complete electrode model of electrical impedance tomography approximation properties and characterization of inclusions in electrical impedance tomography one tries to recover the spatial admittance distribution inside a body from boundary measurements in theoretical considerations it is usually assumed that the boundary data consists of the neumann to dirichlet map when conducting real world measurements the obtainable data is a linear finite dimensional operator mapping electrode currents onto electrode potentials in this paper it is shown that when using the complete electrode model to handle electrode measurements the corresponding current to voltage map can be seen as a discrete approximation of the traditional neumann to dirichlet operator this approximating link is utilized further in the special case of constant background conductivity with inhomogeneities it is demonstrated how inclusions with strictly higher or lower conductivities can be characterized by the limit behavior of the range of a boundary operator determined through electrode measurements when the electrodes get infinitely small and cover all

frequency domain elastic wave modeling by finite differences a tool for crosshole seismic imaging the migration imaging or inversion of wide aperture cross hole data depends on the ability to model wave propagation in complex media for multiple source positions computational costs can be considerably reduced in frequency domain imaging by modeling the frequency domain steady state equations rather than the time domain equations of motion i develop a frequency domain approach in this note that is competitive with time domain modeling when solutions for multiple sources are required or when only a limited number of frequency components of the solution are required

application of the perfectly matched layer pml absorbing boundary condition to elastic wave propagation a method is presented for application of the perfectly matched layer pml absorbing boundary condition abc to the p sv velocity stress finite difference method the pml consists of a nonphysical material containing both passive loss and dependent sources that provides active absorption of fields it has been used in electromagnetic applications where it has provided excellent results for a wide range of angles and frequencies in this work numerical simulations are used to compare the pml and an optimal second order elastic abc peng and toksoz j acoust soc am 95 733 745 1994 reflection factors are used to compare angular performance for continuous wave illumination snapshots of potentials are used to compare performance for broadband illumination these comparisons clearly demonstrate the superiority of the pml formulation within the pml there is a 60 increase in the number of unknowns per grid cell relative to the velocity stress formulation however the high quality of the pml abc allows the use of a smaller grid which can result in a lower overall computational cost

the calderon problem with partial data in this paper we improve an earlier result by bukhgeim and uhlmann 1 by showing that in dimension n 3 the knowledge of the cauchy data for the schr nodinger equation measured on possibly very small subsets of the boundary determines uniquely the potential we follow the general strategy of 1 but use a richer set of solutions to the dirichlet problem this implies a similar result for the problem of electrical impedance tomography which consists in determining the conductivity of a body by making voltage and current measurements at the boundary

determining conductivity by boundary measurements ii interior results abstract in a recent paper the authors showed that an unknown real analytic conductivity gamma may be determined from static boundary measurements in this document they extend this analysis by demonstrating that a similar result holds for piecewise real analytic conductivities in addition for the special case of a layered structure it is shown that a three times continuously differentiable conductivity is identifiable by boundary measurements originator supplied keywords include inversion convergence algorithms estimates coefficients

on block triangular preconditioned iteration methods for solving the helmholtz equation abstract to further improve the efficiency of solving the helmholtz equation in heterogeneous media with large wavenumber the krylov subspace methods incorporated with a class of inexact rotated block triangular preconditioners are presented to solve a block two by two linear system derived from the discrete helmholtz equation we further develop the eigenvalue properties of the preconditioned matrices to discuss the convergence of the corresponding preconditioned iteration methods the superiority of such preconditioned iteration methods is prominent according to the numerical results when comparing with other classical iteration methods we also investigate how the wavenumber influences the performance of the corresponding methods and it is shown that the iteration number of our proposed methods linearly increase with the wavenumber roughly furthermore the computational wave fields which conform the real physical law are exhibited to show the correctness of our proposed numerical modeling algorithm

