposition are significantly more expensive than wavefield extrapolation and are the computational bottleneck for elastic reverse time migration in heterogeneous anisotropic media we have expressed elastic wave mode separation and vector decomposition for anisotropic media as space wavenumber domain operations in the form of fourier integral operators and developed fast algorithms for their implementation using their low rank approximations synthetic data generated from 2d and 3d models demonstrated that these methods are accurate and efficient

scalar reverse time depth migration of prestack elastic seismic data reflected p to p and p to s converted seismic waves in a two component elastic common source gather generated with a p wave source in a two dimensional model can be imaged by two independent scalar reverse time depth migrations the inputs to migration are pure p and s waves that are extracted by divergence and curl calculations during shallow extrapolation of the elastic data recorded at the earth s surface for both p to p and p to s converted reflected waves the imaging time at each point is the p wave traveltime from the source to that point the extracted p wave is reverse time extrapolated and imaged with a p velocity model using a finite difference solution of the scalar wave equation the extracted s wave is reverse time extrapolated and imaged similarly but with an s velocity model converted s wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources synthetic examples show that the algorithm gives satisfactory resul

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

nonliner inversion of seismic reflection data in a laterally invariant medium interpretation of seismic waveforms can be expressed as an optimization problem based on a non linear least squares criterion to find the model which best explains the data an initial model is corrected iteratively using a gradient method conjugate gradient at each iteration computation of the direction of the model perturbation requires the forward propagation of the actual sources and the reverse time propagation of the residuals misfit between the data and the synthetics the two wave fields thus obtained are then correlated an extra forward propagation is required to compute the amplitude of the perturbation along the conjugate gradient direction the number of propagations to be simulated numerically in each iteration equals three times the number of shots since a 2 d finite difference code is employed to solve forward and backward propagation problems the method is general and can handle arbitrary 2 d source receiver configurations and lateral heterogeneities using conventional velocity

the possibilities of compressed sensing based kirchhoff prestack migration an approximate subsurface reflectivity distribution of the earth is usually obtained through the migration process however conventional migration algorithms including those based on the least squares approach yield structure descriptions that are slightly smeared and of low resolution caused by the common migration artifacts due to limited aperture coarse sampling band limited source and low subsurface illumination to alleviate this problem we use the fact that minimizing the l1 norm of a signal promotes its sparsity thus we formulated the kirchhoff migration problem as a compressed sensing cs basis pursuit denoise problem to solve for highly focused migrated images compared with those obtained by standard and least squares migration algorithms the results of various subsurface reflectivity models revealed that solutions computed using the cs based migration provide a more accurate subsurface reflectivity location and amplitude we applied the cs algorithm to image synthetic data from a fault model using dense and sparse acquisition geometries our results suggest that the proposed approach may still provide highly resolved images with a relatively small number of measurements we also evaluated the robustness of the basis pursuit denoise algorithm in the presence of gaussian random observational noise and in the case of imaging the recorded data with inaccurate migration velocities

2d frequency domain elastic full waveform inversion using time domain modeling and a multistep length gradient approach abstractto decouple the parameters in elastic full waveform inversion fwi we evaluated a new multistep length gradient approach to assign individual weights separately for each parameter gradient and search for an optimal step length along the composite gradient direction to perform wavefield extrapolations for the inversion we used parallelized high precision finite element fe modeling in the time domain the inversion was implemented in the frequency domain the data were obtained at every subsurface grid point using the discrete fourier transform at each time domain extrapolation step we also used frequency selection to reduce cycle skipping time windowing to remove the artifacts associated with different source spatial patterns between the test and predicted data and source wavelet estimation at the receivers over the full frequency spectrum by using a fast fourier transform in the inversion the velocity and density reconstructions behaved differently as a low wavenumber tomography for v