p sv wave propagation in heterogeneous media velocity stress finite difference method i present a finite difference method for modeling p sv wave propagation in heterogeneous media this is an extension of the method i previously proposed for modeling sh wave propagation by using velocity and stress in a discrete grid the two components of the velocity cannot be defined at the same node for a complete staggered grid the stability condition and the p wave phase velocity dispersion curve do not depend on the poisson s ratio while the s wave phase velocity dispersion curve behavior is rather insensitive to the poisson s ratio therefore the same code used for elastic media can be used for liquid media where s wave velocity goes to zero and no special treatment is needed for a liquid solid interface typical physical phenomena arising with p sv modeling such as surface waves are in agreement with analytical results the weathered layer and corner edge models show in seismograms the same converted phases obtained by previous authors this method gives stable results for step discontinuities as shown for a liquid layer above an elastic half space the head wave preserves the correct amplitude finally the corner edge model illustrates a more complex geometry for the liquid solid interface as the poisson s ratio v increases from 0 25 to 0 5 the shear converted phases are removed from seismograms and from the time section of the wave field

detecting inclusions in electrical impedance tomography without reference measurements we develop a new variant of the factorization method that can be used to detect inclusions in electrical impedance tomography from either absolute current to voltage measurements at a single nonzero frequency or from frequency difference measurements this eliminates the need for numerically simulated reference measurements at an inclusion free body and thus greatly improves the method s robustness against forward modeling errors e g in the assumed body s shape

assessment of rod behaviour theories used in spectral finite element modelling abstract in this work different theories of rods have been discussed and compared the investigated theories are widely used in spectral finite element modelling of rod behaviour associated with propagation of symmetric longitudinal waves these are various single two mode and three mode theories including the elementary classical love and mindlin herrmann approaches as well as new two three and four mode theories proposed by the authors dispersion curves associated with each theory obtained by the use of hamilton s principle have been presented and discussed in the paper the investigation programme carried out by the authors aimed to show major differences and similarities between the rod theories and to discuss certain numerical aspects of their application great attention has been paid on properties limitations as well as difficulties associated with the use of the theories the results obtained from a wide program on numerical tests allowed the authors to draw certain general conclusions that are valid not only in the field of the spectral finite element method but also in the field of dynamics of engineering rod structures

multidirectional vector based elastic reverse time migration and angle domain common image gathers with approximate wavefield decomposition of p and s waves abstractelastic reverse time migration e rtm has limitations when the migration velocities contain strong contrasts first the traditional scheme of p s wave mode separation is based on helmholtz s equations which ignore the conversion between p and s waves at the current separation time thus it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition second the vector based imaging condition needs to use the reflection image normal and it also cannot give the correct polarity of the pp image in all possible conditions third the angle domain common image gathers adcigs calculated using the poynting vectors pvs do not consider the wave interferences that happen at each reflector therefore smooth models are often used for e rtm we relax this condition by proposing an improved data flow that involves three new contributions the first contribution is an improved system of p s wave mode

reconstructing discontinuities using complex geometrical optics solutions in this paper we provide a framework for constructing general complex geometrical optics solutions for several systems of two variables that can be reduced to a system with the laplacian as the leading order term we apply these special solutions to the problem of reconstructing inclusions inside a domain filled with known conductivity from local boundary measurements computational results demonstrate the versatility of these solutions to determine electrical inclusions

the pseudospectral method accurate representation of interfaces in elastic wave calculations when finite difference methods are used to solve the elastic wave equation in a discontinuous medium the error has two dominant components dispersive errors lead to artificial wave trains errors from interfaces lead to circular wavefronts emanating from each location where the interface appears jagged to the rectangular grid the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy with this method dispersive errors are essentially eliminated the mappings introduced in this paper also eliminate the other dominant error source test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost although the mapping method is described here in connection with the pseudospectral method it can also be used with high order finite difference approximations

hyperbolic geometry and local dirichlet neumann map abstract we show in three dimensions that if we know the dirichlet to neumann dn map associated to the schrodinger equation with a potential around the intersection of a sphere centered outside the convex hull of a bounded domain with smooth boundary we can determine uniquely the corresponding spherical mean of the potential with an appropriate measure this allows one to determine the potential near the boundary where the dn map is measured similar results are valid for electrical impedance tomography

grid dispersion and stability criteria of some common finite element methods for acoustic and elastic wave equations purely numerical methods based on finite element approximation of the acoustic or elastic wave equation are becoming increasingly popular for the generation of synthetic seismograms we present formulas for the grid dispersion and stability criteria for some popular finite element methods fem for wave propagation namely classical and spectral fem we develop an approach based on a generalized eigenvalue formulation to analyze the dispersive behavior of these fems for acoustic or elastic wave propagation that overcomes difficulties caused by irregular node spacing within the element and the use of high order polynomials as is the case for spectral fem analysis reveals that for spectral fem of order four or greater dispersion is less than 0 2 at four to five nodes per wavelength and dispersion is not angle dependent new results can be compared with grid dispersion results of some classical finite difference methods fdm used for acoustic or elastic wave propagation analysis reveals that fdm and classical fem require a larger sampling ratio than a spectral fem to obtain results with the same degree of accuracy the staggered grid fdm is an efficient scheme but the dispersion is angle dependent with larger values along the grid axes on the other hand spectral fem of order four or greater is isotropic with small dispersion making it attractive for simulations with long propagation times

accurate and efficient seismic modeling in random media the optimum method for seismic modeling in random media must 1 be highly accurate to be sensitive to subtle effects of wave propagation 2 allow coarse sampling to model media that are large compared to the scale lengths and wave propagation distances which are long compared to the wavelengths this is necessary to obtain statistically meaningful overall attributes of wavefields high order staggered grid finite difference algorithms and the pseudospectral method combine high accuracy in time and space with coarse sampling investigations for random media reveal that both methods lead to nearly identical wavefields the small differences can be attributed mainly to differences in the numerical dispersion this result is important because it shows that errors of the numerical differentiation which are caused by poor polynomial interpolation near discontinuities do not accumulate but cancel in a random medium where discontinuities are numerous the differentiator can be longer than the medium scale leng

sh wave propagation in heterogeneous media velocity stress finite difference method a new finite difference fd method is presented for modeling sh wave propagation in a generally heterogeneous medium this method uses both velocity and stress in a discrete grid density and shear modulus are similarly discretized avoiding any spatial smoothing therefore boundaries will be correctly modeled under an implicit formulation standard problems quarter plane propagation sedimentary basin propagation are studied to compare this method with other methods finally a more complex example a salt dome inside a two layered medium shows the effect of lateral propagation on seismograms recorded at the surface a corner wave always in phase with the incident wave and a head wave will appear which will pose severe problems of interpretation with the usual vertical migration methods

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

source independent full waveform inversion of seismic data a set of seismic trace data is collected in an input data set that is first fourier transformed in its entirety into the frequency domain a normalized wavefield is obtained for each trace of the input data set in the frequency domain normalization is done with respect to the frequency response of a reference trace selected from the set of seismic trace data the normalized wavefield is source independent complex and dimensionless the normalized wavefield is shown to be uniquely defined as the normalized impulse response provided that a certain condition is met for the source this property allows construction of the inversion algorithm disclosed herein without any source or source coupling information the algorithm minimizes the error between data normalized wavefield and the model normalized wavefield the methodology is applicable to any 3 d seismic problem and damping may be easily included in the process

seismic finite difference modeling with spatially varying time steps numerical seismic modeling by finite difference methods usually works with a global time step size because of stability considerations the time step size is determined essentially by the highest seismic velocity i e the higher the highest velocity the smaller the time step needs to be therefore if large velocity contrasts exist within the numerical grid domains of low velocity are oversampled temporally using different time step sizes in different parts of the numerical grid can reduce computational costs considerably

on uniqueness of recovery of a discontinuous conductivity coefficient on considere le probleme de retrouver le coefficient a de l equation elliptique div a u 0 dans un domaine avec la condition aux limites u sur quand u n est donnee pour tout regulier on demontre l unicite pour a a 0 b discontinu ou est la fonction indicateur d un ensemble ouvert a o est une fonction c 2 donnee et b est une fonction c 2 inconnue