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

multiscale seismic waveform inversion iterative inversion methods have been unsuccessful at inverting seismic data obtained from complicated earth models e g the marmousi model the primary difficulty being the presence of numerous local minima in the objective function the presence of local minima at all scales in the seismic inversion problem prevent iterative methods of inversion from attaining a reasonable degree of convergence to the neighborhood of the global minimum the multigrid method is a technique that improves the performance of iterative inversion by decomposing the problem by scale at long scales there are fewer local minima and those that remain are further apart from each other thus at long scales iterative methods can get closer to the neighborhood of the global minimum we apply the multigrid method to a subsampled low frequency version of the marmousi data set although issues of source estimation source bandwidth and noise are not treated results show that iterative inversion methods perform much better when employed with a decomposition by scale furthermore the method greatly reduces the computational burden of the inversion that will be of importance for 3 d extensions to the method

phase correction in separating p and s waves in elastic data two dimensional elastic data containing reflected p waves and converted s wave generated by a p source may be separated using dilatation and rotation calculation sun 1999 the algorithm is a combination of elastic full wavefield extrapolation sun and mcmechan 1986 chang and mcmechan 1987 1994 and wave type separation using dilatation divergence and rotation curl calculations dellinger and etgen 1990 it includes 1 downward extrapolating the multicomponent elastic data in an elastic velocity model using the elastic wave equation 2 calculating the dilatation to represent pure p waves and calculating the rotation to represent pure s waves at some depth and 3 upward extrapolating the dilatation in a p velocity model and upward extrapolating the rotation in an s velocity model using the acoustic wave equation for each

comparison of two algorithms for isotropic elastic p and s vector decomposition the research leading to this paper is supported by the sponsors of the ut dallas geophysical consortium a portion of the computations was done at the texas advanced computing center this paper is contribution no 1270 from the department of geosciences at the university of texas at dallas special thanks go to j brown a shabelansky and two anonymous reviewers for their useful comments and encouragements

on second order fritz john type optimality conditions in nonsmooth multiobjective programming we study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint we suppose that the functions involved are frechet differentiable and their frechet derivatives are continuous or stable at the point considered we provide necessary second order optimality conditions and also sufficient conditions via a fritz john type lagrange multiplier rule and a set valued second order directional derivative in such a way that our sufficient conditions are close to the necessary conditions some consequences are obtained for parabolic directionally differentiable functions and c1 1 functions in this last case expressed by means of the second order clarke subdifferential some illustrative examples are also given

towards the asymptotic sum capacity of the mimo cellular two way relay channel in this paper we consider the transceiver and relay design for the multiple input multiple output mimo cellular two way relay channel ctwrc where a multi antenna base station bs exchanges information with multiple multiantenna mobile stations via a multi antenna relay station rs we propose a novel two way relaying scheme to approach the sum capacity of the mimo ctwrc a key contribution of this work is a new nonlinear lattice based precoding technique to precompensate the interstream interference so as to achieve efficient interference free lattice decoding at the relay we derive sufficient conditions for the proposed scheme to asymptotically achieve the sum capacity of the mimo ctwrc in the high signal to noise ratio snr regime to fully exploit the potential of the proposed scheme we also investigate the optimal power allocation at the bs and the rs to maximize the weighted sum rate of the mimo ctwrc in the general snr regime it is shown that the problem can be formulated as a monotonic program and a polyblock outer approximation algorithm is developed to find the globally optimal solution with guaranteed convergence we demonstrate by numerical results that the proposed scheme significantly outperforms the existing schemes and closely approaches the sum capacity of the mimo ctwrc in the high snr regime

amplitude balancing in separating p and s waves in 2d and 3d elastic seismic data the reflected p and s waves in elastic displacement component data recorded at the earth s surface are separated by reverse time downward extrapolation of the data in an elastic computational model followed by calculations to give divergence dilatation and curl rotation at a selected reference depth the surface data are then reconstructed by separate forward time upward scalar extrapolations from the reference depth of the magnitude of the divergence and curl wavefields and extraction of the separated p and s waves respectively at the top of the models a p wave amplitude will change by a factor that is inversely proportional to the p velocity when it is transformed from displacement to divergence and an s wave amplitude will change by a factor that is inversely proportional to the s velocity when it is transformed from displacement to curl consequently the ratio of the p to the s wave amplitude the p s amplitude ratio in the form of divergence and curl postseparation is different