3 d finite difference modeling of elastic waves in borehole environments full wavefield sonic logs may be simulated for arbitrarily complicated three dimensional 3 d borehole environments using 3 d elastic finite differences to ensure reliability through large contrasts in poisson s ratio across mud casing lithology contacts a staggered grid finite difference formulation is used cylindrical symmetry is not assumed so the responses of features such as asymmetrical washouts and dipping structure are easily obtained when features are asymmetrical seismic responses vary significantly at various points around the hole circumference at any depth even for simple hole geometries observed responses are complicated because of coupling between waves inside and outside the hole observations are also sensitive to the source receiver separation output formats include fixed time snapshots of displacement divergence and curl and seismograms for the center and edge of a borehole this allows detailed arrival identification and interpretation to our knowledge this is the most

reconstruction of penetrable inclusions in elastic waves by boundary measurements abstract we use ikehata s enclosure method to reconstruct penetrable unknown inclusions in a plane elastic body in time harmonic waves complex geometrical optics solutions with complex polynomial phases are adopted as the probing utility in a situation similar to ours due to the presence of a zeroth order term in the equation some technical assumptions need to be assumed in early researches in a recent work of sini and yoshida they succeeded in abandoning these assumptions by using a different idea to obtain a crucial estimate in particular the boundaries of the inclusions need only to be lipschitz in this work we apply the same idea to our model it s interesting that with more careful treatment we find the boundaries of the inclusions can in fact be assumed to be only continuous

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

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

wave equation datuming before stack this note describes the extension to unstacked seismic data of a computationally efficient form of the kirchhoff integral published several years ago in the previous paper berryhill 1979 a wave equation procedure was developed to change the datum of a collection of zero offset seismic traces from one surface of arbitrary shape to another even when the velocity for wave propagation is not constant this procedure was designated wave equation datuming and its applications to zero offset data were shown to include velocity replacement datum corrections and multilayer forward modeling extending this procedure to unstacked data requires no change in the mathematical algorithm it is necessary only to recognize that operating on a common source group of seismic traces has the effect of extrapolating the receivers from one datum to another and that because of reciprocity operating on a common receiver group changes the datum of the sources two passes through the data common source computations th

probing for electrical inclusions with complex spherical waves let a physical body in 2 or 3 be given assume that the electric conductivity distribution inside consists of conductive inclusions in a known smooth background further assume that a subset is available for boundary measurements it is proved using hyperbolic geometry that certain information about the location of the inclusions can be exactly recovered from static electric measurements on more precisely given a ball b with center outside the convex hull of and satisfying b boundary measurements on with explicitly given dirichlet data are enough to determine whether b intersects the inclusion an approximate detection algorithm is introduced based on the theory numerical experiments in dimension two with simulated noisy data suggest that the algorithm finds the inclusion free domain near and is robust against measurement noise 2007 wiley periodicals inc

a real time algorithm for the location search of discontinuous conductivities with one measurement we consider an inverse problem for finding the anomaly of discontinuous electrical conductivity by one current voltage observation we develop a real time algorithm for determining the location of the anomaly this new idea is based on the observation of the pattern of a simple weighted combination of the input current and the output voltage combined with the size estimation result this algorithm gives a good initial guess for newton type schemes we give the rigorous proof for the location search algorithm both the mathematical analysis and its numerical implementation indicate our location search algorithm is very fast stable and efficient 2001 john wiley sons inc

determining conductivity by boundary measurements a p calderon pose la question est il possible de determiner la conductivite thermique d un objet a partir de mesures statiques de la temperature et du flux de chaleur a la limite on demontre que ces mesures determinent de facon unique la conductivite thermique et toutes ses derivees a la limite

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

on complemented subspaces of non archimedean kothe spaces abstract the generalized power series spaces d f a r are the most known and important examples of nuclear kothe spaces we prove that every kothe montel space over a non archimedean field k has a complemented closed subspace isomorphic to a generalized power series space d f a r it follows that every non normable frechet space e that is not isomorphic to the product of a banach space and the space k n has a closed subspace isomorphic to a generalized power series space d f a r

fourth order finite difference p sv seismograms i describe the properties of a fourth order accurate space second order accurate time two dimensional p sk finite difference scheme based on the madariagavirieux staggered grid formulation the numerical scheme is developed from the first order system of hyperbolic elastic equations of motion and constitutive laws expressed in particle velocities and stresses the madariaga virieux staggered grid scheme has the desirable quality that it can correctly model any variation in material properties including both large and small poisson s ratio materials with minimal numerical dispersion and numerical anisotropy dispersion analysis indicates that the shortest wavelengths in the model need to be sampled at 5 gridpoints wavelength the scheme can be used to accurately simulate wave propagation in mixed acoustic elastic media making it ideal for modeling marine problems explicitly calculating both velocities and stresses makes it relatively simple to initiate a source at the free surface or within a layer and to satisfy free surface boundary conditions benchmark comparisons of finite difference and analytical solutions to lamb s problem are almost identical as are comparisons of finite difference and reflectivity solutions for elastic elastic and acoustic elastic layered models

a direct impedance tomography algorithm for locating small inhomogeneities impedance tomography seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface in its most general form impedance tomography is quite ill posed but when additional a priori information is admitted the situation changes dramatically in this paper we consider the case where the goal is to find a number of small objects inhomogeneities inside an otherwise known conductor taking advantage of the smallness of the inhomogeneities we can use asymptotic analysis to design a direct i e non iterative reconstruction algorithm for the determination of their locations the viability of this direct approach is documented by numerical examples

reconstructing acoustic obstacles by planar and cylindrical waves in this paper we develop a novel method of reconstructing acoustic obstacles in r2 which follows a similar spirit of the linear sampling method originated by colton and kirsch the reconstruction scheme makes use of the near field measurements encoded into the boundary dirichlet to neumann map or the neumann to dirichlet map both the plane waves and cylindrical waves are shown to meet the reconstruction purpose rigorous mathematical justification of the reconstruction scheme is established the mapping properties of the newly introduced function operators involved in the reconstruction scheme are established these results are of significant mathematical interests for their own sake moreover due to the distinct properties of the function operators the indictor function in the proposed reconstruction scheme exhibits completely different behaviors from those having been established for the indictor function in the original linear sampling method for inverse scattering problems numerical experiments

elastic full waveform inversion for surface topography abstractelastic full waveform inversion efwi attempts to find high resolution model parameters that are able to match observed data exactly by minimizing residuals between the observed and predicted data however irregular surface topography can generate problematic efwi results to resolve this issue an efwi method in the curvilinear system is presented to invert velocities for areas with surface topography this method meshes the regions near the surface topography into body fitted grids and areas off surface regions into rectangular grids we derived the gradient formulas with respect to multiple parameters in the curvilinear system and developed a step search method to calculate the corresponding step lengths wiener filtering is performed on a multiscale decomposition method to invert the velocity from a low to a high wavenumber error analysis using a homogeneous model with a surface topography demonstrates the accuracy and validity of our methods numerical experiments with an elastic hill g

two dimensional nonlinear inversion of seismic waveforms numerical results the nonlinear problem of inversion of seismic waveforms can be set up using least squares methods the inverse problem is then reduced to the problem of minimizing a lp nonquadratic function in a space of many 104to106 variables using gradient methods leads to iterative algorithms each iteration implying a forward propagation generated by the actual sources a backward propagation generated by the data residuals acting as if they were sources and a correlation at each point of the space of the two fields thus obtained which gives the updated model the quality of the results of any inverse method depends heavily on the realism of the forward modeling finite difference schemes are a good choice relative to realism because although they are time consuming they give excellent results numerical tests performed with multioffset synthetic data from a two dimensional model prove the feasibility of the approach if only surface recorded reflections are used the high spatial frequency content of the

elastic imaging with exact wavefield extrapolation for application to ocean bottom 4c seismic data abstracta central component of imaging methods is receiver side wavefield backpropagation or extrapolation in which the wavefield from a physical source scattered at any point in the subsurface is estimated from data recorded by receivers located near or at the earth s surface elastic reverse time migration usually accomplishes wavefield extrapolation by simultaneous reversed time injection of the particle displacements or velocities recorded at each receiver location into a wavefield modeling code here we formulate an exact integral expression based on reciprocity theory that uses a combination of velocity stress recordings and quadrupole dipole backpropagating sources rather than the commonly used approximate formula involving only particle velocity data and dipole backpropagating sources the latter approximation results in two types of nonphysical waves in the scattered wavefield estimate first each arrival contained in the data is injected upward and downward rather than unidirectionally as

reconstruction of a penetrable obstacle by complex spherical waves we consider an inverse acoustic scattering problem for identifying a non convex penetrable obstacle in three dimensions in a homogeneous medium we apply the complex geometrical optics solutions with logarithmic phase which is called complex spherical waves to reconstruction problem the reconstruction schemes will be demonstrated in the last section