a hierarchical elastic full waveform inversion scheme based on wavefield separation and the multistep length approach abstractelastic full waveform inversion fwi updates model parameters by minimizing the residuals of the p and s wavefields resulting in more local minima and serious nonlinearity in addition the coupling of different parameters degrades the inversion results to address these problems we have developed a hierarchical elastic fwi scheme based on wavefield separation and a multistep length gradient approach first we have derived the gradients expressed by different wave modes analyzed the crosstalk between various parameters and evaluated the sensitivity of separated p wave separated s wave and p and s wave misfit functions then a practical four stage inversion workflow was developed in the first stage conventional fwi is used to achieve rough estimates of the p and s wave velocities in the second stage we only invert the p wave velocity applying the separated p wavefields when strong s wave energy is involved or we merely update the s wave velocity by matching the separated s wavefiel

improving the efficiency of elastic wave mode separation for heterogeneous tilted transverse isotropic media wave mode separation for ti transversely isotropic models can be carried out by nonstationary filtering the elastic wavefields with localized filters these filters are constructed based on the polarization vectors obtained by solving the christoffel equation using local medium parameters this procedure although accurate is computationally expensive especially in 3d we develop an efficient method for wave mode separation which exploits the same general idea of projecting wavefields onto polarization vectors the method consists of two steps 1 separate wave modes in the wavenumber domain at a number of reference models to obtain the same number of partially separated wavefields then transform all the wavefields to the space domain 2 interpolate the wavefields obtained in step 1 in the space domain using the spatially variable model parameters the new method resembles the phaseshift plus interpolation pspi technique which interpolates the wavefields that are reconstructed at several reference velocities synthetic examples indicate that the separation followed by interpolation is effective for models with complex geology the new technique has the benefits of speed and accuracy

which data residual norm for robust elastic frequency domain full waveform inversion elastic full waveform inversion is an ill posed data fitting procedure that is sensitive to noise inaccuracies of the starting model definitionofmultiparameterclasses andinaccuratemodeling of wavefield amplitudes we have investigated the performance of different minimization functionals as the least squares norm 2 the least absolute values norm 1 and combinations of both the huber and so called hybrid criteria with reference to twonoisyoffshorevalhallmodelandonshoreoverthrustmodel synthetic data sets the four minimization functionals were implemented in 2d elastic frequency domain full waveform inversion fwi where efficient multiscale strategies were designed by successive inversions of a few increasing frequencies for the offshore and onshore case studies the 1 norm provided the most reliable models for p and s wave velocities vp and vs even when strongly decimated data sets that correspond to fewfrequencieswereusedintheinversionandwhenoutlierspolluted the data the 2 norm can provide reliable results in the presence of uniform white noise for vp and vs if the data redundancyisincreasedbyrefiningthefrequencysamplingintervalin the inversion at the expense of computational efficiency the 1 norm and the huber and hybrid criteria unlike the 2 norm allowforsuccessfulimagingofthevsmodelfromnoisydataina soft seabed environment where the p to s waves have a small footprint in the data however the huber and hybrid criteria are sensitive to a threshold criterion that controls the transition between the criteria and that requires tedious trial and error investigations for reliable estimation the 1 norm provides a robust alternativetothe2 normforinvertingdecimateddatasetsinthe frameworkofefficientfrequency domainfwi

second order optimality conditions with the envelope like effect for nonsmooth vector optimization in infinite dimensions abstract second order necessary conditions and sufficient conditions with the envelope like effect for optimality in nonsmooth vector optimization are established we use approximations as generalized derivatives imposing strict differentiability for necessary conditions and differentiability for sufficient conditions and avoiding continuous differentiability convexity conditions are not imposed explicitly the results make it clear when the envelope like effect occurs and improve or include several recent existing ones examples are provided to show advantages of our theorems over some known ones in the literature

conditioning the full waveform inversion gradient to welcome anisotropy abstractmultiparameter full waveform inversion fwi suffers from complex nonlinearity in the objective function compounded by the eventual trade off between the model parameters a hierarchical approach based on frequency and arrival time data decimation to maneuver the complex nonlinearity associated with this problem usually falls short in anisotropic media in place of data decimation i use a model gradient filter approach to access the parts of the gradient more suitable to combat the potential nonlinearity and parameter trade off the filter is based on representing the gradient in the time lag normalized domain in which small scattering angles of the gradient update are initially muted out the model update hierarchical filtering strategy include applying varying degrees of filtering to the different anisotropic parameter updates a feature not easily accessible to simple data decimation using fwi and reflection based fwi when the modeled data are obtained with the single scattering theory al