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

kirchhoff integral extrapolation and migration of nonplanar data the form of the kirchhoff integral commonly used for migration of seismic data assumes that the observation surface is a flat horizontal plane this restriction is not necessary in theory or practice a useful integral expression can be obtained for an arbitrary observation surface by use of the kirchhoff approximation for relating the field on the observation surface to its normal derivative kirchhoff migration is usually carried out by an integral that represents summation over receivers for a given shot summation over shots is not explicitly handled by invoking the principle of reciprocity and applying the kirchhoff integral twice one arrives at an integral expression that explicitly represents shot and receiver summation contains the factors needed to compensate for nonplanar survey geometry and can be used for wave extrapolation as well as migration adaptation of this integral to handling 2 d or line survey data is straightforward and with the usual approximations concerning spherical wave s

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

dispersion relation preserving stereo modeling method beyond nyquist frequency for acoustic wave equation abstractaccording to nyquist shannon sampling theory two sampling points of the displacements per minimal wavelength are required to reconstruct a wavefield however high frequency components above the nyquist frequency can be theoretically recovered by introducing spatial derivatives of the displacements we have developed a new finite difference scheme for solving the acoustic equation named the dispersion relation preserving stereo modeling method which is practical for seismic forward modeling beyond the nyquist frequency the key idea in this model is to approximate the high order spatial derivatives with wavenumber domain optimization in wavefields represented by the wave displacements and their gradients we investigated the theoretical properties of this method including the absolute error of spatial operators numerical error stability criterion and computational efficiency our results indicated that the method can effectively suppress the numerical dispersion on extremely coarse grids wi

unstructured grid method for stress wave propagation in elastic media a new numerical method named unstructured grid method is presented to calculate two dimensional stress wave propagation problems in elastic media this method is based on the dynamic equilibrium equations of the investigated lumps that are formed among the auxiliary unstructured grids the approach to the solution is obtained by calculating the node displacements and the central point stresses of the spatial grids alternately the stability criterion of the method is given and tested numerically analysis of dispersion for numerical phase velocity is studied the rule is provided to determine the spatial step size of discretization mesh so as to ensure the numerical accuracy of the unstructured grid method numerical results are compared with the available exact solutions and with the solutions of central difference method for demonstrating the good accuracy of the new method the unstructured grid method possesses higher calculating speed than the central difference method finally the dynamic behavior of a square plate with a hole under an impact loading is researched

global uniqueness for a two dimensional inverse boundary value problem we show that the coefficient y x of the elliptic equation vie qyvu 0 in a two dimensional domain is uniquely determined by the corresponding dirichlet to neumann map on the boundary and give a reconstruction pro

regularized seismic full waveform inversion with prior model information full waveform inversion fwi delivers high resolution quantitative images and is a promising technique to obtain macroscale physical property model of the subsurface in most geophysical applications prior information such as that collected in wells is available and should be used to increase the image reliability for this we propose to introduce three terms in the definition of the fwi misfit function the data misfit itself the first order tikhonov regularization term acting as a smoothing operator and a prior model norm term this last term is the way to smoothly introduce prior information into the fwi workflow on a selected target of the marmousi synthetic example significant improvement was obtained when using the prior model term for noise free and noisy synthetic data the prior model term may significantly reduce the inversion sensi tivity to incorrect initial conditions the limited range of spatial wavenumber sampling by the acquisition may be compensated with the prior model information for multiple free and multiplecontaminated data prior and initial models play different roles in the inversion scheme the starting model is used for wave propagation and therefore drives the data misfit gradient whereas the prior model is never explicitly used for solving the wave equation and only drives the optimization step as an additional constraint to minimize the total objective function thus the prior model is not required to follow kinematic properties as precisely as the initial model except in zones of poor illumination in addition we investigate the influence of a simple dynamic decreasing weighting of the prior model term once the cycle skipping problem has been solved the impact of the prior model term is gradually reduced within the misfit function to be driven by seismic data only

3d weak dispersion reverse time migration using a stereo modeling operator reliable 3d imaging is a required tool for developing models of complex geologic structures reverse time migration rtm as the most powerful depth imaging method has become the preferred imaging tool because of its ability to handle complex velocity models including steeply dipping interfaces and large velocity contrasts finite difference methods are among the most popular numerical approaches used for rtm however these methods often encounter a serious issue of numerical dispersion which is typically suppressed by reducing the grid interval of the propagation model resulting in large computation and memory requirements in addition even with small grid spacing numerical anisotropy may degrade images or worse provide images that appear to be focused but position events incorrectly recently stereo operators have been developed to approximate the partial differential operator in space these operators have been used to develop several weak dispersion and efficient stereo modeling methods that have been found to be superior to conventional algorithms in suppressing numerical dispersion and numerical anisotropy we generalized one stereo modeling method fourth order nearly analytic central difference nacd from 2d to 3d and applied it to 3d rtm the rtm results for the 3d seg eage phase a classic data set 1 and the seg advanced modeling project model demonstrated that even when using a large grid size the nacd method can handle very complex velocity models and produced better images than can be obtained using the fourth order and eighth order lax wendroff correction lwc schemes we also applied 3d nacd and fourth order lwc to a field data set and illustrated significant improvements in terms of structure imaging horizon layer continuity and positioning we also investigated numerical dispersion and found that not only does the nacd method have superior dispersion characteristics but also that the angular variation of dispersion is significantly less than for lwc

an efficient multiscale method for time domain waveform tomography this efficient multiscale method for time domain waveform tomography incorporates filters that are more efficient than hamming window filters a strategy for choosing optimal frequency bands is proposed to achieve computational efficiency in the time domain a staggered grid explicit finite difference method with fourth order accuracy in space and second order accuracy in time is used for forward modeling and the adjoint calculation the adjoint method is utilized in inverting for an efficient computation of the gradient directions in the multiscale approach multifrequency data and multiple grid sizes are used to overcome somewhat the severe local minima problem of waveform tomography the method is applied successfully to 1d and 2d heterogeneous models it can accurately recover low and high wavenumber components of the velocity models the inversion result for the 2d model demonstrates that the multiscale method is computationally efficient and converges faster than a conventional single scale method

nonlinear fourier analysis for discontinuous conductivities computational results abstract two reconstruction methods of electrical impedance tomography eit are numerically compared for nonsmooth conductivities in the plane based on the use of complex geometrical optics cgo solutions to d bar equations involving the global uniqueness proofs for calderon problem exposed in nachman 1996 43 and astala and paivarinta 2006 6 the astala paivarinta theory based low pass transport matrix method implemented in astala et al 2011 3 and the shortcut method which considers ingredients of both theories the latter method is formally similar to the nachman theory based regularized eit reconstruction algorithm studied in knudsen et al 2009 34 and several references from there new numerical results are presented using parallel computation with size parameters larger than ever leading mainly to two conclusions as follows first both methods can approximate piecewise constant conductivities better and better as the cutoff frequency increases and there seems to be a gibbs like phenomenon producing ringing artifacts second the transport matrix method loses accuracy away from a freely chosen pivot point located outside of the object to be studied whereas the shortcut method produces reconstructions with more uniform quality

inverse conductivity problem with one measurement uniqueness of balls in r 3 we prove that the location and size of an unknown ball d entering the conductivity equation mbox div 1 k 1 chi d nabla u 0 in a bounded domain omega subset bbb r 3 are uniquely determined by any single nonzero cauchy data u pd u nu on bo

a new family of finite difference schemes to solve the heterogeneous acoustic wave equation equivalent staggered grid scheme esg is a new family of schemes based on the finite difference method fdm the method is applied to acoustic wave propagation in variable density media and the results are compared with those from some classic fdm approaches the main feature of this new family is that it is designed to generate results numerically equivalent to those using the standard staggered grid formulations ssg but with the same memory requirements of simple grid schemes hence it results in a reduction of memory usage by 33 in 2d and 50 in 3d problems compared to the memory usage of ssg the first order system of equations in terms of pressure and velocity is not used here instead the formulation is based on applying new central difference operators to the secondorder acoustic wave equation in terms of pressure obtaining the same level of accuracy and stability as the ssg schemes the equivalence between the esg and ssg is mathematically demonstrated and issues concerning the application of seismic sources and the boundary conditions are addressed