improvements to elastic full waveform inversion using cross gradient constraints seismic full waveform inversion fwi is potentially a powerful method for obtaining high resolution subsurface images but the results are often distorted by non linear effects and parameter trade offs such distortions can be particularly severe in the case of multi parameter fwi such as elastic fwi where inversion is performed simultaneously for p and s wave velocities and density the problem can be alleviated by adding additional constraints in form of plausible a priori information a usually well justified constraint includes the structural similarity of different model parameters that is an anomalous body likely exhibits variations in all elastic properties although their magnitudes may be different to consider such type of a priori information we have developed a structurally constrained elastic fwi which is based on minimization of cross products of gradients of different model parameters our synthetic 2d experiments show that the structurally constrained fwi can significantly improve mo

a strategy for nonlinear elastic inversion of seismic reflection data the problem of interpretation of seismic reflection data can be posed with sufficient generality using the concepts of inverse theory in its roughest formulation the inverse problem consists of obtaining the earth model for which the predicted data best fit the observed data if an adequate forward model is used this best model will give the best images of the earth s interior three parameters are needed for describing a perfectly elastic isotropic earth the density x and the lame parameters x and x or the density x and the p wave and s wave velocities x and x the choice of parameters is not neutral in the sense that although theoretically equivalent if they are not adequately chosen the numerical algorithms in the inversion can be inefficient in the long spatial wavelengths of the model adequate parameters are the p wave and s wave velocities while in the short spatial wavelengths p wave impedance s wave impedance and density are adequate the problem of inversion o

elastic full waveform inversion using the second generation wavelet and an adaptive operator length scheme abstractelastic full waveform inversion fwi can generally yield subsurface model parameters of high spatial resolution however the computational and storage burden is tremendous to mitigate this problem we developed the following three strategies first we derived a time space domain adaptive staggered grid finite difference method the size of the calculation was reduced by using smaller operator lengths for high s wave velocities second we evaluated a multiscale elastic fwi scheme based on the second generation wavelet by which scale decompositions of data and model space were implemented with the scale increasing spatial sampling intervals and time steps became larger therefore the minimum computational cost could be achieved by combining the variable operator length scheme with the multiscale scheme third to economize the memory consumption we extended the efficient boundary storage and checkpointing schemes into elastic fwi only wavefields of several parts in the boundary area betwe

approximate gauss newton methods for nonlinear least squares problems the gauss newton algorithm is an iterative method regularly used for solving nonlinear least squares problems it is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting the procedure consists of a sequence of linear least squares approximations to the nonlinear problem each of which is solved by an inner direct or iterative process in comparison with newton s method and its variants the algorithm is attractive because it does not require the evaluation of second order derivatives in the hessian of the objective function in practice the exact gauss newton method is too expensive to apply operationally in meteorological forecasting and various approximations are made in order to reduce computational costs and to solve the problems in real time here we investigate the effects on the convergence of the gauss newton method of two types of approximation used commonly in data assimilation first we examine truncated gauss newton methods where the inner linear least squares problem is not solved exactly and second we examine perturbed gauss newton methods where the true linearized inner problem is approximated by a simplified or perturbed linear least squares problem we give conditions ensuring that the truncated and perturbed gauss newton methods converge and also derive rates of convergence for the iterations the results are illustrated by a simple numerical example a practical application to the problem of data assimilation in a typical meteorological system is presented

second order optimality conditions with the envelope like effect in nonsmooth multiobjective mathematical programming i l stability and set valued directional derivatives abstract second order necessary conditions and sufficient conditions with the envelope like effect for optimality in nonsmooth multiobjective mathematical programming are established we use set valued second order directional derivatives and impose strict differentiability for necessary conditions and l stability for sufficient conditions avoiding continuous differentiability the results improve and sharpen several recent existing ones examples are provided to show advantages of our theorems over some known ones in the literature in part i we consider l stability and second order set valued directional derivatives of vector functions part ii is devoted to second order necessary optimality conditions and sufficient ones

approximations and metric regularity in mathematical programming in banach space this paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in banach spaces these conditions are used to obtain lagrange kuhn tucker multipliers for minimization problems with infinitely many inequality and equality constraints